id sid tid token lemma pos cana-4527 1 1 communications communication NOUN cana-4527 1 2 on on ADP cana-4527 1 3 applied apply VERB cana-4527 1 4 nonlinear nonlinear ADJ cana-4527 1 5 analysis analysis NOUN cana-4527 1 6 issn issn NOUN cana-4527 1 7 : : PUNCT cana-4527 1 8 1074 1074 NUM cana-4527 1 9 - - PUNCT cana-4527 1 10 133x 133x NUM cana-4527 1 11 vol vol NOUN cana-4527 1 12 32 32 NUM cana-4527 1 13 no no NOUN cana-4527 1 14 . . PUNCT cana-4527 2 1 9s 9s NUM cana-4527 2 2 ( ( PUNCT cana-4527 2 3 2025 2025 NUM cana-4527 2 4 ) ) PUNCT cana-4527 2 5 2415 2415 NUM cana-4527 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 2 7 space space NOUN cana-4527 2 8 - - PUNCT cana-4527 2 9 time time NOUN cana-4527 2 10 caputo caputo PROPN cana-4527 2 11 fabrizio fabrizio PROPN cana-4527 2 12 fractional fractional PROPN cana-4527 2 13 differentiation differentiation NOUN cana-4527 2 14 through through ADP cana-4527 2 15 a a DET cana-4527 2 16 proposed propose VERB cana-4527 2 17 fixed fix VERB cana-4527 2 18 transform transform NOUN cana-4527 2 19 - - PUNCT cana-4527 2 20 based base VERB cana-4527 2 21 domain domain NOUN cana-4527 2 22 ( ( PUNCT cana-4527 2 23 ftd ftd NOUN cana-4527 2 24 ) ) PUNCT cana-4527 2 25 approach approach NOUN cana-4527 2 26 for for ADP cana-4527 2 27 groundwater groundwater NOUN cana-4527 2 28 model model NOUN cana-4527 2 29 1uttara 1uttara NUM cana-4527 2 30 deshmukh deshmukh NOUN cana-4527 2 31 , , PUNCT cana-4527 2 32 2gajanan 2gajanan NUM cana-4527 2 33 lomte lomte PROPN cana-4527 2 34 1mgmu 1mgmu NUM cana-4527 2 35 , , PUNCT cana-4527 2 36 chhatrapati chhatrapati NOUN cana-4527 2 37 sambhajinagar sambhajinagar NOUN cana-4527 2 38 , , PUNCT cana-4527 2 39 udeshmukh@mgmu.ac.in udeshmukh@mgmu.ac.in ADV cana-4527 2 40 . . PUNCT cana-4527 3 1 2mgmu 2mgmu NUM cana-4527 3 2 , , PUNCT cana-4527 3 3 chhatrapati chhatrapati NOUN cana-4527 3 4 sambhajinagar,gclomte@gmail.com sambhajinagar,gclomte@gmail.com PROPN cana-4527 3 5 article article NOUN cana-4527 3 6 history history NOUN cana-4527 3 7 : : PUNCT cana-4527 3 8 received receive VERB cana-4527 3 9 : : PUNCT cana-4527 3 10 03 03 NUM cana-4527 3 11 - - PUNCT cana-4527 3 12 03 03 NUM cana-4527 3 13 - - PUNCT cana-4527 3 14 2025 2025 NUM cana-4527 3 15 revised revise VERB cana-4527 3 16 : : PUNCT cana-4527 3 17 03 03 NUM cana-4527 3 18 - - SYM cana-4527 3 19 03 03 NUM cana-4527 3 20 - - PUNCT cana-4527 3 21 2025 2025 NUM cana-4527 3 22 accepted accept VERB cana-4527 3 23 : : PUNCT cana-4527 3 24 08 08 NUM cana-4527 3 25 - - SYM cana-4527 3 26 03 03 NUM cana-4527 3 27 - - PUNCT cana-4527 3 28 2025 2025 NUM cana-4527 3 29 abstract abstract NOUN cana-4527 3 30 : : PUNCT cana-4527 3 31 mathematical mathematical ADJ cana-4527 3 32 models model NOUN cana-4527 3 33 of of ADP cana-4527 3 34 groundwater groundwater NOUN cana-4527 3 35 flow flow NOUN cana-4527 3 36 have have AUX cana-4527 3 37 been be AUX cana-4527 3 38 used use VERB cana-4527 3 39 since since SCONJ cana-4527 3 40 the the DET cana-4527 3 41 late late ADJ cana-4527 3 42 1800s 1800s NUM cana-4527 3 43 . . PUNCT cana-4527 4 1 a a DET cana-4527 4 2 set set NOUN cana-4527 4 3 of of ADP cana-4527 4 4 differential differential ADJ cana-4527 4 5 equations equation NOUN cana-4527 4 6 known know VERB cana-4527 4 7 to to PART cana-4527 4 8 control control VERB cana-4527 4 9 groundwater groundwater NOUN cana-4527 4 10 flow flow NOUN cana-4527 4 11 make make VERB cana-4527 4 12 up up ADP cana-4527 4 13 a a DET cana-4527 4 14 mathematical mathematical ADJ cana-4527 4 15 model model NOUN cana-4527 4 16 . . PUNCT cana-4527 5 1 we we PRON cana-4527 5 2 suggest suggest VERB cana-4527 5 3 the the DET cana-4527 5 4 fixed fix VERB cana-4527 5 5 transform transform NOUN cana-4527 5 6 domain domain NOUN cana-4527 5 7 ( ( PUNCT cana-4527 5 8 ftd ftd NOUN cana-4527 5 9 ) ) PUNCT cana-4527 5 10 approach approach NOUN cana-4527 5 11 to to PART cana-4527 5 12 address address VERB cana-4527 5 13 the the DET cana-4527 5 14 ground ground NOUN cana-4527 5 15 water water NOUN cana-4527 5 16 flow flow NOUN cana-4527 5 17 problem problem NOUN cana-4527 5 18 's 's PART cana-4527 5 19 time time NOUN cana-4527 5 20 - - PUNCT cana-4527 5 21 space space NOUN cana-4527 5 22 fractional fractional ADJ cana-4527 5 23 derivative derivative NOUN cana-4527 5 24 . . PUNCT cana-4527 6 1 the the DET cana-4527 6 2 fractional fractional ADJ cana-4527 6 3 differential differential NOUN cana-4527 6 4 equation equation NOUN cana-4527 6 5 can can AUX cana-4527 6 6 represent represent VERB cana-4527 6 7 groundwater groundwater NOUN cana-4527 6 8 flow flow NOUN cana-4527 6 9 since since SCONJ cana-4527 6 10 it it PRON cana-4527 6 11 is be AUX cana-4527 6 12 not not PART cana-4527 6 13 static static ADJ cana-4527 6 14 and and CCONJ cana-4527 6 15 can can AUX cana-4527 6 16 relate relate VERB cana-4527 6 17 to to ADP cana-4527 6 18 initial initial ADJ cana-4527 6 19 and and CCONJ cana-4527 6 20 boundary boundary ADJ cana-4527 6 21 circumstances circumstance NOUN cana-4527 6 22 . . PUNCT cana-4527 7 1 utilizing utilize VERB cana-4527 7 2 the the DET cana-4527 7 3 darcy darcy PROPN cana-4527 7 4 law law PROPN cana-4527 7 5 and and CCONJ cana-4527 7 6 the the DET cana-4527 7 7 conservation conservation NOUN cana-4527 7 8 of of ADP cana-4527 7 9 mass mass ADJ cana-4527 7 10 law law NOUN cana-4527 7 11 , , PUNCT cana-4527 7 12 this this DET cana-4527 7 13 model model NOUN cana-4527 7 14 considers consider VERB cana-4527 7 15 the the DET cana-4527 7 16 water water NOUN cana-4527 7 17 flow flow NOUN cana-4527 7 18 to to ADP cana-4527 7 19 the the DET cana-4527 7 20 piezometric piezometric ADJ cana-4527 7 21 head head NOUN cana-4527 7 22 as as ADP cana-4527 7 23 a a DET cana-4527 7 24 function function NOUN cana-4527 7 25 of of ADP cana-4527 7 26 the the DET cana-4527 7 27 derivative derivative NOUN cana-4527 7 28 of of ADP cana-4527 7 29 the the DET cana-4527 7 30 fractional fractional ADJ cana-4527 7 31 arrangement arrangement NOUN cana-4527 7 32 . . PUNCT cana-4527 8 1 the the DET cana-4527 8 2 primary primary ADJ cana-4527 8 3 goal goal NOUN cana-4527 8 4 of of ADP cana-4527 8 5 this this DET cana-4527 8 6 study study NOUN cana-4527 8 7 was be AUX cana-4527 8 8 to to PART cana-4527 8 9 provide provide VERB cana-4527 8 10 a a DET cana-4527 8 11 novel novel ADJ cana-4527 8 12 idea idea NOUN cana-4527 8 13 for for ADP cana-4527 8 14 simulating simulate VERB cana-4527 8 15 issues issue NOUN cana-4527 8 16 with with ADP cana-4527 8 17 groundwater groundwater NOUN cana-4527 8 18 flow flow NOUN cana-4527 8 19 . . PUNCT cana-4527 9 1 finally finally ADV cana-4527 9 2 , , PUNCT cana-4527 9 3 two two NUM cana-4527 9 4 cases case NOUN cana-4527 9 5 and and CCONJ cana-4527 9 6 simulations simulation NOUN cana-4527 9 7 for for ADP cana-4527 9 8 these these DET cana-4527 9 9 examples example NOUN cana-4527 9 10 show show VERB cana-4527 9 11 that that SCONJ cana-4527 9 12 the the DET cana-4527 9 13 proposed propose VERB cana-4527 9 14 method method NOUN cana-4527 9 15 works work VERB cana-4527 9 16 . . PUNCT cana-4527 10 1 empirical empirical ADJ cana-4527 10 2 instances instance NOUN cana-4527 10 3 demonstrate demonstrate VERB cana-4527 10 4 that that SCONJ cana-4527 10 5 the the DET cana-4527 10 6 suggested suggest VERB cana-4527 10 7 approach approach NOUN cana-4527 10 8 is be AUX cana-4527 10 9 highly highly ADV cana-4527 10 10 efficient efficient ADJ cana-4527 10 11 and and CCONJ cana-4527 10 12 applicable applicable ADJ cana-4527 10 13 to to ADP cana-4527 10 14 related related ADJ cana-4527 10 15 issues issue NOUN cana-4527 10 16 . . PUNCT cana-4527 11 1 a a DET cana-4527 11 2 numerical numerical ADJ cana-4527 11 3 scheme scheme NOUN cana-4527 11 4 stability stability NOUN cana-4527 11 5 analysis analysis NOUN cana-4527 11 6 is be AUX cana-4527 11 7 provided provide VERB cana-4527 11 8 to to PART cana-4527 11 9 confirm confirm VERB cana-4527 11 10 the the DET cana-4527 11 11 effectiveness effectiveness NOUN cana-4527 11 12 of of ADP cana-4527 11 13 the the DET cana-4527 11 14 proposed propose VERB cana-4527 11 15 framework framework NOUN cana-4527 11 16 . . PUNCT cana-4527 12 1 by by ADP cana-4527 12 2 comparing compare VERB cana-4527 12 3 the the DET cana-4527 12 4 results result NOUN cana-4527 12 5 of of ADP cana-4527 12 6 the the DET cana-4527 12 7 suggested suggest VERB cana-4527 12 8 method method NOUN cana-4527 12 9 with with ADP cana-4527 12 10 those those PRON cana-4527 12 11 of of ADP cana-4527 12 12 barker barker PROPN cana-4527 12 13 's 's PART cana-4527 12 14 fractal fractal ADJ cana-4527 12 15 radial radial ADJ cana-4527 12 16 flow flow NOUN cana-4527 12 17 framework framework NOUN cana-4527 12 18 , , PUNCT cana-4527 12 19 an an DET cana-4527 12 20 association association NOUN cana-4527 12 21 between between ADP cana-4527 12 22 the the DET cana-4527 12 23 non non ADJ cana-4527 12 24 - - ADJ cana-4527 12 25 integral integral ADJ cana-4527 12 26 dimension dimension NOUN cana-4527 12 27 of of ADP cana-4527 12 28 the the DET cana-4527 12 29 flow flow NOUN cana-4527 12 30 and and CCONJ cana-4527 12 31 the the DET cana-4527 12 32 fractional fractional ADJ cana-4527 12 33 order order NOUN cana-4527 12 34 of of ADP cana-4527 12 35 the the DET cana-4527 12 36 derivative derivative NOUN cana-4527 12 37 is be AUX cana-4527 12 38 illustrated illustrate VERB cana-4527 12 39 . . PUNCT cana-4527 13 1 additionally additionally ADV cana-4527 13 2 , , PUNCT cana-4527 13 3 the the DET cana-4527 13 4 proposed propose VERB cana-4527 13 5 method method NOUN cana-4527 13 6 is be AUX cana-4527 13 7 highly highly ADV cana-4527 13 8 effective effective ADJ cana-4527 13 9 and and CCONJ cana-4527 13 10 applicable applicable ADJ cana-4527 13 11 to to ADP cana-4527 13 12 similar similar ADJ cana-4527 13 13 problems problem NOUN cana-4527 13 14 . . PUNCT cana-4527 14 1 keywords keyword NOUN cana-4527 14 2 : : PUNCT cana-4527 14 3 fractional fractional ADJ cana-4527 14 4 derivative derivative NOUN cana-4527 14 5 ; ; PUNCT cana-4527 14 6 time time NOUN cana-4527 14 7 derivative derivative NOUN cana-4527 14 8 ; ; PUNCT cana-4527 14 9 differential differential ADJ cana-4527 14 10 equation equation NOUN cana-4527 14 11 ; ; PUNCT cana-4527 14 12 difference difference NOUN cana-4527 14 13 method method NOUN cana-4527 14 14 ; ; PUNCT cana-4527 14 15 groundwater groundwater NOUN cana-4527 14 16 . . PUNCT cana-4527 15 1 1 1 X cana-4527 15 2 . . X cana-4527 15 3 introduction introduction NOUN cana-4527 15 4 the the DET cana-4527 15 5 advantageous advantageous ADJ cana-4527 15 6 properties property NOUN cana-4527 15 7 of of ADP cana-4527 15 8 fractional fractional ADJ cana-4527 15 9 calculus calculus NOUN cana-4527 15 10 , , PUNCT cana-4527 15 11 including include VERB cana-4527 15 12 non non ADJ cana-4527 15 13 - - ADJ cana-4527 15 14 locality locality ADJ cana-4527 15 15 , , PUNCT cana-4527 15 16 great great ADJ cana-4527 15 17 dependability dependability NOUN cana-4527 15 18 , , PUNCT cana-4527 15 19 inheritance inheritance NOUN cana-4527 15 20 , , PUNCT cana-4527 15 21 and and CCONJ cana-4527 15 22 analyticity analyticity NOUN cana-4527 15 23 , , PUNCT cana-4527 15 24 have have AUX cana-4527 15 25 made make VERB cana-4527 15 26 it it PRON cana-4527 15 27 an an DET cana-4527 15 28 extremely extremely ADV cana-4527 15 29 esteemed esteemed ADJ cana-4527 15 30 framework framework NOUN cana-4527 15 31 for for ADP cana-4527 15 32 administering administer VERB cana-4527 15 33 complex complex ADJ cana-4527 15 34 and and CCONJ cana-4527 15 35 sustainable sustainable ADJ cana-4527 15 36 phenomena phenomenon NOUN cana-4527 15 37 for for ADP cana-4527 15 38 over over ADP cana-4527 15 39 30 30 NUM cana-4527 15 40 years year NOUN cana-4527 15 41 . . PUNCT cana-4527 16 1 the the DET cana-4527 16 2 modified modify VERB cana-4527 16 3 fractional fractional ADJ cana-4527 16 4 concept concept NOUN cana-4527 16 5 addressed address VERB cana-4527 16 6 issues issue NOUN cana-4527 16 7 related relate VERB cana-4527 16 8 to to ADP cana-4527 16 9 processes process NOUN cana-4527 16 10 , , PUNCT cana-4527 16 11 including include VERB cana-4527 16 12 homogeneities homogeneity NOUN cana-4527 16 13 . . PUNCT cana-4527 17 1 the the DET cana-4527 17 2 fundamental fundamental ADJ cana-4527 17 3 foundation foundation NOUN cana-4527 17 4 was be AUX cana-4527 17 5 established establish VERB cana-4527 17 6 by by ADP cana-4527 17 7 several several ADJ cana-4527 17 8 pioneers pioneer NOUN cana-4527 17 9 , , PUNCT cana-4527 17 10 such such ADJ cana-4527 17 11 as as ADP cana-4527 17 12 hadamard hadamard NOUN cana-4527 17 13 [ [ X cana-4527 17 14 2 2 NUM cana-4527 17 15 ] ] PUNCT cana-4527 17 16 , , PUNCT cana-4527 17 17 liouville liouville PROPN cana-4527 17 18 and and CCONJ cana-4527 17 19 caputo caputo PROPN cana-4527 18 1 [ [ X cana-4527 18 2 1 1 NUM cana-4527 18 3 ] ] PUNCT cana-4527 18 4 , , PUNCT cana-4527 18 5 davison davison PROPN cana-4527 18 6 and and CCONJ cana-4527 18 7 essex essex PROPN cana-4527 18 8 and and CCONJ cana-4527 18 9 jumarie jumarie NOUN cana-4527 19 1 [ [ X cana-4527 19 2 5 5 NUM cana-4527 19 3 ] ] PUNCT cana-4527 19 4 , , PUNCT cana-4527 19 5 grunwald grunwald NOUN cana-4527 19 6 and and CCONJ cana-4527 19 7 letnikov letnikov NOUN cana-4527 20 1 [ [ X cana-4527 20 2 3 3 NUM cana-4527 20 3 ] ] PUNCT cana-4527 20 4 , , PUNCT cana-4527 20 5 riez riez NOUN cana-4527 20 6 , , PUNCT cana-4527 20 7 coimbra coimbra PROPN cana-4527 21 1 [ [ X cana-4527 21 2 4 4 NUM cana-4527 21 3 ] ] PUNCT cana-4527 21 4 , , PUNCT cana-4527 21 5 and and CCONJ cana-4527 21 6 others other NOUN cana-4527 21 7 . . PUNCT cana-4527 22 1 fractional fractional ADJ cana-4527 22 2 - - PUNCT cana-4527 22 3 order order NOUN cana-4527 22 4 differential differential ADJ cana-4527 22 5 communications communication NOUN cana-4527 22 6 on on ADP cana-4527 22 7 applied apply VERB cana-4527 22 8 nonlinear nonlinear ADJ cana-4527 22 9 analysis analysis NOUN cana-4527 22 10 issn issn NOUN cana-4527 22 11 : : PUNCT cana-4527 22 12 1074 1074 NUM cana-4527 22 13 - - PUNCT cana-4527 22 14 133x 133x NUM cana-4527 22 15 vol vol NOUN cana-4527 22 16 32 32 NUM cana-4527 22 17 no no NOUN cana-4527 22 18 . . PUNCT cana-4527 23 1 9s 9s NUM cana-4527 23 2 ( ( PUNCT cana-4527 23 3 2025 2025 NUM cana-4527 23 4 ) ) PUNCT cana-4527 23 5 2416 2416 NUM cana-4527 23 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 23 7 equations equation NOUN cana-4527 23 8 provide provide VERB cana-4527 23 9 valuable valuable ADJ cana-4527 23 10 insights insight NOUN cana-4527 23 11 into into ADP cana-4527 23 12 various various ADJ cana-4527 23 13 fundamental fundamental ADJ cana-4527 23 14 interactions interaction NOUN cana-4527 23 15 in in ADP cana-4527 23 16 electrochemistry electrochemistry NOUN cana-4527 23 17 , , PUNCT cana-4527 23 18 acoustics acoustic NOUN cana-4527 23 19 , , PUNCT cana-4527 23 20 electromagnetics electromagnetic NOUN cana-4527 23 21 , , PUNCT cana-4527 23 22 viscoelasticity viscoelasticity NOUN cana-4527 23 23 , , PUNCT cana-4527 23 24 and and CCONJ cana-4527 23 25 material material NOUN cana-4527 23 26 science science NOUN cana-4527 23 27 [ [ X cana-4527 23 28 6 6 NUM cana-4527 23 29 , , PUNCT cana-4527 23 30 7 7 NUM cana-4527 23 31 , , PUNCT cana-4527 23 32 8 8 NUM cana-4527 23 33 , , PUNCT cana-4527 23 34 9 9 NUM cana-4527 23 35 ] ] PUNCT cana-4527 23 36 . . PUNCT cana-4527 24 1 many many ADJ cana-4527 24 2 new new ADJ cana-4527 24 3 ideas idea NOUN cana-4527 24 4 about about ADP cana-4527 24 5 derivatives derivative NOUN cana-4527 24 6 of of ADP cana-4527 24 7 fractional fractional ADJ cana-4527 24 8 order order NOUN cana-4527 24 9 have have AUX cana-4527 24 10 been be AUX cana-4527 24 11 created create VERB cana-4527 24 12 . . PUNCT cana-4527 25 1 these these PRON cana-4527 25 2 comprise comprise VERB cana-4527 25 3 the the DET cana-4527 25 4 following following NOUN cana-4527 25 5 : : PUNCT cana-4527 25 6 the the DET cana-4527 25 7 generalized generalize VERB cana-4527 25 8 riemann riemann PROPN cana-4527 25 9 - - PUNCT cana-4527 25 10 liouville liouville PROPN cana-4527 25 11 and and CCONJ cana-4527 25 12 generalized generalize VERB cana-4527 25 13 caputo caputo PROPN cana-4527 25 14 derivative derivative NOUN cana-4527 25 15 by by ADP cana-4527 25 16 atangana atangana PROPN cana-4527 26 1 [ [ X cana-4527 26 2 12 12 NUM cana-4527 26 3 ] ] PUNCT cana-4527 26 4 , , PUNCT cana-4527 26 5 the the DET cana-4527 26 6 losada losada PROPN cana-4527 26 7 and and CCONJ cana-4527 26 8 nieto nieto PROPN cana-4527 27 1 [ [ X cana-4527 27 2 11 11 NUM cana-4527 27 3 ] ] PUNCT cana-4527 27 4 , , PUNCT cana-4527 27 5 the the DET cana-4527 27 6 yang yang PROPN cana-4527 27 7 - - PUNCT cana-4527 27 8 srivastava srivastava PROPN cana-4527 27 9 - - PUNCT cana-4527 27 10 machado machado PROPN cana-4527 27 11 derivative derivative NOUN cana-4527 27 12 by by ADP cana-4527 27 13 yang yang PROPN cana-4527 28 1 [ [ X cana-4527 28 2 11 11 NUM cana-4527 28 3 ] ] PUNCT cana-4527 28 4 , , PUNCT cana-4527 28 5 and and CCONJ cana-4527 28 6 the the DET cana-4527 28 7 caputo caputo PROPN cana-4527 28 8 - - PUNCT cana-4527 28 9 fabrizio fabrizio PROPN cana-4527 28 10 derived derive VERB cana-4527 28 11 form form NOUN cana-4527 28 12 by by ADP cana-4527 28 13 caputo caputo PROPN cana-4527 28 14 and and CCONJ cana-4527 28 15 fabrizio fabrizio PROPN cana-4527 28 16 [ [ X cana-4527 28 17 10 10 NUM cana-4527 28 18 ] ] PUNCT cana-4527 28 19 . . PUNCT cana-4527 29 1 the the DET cana-4527 29 2 caputo caputo PROPN cana-4527 29 3 - - PUNCT cana-4527 29 4 fabrizio fabrizio PROPN cana-4527 29 5 fractional fractional ADJ cana-4527 29 6 derivative derivative NOUN cana-4527 29 7 for for ADP cana-4527 29 8 the the DET cana-4527 29 9 mean mean ADJ cana-4527 29 10 square square ADJ cana-4527 29 11 displacement displacement NOUN cana-4527 29 12 exhibits exhibit VERB cana-4527 29 13 crossover crossover NOUN cana-4527 29 14 behaviours behaviour NOUN cana-4527 29 15 , , PUNCT cana-4527 29 16 whereas whereas SCONJ cana-4527 29 17 the the DET cana-4527 29 18 riemann riemann PROPN cana-4527 29 19 - - PUNCT cana-4527 29 20 liouville liouville NOUN cana-4527 29 21 is be AUX cana-4527 29 22 scale scale NOUN cana-4527 29 23 invariant invariant ADJ cana-4527 29 24 . . PUNCT cana-4527 30 1 while while SCONJ cana-4527 30 2 a a DET cana-4527 30 3 gaussian gaussian NOUN cana-4527 30 4 to to ADP cana-4527 30 5 non non ADJ cana-4527 30 6 - - ADJ cana-4527 30 7 gaussian gaussian ADJ cana-4527 30 8 crossover crossover NOUN cana-4527 30 9 dispersion dispersion NOUN cana-4527 30 10 is be AUX cana-4527 30 11 possible possible ADJ cana-4527 30 12 , , PUNCT cana-4527 30 13 the the DET cana-4527 30 14 caputo caputo PROPN cana-4527 30 15 - - PUNCT cana-4527 30 16 fabrizio fabrizio PROPN cana-4527 30 17 kernel kernel PROPN cana-4527 30 18 remains remain VERB cana-4527 30 19 constant constant ADJ cana-4527 30 20 throughout throughout ADP cana-4527 30 21 the the DET cana-4527 30 22 transition transition NOUN cana-4527 30 23 . . PUNCT cana-4527 31 1 more more ADV cana-4527 31 2 favourable favourable ADJ cana-4527 31 3 attributes attribute NOUN cana-4527 31 4 are be AUX cana-4527 31 5 demonstrated demonstrate VERB cana-4527 31 6 by by ADP cana-4527 31 7 the the DET cana-4527 31 8 new new PROPN cana-4527 31 9 caputo caputo PROPN cana-4527 31 10 - - PUNCT cana-4527 31 11 fabrizio fabrizio PROPN cana-4527 31 12 fractional fractional ADJ cana-4527 31 13 derivative derivative NOUN cana-4527 31 14 than than ADP cana-4527 31 15 its its PRON cana-4527 31 16 antecedent antecedent NOUN cana-4527 31 17 . . PUNCT cana-4527 32 1 it it PRON cana-4527 32 2 can can AUX cana-4527 32 3 disclose disclose VERB cana-4527 32 4 distinctions distinction NOUN cana-4527 32 5 and and CCONJ cana-4527 32 6 patterns pattern NOUN cana-4527 32 7 in in ADP cana-4527 32 8 substances substance NOUN cana-4527 32 9 on on ADP cana-4527 32 10 many many ADJ cana-4527 32 11 levels level NOUN cana-4527 32 12 that that PRON cana-4527 32 13 are be AUX cana-4527 32 14 not not PART cana-4527 32 15 attainable attainable ADJ cana-4527 32 16 through through ADP cana-4527 32 17 conventional conventional ADJ cana-4527 32 18 local local ADJ cana-4527 32 19 theories theory NOUN cana-4527 32 20 or or CCONJ cana-4527 32 21 popular popular ADJ cana-4527 32 22 fractional fractional ADJ cana-4527 32 23 order order NOUN cana-4527 32 24 derivatives derivative NOUN cana-4527 32 25 . . PUNCT cana-4527 33 1 in in ADP cana-4527 33 2 addition addition NOUN cana-4527 33 3 , , PUNCT cana-4527 33 4 these these DET cana-4527 33 5 varieties variety NOUN cana-4527 33 6 of of ADP cana-4527 33 7 materials material NOUN cana-4527 33 8 are be AUX cana-4527 33 9 used use VERB cana-4527 33 10 to to PART cana-4527 33 11 study study VERB cana-4527 33 12 the the DET cana-4527 33 13 large large ADJ cana-4527 33 14 - - PUNCT cana-4527 33 15 scale scale NOUN cana-4527 33 16 behaviours behaviour NOUN cana-4527 33 17 of of ADP cana-4527 33 18 some some DET cana-4527 33 19 materials material NOUN cana-4527 33 20 linked link VERB cana-4527 33 21 to to ADP cana-4527 33 22 atom atom NOUN cana-4527 33 23 - - PUNCT cana-4527 33 24 to to ADP cana-4527 33 25 - - PUNCT cana-4527 33 26 atom atom NOUN cana-4527 33 27 nonlocal nonlocal ADJ cana-4527 33 28 communications communication NOUN cana-4527 33 29 , , PUNCT cana-4527 33 30 which which PRON cana-4527 33 31 are be AUX cana-4527 33 32 known know VERB cana-4527 33 33 to to PART cana-4527 33 34 be be AUX cana-4527 33 35 important important ADJ cana-4527 33 36 material material NOUN cana-4527 33 37 properties property NOUN cana-4527 33 38 [ [ X cana-4527 33 39 13 13 NUM cana-4527 33 40 ] ] PUNCT cana-4527 33 41 . . PUNCT cana-4527 34 1 in in ADP cana-4527 34 2 recent recent ADJ cana-4527 34 3 years year NOUN cana-4527 34 4 , , PUNCT cana-4527 34 5 many many ADJ cana-4527 34 6 studies study NOUN cana-4527 34 7 have have AUX cana-4527 34 8 been be AUX cana-4527 34 9 conducted conduct VERB cana-4527 34 10 on on ADP cana-4527 34 11 fractional fractional ADJ cana-4527 34 12 - - PUNCT cana-4527 34 13 order order NOUN cana-4527 34 14 ordinary ordinary ADJ cana-4527 34 15 differential differential ADJ cana-4527 34 16 equations equation NOUN cana-4527 34 17 , , PUNCT cana-4527 34 18 partial partial ADJ cana-4527 34 19 differential differential NOUN cana-4527 34 20 equations equation NOUN cana-4527 34 21 , , PUNCT cana-4527 34 22 and and CCONJ cana-4527 34 23 partial partial ADJ cana-4527 34 24 integral integral ADJ cana-4527 34 25 differential differential NOUN cana-4527 34 26 equations equation NOUN cana-4527 34 27 . . PUNCT cana-4527 35 1 numerous numerous ADJ cana-4527 35 2 fields field NOUN cana-4527 35 3 use use VERB cana-4527 35 4 these these DET cana-4527 35 5 equations equation NOUN cana-4527 35 6 , , PUNCT cana-4527 35 7 including include VERB cana-4527 35 8 finance finance NOUN cana-4527 35 9 , , PUNCT cana-4527 35 10 economics economic NOUN cana-4527 35 11 , , PUNCT cana-4527 35 12 engineering engineering NOUN cana-4527 35 13 , , PUNCT cana-4527 35 14 and and CCONJ cana-4527 35 15 biology biology NOUN cana-4527 35 16 [ [ X cana-4527 35 17 14 14 NUM cana-4527 35 18 ] ] PUNCT cana-4527 35 19 . . PUNCT cana-4527 36 1 consequently consequently ADV cana-4527 36 2 , , PUNCT cana-4527 36 3 much much ADJ cana-4527 36 4 emphasis emphasis NOUN cana-4527 36 5 has have AUX cana-4527 36 6 been be AUX cana-4527 36 7 placed place VERB cana-4527 36 8 on on ADP cana-4527 36 9 the the DET cana-4527 36 10 solutions solution NOUN cana-4527 36 11 to to ADP cana-4527 36 12 these these DET cana-4527 36 13 equations equation NOUN cana-4527 36 14 of of ADP cana-4527 36 15 physical physical ADJ cana-4527 36 16 importance importance NOUN cana-4527 36 17 [ [ X cana-4527 36 18 15 15 NUM cana-4527 36 19 , , PUNCT cana-4527 36 20 16 16 NUM cana-4527 36 21 , , PUNCT cana-4527 36 22 17 17 NUM cana-4527 36 23 , , PUNCT cana-4527 36 24 18 18 NUM cana-4527 36 25 ] ] PUNCT cana-4527 36 26 . . PUNCT cana-4527 37 1 it it PRON cana-4527 37 2 is be AUX cana-4527 37 3 widely widely ADV cana-4527 37 4 known know VERB cana-4527 37 5 that that SCONJ cana-4527 37 6 most most ADJ cana-4527 37 7 of of ADP cana-4527 37 8 these these DET cana-4527 37 9 non non ADJ cana-4527 37 10 - - ADJ cana-4527 37 11 linear linear ADJ cana-4527 37 12 fractional fractional ADJ cana-4527 37 13 order order NOUN cana-4527 37 14 differential differential NOUN cana-4527 37 15 equations equation NOUN cana-4527 37 16 have have VERB cana-4527 37 17 analytical analytical ADJ cana-4527 37 18 solutions solution NOUN cana-4527 37 19 ; ; PUNCT cana-4527 37 20 however however ADV cana-4527 37 21 , , PUNCT cana-4527 37 22 they they PRON cana-4527 37 23 are be AUX cana-4527 37 24 either either CCONJ cana-4527 37 25 extremely extremely ADV cana-4527 37 26 complex complex ADJ cana-4527 37 27 to to PART cana-4527 37 28 use use VERB cana-4527 37 29 or or CCONJ cana-4527 37 30 completely completely ADV cana-4527 37 31 missing missing ADJ cana-4527 37 32 in in ADP cana-4527 37 33 closed closed ADJ cana-4527 37 34 - - PUNCT cana-4527 37 35 form form NOUN cana-4527 37 36 solutions solution NOUN cana-4527 37 37 . . PUNCT cana-4527 38 1 approximate approximate ADJ cana-4527 38 2 and and CCONJ cana-4527 38 3 computational computational ADJ cana-4527 38 4 strategies strategy NOUN cana-4527 38 5 are be AUX cana-4527 38 6 introduced introduce VERB cana-4527 38 7 in in ADP cana-4527 38 8 this this DET cana-4527 38 9 case case NOUN cana-4527 38 10 . . PUNCT cana-4527 39 1 many many ADJ cana-4527 39 2 techniques technique NOUN cana-4527 39 3 were be AUX cana-4527 39 4 employed employ VERB cana-4527 39 5 , , PUNCT cana-4527 39 6 including include VERB cana-4527 39 7 the the DET cana-4527 39 8 spectral spectral ADJ cana-4527 39 9 methodology methodology NOUN cana-4527 39 10 [ [ X cana-4527 39 11 20 20 NUM cana-4527 39 12 ] ] PUNCT cana-4527 39 13 , , PUNCT cana-4527 39 14 the the DET cana-4527 39 15 variational variational ADJ cana-4527 39 16 iteration iteration NOUN cana-4527 39 17 methodology methodology NOUN cana-4527 40 1 [ [ X cana-4527 40 2 21 21 NUM cana-4527 40 3 ] ] PUNCT cana-4527 40 4 , , PUNCT cana-4527 40 5 the the DET cana-4527 40 6 adomian adomian NOUN cana-4527 40 7 decomposition decomposition NOUN cana-4527 40 8 methodology methodology NOUN cana-4527 40 9 [ [ X cana-4527 40 10 22 22 NUM cana-4527 40 11 , , PUNCT cana-4527 40 12 23 23 NUM cana-4527 40 13 ] ] PUNCT cana-4527 40 14 , , PUNCT cana-4527 40 15 and and CCONJ cana-4527 40 16 the the DET cana-4527 40 17 finite finite ADJ cana-4527 40 18 difference difference NOUN cana-4527 40 19 method method NOUN cana-4527 40 20 [ [ X cana-4527 40 21 19 19 NUM cana-4527 40 22 ] ] PUNCT cana-4527 40 23 . . PUNCT cana-4527 41 1 to to PART cana-4527 41 2 promote promote VERB cana-4527 41 3 worldwide worldwide ADJ cana-4527 41 4 research research NOUN cana-4527 41 5 and and CCONJ cana-4527 41 6 useful useful ADJ cana-4527 41 7 applications application NOUN cana-4527 41 8 of of ADP cana-4527 41 9 related related ADJ cana-4527 41 10 approaches approach NOUN cana-4527 41 11 , , PUNCT cana-4527 41 12 the the DET cana-4527 41 13 regional regional ADJ cana-4527 41 14 groundwater groundwater NOUN cana-4527 41 15 flow flow NOUN cana-4527 41 16 commission commission NOUN cana-4527 41 17 ( ( PUNCT cana-4527 41 18 rgfc rgfc NOUN cana-4527 41 19 ) ) PUNCT cana-4527 41 20 was be AUX cana-4527 41 21 established establish VERB cana-4527 41 22 in in ADP cana-4527 41 23 2011 2011 NUM cana-4527 41 24 following follow VERB cana-4527 41 25 the the DET cana-4527 41 26 international international ADJ cana-4527 41 27 association association NOUN cana-4527 41 28 of of ADP cana-4527 41 29 hydrogeologists hydrogeologist NOUN cana-4527 41 30 ( ( PUNCT cana-4527 41 31 iah iah PROPN cana-4527 41 32 ) ) PUNCT cana-4527 41 33 's 's PART cana-4527 41 34 recognition recognition NOUN cana-4527 41 35 of of ADP cana-4527 41 36 the the DET cana-4527 41 37 issue issue NOUN cana-4527 41 38 of of ADP cana-4527 41 39 " " PUNCT cana-4527 41 40 groundwater groundwater NOUN cana-4527 41 41 flow flow NOUN cana-4527 41 42 . . PUNCT cana-4527 41 43 " " PUNCT cana-4527 42 1 according accord VERB cana-4527 42 2 to to ADP cana-4527 42 3 an an DET cana-4527 42 4 evaluation evaluation NOUN cana-4527 42 5 of of ADP cana-4527 42 6 the the DET cana-4527 42 7 future future NOUN cana-4527 42 8 of of ADP cana-4527 42 9 groundwater groundwater NOUN cana-4527 42 10 science science NOUN cana-4527 42 11 and and CCONJ cana-4527 42 12 research research NOUN cana-4527 42 13 , , PUNCT cana-4527 42 14 new new ADJ cana-4527 42 15 instruments instrument NOUN cana-4527 42 16 for for ADP cana-4527 42 17 contemporary contemporary ADJ cana-4527 42 18 hydrogeology hydrogeology NOUN cana-4527 42 19 and and CCONJ cana-4527 42 20 basin basin NOUN cana-4527 42 21 - - PUNCT cana-4527 42 22 wide wide ADJ cana-4527 42 23 investigations investigation NOUN cana-4527 42 24 of of ADP cana-4527 42 25 regional regional ADJ cana-4527 42 26 groundwater groundwater NOUN cana-4527 42 27 flow flow NOUN cana-4527 42 28 systems system NOUN cana-4527 42 29 are be AUX cana-4527 42 30 required require VERB cana-4527 42 31 . . PUNCT cana-4527 43 1 this this DET cana-4527 43 2 study study NOUN cana-4527 43 3 will will AUX cana-4527 43 4 concentrate concentrate VERB cana-4527 43 5 on on ADP cana-4527 43 6 the the DET cana-4527 43 7 suggested suggest VERB cana-4527 43 8 fixed fix VERB cana-4527 43 9 transform transform NOUN cana-4527 43 10 domain domain NOUN cana-4527 43 11 ( ( PUNCT cana-4527 43 12 ftd ftd NOUN cana-4527 43 13 ) ) PUNCT cana-4527 43 14 approach approach NOUN cana-4527 43 15 to to ADP cana-4527 43 16 solving solve VERB cana-4527 43 17 fractional fractional ADJ cana-4527 43 18 differential differential ADJ cana-4527 43 19 equations equation NOUN cana-4527 43 20 based base VERB cana-4527 43 21 on on ADP cana-4527 43 22 groundwater groundwater NOUN cana-4527 43 23 flow flow NOUN cana-4527 43 24 . . PUNCT cana-4527 44 1 this this DET cana-4527 44 2 transformation transformation NOUN cana-4527 44 3 can can AUX cana-4527 44 4 be be AUX cana-4527 44 5 helpful helpful ADJ cana-4527 44 6 in in ADP cana-4527 44 7 differential differential ADJ cana-4527 44 8 equations equation NOUN cana-4527 44 9 since since SCONJ cana-4527 44 10 it it PRON cana-4527 44 11 can can AUX cana-4527 44 12 make make VERB cana-4527 44 13 the the DET cana-4527 44 14 equations equation NOUN cana-4527 44 15 easier easy ADJ cana-4527 44 16 to to PART cana-4527 44 17 solve solve VERB cana-4527 44 18 . . PUNCT cana-4527 45 1 additionally additionally ADV cana-4527 45 2 , , PUNCT cana-4527 45 3 it it PRON cana-4527 45 4 creates create VERB cana-4527 45 5 additional additional ADJ cana-4527 45 6 complications complication NOUN cana-4527 45 7 when when SCONJ cana-4527 45 8 boundary boundary ADJ cana-4527 45 9 conditions condition NOUN cana-4527 45 10 are be AUX cana-4527 45 11 added add VERB cana-4527 45 12 to to ADP cana-4527 45 13 fractional fractional ADJ cana-4527 45 14 differential differential ADJ cana-4527 45 15 equations equation NOUN cana-4527 45 16 . . PUNCT cana-4527 46 1 given give VERB cana-4527 46 2 that that SCONJ cana-4527 46 3 the the DET cana-4527 46 4 suggested suggest VERB cana-4527 46 5 approach approach NOUN cana-4527 46 6 is be AUX cana-4527 46 7 limited limit VERB cana-4527 46 8 to to ADP cana-4527 46 9 handling handle VERB cana-4527 46 10 the the DET cana-4527 46 11 fractional fractional ADJ cana-4527 46 12 space space NOUN cana-4527 46 13 operator operator NOUN cana-4527 46 14 and and CCONJ cana-4527 46 15 the the DET cana-4527 46 16 beginning beginning ADJ cana-4527 46 17 conditions condition NOUN cana-4527 46 18 . . PUNCT cana-4527 47 1 these these DET cana-4527 47 2 characteristics characteristic NOUN cana-4527 47 3 allow allow VERB cana-4527 47 4 for for ADP cana-4527 47 5 reduced reduce VERB cana-4527 47 6 computing computing NOUN cana-4527 47 7 time time NOUN cana-4527 47 8 and and CCONJ cana-4527 47 9 rapid rapid ADJ cana-4527 47 10 method method NOUN cana-4527 47 11 execution execution NOUN cana-4527 47 12 . . PUNCT cana-4527 48 1 numerical numerical ADJ cana-4527 48 2 experiments experiment NOUN cana-4527 48 3 show show VERB cana-4527 48 4 the the DET cana-4527 48 5 precision precision NOUN cana-4527 48 6 and and CCONJ cana-4527 48 7 efficacy efficacy NOUN cana-4527 48 8 of of ADP cana-4527 48 9 the the DET cana-4527 48 10 proposed propose VERB cana-4527 48 11 methodology methodology NOUN cana-4527 48 12 . . PUNCT cana-4527 49 1 2 2 X cana-4527 49 2 . . X cana-4527 49 3 related relate VERB cana-4527 49 4 work work NOUN cana-4527 49 5 many many ADJ cana-4527 49 6 scientific scientific ADJ cana-4527 49 7 and and CCONJ cana-4527 49 8 engineering engineering NOUN cana-4527 49 9 investigations investigation NOUN cana-4527 49 10 regarding regard VERB cana-4527 49 11 numerical numerical ADJ cana-4527 49 12 simulations simulation NOUN cana-4527 49 13 and and CCONJ cana-4527 49 14 mathematical mathematical ADJ cana-4527 49 15 modelling modelling NOUN cana-4527 49 16 of of ADP cana-4527 49 17 dynamical dynamical ADJ cana-4527 49 18 systems system NOUN cana-4527 49 19 have have AUX cana-4527 49 20 been be AUX cana-4527 49 21 implemented implement VERB cana-4527 49 22 . . PUNCT cana-4527 50 1 fractional fractional ADJ cana-4527 50 2 order order NOUN cana-4527 50 3 operators operator NOUN cana-4527 50 4 are be AUX cana-4527 50 5 a a DET cana-4527 50 6 wellresearched wellresearched ADJ cana-4527 50 7 and and CCONJ cana-4527 50 8 practical practical ADJ cana-4527 50 9 approach approach NOUN cana-4527 50 10 to to ADP cana-4527 50 11 enhancing enhance VERB cana-4527 50 12 the the DET cana-4527 50 13 capabilities capability NOUN cana-4527 50 14 of of ADP cana-4527 50 15 traditional traditional ADJ cana-4527 50 16 models model NOUN cana-4527 50 17 . . PUNCT cana-4527 51 1 the the DET cana-4527 51 2 history history NOUN cana-4527 51 3 of of ADP cana-4527 51 4 communications communication NOUN cana-4527 51 5 on on ADP cana-4527 51 6 applied apply VERB cana-4527 51 7 nonlinear nonlinear ADJ cana-4527 51 8 analysis analysis NOUN cana-4527 51 9 issn issn NOUN cana-4527 51 10 : : PUNCT cana-4527 51 11 1074 1074 NUM cana-4527 51 12 - - PUNCT cana-4527 51 13 133x 133x NUM cana-4527 51 14 vol vol NOUN cana-4527 51 15 32 32 NUM cana-4527 51 16 no no NOUN cana-4527 51 17 . . PUNCT cana-4527 52 1 9s 9s NUM cana-4527 52 2 ( ( PUNCT cana-4527 52 3 2025 2025 NUM cana-4527 52 4 ) ) PUNCT cana-4527 52 5 2417 2417 NUM cana-4527 52 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 52 7 fractional fractional ADJ cana-4527 52 8 order order NOUN cana-4527 52 9 operators operator NOUN cana-4527 52 10 includes include VERB cana-4527 52 11 solitary solitary ADJ cana-4527 52 12 and and CCONJ cana-4527 52 13 non non ADJ cana-4527 52 14 - - ADJ cana-4527 52 15 singular singular ADJ cana-4527 52 16 kernels kernel NOUN cana-4527 52 17 and and CCONJ cana-4527 52 18 local local ADJ cana-4527 52 19 and and CCONJ cana-4527 52 20 nonlocal nonlocal ADJ cana-4527 52 21 kernels kernel NOUN cana-4527 52 22 . . PUNCT cana-4527 53 1 a a DET cana-4527 53 2 wide wide ADJ cana-4527 53 3 range range NOUN cana-4527 53 4 of of ADP cana-4527 53 5 fractional fractional ADJ cana-4527 53 6 differential differential ADJ cana-4527 53 7 equations equation NOUN cana-4527 53 8 ( ( PUNCT cana-4527 53 9 fdes fde NOUN cana-4527 53 10 ) ) PUNCT cana-4527 53 11 have have AUX cana-4527 53 12 been be AUX cana-4527 53 13 studied study VERB cana-4527 53 14 by by ADP cana-4527 53 15 specialists specialist NOUN cana-4527 53 16 , , PUNCT cana-4527 53 17 including include VERB cana-4527 53 18 sequential sequential ADJ cana-4527 53 19 , , PUNCT cana-4527 53 20 hybrid hybrid NOUN cana-4527 53 21 , , PUNCT cana-4527 53 22 mixed mixed ADJ cana-4527 53 23 , , PUNCT cana-4527 53 24 and and CCONJ cana-4527 53 25 many many ADJ cana-4527 53 26 other other ADJ cana-4527 53 27 classes class NOUN cana-4527 53 28 that that PRON cana-4527 53 29 have have AUX cana-4527 53 30 not not PART cana-4527 53 31 yet yet ADV cana-4527 53 32 been be AUX cana-4527 53 33 fully fully ADV cana-4527 53 34 investigated investigate VERB cana-4527 53 35 in in ADP cana-4527 53 36 this this DET cana-4527 53 37 field field NOUN cana-4527 53 38 . . PUNCT cana-4527 54 1 plot plot NOUN cana-4527 54 2 perturbation perturbation NOUN cana-4527 54 3 techniques technique NOUN cana-4527 54 4 are be AUX cana-4527 54 5 extremely extremely ADV cana-4527 54 6 helpful helpful ADJ cana-4527 54 7 in in ADP cana-4527 54 8 non non ADJ cana-4527 54 9 - - ADJ cana-4527 54 10 linear linear ADJ cana-4527 54 11 analysis analysis NOUN cana-4527 54 12 because because SCONJ cana-4527 54 13 they they PRON cana-4527 54 14 facilitate facilitate VERB cana-4527 54 15 comprehension comprehension NOUN cana-4527 54 16 of of ADP cana-4527 54 17 system system NOUN cana-4527 54 18 dynamics dynamic NOUN cana-4527 54 19 represented represent VERB cana-4527 54 20 by by ADP cana-4527 54 21 a a DET cana-4527 54 22 diverse diverse ADJ cana-4527 54 23 array array NOUN cana-4527 54 24 of of ADP cana-4527 54 25 mathematical mathematical ADJ cana-4527 54 26 approaches approach NOUN cana-4527 54 27 . . PUNCT cana-4527 55 1 solving solve VERB cana-4527 55 2 or or CCONJ cana-4527 55 3 evaluating evaluate VERB cana-4527 55 4 a a DET cana-4527 55 5 differential differential ADJ cana-4527 55 6 equation equation NOUN cana-4527 55 7 that that PRON cana-4527 55 8 characterizes characterize VERB cana-4527 55 9 a a DET cana-4527 55 10 certain certain ADJ cana-4527 55 11 dynamical dynamical ADJ cana-4527 55 12 system system NOUN cana-4527 55 13 can can AUX cana-4527 55 14 often often ADV cana-4527 55 15 be be AUX cana-4527 55 16 difficult difficult ADJ cana-4527 55 17 . . PUNCT cana-4527 56 1 to to PART cana-4527 56 2 determine determine VERB cana-4527 56 3 whether whether SCONJ cana-4527 56 4 any any DET cana-4527 56 5 unique unique ADJ cana-4527 56 6 answers answer NOUN cana-4527 56 7 ( ( PUNCT cana-4527 56 8 eus eus NOUN cana-4527 56 9 ) ) PUNCT cana-4527 56 10 existed exist VERB cana-4527 56 11 in in ADP cana-4527 56 12 the the DET cana-4527 56 13 riemann riemann PROPN cana-4527 56 14 - - PUNCT cana-4527 56 15 liouville liouville NOUN cana-4527 56 16 meaning meaning NOUN cana-4527 56 17 of of ADP cana-4527 56 18 the the DET cana-4527 56 19 derivatives derivative NOUN cana-4527 56 20 , , PUNCT cana-4527 56 21 zhao zhao PROPN cana-4527 56 22 et et PROPN cana-4527 56 23 al al PROPN cana-4527 56 24 . . PUNCT cana-4527 57 1 [ [ X cana-4527 57 2 24 24 NUM cana-4527 57 3 ] ] PUNCT cana-4527 57 4 examined examine VERB cana-4527 57 5 the the DET cana-4527 57 6 second second ADJ cana-4527 57 7 kind kind NOUN cana-4527 57 8 of of ADP cana-4527 57 9 quadratic quadratic ADJ cana-4527 57 10 perturbation perturbation NOUN cana-4527 57 11 problem problem NOUN cana-4527 57 12 . . PUNCT cana-4527 58 1 in in ADP cana-4527 58 2 the the DET cana-4527 58 3 study study NOUN cana-4527 58 4 of of ADP cana-4527 58 5 sitho sitho PROPN cana-4527 58 6 et et PROPN cana-4527 58 7 al al PROPN cana-4527 58 8 . . PUNCT cana-4527 59 1 [ [ X cana-4527 59 2 25 25 NUM cana-4527 59 3 ] ] PUNCT cana-4527 59 4 , , PUNCT cana-4527 59 5 fractional fractional ADJ cana-4527 59 6 integro integro ADJ cana-4527 59 7 - - PUNCT cana-4527 59 8 differential differential NOUN cana-4527 59 9 equations equation NOUN cana-4527 59 10 for for ADP cana-4527 59 11 the the DET cana-4527 59 12 eus eus NOUN cana-4527 59 13 were be AUX cana-4527 59 14 investigated investigate VERB cana-4527 59 15 along along ADP cana-4527 59 16 with with ADP cana-4527 59 17 possible possible ADJ cana-4527 59 18 applications application NOUN cana-4527 59 19 in in ADP cana-4527 59 20 situations situation NOUN cana-4527 59 21 where where SCONJ cana-4527 59 22 the the DET cana-4527 59 23 derivative derivative NOUN cana-4527 59 24 was be AUX cana-4527 59 25 in in ADP cana-4527 59 26 the the DET cana-4527 59 27 r r NOUN cana-4527 59 28 - - PUNCT cana-4527 59 29 l l NOUN cana-4527 59 30 sense sense NOUN cana-4527 59 31 . . PUNCT cana-4527 60 1 in in ADP cana-4527 60 2 work work NOUN cana-4527 60 3 [ [ X cana-4527 60 4 26 26 NUM cana-4527 60 5 ] ] PUNCT cana-4527 60 6 , , PUNCT cana-4527 60 7 awadalla awadalla NOUN cana-4527 60 8 and and CCONJ cana-4527 60 9 abuasbeh abuasbeh PROPN cana-4527 60 10 investigated investigate VERB cana-4527 60 11 a a DET cana-4527 60 12 second second ADJ cana-4527 60 13 - - PUNCT cana-4527 60 14 class class NOUN cana-4527 60 15 perturbed perturb VERB cana-4527 60 16 sequential sequential ADJ cana-4527 60 17 fde fde NOUN cana-4527 60 18 for for ADP cana-4527 60 19 the the DET cana-4527 60 20 eus eus PROPN cana-4527 60 21 for for ADP cana-4527 60 22 caputo caputo PROPN cana-4527 60 23 - - PUNCT cana-4527 60 24 hadamard hadamard PROPN cana-4527 60 25 operators operator NOUN cana-4527 60 26 . . PUNCT cana-4527 61 1 a a DET cana-4527 61 2 group group NOUN cana-4527 61 3 of of ADP cana-4527 61 4 hybridized hybridize VERB cana-4527 61 5 fdes fde NOUN cana-4527 61 6 were be AUX cana-4527 61 7 studied study VERB cana-4527 61 8 by by ADP cana-4527 61 9 gul gul PROPN cana-4527 61 10 et et PROPN cana-4527 61 11 al al PROPN cana-4527 61 12 . . PUNCT cana-4527 62 1 [ [ X cana-4527 62 2 27 27 NUM cana-4527 62 3 ] ] PUNCT cana-4527 62 4 , , PUNCT cana-4527 62 5 who who PRON cana-4527 62 6 applied apply VERB cana-4527 62 7 their their PRON cana-4527 62 8 results result NOUN cana-4527 62 9 to to ADP cana-4527 62 10 dynamical dynamical ADJ cana-4527 62 11 issues issue NOUN cana-4527 62 12 . . PUNCT cana-4527 63 1 their their PRON cana-4527 63 2 function function NOUN cana-4527 63 3 was be AUX cana-4527 63 4 the the DET cana-4527 63 5 derivative derivative NOUN cana-4527 63 6 of of ADP cana-4527 63 7 caputo caputo PROPN cana-4527 63 8 . . PUNCT cana-4527 64 1 khan khan PROPN cana-4527 64 2 et et PROPN cana-4527 64 3 al al PROPN cana-4527 64 4 . . PUNCT cana-4527 65 1 [ [ X cana-4527 65 2 28 28 NUM cana-4527 65 3 ] ] PUNCT cana-4527 65 4 used use VERB cana-4527 65 5 the the DET cana-4527 65 6 leray leray ADJ cana-4527 65 7 - - PUNCT cana-4527 65 8 schauder schauder NOUN cana-4527 65 9 and and CCONJ cana-4527 65 10 banach banach ADV cana-4527 65 11 alternative alternative ADJ cana-4527 65 12 theorems theorem NOUN cana-4527 65 13 to to PART cana-4527 65 14 analyze analyze VERB cana-4527 65 15 several several ADJ cana-4527 65 16 mixed mixed ADJ cana-4527 65 17 fdes fde NOUN cana-4527 65 18 for for ADP cana-4527 65 19 eus eus PROPN cana-4527 65 20 and and CCONJ cana-4527 65 21 hyers hyers PROPN cana-4527 65 22 - - PUNCT cana-4527 65 23 ulam ulam PROPN cana-4527 65 24 stability stability NOUN cana-4527 65 25 . . PUNCT cana-4527 66 1 they they PRON cana-4527 66 2 employed employ VERB cana-4527 66 3 two two NUM cana-4527 66 4 distinct distinct ADJ cana-4527 66 5 forms form NOUN cana-4527 66 6 of of ADP cana-4527 66 7 fractional fractional ADJ cana-4527 66 8 ordering ordering NOUN cana-4527 66 9 : : PUNCT cana-4527 66 10 the the DET cana-4527 66 11 ab ab ADJ cana-4527 66 12 - - PUNCT cana-4527 66 13 fractional fractional ADJ cana-4527 66 14 operator operator NOUN cana-4527 66 15 and and CCONJ cana-4527 66 16 caputo caputo PROPN cana-4527 66 17 's 's PART cana-4527 66 18 derivative derivative NOUN cana-4527 66 19 of of ADP cana-4527 66 20 fractions fraction NOUN cana-4527 66 21 . . PUNCT cana-4527 67 1 losada losada PROPN cana-4527 67 2 and and CCONJ cana-4527 67 3 nieto nieto PROPN cana-4527 68 1 [ [ X cana-4527 68 2 29 29 NUM cana-4527 68 3 ] ] PUNCT cana-4527 68 4 looked look VERB cana-4527 68 5 into into ADP cana-4527 68 6 sequential sequential ADJ cana-4527 68 7 fdes fde NOUN cana-4527 68 8 for for ADP cana-4527 68 9 the the DET cana-4527 68 10 eus eus NOUN cana-4527 68 11 with with ADP cana-4527 68 12 a a DET cana-4527 68 13 non non ADJ cana-4527 68 14 - - ADJ cana-4527 68 15 singular singular ADJ cana-4527 68 16 kernel kernel NOUN cana-4527 68 17 . . PUNCT cana-4527 69 1 the the DET cana-4527 69 2 description description NOUN cana-4527 69 3 of of ADP cana-4527 69 4 a a DET cana-4527 69 5 fractional fractional ADJ cana-4527 69 6 derivative derivative NOUN cana-4527 69 7 with with ADP cana-4527 69 8 a a DET cana-4527 69 9 kernel kernel NOUN cana-4527 69 10 that that PRON cana-4527 69 11 is be AUX cana-4527 69 12 not not PART cana-4527 69 13 a a DET cana-4527 69 14 singular singular ADJ cana-4527 69 15 number number NOUN cana-4527 69 16 is be AUX cana-4527 69 17 provided provide VERB cana-4527 69 18 by by ADP cana-4527 69 19 caputo caputo PROPN cana-4527 69 20 , , PUNCT cana-4527 69 21 m. m. NOUN cana-4527 69 22 fabrizio fabrizio PROPN cana-4527 69 23 [ [ X cana-4527 69 24 30 30 NUM cana-4527 69 25 ] ] PUNCT cana-4527 69 26 . . PUNCT cana-4527 70 1 atangana atangana PROPN cana-4527 70 2 and and CCONJ cana-4527 70 3 baleanu baleanu NOUN cana-4527 71 1 [ [ X cana-4527 71 2 31 31 NUM cana-4527 71 3 ] ] PUNCT cana-4527 71 4 discussed discuss VERB cana-4527 71 5 the the DET cana-4527 71 6 idea idea NOUN cana-4527 71 7 of of ADP cana-4527 71 8 the the DET cana-4527 71 9 ab ab ADJ cana-4527 71 10 - - PUNCT cana-4527 71 11 fractional fractional ADJ cana-4527 71 12 derivative derivative NOUN cana-4527 71 13 and and CCONJ cana-4527 71 14 how how SCONJ cana-4527 71 15 it it PRON cana-4527 71 16 can can AUX cana-4527 71 17 be be AUX cana-4527 71 18 used use VERB cana-4527 71 19 . . PUNCT cana-4527 72 1 the the DET cana-4527 72 2 concept concept NOUN cana-4527 72 3 in in ADP cana-4527 72 4 [ [ X cana-4527 72 5 31 31 NUM cana-4527 72 6 ] ] PUNCT cana-4527 72 7 was be AUX cana-4527 72 8 expanded expand VERB cana-4527 72 9 upon upon SCONJ cana-4527 72 10 by by ADP cana-4527 72 11 al al PROPN cana-4527 72 12 - - PUNCT cana-4527 72 13 refai refai PROPN cana-4527 72 14 and and CCONJ cana-4527 72 15 baleanu baleanu NOUN cana-4527 73 1 [ [ X cana-4527 73 2 32 32 NUM cana-4527 73 3 ] ] PUNCT cana-4527 73 4 , , PUNCT cana-4527 73 5 who who PRON cana-4527 73 6 also also ADV cana-4527 73 7 resolved resolve VERB cana-4527 73 8 the the DET cana-4527 73 9 ab ab ADJ cana-4527 73 10 - - PUNCT cana-4527 73 11 operator operator NOUN cana-4527 73 12 setup setup NOUN cana-4527 73 13 issue issue NOUN cana-4527 73 14 . . PUNCT cana-4527 74 1 dhage dhage NOUN cana-4527 75 1 [ [ X cana-4527 75 2 34 34 NUM cana-4527 75 3 ] ] PUNCT cana-4527 75 4 and and CCONJ cana-4527 75 5 dhage dhage NOUN cana-4527 75 6 et et PROPN cana-4527 75 7 al al PROPN cana-4527 75 8 . . PUNCT cana-4527 76 1 [ [ X cana-4527 76 2 33 33 NUM cana-4527 76 3 ] ] PUNCT cana-4527 76 4 looked look VERB cana-4527 76 5 into into ADP cana-4527 76 6 mixed mixed ADJ cana-4527 76 7 classes class NOUN cana-4527 76 8 of of ADP cana-4527 76 9 fractional fractional ADJ cana-4527 76 10 differential differential ADJ cana-4527 76 11 equations equation NOUN cana-4527 76 12 to to PART cana-4527 76 13 see see VERB cana-4527 76 14 if if SCONJ cana-4527 76 15 there there PRON cana-4527 76 16 were be VERB cana-4527 76 17 any any DET cana-4527 76 18 solutions solution NOUN cana-4527 76 19 and and CCONJ cana-4527 76 20 only only ADV cana-4527 76 21 one one NUM cana-4527 76 22 . . PUNCT cana-4527 77 1 the the DET cana-4527 77 2 ab ab PROPN cana-4527 77 3 operator operator NOUN cana-4527 77 4 for for ADP cana-4527 77 5 the the DET cana-4527 77 6 derivative derivative NOUN cana-4527 77 7 was be AUX cana-4527 77 8 modified modify VERB cana-4527 77 9 by by ADP cana-4527 77 10 the the DET cana-4527 77 11 concept concept NOUN cana-4527 77 12 of of ADP cana-4527 77 13 the the DET cana-4527 77 14 inverse inverse NOUN cana-4527 77 15 operator operator NOUN cana-4527 77 16 of of ADP cana-4527 77 17 fractional fractional ADJ cana-4527 77 18 order order NOUN cana-4527 77 19 , , PUNCT cana-4527 77 20 which which PRON cana-4527 77 21 was be AUX cana-4527 77 22 first first ADV cana-4527 77 23 presented present VERB cana-4527 77 24 by by ADP cana-4527 77 25 al al PROPN cana-4527 77 26 - - PUNCT cana-4527 77 27 refai refai PROPN cana-4527 77 28 [ [ X cana-4527 77 29 35 35 NUM cana-4527 77 30 ] ] PUNCT cana-4527 77 31 and and CCONJ cana-4527 77 32 demonstrated demonstrate VERB cana-4527 77 33 to to PART cana-4527 77 34 have have VERB cana-4527 77 35 a a DET cana-4527 77 36 wide wide ADJ cana-4527 77 37 range range NOUN cana-4527 77 38 of of ADP cana-4527 77 39 applications application NOUN cana-4527 77 40 . . PUNCT cana-4527 78 1 khan khan PROPN cana-4527 78 2 et et PROPN cana-4527 78 3 al al PROPN cana-4527 78 4 . . PUNCT cana-4527 79 1 [ [ X cana-4527 79 2 36 36 NUM cana-4527 79 3 ] ] PUNCT cana-4527 79 4 used use VERB cana-4527 79 5 the the DET cana-4527 79 6 problem problem NOUN cana-4527 79 7 of of ADP cana-4527 79 8 imperfect imperfect ADJ cana-4527 79 9 tests test NOUN cana-4527 79 10 to to PART cana-4527 79 11 show show VERB cana-4527 79 12 some some DET cana-4527 79 13 simulations simulation NOUN cana-4527 79 14 for for ADP cana-4527 79 15 a a DET cana-4527 79 16 disease disease NOUN cana-4527 79 17 model model NOUN cana-4527 79 18 . . PUNCT cana-4527 80 1 shi shi PROPN cana-4527 80 2 and and CCONJ cana-4527 80 3 cui cui VERB cana-4527 80 4 [ [ X cana-4527 80 5 37 37 NUM cana-4527 80 6 ] ] PUNCT cana-4527 80 7 picked pick VERB cana-4527 80 8 up up ADP cana-4527 80 9 hepatitis hepatitis PROPN cana-4527 80 10 c c PROPN cana-4527 80 11 and and CCONJ cana-4527 80 12 were be AUX cana-4527 80 13 given give VERB cana-4527 80 14 the the DET cana-4527 80 15 conditions condition NOUN cana-4527 80 16 they they PRON cana-4527 80 17 needed need VERB cana-4527 80 18 to to PART cana-4527 80 19 stay stay VERB cana-4527 80 20 stable stable ADJ cana-4527 80 21 . . PUNCT cana-4527 81 1 subramanian subramanian PROPN cana-4527 82 1 [ [ X cana-4527 82 2 38 38 NUM cana-4527 82 3 , , PUNCT cana-4527 82 4 39 39 NUM cana-4527 82 5 ] ] PUNCT cana-4527 82 6 discussed discuss VERB cana-4527 82 7 the the DET cana-4527 82 8 existence existence NOUN cana-4527 82 9 of of ADP cana-4527 82 10 a a DET cana-4527 82 11 solution solution NOUN cana-4527 82 12 for for ADP cana-4527 82 13 a a DET cana-4527 82 14 connected connected ADJ cana-4527 82 15 system system NOUN cana-4527 82 16 with with ADP cana-4527 82 17 integro integro ADJ cana-4527 82 18 - - PUNCT cana-4527 82 19 differential differential NOUN cana-4527 82 20 equations equation NOUN cana-4527 82 21 and and CCONJ cana-4527 82 22 sequential sequential ADJ cana-4527 82 23 fractional fractional ADJ cana-4527 82 24 operators operator NOUN cana-4527 82 25 . . PUNCT cana-4527 83 1 a a DET cana-4527 83 2 family family NOUN cana-4527 83 3 of of ADP cana-4527 83 4 fractional fractional ADJ cana-4527 83 5 order order NOUN cana-4527 83 6 rapid rapid ADJ cana-4527 83 7 integro integro ADJ cana-4527 83 8 - - PUNCT cana-4527 83 9 differential differential NOUN cana-4527 83 10 equations equation NOUN cana-4527 83 11 with with ADP cana-4527 83 12 boundary boundary ADJ cana-4527 83 13 conditions condition NOUN cana-4527 83 14 was be AUX cana-4527 83 15 examined examine VERB cana-4527 83 16 by by ADP cana-4527 83 17 zada zada PROPN cana-4527 83 18 et et PROPN cana-4527 83 19 al al PROPN cana-4527 83 20 . . PUNCT cana-4527 84 1 [ [ X cana-4527 84 2 40 40 NUM cana-4527 84 3 ] ] PUNCT cana-4527 84 4 in in ADP cana-4527 84 5 terms term NOUN cana-4527 84 6 of of ADP cana-4527 84 7 ulam ulam NOUN cana-4527 84 8 - - PUNCT cana-4527 84 9 hyers hyer NOUN cana-4527 84 10 stability stability NOUN cana-4527 84 11 . . PUNCT cana-4527 85 1 3 3 X cana-4527 85 2 . . X cana-4527 85 3 problem problem NOUN cana-4527 85 4 statement statement NOUN cana-4527 85 5 the the DET cana-4527 85 6 groundwater groundwater NOUN cana-4527 85 7 problem problem NOUN cana-4527 85 8 is be AUX cana-4527 85 9 perhaps perhaps ADV cana-4527 85 10 the the DET cana-4527 85 11 hardest hard ADJ cana-4527 85 12 to to PART cana-4527 85 13 represent represent VERB cana-4527 85 14 mathematically mathematically ADV cana-4527 85 15 among among ADP cana-4527 85 16 the the DET cana-4527 85 17 realworld realworld PROPN cana-4527 85 18 issues issue NOUN cana-4527 85 19 . . PUNCT cana-4527 86 1 the the DET cana-4527 86 2 behaviour behaviour NOUN cana-4527 86 3 of of ADP cana-4527 86 4 the the DET cana-4527 86 5 medium medium NOUN cana-4527 86 6 through through ADP cana-4527 86 7 which which PRON cana-4527 86 8 the the DET cana-4527 86 9 water water NOUN cana-4527 86 10 travels travel VERB cana-4527 86 11 must must AUX cana-4527 86 12 be be AUX cana-4527 86 13 precisely precisely ADV cana-4527 86 14 understood understand VERB cana-4527 86 15 to to PART cana-4527 86 16 model model VERB cana-4527 86 17 this this DET cana-4527 86 18 problem problem NOUN cana-4527 86 19 accurately accurately ADV cana-4527 86 20 . . PUNCT cana-4527 87 1 however however ADV cana-4527 87 2 , , PUNCT cana-4527 87 3 the the DET cana-4527 87 4 medium medium NOUN cana-4527 87 5 in in ADP cana-4527 87 6 which which PRON cana-4527 87 7 the the DET cana-4527 87 8 flux flux NOUN cana-4527 87 9 happens happen VERB cana-4527 87 10 might might AUX cana-4527 87 11 change change VERB cana-4527 87 12 throughout throughout ADP cana-4527 87 13 time time NOUN cana-4527 87 14 and and CCONJ cana-4527 87 15 from from ADP cana-4527 87 16 one one NUM cana-4527 87 17 era era NOUN cana-4527 87 18 to to ADP cana-4527 87 19 the the DET cana-4527 87 20 next next ADJ cana-4527 87 21 . . PUNCT cana-4527 88 1 for for ADP cana-4527 88 2 instance instance NOUN cana-4527 88 3 , , PUNCT cana-4527 88 4 an an DET cana-4527 88 5 aquifer aquifer NOUN cana-4527 88 6 's 's PART cana-4527 88 7 hydraulic hydraulic ADJ cana-4527 88 8 conductivity conductivity NOUN cana-4527 88 9 might might AUX cana-4527 88 10 change change VERB cana-4527 88 11 in in ADP cana-4527 88 12 different different ADJ cana-4527 88 13 directions direction NOUN cana-4527 88 14 . . PUNCT cana-4527 89 1 many many ADJ cana-4527 89 2 academics academic NOUN cana-4527 89 3 have have AUX cana-4527 89 4 collaborated collaborate VERB cana-4527 89 5 to to PART cana-4527 89 6 produce produce VERB cana-4527 89 7 a a DET cana-4527 89 8 more more ADV cana-4527 89 9 precise precise ADJ cana-4527 89 10 framework framework NOUN cana-4527 89 11 that that PRON cana-4527 89 12 can can AUX cana-4527 89 13 be be AUX cana-4527 89 14 used use VERB cana-4527 89 15 to to PART cana-4527 89 16 predict predict VERB cana-4527 89 17 the the DET cana-4527 89 18 amount amount NOUN cana-4527 89 19 of of ADP cana-4527 89 20 water water NOUN cana-4527 89 21 that that PRON cana-4527 89 22 will will AUX cana-4527 89 23 flow flow VERB cana-4527 89 24 through through ADP cana-4527 89 25 the the DET cana-4527 89 26 aquifer aquifer NOUN cana-4527 89 27 . . PUNCT cana-4527 90 1 nevertheless nevertheless ADV cana-4527 90 2 , , PUNCT cana-4527 90 3 the the DET cana-4527 90 4 findings finding NOUN cana-4527 90 5 continue continue VERB cana-4527 90 6 to to PART cana-4527 90 7 exhibit exhibit VERB cana-4527 90 8 certain certain ADJ cana-4527 90 9 deficiencies deficiency NOUN cana-4527 90 10 . . PUNCT cana-4527 91 1 by by ADP cana-4527 91 2 utilizing utilize VERB cana-4527 91 3 the the DET cana-4527 91 4 proposed propose VERB cana-4527 91 5 fixed fix VERB cana-4527 91 6 transform transform NOUN cana-4527 91 7 domain domain NOUN cana-4527 91 8 approach approach NOUN cana-4527 91 9 , , PUNCT cana-4527 91 10 we we PRON cana-4527 91 11 demonstrate demonstrate VERB cana-4527 91 12 that that SCONJ cana-4527 91 13 the the DET cana-4527 91 14 fractionally fractionally ADV cana-4527 91 15 ordered order VERB cana-4527 91 16 derivative derivative ADJ cana-4527 91 17 model model NOUN cana-4527 91 18 of of ADP cana-4527 91 19 real real ADJ cana-4527 91 20 problems problem NOUN cana-4527 91 21 yields yield VERB cana-4527 91 22 superior superior ADJ cana-4527 91 23 results result NOUN cana-4527 91 24 to to ADP cana-4527 91 25 experimental experimental ADJ cana-4527 91 26 data datum NOUN cana-4527 91 27 in in ADP cana-4527 91 28 its its PRON cana-4527 91 29 mathematical mathematical ADJ cana-4527 91 30 representation representation NOUN cana-4527 91 31 . . PUNCT cana-4527 92 1 communications communication NOUN cana-4527 92 2 on on ADP cana-4527 92 3 applied apply VERB cana-4527 92 4 nonlinear nonlinear ADJ cana-4527 92 5 analysis analysis NOUN cana-4527 92 6 issn issn NOUN cana-4527 92 7 : : PUNCT cana-4527 92 8 1074 1074 NUM cana-4527 92 9 - - PUNCT cana-4527 92 10 133x 133x NUM cana-4527 92 11 vol vol NOUN cana-4527 92 12 32 32 NUM cana-4527 92 13 no no NOUN cana-4527 92 14 . . PUNCT cana-4527 93 1 9s 9s NUM cana-4527 93 2 ( ( PUNCT cana-4527 93 3 2025 2025 NUM cana-4527 93 4 ) ) PUNCT cana-4527 93 5 2418 2418 NUM cana-4527 93 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 93 7 4 4 X cana-4527 93 8 . . PUNCT cana-4527 93 9 proposed propose VERB cana-4527 93 10 methodology methodology NOUN cana-4527 93 11 the the DET cana-4527 93 12 groundwater groundwater NOUN cana-4527 93 13 flow flow NOUN cana-4527 93 14 model model NOUN cana-4527 93 15 in in ADP cana-4527 93 16 this this DET cana-4527 93 17 study study NOUN cana-4527 93 18 will will AUX cana-4527 93 19 be be AUX cana-4527 93 20 established establish VERB cana-4527 93 21 using use VERB cana-4527 93 22 the the DET cana-4527 93 23 caputo caputo PROPN cana-4527 93 24 - - PUNCT cana-4527 93 25 fabrizio fabrizio PROPN cana-4527 93 26 derivative derivative NOUN cana-4527 93 27 . . PUNCT cana-4527 94 1 for for ADP cana-4527 94 2 this this PRON cana-4527 94 3 , , PUNCT cana-4527 94 4 the the DET cana-4527 94 5 issue issue NOUN cana-4527 94 6 of of ADP cana-4527 94 7 water water NOUN cana-4527 94 8 flow flow NOUN cana-4527 94 9 inside inside ADP cana-4527 94 10 a a DET cana-4527 94 11 finite finite ADJ cana-4527 94 12 aquifer aquifer NOUN cana-4527 94 13 is be AUX cana-4527 94 14 used use VERB cana-4527 94 15 . . PUNCT cana-4527 95 1 4.1 4.1 NUM cana-4527 95 2 . . PUNCT cana-4527 96 1 caputo caputo PROPN cana-4527 96 2 derivative derivative PROPN cana-4527 96 3 : : PUNCT cana-4527 96 4 definition definition NOUN cana-4527 96 5 -4.1.1 -4.1.1 NUM cana-4527 96 6 : : PUNCT cana-4527 96 7 the the DET cana-4527 96 8 following follow VERB cana-4527 96 9 is be AUX cana-4527 96 10 the the DET cana-4527 96 11 definition definition NOUN cana-4527 96 12 of of ADP cana-4527 96 13 the the DET cana-4527 96 14 caputo caputo PROPN cana-4527 96 15 derivative derivative NOUN cana-4527 96 16 of of ADP cana-4527 96 17 order order NOUN cana-4527 96 18 ∝ ∝ PRON cana-4527 96 19 for for ADP cana-4527 96 20 the the DET cana-4527 96 21 function function NOUN cana-4527 96 22 f(t f(t PROPN cana-4527 96 23 ) ) PUNCT cana-4527 96 24 , , PUNCT cana-4527 96 25 if if SCONJ cana-4527 96 26 ∝ ∝ PROPN cana-4527 96 27 > > X cana-4527 96 28 0 0 NUM cana-4527 96 29 : : PUNCT cana-4527 96 30 π·βˆπ‘“(𝑑 π·βˆπ‘“(𝑑 ADJ cana-4527 96 31 ) ) PUNCT cana-4527 96 32 = = SYM cana-4527 96 33 1 1 NUM cana-4527 96 34 Ξ³(nβˆ’βˆ Ξ³(nβˆ’βˆ NOUN cana-4527 96 35 ) ) PUNCT cana-4527 96 36 ∫ ∫ PROPN cana-4527 96 37 𝑓𝑛(𝜏 𝑓𝑛(𝜏 NOUN cana-4527 96 38 ) ) PUNCT cana-4527 96 39 ( ( PUNCT cana-4527 96 40 𝑑 𝑑 NOUN cana-4527 96 41 βˆ’ βˆ’ PROPN cana-4527 96 42 𝜏)𝛼+1βˆ’π‘› 𝜏)𝛼+1βˆ’π‘› NOUN cana-4527 96 43 π‘₯ π‘₯ NOUN cana-4527 96 44 0 0 NUM cana-4527 96 45 π‘‘πœ π‘‘πœ NOUN cana-4527 96 46 ( ( PUNCT cana-4527 96 47 4.1 4.1 NUM cana-4527 96 48 ) ) PUNCT cana-4527 96 49 where where SCONJ cana-4527 96 50 in in ADP cana-4527 96 51 -1 -1 PROPN cana-4527 96 52 < < X cana-4527 96 53 ∝ ∝ PROPN cana-4527 96 54 < < X cana-4527 96 55 n n CCONJ cana-4527 96 56 , , PUNCT cana-4527 96 57 n n PROPN cana-4527 96 58 ∈ ∈ PROPN cana-4527 96 59 n n CCONJ cana-4527 96 60 , , PUNCT cana-4527 96 61 & & CCONJ cana-4527 96 62 Ξ³ Ξ³ PROPN cana-4527 96 63 ( ( PUNCT cana-4527 96 64 . . PUNCT cana-4527 96 65 ) ) PUNCT cana-4527 97 1 i i PRON cana-4527 97 2 d d PROPN cana-4527 97 3 represent represent VERB cana-4527 97 4 the the DET cana-4527 97 5 gamma gamma PROPN cana-4527 97 6 function function NOUN cana-4527 97 7 . . PUNCT cana-4527 98 1 it it PRON cana-4527 98 2 has have VERB cana-4527 98 3 linear linear ADJ cana-4527 98 4 property property NOUN cana-4527 98 5 . . PUNCT cana-4527 99 1 𝐷∝(π‘˜1𝑔1(𝑑 𝐷∝(π‘˜1𝑔1(𝑑 X cana-4527 99 2 ) ) PUNCT cana-4527 100 1 + + NUM cana-4527 100 2 π‘˜1𝑔1(𝑑 π‘˜1𝑔1(𝑑 NOUN cana-4527 100 3 ) ) PUNCT cana-4527 100 4 ) ) PUNCT cana-4527 101 1 = = PUNCT cana-4527 101 2 π‘˜1π·βˆπ‘”1(𝑑 π‘˜1π·βˆπ‘”1(𝑑 PROPN cana-4527 101 3 ) ) PUNCT cana-4527 102 1 + + CCONJ cana-4527 103 1 π‘˜2π·βˆπ‘”2(𝑑 π‘˜2π·βˆπ‘”2(𝑑 NOUN cana-4527 103 2 ) ) PUNCT cana-4527 103 3 , , PUNCT cana-4527 103 4 where where SCONJ cana-4527 103 5 π‘˜1 π‘˜1 NOUN cana-4527 103 6 and and CCONJ cana-4527 103 7 π‘˜2 π‘˜2 PROPN cana-4527 103 8 in in ADP cana-4527 103 9 this this DET cana-4527 103 10 case case NOUN cana-4527 103 11 are be AUX cana-4527 103 12 constants constant NOUN cana-4527 103 13 . . PUNCT cana-4527 104 1 remark remark NOUN cana-4527 104 2 1 1 NUM cana-4527 104 3 : : PUNCT cana-4527 104 4 assume assume VERB cana-4527 104 5 that that SCONJ cana-4527 104 6 ( ( PUNCT cana-4527 104 7 0 0 NUM cana-4527 104 8 < < X cana-4527 104 9 Ξ± Ξ± PROPN cana-4527 104 10 ≀ ≀ NUM cana-4527 104 11 1 1 NUM cana-4527 104 12 ) ) PUNCT cana-4527 104 13 . . PUNCT cana-4527 105 1 the the DET cana-4527 105 2 formula formula NOUN cana-4527 105 3 that that PRON cana-4527 105 4 follows follow VERB cana-4527 105 5 can can AUX cana-4527 105 6 be be AUX cana-4527 105 7 obtained obtain VERB cana-4527 105 8 from from ADP cana-4527 105 9 definition definition NOUN cana-4527 105 10 4.1.1 4.1.1 NUM cana-4527 105 11 . . PUNCT cana-4527 106 1 𝐷𝛼𝑑𝛾 𝐷𝛼𝑑𝛾 PROPN cana-4527 106 2 = = PUNCT cana-4527 106 3 { { PUNCT cana-4527 106 4 0 0 NUM cana-4527 106 5 𝛾 𝛾 NOUN cana-4527 106 6 = = SYM cana-4527 106 7 0 0 NUM cana-4527 106 8 Ξ³(Ξ³ Ξ³(Ξ³ PROPN cana-4527 106 9 + + CCONJ cana-4527 106 10 1 1 X cana-4527 106 11 ) ) PUNCT cana-4527 106 12 Ξ³ Ξ³ NOUN cana-4527 106 13 ( ( PUNCT cana-4527 106 14 Ξ³ Ξ³ X cana-4527 106 15 + + PROPN cana-4527 106 16 1 1 NUM cana-4527 106 17 βˆ’ βˆ’ PROPN cana-4527 106 18 Ξ± Ξ± X cana-4527 106 19 ) ) PUNCT cana-4527 106 20 π‘‘π›Ύβˆ’π›Ό π‘‘π›Ύβˆ’π›Ό NOUN cana-4527 106 21 𝛾 𝛾 NOUN cana-4527 106 22 = = SYM cana-4527 106 23 1,2,3 1,2,3 NUM cana-4527 106 24 … … PUNCT cana-4527 106 25 … … PUNCT cana-4527 106 26 … … PUNCT cana-4527 106 27 ( ( PUNCT cana-4527 106 28 4.2 4.2 NUM cana-4527 106 29 ) ) PUNCT cana-4527 106 30 definition definition NOUN cana-4527 106 31 -4.1 -4.1 NOUN cana-4527 106 32 . . PROPN cana-4527 106 33 2 2 NUM cana-4527 106 34 : : PUNCT cana-4527 106 35 let let VERB cana-4527 106 36 𝐴 𝐴 PROPN cana-4527 106 37 = = SYM cana-4527 106 38 { { PUNCT cana-4527 106 39 𝑓(𝑑 𝑓(𝑑 PROPN cana-4527 106 40 ) ) PUNCT cana-4527 106 41 ; ; PUNCT cana-4527 106 42 βˆƒ βˆƒ PROPN cana-4527 106 43 𝜏1,𝜏2 𝜏1,𝜏2 PROPN cana-4527 106 44 > > X cana-4527 106 45 0 0 X cana-4527 106 46 . . PUNCT cana-4527 107 1 such such ADJ cana-4527 107 2 that that SCONJ cana-4527 107 3 , , PUNCT cana-4527 107 4 |𝑓(𝑧)| |𝑓(𝑧)| ADJ cana-4527 107 5 < < X cana-4527 107 6 𝑀𝑒 𝑀𝑒 PROPN cana-4527 107 7 |𝑑| |𝑑| ADP cana-4527 107 8 πœπ‘— πœπ‘— NOUN cana-4527 107 9 , , PUNCT cana-4527 107 10 𝑖𝑓 𝑖𝑓 NUM cana-4527 107 11 𝑑 𝑑 PROPN cana-4527 107 12 ∈ ∈ PROPN cana-4527 107 13 ( ( PUNCT cana-4527 107 14 βˆ’1)𝑗 βˆ’1)𝑗 PUNCT cana-4527 107 15 Γ— Γ— NOUN cana-4527 107 16 [ [ X cana-4527 107 17 0 0 NUM cana-4527 107 18 , , PUNCT cana-4527 107 19 ∞ ∞ PROPN cana-4527 107 20 ] ] X cana-4527 107 21 } } PUNCT cana-4527 107 22 f(u)= f(u)= ADJ cana-4527 107 23 ~[f(t ~[f(t NOUN cana-4527 107 24 ) ) PUNCT cana-4527 107 25 ] ] PUNCT cana-4527 107 26 is be AUX cana-4527 107 27 the the DET cana-4527 107 28 suggested suggest VERB cana-4527 107 29 transform transform NOUN cana-4527 107 30 of of ADP cana-4527 107 31 f∈a f∈a NOUN cana-4527 107 32 and and CCONJ cana-4527 107 33 has have VERB cana-4527 107 34 the the DET cana-4527 107 35 following follow VERB cana-4527 107 36 definition definition NOUN cana-4527 107 37 : : PUNCT cana-4527 107 38 𝐹(𝑒 𝐹(𝑒 X cana-4527 107 39 ) ) PUNCT cana-4527 107 40 = = SYM cana-4527 108 1 ~ ~ PUNCT cana-4527 109 1 [ [ X cana-4527 109 2 𝑓(𝑑 𝑓(𝑑 NOUN cana-4527 109 3 ) ) PUNCT cana-4527 109 4 ] ] PUNCT cana-4527 109 5 = = SYM cana-4527 109 6 ∫ ∫ PROPN cana-4527 109 7 𝑓(𝑒𝑑)π‘’βˆ’π‘‘π‘‘π‘‘ 𝑓(𝑒𝑑)π‘’βˆ’π‘‘π‘‘π‘‘ PROPN cana-4527 109 8 , , PUNCT cana-4527 109 9 𝑒 𝑒 PROPN cana-4527 109 10 ∈ ∈ PROPN cana-4527 109 11 ( ( PUNCT cana-4527 109 12 𝜏1 𝜏1 NOUN cana-4527 109 13 , , PUNCT cana-4527 109 14 𝜏2 𝜏2 PROPN cana-4527 109 15 ) ) PUNCT cana-4527 109 16 ( ( PUNCT cana-4527 109 17 4 4 NUM cana-4527 109 18 . . NOUN cana-4527 109 19 3 3 NUM cana-4527 109 20 ) ) PUNCT cana-4527 109 21 ∞ ∞ NUM cana-4527 109 22 0 0 NUM cana-4527 109 23 when when SCONJ cana-4527 109 24 u u NOUN cana-4527 109 25 is be AUX cana-4527 109 26 a a DET cana-4527 109 27 parameter parameter NOUN cana-4527 109 28 , , PUNCT cana-4527 109 29 it it PRON cana-4527 109 30 might might AUX cana-4527 109 31 be be AUX cana-4527 109 32 real real ADJ cana-4527 109 33 or or CCONJ cana-4527 109 34 complex complex ADJ cana-4527 109 35 and and CCONJ cana-4527 109 36 is be AUX cana-4527 109 37 not not PART cana-4527 109 38 dependent dependent ADJ cana-4527 109 39 on on ADP cana-4527 109 40 t. t. PROPN cana-4527 109 41 the the DET cana-4527 109 42 connection connection NOUN cana-4527 109 43 [ [ X cana-4527 109 44 41 41 NUM cana-4527 109 45 ] ] PUNCT cana-4527 109 46 , , PUNCT cana-4527 109 47 states state VERB cana-4527 109 48 that that SCONJ cana-4527 109 49 g(1 g(1 NOUN cana-4527 109 50 / / SYM cana-4527 109 51 j)=jl(j j)=jl(j PROPN cana-4527 109 52 ) ) PUNCT cana-4527 109 53 and and CCONJ cana-4527 109 54 l(1 l(1 NOUN cana-4527 109 55 / / SYM cana-4527 109 56 u)=u u)=u ADP cana-4527 109 57 g g NOUN cana-4527 109 58 ( ( PUNCT cana-4527 109 59 u u NOUN cana-4527 109 60 ) ) PUNCT cana-4527 109 61 represents represent VERB cana-4527 109 62 the the DET cana-4527 109 63 relationship relationship NOUN cana-4527 109 64 between between ADP cana-4527 109 65 the the DET cana-4527 109 66 sumudu sumudu NOUN cana-4527 109 67 transform transform VERB cana-4527 109 68 and and CCONJ cana-4527 109 69 laplace laplace NOUN cana-4527 109 70 transformation transformation NOUN cana-4527 109 71 , , PUNCT cana-4527 109 72 respectively respectively ADV cana-4527 109 73 . . PUNCT cana-4527 110 1 𝐽 𝐽 PROPN cana-4527 111 1 [ [ X cana-4527 111 2 𝐷𝛼𝑓(𝑑 𝐷𝛼𝑓(𝑑 NOUN cana-4527 111 3 ) ) PUNCT cana-4527 111 4 ] ] PUNCT cana-4527 112 1 = = PUNCT cana-4527 112 2 𝐹(𝑒 𝐹(𝑒 NUM cana-4527 112 3 ) ) PUNCT cana-4527 112 4 𝑒𝛼 𝑒𝛼 ADJ cana-4527 112 5 βˆ’ βˆ’ PROPN cana-4527 112 6 βˆ‘ βˆ‘ PROPN cana-4527 112 7 π‘“π‘˜(0 π‘“π‘˜(0 NOUN cana-4527 112 8 ) ) PUNCT cana-4527 112 9 π‘’π›Όβˆ’π‘˜ π‘’π›Όβˆ’π‘˜ NOUN cana-4527 113 1 π‘›βˆ’1 π‘›βˆ’1 NUM cana-4527 113 2 π‘˜=0 π‘˜=0 PROPN cana-4527 113 3 , , PUNCT cana-4527 113 4 𝑛 𝑛 DET cana-4527 113 5 βˆ’ βˆ’ NUM cana-4527 113 6 1 1 NUM cana-4527 113 7 < < X cana-4527 113 8 βˆβ‰€ βˆβ‰€ X cana-4527 113 9 𝑛 𝑛 X cana-4527 113 10 , , PUNCT cana-4527 113 11 𝑛 𝑛 PRON cana-4527 113 12 ∈ ∈ NOUN cana-4527 113 13 𝑁 𝑁 PROPN cana-4527 113 14 ( ( PUNCT cana-4527 113 15 4.4 4.4 NUM cana-4527 113 16 ) ) PUNCT cana-4527 113 17 where where SCONJ cana-4527 113 18 𝐹(𝑒 𝐹(𝑒 ADP cana-4527 113 19 ) ) PUNCT cana-4527 113 20 = = SYM cana-4527 113 21 ~ ~ PUNCT cana-4527 114 1 [ [ X cana-4527 114 2 𝑓(𝑑 𝑓(𝑑 NOUN cana-4527 114 3 ) ) PUNCT cana-4527 114 4 ] ] PUNCT cana-4527 114 5 communications communication NOUN cana-4527 114 6 on on ADP cana-4527 114 7 applied apply VERB cana-4527 114 8 nonlinear nonlinear ADJ cana-4527 114 9 analysis analysis NOUN cana-4527 114 10 issn issn NOUN cana-4527 114 11 : : PUNCT cana-4527 114 12 1074 1074 NUM cana-4527 114 13 - - PUNCT cana-4527 114 14 133x 133x NUM cana-4527 114 15 vol vol NOUN cana-4527 114 16 32 32 NUM cana-4527 114 17 no no NOUN cana-4527 114 18 . . PUNCT cana-4527 115 1 9s 9s NUM cana-4527 115 2 ( ( PUNCT cana-4527 115 3 2025 2025 NUM cana-4527 115 4 ) ) PUNCT cana-4527 115 5 2419 2419 NUM cana-4527 115 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 115 7 as as ADP cana-4527 115 8 part part NOUN cana-4527 115 9 of of ADP cana-4527 115 10 their their PRON cana-4527 115 11 groundwater groundwater NOUN cana-4527 115 12 studies study NOUN cana-4527 115 13 , , PUNCT cana-4527 115 14 numerous numerous ADJ cana-4527 115 15 academics academic NOUN cana-4527 115 16 are be AUX cana-4527 115 17 looking look VERB cana-4527 115 18 into into ADP cana-4527 115 19 and and CCONJ cana-4527 115 20 solving solve VERB cana-4527 115 21 the the DET cana-4527 115 22 standard standard ADJ cana-4527 115 23 equation equation NOUN cana-4527 115 24 for for ADP cana-4527 115 25 shallow shallow ADJ cana-4527 115 26 water water NOUN cana-4527 116 1 [ [ X cana-4527 116 2 42 42 NUM cana-4527 116 3 ] ] PUNCT cana-4527 116 4 . . PUNCT cana-4527 117 1 the the DET cana-4527 117 2 equations equation NOUN cana-4527 117 3 that that PRON cana-4527 117 4 were be AUX cana-4527 117 5 examined examine VERB cana-4527 117 6 and and CCONJ cana-4527 117 7 resolved resolve VERB cana-4527 117 8 in in ADP cana-4527 117 9 this this DET cana-4527 117 10 investigation investigation NOUN cana-4527 117 11 regulate regulate VERB cana-4527 117 12 groundwater groundwater NOUN cana-4527 117 13 flow flow NOUN cana-4527 117 14 . . PUNCT cana-4527 118 1 this this DET cana-4527 118 2 derivative derivative NOUN cana-4527 118 3 uses use VERB cana-4527 118 4 the the DET cana-4527 118 5 concept concept NOUN cana-4527 118 6 of of ADP cana-4527 118 7 continuity continuity NOUN cana-4527 118 8 low low ADJ cana-4527 118 9 without without ADP cana-4527 118 10 recourse recourse NOUN cana-4527 118 11 and and CCONJ cana-4527 118 12 the the DET cana-4527 118 13 darcy darcy PROPN cana-4527 118 14 law law NOUN cana-4527 118 15 . . PUNCT cana-4527 119 1 𝑆0πœ•π‘‘πœ‘ 𝑆0πœ•π‘‘πœ‘ PROPN cana-4527 119 2 ( ( PUNCT cana-4527 119 3 π‘Ÿ π‘Ÿ NOUN cana-4527 119 4 , , PUNCT cana-4527 119 5 𝑑 𝑑 NOUN cana-4527 119 6 ) ) PUNCT cana-4527 119 7 = = PUNCT cana-4527 119 8 π‘˜ π‘˜ PRON cana-4527 119 9 π‘Ÿπ‘›βˆ’1 π‘Ÿπ‘›βˆ’1 PROPN cana-4527 119 10 πœ•π‘Ÿ[π‘Ÿπ‘›βˆ’1πœ•π‘Ÿπœ‘ πœ•π‘Ÿ[π‘Ÿπ‘›βˆ’1πœ•π‘Ÿπœ‘ PROPN cana-4527 119 11 ( ( PUNCT cana-4527 119 12 π‘Ÿ π‘Ÿ NOUN cana-4527 119 13 , , PUNCT cana-4527 119 14 𝑑 𝑑 NOUN cana-4527 119 15 ) ) PUNCT cana-4527 119 16 ] ] PUNCT cana-4527 119 17 ( ( PUNCT cana-4527 119 18 4.5 4.5 NUM cana-4527 119 19 ) ) PUNCT cana-4527 119 20 πœ‘ πœ‘ NOUN cana-4527 119 21 ( ( PUNCT cana-4527 119 22 π‘Ÿ π‘Ÿ NOUN cana-4527 119 23 , , PUNCT cana-4527 119 24 0 0 NUM cana-4527 119 25 ) ) PUNCT cana-4527 119 26 = = SYM cana-4527 119 27 π‘Žπ‘Ÿ π‘Žπ‘Ÿ NOUN cana-4527 119 28 , , PUNCT cana-4527 119 29 𝑄 𝑄 PROPN cana-4527 119 30 = = PUNCT cana-4527 119 31 2πœ‹π‘› 2πœ‹π‘› ADJ cana-4527 119 32 2⁄ 2⁄ NUM cana-4527 119 33 Ξ³(𝑛 Ξ³(𝑛 PROPN cana-4527 119 34 2⁄ 2⁄ NUM cana-4527 119 35 ) ) PUNCT cana-4527 119 36 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 119 37 π‘›βˆ’1π‘˜π‘‘3βˆ’π‘›πœ•π‘Ÿπœ‘ π‘›βˆ’1π‘˜π‘‘3βˆ’π‘›πœ•π‘Ÿπœ‘ CCONJ cana-4527 119 38 ( ( PUNCT cana-4527 119 39 π‘Ÿπ‘ π‘Ÿπ‘ NOUN cana-4527 119 40 , , PUNCT cana-4527 119 41 𝑑 𝑑 PROPN cana-4527 119 42 ) ) PUNCT cana-4527 119 43 ( ( PUNCT cana-4527 119 44 4.6 4.6 NUM cana-4527 119 45 ) ) PUNCT cana-4527 119 46 where where SCONJ cana-4527 119 47 n n PRON cana-4527 119 48 denotes denote VERB cana-4527 119 49 the the DET cana-4527 119 50 radial radial ADJ cana-4527 119 51 dimension dimension NOUN cana-4527 119 52 for for ADP cana-4527 119 53 n n NOUN cana-4527 119 54 = = SYM cana-4527 119 55 1 1 NUM cana-4527 119 56 , , PUNCT cana-4527 119 57 2 2 NUM cana-4527 119 58 , , PUNCT cana-4527 119 59 or or CCONJ cana-4527 119 60 3 3 NUM cana-4527 119 61 . . PUNCT cana-4527 120 1 the the DET cana-4527 120 2 scalar scalar ADJ cana-4527 120 3 quantities quantity NOUN cana-4527 120 4 represent represent VERB cana-4527 120 5 the the DET cana-4527 120 6 hydraulic hydraulic ADJ cana-4527 120 7 conductivity conductivity NOUN cana-4527 120 8 and and CCONJ cana-4527 120 9 the the DET cana-4527 120 10 recommended recommend VERB cana-4527 120 11 storage storage NOUN cana-4527 120 12 capacity capacity NOUN cana-4527 120 13 𝑆0 𝑆0 NOUN cana-4527 120 14 and and CCONJ cana-4527 120 15 k k NOUN cana-4527 120 16 , , PUNCT cana-4527 120 17 respectively respectively ADV cana-4527 120 18 . . PUNCT cana-4527 121 1 furthermore furthermore ADV cana-4527 121 2 , , PUNCT cana-4527 121 3 Ο†(r Ο†(r ADJ cana-4527 121 4 , , PUNCT cana-4527 121 5 0 0 NUM cana-4527 121 6 ) ) PUNCT cana-4527 121 7 = = SYM cana-4527 121 8 ar ar NOUN cana-4527 121 9 suggests suggest VERB cana-4527 121 10 that that SCONJ cana-4527 121 11 the the DET cana-4527 121 12 aquifer aquifer NOUN cana-4527 121 13 's 's PART cana-4527 121 14 primary primary ADJ cana-4527 121 15 hydraulic hydraulic ADJ cana-4527 121 16 head head NOUN cana-4527 121 17 , , PUNCT cana-4527 121 18 with with ADP cana-4527 121 19 a a DET cana-4527 121 20 positive positive ADJ cana-4527 121 21 gradient gradient NOUN cana-4527 121 22 a a PRON cana-4527 121 23 , , PUNCT cana-4527 121 24 is be AUX cana-4527 121 25 a a DET cana-4527 121 26 linear linear ADJ cana-4527 121 27 function function NOUN cana-4527 121 28 of of ADP cana-4527 121 29 space space NOUN cana-4527 121 30 , , PUNCT cana-4527 121 31 with with ADP cana-4527 121 32 lim lim PROPN cana-4527 121 33 π‘Ÿβ†’βˆž π‘Ÿβ†’βˆž NUM cana-4527 121 34 Ο†(r Ο†(r PROPN cana-4527 121 35 , , PUNCT cana-4527 121 36 t t PROPN cana-4527 121 37 ) ) PUNCT cana-4527 121 38 = = PUNCT cana-4527 122 1 π‘Žπ‘Ÿ π‘Žπ‘Ÿ ADP cana-4527 122 2 meaning mean VERB cana-4527 122 3 that that SCONJ cana-4527 122 4 the the DET cana-4527 122 5 water water NOUN cana-4527 122 6 level level NOUN cana-4527 122 7 of of ADP cana-4527 122 8 the the DET cana-4527 122 9 aquifer aquifer NOUN cana-4527 122 10 is be AUX cana-4527 122 11 either either CCONJ cana-4527 122 12 at at ADP cana-4527 122 13 or or CCONJ cana-4527 122 14 at at ADP cana-4527 122 15 the the DET cana-4527 122 16 start start NOUN cana-4527 122 17 of of ADP cana-4527 122 18 the the DET cana-4527 122 19 pumping pumping NOUN cana-4527 122 20 test test NOUN cana-4527 122 21 . . PUNCT cana-4527 123 1 𝑆0πœ•π‘‘πœ‘ 𝑆0πœ•π‘‘πœ‘ PROPN cana-4527 123 2 ( ( PUNCT cana-4527 123 3 π‘Ÿ π‘Ÿ NOUN cana-4527 123 4 , , PUNCT cana-4527 123 5 𝑑 𝑑 NOUN cana-4527 123 6 ) ) PUNCT cana-4527 123 7 = = PUNCT cana-4527 123 8 π‘˜ π‘˜ DET cana-4527 123 9 π‘Ÿπ‘›βˆ’1 π‘Ÿπ‘›βˆ’1 PROPN cana-4527 123 10 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 123 11 [ [ X cana-4527 123 12 π‘Ÿπ‘›βˆ’1πœ•π‘Ÿ π‘Ÿπ‘›βˆ’1πœ•π‘Ÿ NOUN cana-4527 123 13 βˆπœ‘ βˆπœ‘ X cana-4527 123 14 ( ( PUNCT cana-4527 123 15 π‘Ÿ π‘Ÿ NOUN cana-4527 123 16 , , PUNCT cana-4527 123 17 𝑑 𝑑 NOUN cana-4527 123 18 ) ) PUNCT cana-4527 123 19 ] ] PUNCT cana-4527 123 20 ( ( PUNCT cana-4527 123 21 4.7 4.7 NUM cana-4527 123 22 ) ) PUNCT cana-4527 123 23 this this DET cana-4527 123 24 physical physical ADJ cana-4527 123 25 parameter parameter NOUN cana-4527 123 26 , , PUNCT cana-4527 123 27 which which PRON cana-4527 123 28 describes describe VERB cana-4527 123 29 the the DET cana-4527 123 30 flow flow NOUN cana-4527 123 31 across across ADP cana-4527 123 32 geological geological ADJ cana-4527 123 33 formations formation NOUN cana-4527 123 34 , , PUNCT cana-4527 123 35 is be AUX cana-4527 123 36 contained contain VERB cana-4527 123 37 in in ADP cana-4527 123 38 the the DET cana-4527 123 39 binary binary ADJ cana-4527 123 40 integral integral ADJ cana-4527 123 41 equation equation NOUN cana-4527 123 42 [ [ X cana-4527 123 43 43 43 NUM cana-4527 123 44 ] ] PUNCT cana-4527 123 45 . . PUNCT cana-4527 124 1 the the DET cana-4527 124 2 boundary boundary ADJ cana-4527 124 3 condition condition NOUN cana-4527 124 4 is be AUX cana-4527 124 5 given give VERB cana-4527 124 6 a a DET cana-4527 124 7 more more ADV cana-4527 124 8 general general ADJ cana-4527 124 9 form form NOUN cana-4527 124 10 at at ADP cana-4527 124 11 the the DET cana-4527 124 12 well well NOUN cana-4527 124 13 by by ADP cana-4527 124 14 utilizing utilize VERB cana-4527 124 15 the the DET cana-4527 124 16 same same ADJ cana-4527 124 17 transformation transformation NOUN cana-4527 124 18 in in ADP cana-4527 124 19 ( ( PUNCT cana-4527 124 20 4.6 4.6 NUM cana-4527 124 21 ): ): PUNCT cana-4527 124 22 𝑄 𝑄 PROPN cana-4527 124 23 = = SYM cana-4527 124 24 ( ( PUNCT cana-4527 124 25 βˆ’1)π‘š+1 βˆ’1)π‘š+1 PROPN cana-4527 124 26 2πœ‹π‘›/2 2πœ‹π‘›/2 PROPN cana-4527 124 27 Ξ³ Ξ³ X cana-4527 124 28 ( ( PUNCT cana-4527 124 29 n n PROPN cana-4527 124 30 2 2 NUM cana-4527 124 31 ) ) PUNCT cana-4527 124 32 Ξ³(p Ξ³(p PROPN cana-4527 124 33 ) ) PUNCT cana-4527 124 34 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 124 35 π‘›βˆ’1 π‘›βˆ’1 NUM cana-4527 124 36 π‘˜π‘‘3βˆ’π‘›+βˆπœ•π‘Ÿ π‘˜π‘‘3βˆ’π‘›+βˆπœ•π‘Ÿ NOUN cana-4527 124 37 π›Όπœ‘ π›Όπœ‘ NOUN cana-4527 124 38 ( ( PUNCT cana-4527 124 39 π‘Ÿπ‘ π‘Ÿπ‘ PROPN cana-4527 124 40 , , PUNCT cana-4527 124 41 𝑑 𝑑 PROPN cana-4527 124 42 ) ) PUNCT cana-4527 124 43 ( ( PUNCT cana-4527 124 44 4.8 4.8 NUM cana-4527 124 45 ) ) PUNCT cana-4527 124 46 the the DET cana-4527 124 47 initial initial ADJ cana-4527 124 48 condition condition NOUN cana-4527 124 49 in in ADP cana-4527 124 50 ( ( PUNCT cana-4527 124 51 4.6 4.6 NUM cana-4527 124 52 ) ) PUNCT cana-4527 124 53 and and CCONJ cana-4527 124 54 the the DET cana-4527 124 55 relationships relationship NOUN cana-4527 124 56 in in ADP cana-4527 124 57 ( ( PUNCT cana-4527 124 58 4.7 4.7 NUM cana-4527 124 59 ) ) PUNCT cana-4527 124 60 and and CCONJ cana-4527 124 61 ( ( PUNCT cana-4527 124 62 4.8 4.8 NUM cana-4527 124 63 ) ) PUNCT cana-4527 124 64 represent represent VERB cana-4527 124 65 an an DET cana-4527 124 66 entire entire ADJ cana-4527 124 67 set set NOUN cana-4527 124 68 of of ADP cana-4527 124 69 equations equation NOUN cana-4527 124 70 for for ADP cana-4527 124 71 an an DET cana-4527 124 72 existential existential ADJ cana-4527 124 73 solution solution NOUN cana-4527 124 74 . . PUNCT cana-4527 125 1 the the DET cana-4527 125 2 groundwater groundwater NOUN cana-4527 125 3 model model NOUN cana-4527 125 4 equation equation NOUN cana-4527 125 5 can can AUX cana-4527 125 6 be be AUX cana-4527 125 7 solved solve VERB cana-4527 125 8 in in ADP cana-4527 125 9 various various ADJ cana-4527 125 10 ways way NOUN cana-4527 125 11 , , PUNCT cana-4527 125 12 including include VERB cana-4527 125 13 the the DET cana-4527 125 14 frobenius frobenius NOUN cana-4527 125 15 and and CCONJ cana-4527 125 16 finite finite ADJ cana-4527 125 17 element element NOUN cana-4527 125 18 methods method NOUN cana-4527 125 19 . . PUNCT cana-4527 126 1 the the DET cana-4527 126 2 method method NOUN cana-4527 126 3 suggested suggest VERB cana-4527 126 4 in in ADP cana-4527 126 5 this this DET cana-4527 126 6 paper paper NOUN cana-4527 126 7 will will AUX cana-4527 126 8 be be AUX cana-4527 126 9 applied apply VERB cana-4527 126 10 to to PART cana-4527 126 11 solve solve VERB cana-4527 126 12 the the DET cana-4527 126 13 fractional fractional ADJ cana-4527 126 14 groundwater groundwater NOUN cana-4527 126 15 model model NOUN cana-4527 126 16 equation equation NOUN cana-4527 126 17 . . PUNCT cana-4527 127 1 4.2 4.2 NUM cana-4527 127 2 . . PUNCT cana-4527 128 1 the the DET cana-4527 128 2 generalised generalise VERB cana-4527 128 3 flow flow NOUN cana-4527 128 4 equation equation NOUN cana-4527 128 5 's 's PART cana-4527 128 6 solution solution NOUN cana-4527 128 7 examine examine VERB cana-4527 128 8 the the DET cana-4527 128 9 caputo caputo PROPN cana-4527 128 10 derivation derivation NOUN cana-4527 128 11 of of ADP cana-4527 128 12 an an DET cana-4527 128 13 order order NOUN cana-4527 128 14 Ξ± Ξ± PRON cana-4527 128 15 regulated regulated ADJ cana-4527 128 16 groundwater groundwater NOUN cana-4527 128 17 flow flow NOUN cana-4527 128 18 , , PUNCT cana-4527 128 19 where where SCONJ cana-4527 128 20 Ξ± Ξ± NOUN cana-4527 128 21 is be AUX cana-4527 128 22 larger large ADJ cana-4527 128 23 than than ADP cana-4527 128 24 or or CCONJ cana-4527 128 25 equal equal ADJ cana-4527 128 26 to to ADP cana-4527 128 27 1 1 NUM cana-4527 128 28 and and CCONJ cana-4527 128 29 lies lie VERB cana-4527 128 30 between between ADP cana-4527 128 31 0 0 NUM cana-4527 128 32 and and CCONJ cana-4527 128 33 1 1 NUM cana-4527 128 34 . . NOUN cana-4527 128 35 example example NOUN cana-4527 128 36 1 1 NUM cana-4527 128 37 : : PUNCT cana-4527 128 38 to to PART cana-4527 128 39 determine determine VERB cana-4527 128 40 the the DET cana-4527 128 41 following follow VERB cana-4527 128 42 time time NOUN cana-4527 128 43 derivatives derivative NOUN cana-4527 128 44 in in ADP cana-4527 128 45 fractions fraction NOUN cana-4527 128 46 , , PUNCT cana-4527 128 47 the the DET cana-4527 128 48 flow flow NOUN cana-4527 128 49 of of ADP cana-4527 128 50 groundwater groundwater NOUN cana-4527 128 51 equation equation NOUN cana-4527 128 52 needs need VERB cana-4527 128 53 to to PART cana-4527 128 54 be be AUX cana-4527 128 55 examined examine VERB cana-4527 128 56 . . PUNCT cana-4527 129 1 𝐾 𝐾 PROPN cana-4527 129 2 πœ•π‘Ÿ πœ•π‘Ÿ VERB cana-4527 129 3 2 2 NUM cana-4527 129 4 πœ‘ πœ‘ NOUN cana-4527 129 5 ( ( PUNCT cana-4527 129 6 π‘Ÿ π‘Ÿ NOUN cana-4527 129 7 , , PUNCT cana-4527 129 8 𝑑 𝑑 NOUN cana-4527 129 9 ) ) PUNCT cana-4527 129 10 + + CCONJ cana-4527 129 11 π‘˜ π‘˜ ADP cana-4527 129 12 π‘Ÿ π‘Ÿ NOUN cana-4527 129 13 πœ•π‘Ÿπœ‘(π‘Ÿ πœ•π‘Ÿπœ‘(π‘Ÿ SYM cana-4527 129 14 , , PUNCT cana-4527 129 15 𝑑 𝑑 NOUN cana-4527 129 16 ) ) PUNCT cana-4527 129 17 βˆ’ βˆ’ PROPN cana-4527 130 1 𝑆0πœ•π‘‘ 𝑆0πœ•π‘‘ ADJ cana-4527 130 2 π›Όπœ‘ π›Όπœ‘ NOUN cana-4527 130 3 ( ( PUNCT cana-4527 130 4 π‘Ÿ π‘Ÿ NOUN cana-4527 130 5 , , PUNCT cana-4527 130 6 𝑑 𝑑 PROPN cana-4527 130 7 ) ) PUNCT cana-4527 130 8 = = SYM cana-4527 130 9 0 0 NUM cana-4527 130 10 ( ( PUNCT cana-4527 130 11 4.9 4.9 NUM cana-4527 130 12 ) ) PUNCT cana-4527 130 13 subject subject NOUN cana-4527 130 14 eq eq ADP cana-4527 130 15 . . PUNCT cana-4527 130 16 ( ( PUNCT cana-4527 130 17 4.6 4.6 NUM cana-4527 130 18 ) ) PUNCT cana-4527 130 19 . . PUNCT cana-4527 131 1 here here ADV cana-4527 131 2 , , PUNCT cana-4527 131 3 the the DET cana-4527 131 4 approach approach NOUN cana-4527 131 5 relies rely VERB cana-4527 131 6 on on ADP cana-4527 131 7 using use VERB cana-4527 131 8 transform transform NOUN cana-4527 131 9 ( ( PUNCT cana-4527 131 10 4.9 4.9 NUM cana-4527 131 11 ) ) PUNCT cana-4527 131 12 to to PART cana-4527 131 13 obtain obtain VERB cana-4527 131 14 communications communication NOUN cana-4527 131 15 on on ADP cana-4527 131 16 applied apply VERB cana-4527 131 17 nonlinear nonlinear ADJ cana-4527 131 18 analysis analysis NOUN cana-4527 131 19 issn issn NOUN cana-4527 131 20 : : PUNCT cana-4527 131 21 1074 1074 NUM cana-4527 131 22 - - PUNCT cana-4527 131 23 133x 133x NUM cana-4527 131 24 vol vol NOUN cana-4527 131 25 32 32 NUM cana-4527 131 26 no no NOUN cana-4527 131 27 . . PUNCT cana-4527 132 1 9s 9s NUM cana-4527 132 2 ( ( PUNCT cana-4527 132 3 2025 2025 NUM cana-4527 132 4 ) ) PUNCT cana-4527 132 5 2420 2420 NUM cana-4527 132 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 133 1 𝑠 𝑠 PROPN cana-4527 133 2 [ [ X cana-4527 133 3 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 133 4 , , PUNCT cana-4527 133 5 𝑑 𝑑 NOUN cana-4527 133 6 ) ) PUNCT cana-4527 133 7 ] ] PUNCT cana-4527 133 8 = = PUNCT cana-4527 133 9 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 133 10 , , PUNCT cana-4527 133 11 0 0 NUM cana-4527 133 12 ) ) PUNCT cana-4527 133 13 + + CCONJ cana-4527 133 14 πœ—π›Όπ½ πœ—π›Όπ½ NOUN cana-4527 133 15 [ [ PUNCT cana-4527 133 16 π‘˜ π‘˜ PRON cana-4527 133 17 𝑆0 𝑆0 NOUN cana-4527 133 18 πœ•π‘Ÿ πœ•π‘Ÿ ADV cana-4527 133 19 2πœ‘(π‘Ÿ 2πœ‘(π‘Ÿ NUM cana-4527 133 20 , , PUNCT cana-4527 133 21 𝑑 𝑑 X cana-4527 133 22 ) ) PUNCT cana-4527 133 23 + + CCONJ cana-4527 133 24 π‘˜ π‘˜ X cana-4527 133 25 𝑆0π‘Ÿ 𝑆0π‘Ÿ ADP cana-4527 133 26 πœ•π‘Ÿπœ‘(π‘Ÿ πœ•π‘Ÿπœ‘(π‘Ÿ NUM cana-4527 133 27 , , PUNCT cana-4527 133 28 𝑑 𝑑 NOUN cana-4527 133 29 ) ) PUNCT cana-4527 133 30 ] ] PUNCT cana-4527 133 31 ( ( PUNCT cana-4527 133 32 4.10 4.10 NUM cana-4527 133 33 ) ) PUNCT cana-4527 133 34 applying apply VERB cana-4527 133 35 the the DET cana-4527 133 36 inverse inverse NOUN cana-4527 133 37 operator operator NOUN cana-4527 133 38 yields yield NOUN cana-4527 133 39 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 133 40 , , PUNCT cana-4527 133 41 𝑑 𝑑 NOUN cana-4527 133 42 ) ) PUNCT cana-4527 133 43 = = SYM cana-4527 133 44 πœ‘ πœ‘ PROPN cana-4527 133 45 ( ( PUNCT cana-4527 133 46 π‘Ÿ π‘Ÿ NOUN cana-4527 133 47 , , PUNCT cana-4527 133 48 0 0 NUM cana-4527 133 49 ) ) PUNCT cana-4527 134 1 + + CCONJ cana-4527 134 2 π‘†βˆ’1[𝑣𝛼𝐽 π‘†βˆ’1[𝑣𝛼𝐽 X cana-4527 134 3 [ [ PUNCT cana-4527 134 4 π‘˜ π‘˜ PRON cana-4527 134 5 𝑆0 𝑆0 NOUN cana-4527 134 6 πœ•π‘Ÿ πœ•π‘Ÿ ADV cana-4527 134 7 2πœ‘(π‘Ÿ 2πœ‘(π‘Ÿ NUM cana-4527 134 8 , , PUNCT cana-4527 134 9 𝑑 𝑑 X cana-4527 134 10 ) ) PUNCT cana-4527 134 11 + + CCONJ cana-4527 134 12 π‘˜ π‘˜ PRON cana-4527 134 13 𝑆0π‘Ÿ 𝑆0π‘Ÿ NOUN cana-4527 134 14 πœ•π‘Ÿπœ‘ πœ•π‘Ÿπœ‘ SYM cana-4527 134 15 ( ( PUNCT cana-4527 134 16 π‘Ÿ π‘Ÿ NOUN cana-4527 134 17 , , PUNCT cana-4527 134 18 𝑑 𝑑 NOUN cana-4527 134 19 ) ) PUNCT cana-4527 134 20 ] ] PUNCT cana-4527 134 21 ( ( PUNCT cana-4527 134 22 4 4 NUM cana-4527 134 23 . . NOUN cana-4527 134 24 11 11 NUM cana-4527 134 25 ) ) PUNCT cana-4527 134 26 assuming assume VERB cana-4527 134 27 the the DET cana-4527 134 28 adomian adomian NOUN cana-4527 134 29 decomposition decomposition NOUN cana-4527 134 30 approach approach NOUN cana-4527 134 31 for for ADP cana-4527 134 32 the the DET cana-4527 134 33 series series NOUN cana-4527 134 34 solution solution NOUN cana-4527 134 35 , , PUNCT cana-4527 134 36 ( ( PUNCT cana-4527 134 37 4.11 4.11 NUM cana-4527 134 38 ) ) PUNCT cana-4527 134 39 is be AUX cana-4527 134 40 πœ‘ πœ‘ X cana-4527 134 41 ( ( PUNCT cana-4527 134 42 π‘Ÿ π‘Ÿ NOUN cana-4527 134 43 , , PUNCT cana-4527 134 44 𝑑 𝑑 NOUN cana-4527 134 45 ) ) PUNCT cana-4527 134 46 = = PUNCT cana-4527 134 47 βˆ‘ βˆ‘ PUNCT cana-4527 134 48 πœ‘π‘– πœ‘π‘– ADV cana-4527 134 49 ∞ ∞ PROPN cana-4527 134 50 𝑖=0 𝑖=0 SYM cana-4527 134 51 ( ( PUNCT cana-4527 134 52 π‘Ÿ π‘Ÿ NOUN cana-4527 134 53 , , PUNCT cana-4527 134 54 𝑑 𝑑 X cana-4527 134 55 ) ) PUNCT cana-4527 134 56 ( ( PUNCT cana-4527 134 57 4.12 4.12 NUM cana-4527 134 58 ) ) PUNCT cana-4527 134 59 often often ADV cana-4527 134 60 , , PUNCT cana-4527 134 61 the the DET cana-4527 134 62 elements element NOUN cana-4527 134 63 ψi ψi ADP cana-4527 134 64 ( ( PUNCT cana-4527 134 65 r r NOUN cana-4527 134 66 , , PUNCT cana-4527 134 67 t t PROPN cana-4527 134 68 ) ) PUNCT cana-4527 134 69 are be AUX cana-4527 134 70 chosen choose VERB cana-4527 134 71 . . PUNCT cana-4527 135 1 substitute substitute NOUN cana-4527 135 2 ( ( PUNCT cana-4527 135 3 4.12 4.12 NUM cana-4527 135 4 ) ) PUNCT cana-4527 135 5 into into ADP cana-4527 135 6 both both DET cana-4527 135 7 sides side NOUN cana-4527 135 8 of of ADP cana-4527 135 9 ( ( PUNCT cana-4527 135 10 4.11 4.11 NUM cana-4527 135 11 ) ) PUNCT cana-4527 135 12 to to PART cana-4527 135 13 reach reach VERB cana-4527 135 14 the the DET cana-4527 135 15 desired desire VERB cana-4527 135 16 result result NOUN cana-4527 135 17 . . PUNCT cana-4527 136 1 βˆ‘ βˆ‘ ADV cana-4527 136 2 πœ‘π‘– πœ‘π‘– ADV cana-4527 136 3 ∞ ∞ PROPN cana-4527 136 4 𝑖=0 𝑖=0 SYM cana-4527 136 5 ( ( PUNCT cana-4527 136 6 π‘Ÿ π‘Ÿ NOUN cana-4527 136 7 , , PUNCT cana-4527 136 8 𝑑 𝑑 X cana-4527 136 9 ) ) PUNCT cana-4527 136 10 = = PUNCT cana-4527 136 11 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 136 12 , , PUNCT cana-4527 136 13 0 0 NUM cana-4527 136 14 ) ) PUNCT cana-4527 137 1 + + CCONJ cana-4527 137 2 π‘†βˆ’1[𝑣𝛼𝑆 π‘†βˆ’1[𝑣𝛼𝑆 PROPN cana-4527 137 3 [ [ PUNCT cana-4527 137 4 π‘˜ π‘˜ PRON cana-4527 137 5 𝑆0 𝑆0 NOUN cana-4527 137 6 πœ•π‘Ÿ πœ•π‘Ÿ ADP cana-4527 137 7 2 2 NUM cana-4527 137 8 βˆ‘ βˆ‘ NOUN cana-4527 137 9 πœ‘π‘– πœ‘π‘– ADV cana-4527 137 10 ∞ ∞ PROPN cana-4527 137 11 𝑖=0 𝑖=0 SYM cana-4527 137 12 ( ( PUNCT cana-4527 137 13 π‘Ÿ π‘Ÿ NOUN cana-4527 137 14 , , PUNCT cana-4527 137 15 𝑑 𝑑 NOUN cana-4527 137 16 ) ) PUNCT cana-4527 137 17 + + CCONJ cana-4527 137 18 π‘˜ π‘˜ DET cana-4527 137 19 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 137 20 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 137 21 βˆ‘ βˆ‘ ADP cana-4527 137 22 πœ‘π‘–(π‘Ÿ πœ‘π‘–(π‘Ÿ PROPN cana-4527 137 23 , , PUNCT cana-4527 137 24 𝑑 𝑑 PROPN cana-4527 137 25 ) ) PUNCT cana-4527 137 26 ] ] PUNCT cana-4527 137 27 ] ] X cana-4527 137 28 ( ( PUNCT cana-4527 137 29 4 4 NUM cana-4527 137 30 βˆ™ βˆ™ NOUN cana-4527 137 31 13 13 NUM cana-4527 137 32 ) ) PUNCT cana-4527 137 33 ∞ ∞ NUM cana-4527 137 34 𝑖=0 𝑖=0 PUNCT cana-4527 138 1 next next ADJ cana-4527 138 2 , , PUNCT cana-4527 138 3 recursive recursive ADJ cana-4527 138 4 relationships relationship NOUN cana-4527 138 5 are be AUX cana-4527 138 6 shown show VERB cana-4527 138 7 as as ADP cana-4527 138 8 πœ‘0(π‘Ÿ πœ‘0(π‘Ÿ NOUN cana-4527 138 9 , , PUNCT cana-4527 138 10 𝑑 𝑑 NOUN cana-4527 138 11 ) ) PUNCT cana-4527 138 12 = = SYM cana-4527 138 13 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-4527 138 14 , , PUNCT cana-4527 138 15 πœ‘π‘–+1(π‘Ÿ πœ‘π‘–+1(π‘Ÿ PROPN cana-4527 138 16 , , PUNCT cana-4527 138 17 𝑑 𝑑 NOUN cana-4527 138 18 ) ) PUNCT cana-4527 138 19 = = PUNCT cana-4527 139 1 π½βˆ’1 π½βˆ’1 PROPN cana-4527 140 1 [ [ X cana-4527 140 2 𝑣𝛼𝑆 𝑣𝛼𝑆 X cana-4527 140 3 [ [ PUNCT cana-4527 140 4 π‘˜ π‘˜ DET cana-4527 140 5 𝑆0 𝑆0 NOUN cana-4527 140 6 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 140 7 2πœ‘π‘–(π‘Ÿ 2πœ‘π‘–(π‘Ÿ NUM cana-4527 140 8 , , PUNCT cana-4527 140 9 𝑑 𝑑 X cana-4527 140 10 ) ) PUNCT cana-4527 140 11 + + CCONJ cana-4527 140 12 π‘˜ π‘˜ DET cana-4527 140 13 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 140 14 πœ•π‘Ÿπœ‘π‘–(π‘Ÿ πœ•π‘Ÿπœ‘π‘–(π‘Ÿ NOUN cana-4527 140 15 , , PUNCT cana-4527 140 16 𝑑 𝑑 PROPN cana-4527 140 17 ) ) PUNCT cana-4527 140 18 ] ] PUNCT cana-4527 140 19 ] ] PUNCT cana-4527 140 20 𝑖 𝑖 X cana-4527 140 21 β‰₯ β‰₯ NUM cana-4527 140 22 1 1 NUM cana-4527 140 23 ( ( PUNCT cana-4527 140 24 4.14 4.14 NUM cana-4527 140 25 ) ) PUNCT cana-4527 140 26 the the DET cana-4527 140 27 remaining remain VERB cana-4527 140 28 components component NOUN cana-4527 140 29 , , PUNCT cana-4527 140 30 i i PRON cana-4527 140 31 > > X cana-4527 140 32 1 1 NUM cana-4527 140 33 , , PUNCT cana-4527 140 34 are be AUX cana-4527 140 35 determined determine VERB cana-4527 140 36 by by ADP cana-4527 140 37 applying apply VERB cana-4527 140 38 the the DET cana-4527 140 39 preconditions precondition NOUN cana-4527 140 40 because because SCONJ cana-4527 140 41 component component NOUN cana-4527 140 42 Ο†0(r Ο†0(r PROPN cana-4527 140 43 , , PUNCT cana-4527 140 44 t t PROPN cana-4527 140 45 ) ) PUNCT cana-4527 140 46 is be AUX cana-4527 140 47 defined define VERB cana-4527 140 48 . . PUNCT cana-4527 141 1 πœ‘0(π‘Ÿ πœ‘0(π‘Ÿ NUM cana-4527 141 2 , , PUNCT cana-4527 141 3 𝑑 𝑑 NOUN cana-4527 141 4 ) ) PUNCT cana-4527 141 5 = = SYM cana-4527 141 6 πœ‘ πœ‘ PROPN cana-4527 141 7 ( ( PUNCT cana-4527 141 8 π‘Ÿ π‘Ÿ NOUN cana-4527 141 9 , , PUNCT cana-4527 141 10 0 0 NUM cana-4527 141 11 ) ) PUNCT cana-4527 141 12 = = SYM cana-4527 141 13 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-4527 141 14 , , PUNCT cana-4527 141 15 πœ‘1(π‘Ÿ πœ‘1(π‘Ÿ PROPN cana-4527 141 16 , , PUNCT cana-4527 141 17 𝑑 𝑑 X cana-4527 141 18 ) ) PUNCT cana-4527 141 19 = = PUNCT cana-4527 142 1 π½βˆ’1[𝑣𝛼𝑆 π½βˆ’1[𝑣𝛼𝑆 PROPN cana-4527 142 2 [ [ PUNCT cana-4527 142 3 π‘˜ π‘˜ DET cana-4527 142 4 𝑆0 𝑆0 NOUN cana-4527 142 5 πœ•π‘Ÿ πœ•π‘Ÿ ADP cana-4527 142 6 2πœ‘0(π‘Ÿ 2πœ‘0(π‘Ÿ NUM cana-4527 142 7 , , PUNCT cana-4527 142 8 𝑑 𝑑 NOUN cana-4527 142 9 ) ) PUNCT cana-4527 142 10 + + CCONJ cana-4527 142 11 π‘˜ π‘˜ DET cana-4527 142 12 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 142 13 πœ•π‘Ÿπœ‘0(π‘Ÿ πœ•π‘Ÿπœ‘0(π‘Ÿ NOUN cana-4527 142 14 , , PUNCT cana-4527 142 15 𝑑 𝑑 NOUN cana-4527 142 16 ) ) PUNCT cana-4527 142 17 ] ] PUNCT cana-4527 142 18 ] ] PUNCT cana-4527 143 1 = = PUNCT cana-4527 143 2 π‘†βˆ’1[πœ—βˆπ‘† π‘†βˆ’1[πœ—βˆπ‘† PROPN cana-4527 144 1 [ [ PUNCT cana-4527 144 2 π‘π‘˜ π‘π‘˜ ADP cana-4527 144 3 𝑆0π‘Ÿ 𝑆0π‘Ÿ SYM cana-4527 144 4 ] ] PUNCT cana-4527 144 5 = = PUNCT cana-4527 144 6 π‘π‘˜π‘‘βˆ π‘π‘˜π‘‘βˆ NUM cana-4527 144 7 π‘Ÿπ‘†0Ξ³(1+∝ π‘Ÿπ‘†0Ξ³(1+∝ NUM cana-4527 144 8 ) ) PUNCT cana-4527 144 9 πœ‘2(π‘Ÿ πœ‘2(π‘Ÿ NOUN cana-4527 144 10 , , PUNCT cana-4527 144 11 𝑑 𝑑 X cana-4527 144 12 ) ) PUNCT cana-4527 144 13 = = PUNCT cana-4527 144 14 π½βˆ’1[πœ—π›Όπ‘† π½βˆ’1[πœ—π›Όπ‘† X cana-4527 145 1 [ [ PUNCT cana-4527 145 2 π‘˜ π‘˜ DET cana-4527 145 3 𝑆0 𝑆0 NOUN cana-4527 145 4 πœ•π‘Ÿ πœ•π‘Ÿ CCONJ cana-4527 145 5 2πœ‘1(π‘Ÿ 2πœ‘1(π‘Ÿ NUM cana-4527 145 6 , , PUNCT cana-4527 145 7 𝑑 𝑑 NOUN cana-4527 145 8 ) ) PUNCT cana-4527 145 9 + + CCONJ cana-4527 145 10 π‘˜ π‘˜ DET cana-4527 145 11 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 145 12 πœ•π‘Ÿπœ‘1(π‘Ÿ πœ•π‘Ÿπœ‘1(π‘Ÿ PROPN cana-4527 145 13 , , PUNCT cana-4527 145 14 𝑑 𝑑 NOUN cana-4527 145 15 ) ) PUNCT cana-4527 145 16 ] ] PUNCT cana-4527 145 17 ] ] PUNCT cana-4527 145 18 = = PUNCT cana-4527 146 1 π½βˆ’1[π‘£βˆπ½ π½βˆ’1[π‘£βˆπ½ NOUN cana-4527 146 2 [ [ PUNCT cana-4527 146 3 π‘˜ π‘˜ DET cana-4527 146 4 𝑆0 𝑆0 NOUN cana-4527 146 5 ( ( PUNCT cana-4527 146 6 2π‘π‘˜π‘‘βˆž 2π‘π‘˜π‘‘βˆž NUM cana-4527 146 7 π‘Ÿ3𝑆0Ξ³(1+∝ π‘Ÿ3𝑆0Ξ³(1+∝ NOUN cana-4527 146 8 ) ) PUNCT cana-4527 146 9 ) ) PUNCT cana-4527 147 1 + + CCONJ cana-4527 147 2 π‘˜ π‘˜ X cana-4527 147 3 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 147 4 ( ( PUNCT cana-4527 147 5 βˆ’π‘π‘˜π‘‘βˆ βˆ’π‘π‘˜π‘‘βˆ NOUN cana-4527 147 6 π‘Ÿ2𝑆0Ξ³(1+∝ π‘Ÿ2𝑆0Ξ³(1+∝ NUM cana-4527 147 7 ) ) PUNCT cana-4527 147 8 ) ) PUNCT cana-4527 148 1 ] ] PUNCT cana-4527 148 2 ] ] X cana-4527 148 3 = = PUNCT cana-4527 148 4 π½βˆ’1[πœ—βˆπ½ π½βˆ’1[πœ—βˆπ½ PUNCT cana-4527 148 5 [ [ PUNCT cana-4527 148 6 π‘π‘˜2𝑑𝛼 π‘π‘˜2𝑑𝛼 X cana-4527 148 7 π‘Ÿ3𝑆0 π‘Ÿ3𝑆0 NUM cana-4527 148 8 2Ξ³(1 2Ξ³(1 PROPN cana-4527 149 1 + + CCONJ cana-4527 149 2 Ξ± Ξ± X cana-4527 149 3 ) ) PUNCT cana-4527 149 4 ] ] PUNCT cana-4527 149 5 ] ] X cana-4527 149 6 = = PUNCT cana-4527 149 7 π½βˆ’1 π½βˆ’1 PROPN cana-4527 149 8 [ [ PUNCT cana-4527 149 9 π‘π‘˜2πœ—2∝ π‘π‘˜2πœ—2∝ PROPN cana-4527 149 10 π‘Ÿ3𝑆0 π‘Ÿ3𝑆0 NUM cana-4527 149 11 2 2 NUM cana-4527 149 12 ] ] PUNCT cana-4527 149 13 = = SYM cana-4527 149 14 π‘π‘˜2𝑑2∝ π‘π‘˜2𝑑2∝ PROPN cana-4527 149 15 π‘Ÿ3𝑆0 π‘Ÿ3𝑆0 PUNCT cana-4527 149 16 2Ξ³(1 2Ξ³(1 NUM cana-4527 149 17 + + CCONJ cana-4527 149 18 2∝ 2∝ NUM cana-4527 149 19 ) ) PUNCT cana-4527 149 20 ( ( PUNCT cana-4527 149 21 4.15 4.15 NUM cana-4527 149 22 ) ) PUNCT cana-4527 149 23 communications communication NOUN cana-4527 149 24 on on ADP cana-4527 149 25 applied apply VERB cana-4527 149 26 nonlinear nonlinear ADJ cana-4527 149 27 analysis analysis NOUN cana-4527 149 28 issn issn NOUN cana-4527 149 29 : : PUNCT cana-4527 149 30 1074 1074 NUM cana-4527 149 31 - - PUNCT cana-4527 149 32 133x 133x NUM cana-4527 149 33 vol vol NOUN cana-4527 149 34 32 32 NUM cana-4527 149 35 no no NOUN cana-4527 149 36 . . PUNCT cana-4527 150 1 9s 9s NUM cana-4527 150 2 ( ( PUNCT cana-4527 150 3 2025 2025 NUM cana-4527 150 4 ) ) PUNCT cana-4527 150 5 2421 2421 NUM cana-4527 150 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 150 7 in in ADP cana-4527 150 8 this this DET cana-4527 150 9 instance instance NOUN cana-4527 150 10 , , PUNCT cana-4527 150 11 we we PRON cana-4527 150 12 calculated calculate VERB cana-4527 150 13 the the DET cana-4527 150 14 values value NOUN cana-4527 150 15 of of ADP cana-4527 150 16 the the DET cana-4527 150 17 four four NUM cana-4527 150 18 components component NOUN cana-4527 150 19 of of ADP cana-4527 150 20 eq eq PROPN cana-4527 150 21 . . PUNCT cana-4527 151 1 ( ( PUNCT cana-4527 151 2 4.12 4.12 NUM cana-4527 151 3 ) ) PUNCT cana-4527 151 4 to to PART cana-4527 151 5 obtain obtain VERB cana-4527 151 6 ψ(r ψ(r PROPN cana-4527 151 7 , , PUNCT cana-4527 151 8 t t PROPN cana-4527 151 9 ) ) PUNCT cana-4527 151 10 , , PUNCT cana-4527 151 11 resulting result VERB cana-4527 151 12 in in ADP cana-4527 151 13 the the DET cana-4527 151 14 following follow VERB cana-4527 151 15 expansion expansion NOUN cana-4527 151 16 . . PUNCT cana-4527 152 1 πœ‘ πœ‘ X cana-4527 152 2 ( ( PUNCT cana-4527 152 3 π‘Ÿ π‘Ÿ NOUN cana-4527 152 4 , , PUNCT cana-4527 152 5 𝑑 𝑑 X cana-4527 152 6 ) ) PUNCT cana-4527 152 7 = = SYM cana-4527 152 8 π‘π‘Ÿ π‘π‘Ÿ X cana-4527 153 1 + + NUM cana-4527 153 2 π‘π‘˜π‘‘π›Ό π‘π‘˜π‘‘π›Ό ADJ cana-4527 153 3 π‘Ÿπ‘†0Ξ³(1 π‘Ÿπ‘†0Ξ³(1 NOUN cana-4527 153 4 + + CCONJ cana-4527 153 5 Ξ± Ξ± X cana-4527 153 6 ) ) PUNCT cana-4527 153 7 + + NUM cana-4527 153 8 π‘π‘˜2𝑑2𝛼 π‘π‘˜2𝑑2𝛼 NOUN cana-4527 153 9 π‘Ÿ3𝑆0 π‘Ÿ3𝑆0 NUM cana-4527 153 10 2Ξ³(1 2Ξ³(1 PROPN cana-4527 153 11 + + CCONJ cana-4527 153 12 2Ξ± 2Ξ± NOUN cana-4527 153 13 ) ) PUNCT cana-4527 153 14 + + NUM cana-4527 153 15 9π‘π‘˜3πœ—3𝛼 9π‘π‘˜3πœ—3𝛼 NUM cana-4527 153 16 π‘Ÿ5𝑆0 π‘Ÿ5𝑆0 NUM cana-4527 153 17 3Ξ³(1 3Ξ³(1 NUM cana-4527 154 1 + + CCONJ cana-4527 154 2 3 3 NUM cana-4527 154 3 Ξ± Ξ± NOUN cana-4527 154 4 ) ) PUNCT cana-4527 154 5 ( ( PUNCT cana-4527 154 6 4.16 4.16 NUM cana-4527 154 7 ) ) PUNCT cana-4527 154 8 additionally additionally ADV cana-4527 154 9 , , PUNCT cana-4527 154 10 by by ADP cana-4527 154 11 using use VERB cana-4527 154 12 the the DET cana-4527 154 13 boundary boundary ADJ cana-4527 154 14 condition condition NOUN cana-4527 154 15 on on ADP cana-4527 154 16 ( ( PUNCT cana-4527 154 17 4.16 4.16 NUM cana-4527 154 18 ) ) PUNCT cana-4527 154 19 , , PUNCT cana-4527 154 20 we we PRON cana-4527 154 21 may may AUX cana-4527 154 22 determine determine VERB cana-4527 154 23 a a DET cana-4527 154 24 constant constant ADJ cana-4527 154 25 number number NOUN cana-4527 154 26 ( ( PUNCT cana-4527 154 27 a a NOUN cana-4527 154 28 ) ) PUNCT cana-4527 154 29 to to PART cana-4527 154 30 acquire acquire VERB cana-4527 154 31 . . PUNCT cana-4527 155 1 𝑝 𝑝 X cana-4527 156 1 = = SYM cana-4527 156 2 𝑄 𝑄 PROPN cana-4527 156 3 2πœ‹π‘Ÿπ‘π‘˜π‘‘π‘“(π‘Ÿπ‘ 2πœ‹π‘Ÿπ‘π‘˜π‘‘π‘“(π‘Ÿπ‘ NUM cana-4527 156 4 , , PUNCT cana-4527 156 5 𝑑 𝑑 X cana-4527 156 6 ) ) PUNCT cana-4527 156 7 ( ( PUNCT cana-4527 156 8 4.17 4.17 NUM cana-4527 156 9 ) ) PUNCT cana-4527 156 10 where where SCONJ cana-4527 156 11 𝑓(π‘Ÿπ‘ 𝑓(π‘Ÿπ‘ NOUN cana-4527 156 12 , , PUNCT cana-4527 156 13 𝑑 𝑑 NOUN cana-4527 156 14 ) ) PUNCT cana-4527 156 15 = = SYM cana-4527 156 16 1 1 NUM cana-4527 156 17 βˆ’ βˆ’ NUM cana-4527 157 1 π‘˜π‘‘βˆ π‘˜π‘‘βˆ NOUN cana-4527 157 2 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 157 3 2𝑆0Ξ³(∝ 2𝑆0Ξ³(∝ NUM cana-4527 157 4 +1 +1 NOUN cana-4527 157 5 ) ) PUNCT cana-4527 158 1 + + CCONJ cana-4527 159 1 3π‘˜π‘‘2∝ 3π‘˜π‘‘2∝ NUM cana-4527 159 2 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 159 3 4𝑆0 4𝑆0 NUM cana-4527 159 4 2Ξ³(2 2Ξ³(2 NUM cana-4527 159 5 ∝ ∝ PROPN cana-4527 159 6 +1 +1 PROPN cana-4527 159 7 ) ) PUNCT cana-4527 160 1 + + CCONJ cana-4527 160 2 β‹―. β‹―. PROPN cana-4527 160 3 ( ( PUNCT cana-4527 160 4 4.18 4.18 NUM cana-4527 160 5 ) ) PUNCT cana-4527 160 6 example example NOUN cana-4527 160 7 2 2 NUM cana-4527 160 8 : : PUNCT cana-4527 160 9 let let VERB cana-4527 160 10 us we PRON cana-4527 160 11 examine examine VERB cana-4527 160 12 the the DET cana-4527 160 13 groundwater groundwater NOUN cana-4527 160 14 equation equation NOUN cana-4527 160 15 , , PUNCT cana-4527 160 16 including include VERB cana-4527 160 17 fractional fractional ADJ cana-4527 160 18 derivatives derivative NOUN cana-4527 160 19 in in ADP cana-4527 160 20 both both DET cana-4527 160 21 space space NOUN cana-4527 160 22 and and CCONJ cana-4527 160 23 time time NOUN cana-4527 160 24 . . PUNCT cana-4527 161 1 𝑆0πœ•π‘‘πœ‘ 𝑆0πœ•π‘‘πœ‘ PROPN cana-4527 161 2 ( ( PUNCT cana-4527 161 3 π‘Ÿ π‘Ÿ NOUN cana-4527 161 4 , , PUNCT cana-4527 161 5 𝑑 𝑑 X cana-4527 161 6 ) ) PUNCT cana-4527 161 7 = = PUNCT cana-4527 161 8 π‘˜ π‘˜ ADP cana-4527 161 9 π‘Ÿ π‘Ÿ NOUN cana-4527 161 10 πœ•π‘Ÿ πœ•π‘Ÿ NOUN cana-4527 161 11 βˆπœ‘(π‘Ÿ βˆπœ‘(π‘Ÿ VERB cana-4527 161 12 , , PUNCT cana-4527 161 13 𝑑 𝑑 X cana-4527 161 14 ) ) PUNCT cana-4527 162 1 + + CCONJ cana-4527 162 2 π‘˜πœ•π‘Ÿ π‘˜πœ•π‘Ÿ PROPN cana-4527 162 3 πœ‡πœ‘(π‘Ÿ πœ‡πœ‘(π‘Ÿ PROPN cana-4527 162 4 , , PUNCT cana-4527 162 5 𝑑 𝑑 PROPN cana-4527 162 6 ) ) PUNCT cana-4527 162 7 , , PUNCT cana-4527 162 8 0 0 PUNCT cana-4527 162 9 < < X cana-4527 162 10 𝛼 𝛼 X cana-4527 162 11 ≀ ≀ NUM cana-4527 162 12 1 1 NUM cana-4527 162 13 < < X cana-4527 162 14 πœ‡ πœ‡ ADP cana-4527 162 15 ≀ ≀ ADJ cana-4527 162 16 2 2 NUM cana-4527 162 17 ( ( PUNCT cana-4527 162 18 4.19 4.19 NUM cana-4527 162 19 ) ) PUNCT cana-4527 162 20 eq eq NOUN cana-4527 162 21 . . PUNCT cana-4527 163 1 ( ( PUNCT cana-4527 163 2 4.6 4.6 NUM cana-4527 163 3 ) ) PUNCT cana-4527 163 4 dictates dictate VERB cana-4527 163 5 that that SCONJ cana-4527 163 6 the the DET cana-4527 163 7 gradient gradient NOUN cana-4527 163 8 is be AUX cana-4527 163 9 the the DET cana-4527 163 10 same same ADJ cana-4527 163 11 as as ADP cana-4527 163 12 zero zero NUM cana-4527 163 13 . . PUNCT cana-4527 164 1 βˆ‚Ξ±r βˆ‚Ξ±r PROPN cana-4527 164 2 Ο†(r Ο†(r ADV cana-4527 164 3 , , PUNCT cana-4527 164 4 0 0 NUM cana-4527 164 5 ) ) PUNCT cana-4527 165 1 = = VERB cana-4527 165 2 constant constant ADJ cana-4527 165 3 is be AUX cana-4527 165 4 the the DET cana-4527 165 5 expected expect VERB cana-4527 165 6 fractional fractional ADJ cana-4527 165 7 change change NOUN cana-4527 165 8 in in ADP cana-4527 165 9 drag drag NOUN cana-4527 165 10 for for ADP cana-4527 165 11 t t NOUN cana-4527 165 12 = = SYM cana-4527 165 13 0 0 PUNCT cana-4527 165 14 since since SCONJ cana-4527 165 15 the the DET cana-4527 165 16 aquifer aquifer NOUN cana-4527 165 17 's 's PART cana-4527 165 18 water water NOUN cana-4527 165 19 level level NOUN cana-4527 165 20 is be AUX cana-4527 165 21 assumed assume VERB cana-4527 165 22 to to PART cana-4527 165 23 be be AUX cana-4527 165 24 constant constant ADJ cana-4527 165 25 before before ADP cana-4527 165 26 pumping pump VERB cana-4527 165 27 . . PUNCT cana-4527 166 1 use use VERB cana-4527 166 2 the the DET cana-4527 166 3 sumudu sumudu NOUN cana-4527 166 4 transform transform NOUN cana-4527 166 5 on on ADP cana-4527 166 6 equation equation NOUN cana-4527 166 7 ( ( PUNCT cana-4527 166 8 4.19 4.19 NUM cana-4527 166 9 ) ) PUNCT cana-4527 166 10 . . PUNCT cana-4527 167 1 the the DET cana-4527 167 2 result result NOUN cana-4527 167 3 is be AUX cana-4527 167 4 : : PUNCT cana-4527 167 5 𝐽[πœ•π‘‘πœ‘(π‘Ÿ 𝐽[πœ•π‘‘πœ‘(π‘Ÿ NUM cana-4527 167 6 , , PUNCT cana-4527 167 7 𝑑 𝑑 NOUN cana-4527 167 8 ) ) PUNCT cana-4527 167 9 ] ] PUNCT cana-4527 168 1 = = PUNCT cana-4527 168 2 𝐽 𝐽 PROPN cana-4527 168 3 [ [ PUNCT cana-4527 168 4 π‘˜ π‘˜ PROPN cana-4527 168 5 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 168 6 πœ•π‘Ÿ πœ•π‘Ÿ X cana-4527 168 7 βˆπœ‘ βˆπœ‘ X cana-4527 168 8 ( ( PUNCT cana-4527 168 9 π‘Ÿ π‘Ÿ NOUN cana-4527 168 10 , , PUNCT cana-4527 168 11 𝑑 𝑑 NOUN cana-4527 168 12 ) ) PUNCT cana-4527 168 13 + + CCONJ cana-4527 168 14 π‘˜ π‘˜ DET cana-4527 168 15 𝑆0 𝑆0 NOUN cana-4527 168 16 πœ•π‘Ÿ πœ•π‘Ÿ CCONJ cana-4527 168 17 πœ‡πœ‘ πœ‡πœ‘ ADP cana-4527 168 18 ( ( PUNCT cana-4527 168 19 π‘Ÿ π‘Ÿ NOUN cana-4527 168 20 , , PUNCT cana-4527 168 21 𝑑 𝑑 NOUN cana-4527 168 22 ) ) PUNCT cana-4527 168 23 , , PUNCT cana-4527 168 24 πœ—βˆ’1(𝐽[πœ‘(π‘Ÿ πœ—βˆ’1(𝐽[πœ‘(π‘Ÿ PROPN cana-4527 168 25 , , PUNCT cana-4527 168 26 𝑑 𝑑 PROPN cana-4527 168 27 ) ) PUNCT cana-4527 168 28 ] ] PUNCT cana-4527 168 29 βˆ’ βˆ’ PROPN cana-4527 168 30 πœ‘ πœ‘ X cana-4527 168 31 ( ( PUNCT cana-4527 168 32 π‘Ÿ π‘Ÿ NOUN cana-4527 168 33 , , PUNCT cana-4527 168 34 0 0 NUM cana-4527 168 35 ) ) PUNCT cana-4527 168 36 ) ) PUNCT cana-4527 169 1 = = PUNCT cana-4527 169 2 𝐽 𝐽 PROPN cana-4527 169 3 [ [ PUNCT cana-4527 169 4 π‘˜ π‘˜ PROPN cana-4527 169 5 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 169 6 πœ•π‘Ÿ πœ•π‘Ÿ X cana-4527 169 7 βˆπœ‘ βˆπœ‘ X cana-4527 169 8 ( ( PUNCT cana-4527 169 9 π‘Ÿ π‘Ÿ NOUN cana-4527 169 10 , , PUNCT cana-4527 169 11 𝑑 𝑑 NOUN cana-4527 169 12 ) ) PUNCT cana-4527 169 13 + + CCONJ cana-4527 169 14 π‘˜ π‘˜ DET cana-4527 169 15 𝑆0 𝑆0 NOUN cana-4527 169 16 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 169 17 πœ‡πœ‘(π‘Ÿ πœ‡πœ‘(π‘Ÿ PROPN cana-4527 169 18 , , PUNCT cana-4527 169 19 𝑑 𝑑 X cana-4527 169 20 ) ) PUNCT cana-4527 169 21 so so SCONJ cana-4527 169 22 𝐽[πœ‘(π‘Ÿ 𝐽[πœ‘(π‘Ÿ PROPN cana-4527 169 23 , , PUNCT cana-4527 169 24 𝑑 𝑑 PROPN cana-4527 169 25 ) ) PUNCT cana-4527 169 26 ] ] PUNCT cana-4527 169 27 = = PUNCT cana-4527 169 28 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 169 29 , , PUNCT cana-4527 169 30 0 0 NUM cana-4527 169 31 ) ) PUNCT cana-4527 170 1 + + CCONJ cana-4527 170 2 πœ—π½ πœ—π½ ADJ cana-4527 170 3 [ [ PUNCT cana-4527 170 4 π‘˜ π‘˜ DET cana-4527 170 5 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 170 6 πœ•π‘Ÿ πœ•π‘Ÿ NOUN cana-4527 170 7 βˆπœ‘(π‘Ÿ βˆπœ‘(π‘Ÿ VERB cana-4527 170 8 , , PUNCT cana-4527 170 9 𝑑 𝑑 X cana-4527 170 10 ) ) PUNCT cana-4527 170 11 + + CCONJ cana-4527 170 12 π‘˜ π‘˜ DET cana-4527 170 13 𝑆0 𝑆0 NOUN cana-4527 170 14 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 170 15 πœ‡πœ‘(π‘Ÿ πœ‡πœ‘(π‘Ÿ PROPN cana-4527 170 16 , , PUNCT cana-4527 170 17 𝑑 𝑑 X cana-4527 170 18 ) ) PUNCT cana-4527 170 19 ( ( PUNCT cana-4527 170 20 4.20 4.20 NUM cana-4527 170 21 ) ) PUNCT cana-4527 170 22 the the DET cana-4527 170 23 inverse inverse NOUN cana-4527 170 24 operator operator NOUN cana-4527 170 25 is be AUX cana-4527 170 26 applied apply VERB cana-4527 170 27 to to ADP cana-4527 170 28 eq eq PROPN cana-4527 170 29 . . PUNCT cana-4527 171 1 ( ( PUNCT cana-4527 171 2 4 4 NUM cana-4527 171 3 . . NOUN cana-4527 171 4 20 20 NUM cana-4527 171 5 ) ) PUNCT cana-4527 171 6 to to PART cana-4527 171 7 obtain obtain VERB cana-4527 171 8 πœ‘ πœ‘ PROPN cana-4527 171 9 ( ( PUNCT cana-4527 171 10 π‘Ÿ π‘Ÿ NOUN cana-4527 171 11 , , PUNCT cana-4527 171 12 𝑑 𝑑 X cana-4527 171 13 ) ) PUNCT cana-4527 171 14 = = PUNCT cana-4527 171 15 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 171 16 , , PUNCT cana-4527 171 17 0 0 NUM cana-4527 171 18 ) ) PUNCT cana-4527 172 1 + + CCONJ cana-4527 172 2 π½βˆ’1 π½βˆ’1 ADP cana-4527 173 1 [ [ X cana-4527 173 2 πœ—π‘† πœ—π‘† X cana-4527 173 3 [ [ PUNCT cana-4527 173 4 π‘˜ π‘˜ DET cana-4527 173 5 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 173 6 πœ•π‘Ÿ πœ•π‘Ÿ NOUN cana-4527 173 7 βˆπœ‘(π‘Ÿ βˆπœ‘(π‘Ÿ VERB cana-4527 173 8 , , PUNCT cana-4527 173 9 𝑑 𝑑 X cana-4527 173 10 ) ) PUNCT cana-4527 174 1 + + CCONJ cana-4527 174 2 π‘˜ π‘˜ DET cana-4527 174 3 𝑆0 𝑆0 NOUN cana-4527 174 4 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 174 5 πœ‡πœ‘(π‘Ÿ πœ‡πœ‘(π‘Ÿ PROPN cana-4527 174 6 , , PUNCT cana-4527 174 7 𝑑 𝑑 X cana-4527 174 8 ) ) PUNCT cana-4527 174 9 ] ] X cana-4527 174 10 ] ] X cana-4527 174 11 ( ( PUNCT cana-4527 174 12 4 4 NUM cana-4527 174 13 . . NOUN cana-4527 174 14 21 21 NUM cana-4527 174 15 ) ) PUNCT cana-4527 174 16 substituting substitute VERB cana-4527 174 17 ( ( PUNCT cana-4527 174 18 4.12 4.12 NUM cana-4527 174 19 ) ) PUNCT cana-4527 174 20 into into ADP cana-4527 174 21 both both DET cana-4527 174 22 sides side NOUN cana-4527 174 23 of of ADP cana-4527 174 24 ( ( PUNCT cana-4527 174 25 4 4 NUM cana-4527 174 26 . . NOUN cana-4527 174 27 21 21 NUM cana-4527 174 28 ) ) PUNCT cana-4527 174 29 results result NOUN cana-4527 174 30 in in ADP cana-4527 174 31 βˆ‘ βˆ‘ PROPN cana-4527 174 32 πœ‘π‘–(π‘Ÿ πœ‘π‘–(π‘Ÿ PROPN cana-4527 174 33 , , PUNCT cana-4527 174 34 𝑑 𝑑 X cana-4527 174 35 ) ) PUNCT cana-4527 174 36 = = PUNCT cana-4527 174 37 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 174 38 , , PUNCT cana-4527 174 39 0 0 NUM cana-4527 174 40 ) ) PUNCT cana-4527 175 1 + + CCONJ cana-4527 175 2 π½βˆ’1[πœ—π½ π½βˆ’1[πœ—π½ NUM cana-4527 175 3 [ [ PUNCT cana-4527 175 4 π‘˜ π‘˜ PRON cana-4527 175 5 π‘Ÿπ‘ 0 π‘Ÿπ‘ 0 NOUN cana-4527 175 6 ∞ ∞ NUM cana-4527 175 7 𝑖=0 𝑖=0 PROPN cana-4527 175 8 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 175 9 𝛼(βˆ‘ 𝛼(βˆ‘ NOUN cana-4527 175 10 πœ‘π‘–(π‘Ÿ πœ‘π‘–(π‘Ÿ ADP cana-4527 175 11 , , PUNCT cana-4527 175 12 𝑑 𝑑 NOUN cana-4527 175 13 ) ) PUNCT cana-4527 175 14 ) ) PUNCT cana-4527 176 1 + + CCONJ cana-4527 176 2 π‘˜ π‘˜ DET cana-4527 176 3 𝑆0 𝑆0 NOUN cana-4527 176 4 πœ•π‘Ÿ πœ•π‘Ÿ ADJ cana-4527 176 5 πœ‡(βˆ‘ πœ‡(βˆ‘ NOUN cana-4527 176 6 πœ‘π‘–(π‘Ÿ πœ‘π‘–(π‘Ÿ ADP cana-4527 176 7 , , PUNCT cana-4527 176 8 𝑑 𝑑 NOUN cana-4527 176 9 ) ) PUNCT cana-4527 176 10 ) ) PUNCT cana-4527 177 1 ] ] PUNCT cana-4527 177 2 ] ] X cana-4527 177 3 ( ( PUNCT cana-4527 177 4 4 4 NUM cana-4527 177 5 . . NOUN cana-4527 177 6 22 22 NUM cana-4527 177 7 ) ) PUNCT cana-4527 177 8 ∞ ∞ NUM cana-4527 177 9 𝑖=0 𝑖=0 PROPN cana-4527 177 10 ∞ ∞ PROPN cana-4527 177 11 𝑖=0 𝑖=0 PROPN cana-4527 177 12 communications communication NOUN cana-4527 177 13 on on ADP cana-4527 177 14 applied apply VERB cana-4527 177 15 nonlinear nonlinear ADJ cana-4527 177 16 analysis analysis NOUN cana-4527 177 17 issn issn NOUN cana-4527 177 18 : : PUNCT cana-4527 177 19 1074 1074 NUM cana-4527 177 20 - - PUNCT cana-4527 177 21 133x 133x NUM cana-4527 177 22 vol vol NOUN cana-4527 177 23 32 32 NUM cana-4527 177 24 no no NOUN cana-4527 177 25 . . PUNCT cana-4527 178 1 9s 9s NUM cana-4527 178 2 ( ( PUNCT cana-4527 178 3 2025 2025 NUM cana-4527 178 4 ) ) PUNCT cana-4527 178 5 2422 2422 NUM cana-4527 178 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 179 1 the the DET cana-4527 179 2 recursive recursive ADJ cana-4527 179 3 relationships relationship NOUN cana-4527 179 4 are be AUX cana-4527 179 5 presented present VERB cana-4527 179 6 as as ADP cana-4527 179 7 πœ‘0(π‘Ÿ πœ‘0(π‘Ÿ PROPN cana-4527 179 8 , , PUNCT cana-4527 179 9 𝑑 𝑑 NOUN cana-4527 179 10 ) ) PUNCT cana-4527 180 1 = = SYM cana-4527 180 2 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-4527 180 3 ( ( PUNCT cana-4527 180 4 4 4 NUM cana-4527 180 5 . . NOUN cana-4527 180 6 23 23 NUM cana-4527 180 7 ) ) PUNCT cana-4527 180 8 πœ‘π‘–+1(π‘Ÿ πœ‘π‘–+1(π‘Ÿ PROPN cana-4527 180 9 , , PUNCT cana-4527 180 10 𝑑 𝑑 NOUN cana-4527 180 11 ) ) PUNCT cana-4527 180 12 = = SYM cana-4527 180 13 𝐽 𝐽 PROPN cana-4527 181 1 [ [ X cana-4527 181 2 πœ—π½ πœ—π½ X cana-4527 181 3 [ [ PUNCT cana-4527 181 4 π‘˜ π‘˜ PRON cana-4527 181 5 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 181 6 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 181 7 βˆžπœ‘1(π‘Ÿ βˆžπœ‘1(π‘Ÿ PROPN cana-4527 181 8 , , PUNCT cana-4527 181 9 𝑑 𝑑 PROPN cana-4527 181 10 ) ) PUNCT cana-4527 181 11 + + CCONJ cana-4527 181 12 π‘˜ π‘˜ DET cana-4527 181 13 𝑆0 𝑆0 NOUN cana-4527 181 14 πœ•π‘Ÿ πœ•π‘Ÿ ADJ cana-4527 181 15 πœ‡πœ‘π‘–(π‘Ÿ πœ‡πœ‘π‘–(π‘Ÿ PROPN cana-4527 181 16 , , PUNCT cana-4527 181 17 𝑑 𝑑 PROPN cana-4527 181 18 ) ) PUNCT cana-4527 181 19 ] ] PUNCT cana-4527 181 20 ] ] PUNCT cana-4527 181 21 , , PUNCT cana-4527 181 22 𝑖 𝑖 X cana-4527 181 23 β‰₯ β‰₯ NOUN cana-4527 181 24 1 1 NUM cana-4527 181 25 ( ( PUNCT cana-4527 181 26 4 4 NUM cana-4527 181 27 . . NOUN cana-4527 181 28 24 24 NUM cana-4527 181 29 ) ) PUNCT cana-4527 181 30 the the DET cana-4527 181 31 equations equation NOUN cana-4527 181 32 below below ADP cana-4527 181 33 obtained obtain VERB cana-4527 181 34 by by ADP cana-4527 181 35 using use VERB cana-4527 181 36 that that DET cana-4527 181 37 βˆ‚Ξ±r βˆ‚Ξ±r PROPN cana-4527 181 38 ψ(r ψ(r PROPN cana-4527 181 39 , , PUNCT cana-4527 181 40 0 0 NUM cana-4527 181 41 ) ) PUNCT cana-4527 181 42 = = VERB cana-4527 181 43 constant constant ADJ cana-4527 181 44 = = PUNCT cana-4527 181 45 a a PRON cana-4527 181 46 with with ADP cana-4527 181 47 ( ( PUNCT cana-4527 181 48 4 4 NUM cana-4527 181 49 . . NUM cana-4527 181 50 24 24 NUM cana-4527 181 51 ) ) PUNCT cana-4527 181 52 πœ‘1(π‘Ÿ πœ‘1(π‘Ÿ PROPN cana-4527 181 53 , , PUNCT cana-4527 181 54 𝑑 𝑑 X cana-4527 181 55 ) ) PUNCT cana-4527 181 56 = = PUNCT cana-4527 181 57 π½βˆ’1 π½βˆ’1 PROPN cana-4527 181 58 [ [ X cana-4527 181 59 πœ—π½ πœ—π½ X cana-4527 181 60 [ [ PUNCT cana-4527 181 61 π‘˜ π‘˜ DET cana-4527 181 62 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 181 63 πœ•π‘Ÿ πœ•π‘Ÿ PROPN cana-4527 181 64 βˆπœ‘0(π‘Ÿ βˆπœ‘0(π‘Ÿ NOUN cana-4527 181 65 , , PUNCT cana-4527 181 66 𝑑 𝑑 X cana-4527 181 67 ) ) PUNCT cana-4527 181 68 + + CCONJ cana-4527 181 69 π‘˜ π‘˜ DET cana-4527 181 70 𝑆0 𝑆0 NOUN cana-4527 181 71 πœ•π‘Ÿ πœ•π‘Ÿ CCONJ cana-4527 181 72 πœ‡πœ‘0(π‘Ÿ πœ‡πœ‘0(π‘Ÿ PROPN cana-4527 181 73 , , PUNCT cana-4527 181 74 𝑑 𝑑 NOUN cana-4527 181 75 ) ) PUNCT cana-4527 181 76 ] ] PUNCT cana-4527 181 77 ] ] PUNCT cana-4527 181 78 = = PUNCT cana-4527 181 79 π‘†βˆ’1 π‘†βˆ’1 X cana-4527 182 1 [ [ X cana-4527 182 2 πœ—π‘† πœ—π‘† X cana-4527 182 3 [ [ PUNCT cana-4527 182 4 π‘π‘˜ π‘π‘˜ X cana-4527 182 5 𝑆0π‘Ÿ 𝑆0π‘Ÿ PROPN cana-4527 182 6 ] ] X cana-4527 182 7 ] ] X cana-4527 182 8 = = SYM cana-4527 182 9 π‘†βˆ’1 π‘†βˆ’1 X cana-4527 182 10 [ [ PUNCT cana-4527 182 11 π‘π‘˜πœ— π‘π‘˜πœ— X cana-4527 182 12 𝑆0π‘Ÿ 𝑆0π‘Ÿ PUNCT cana-4527 182 13 ] ] PUNCT cana-4527 183 1 = = PUNCT cana-4527 183 2 π‘π‘˜π‘‘ π‘π‘˜π‘‘ NOUN cana-4527 183 3 π‘Ÿπ‘ 0 π‘Ÿπ‘ 0 NOUN cana-4527 183 4 πœ‘2(π‘Ÿ πœ‘2(π‘Ÿ NOUN cana-4527 183 5 , , PUNCT cana-4527 183 6 𝑑 𝑑 X cana-4527 183 7 ) ) PUNCT cana-4527 183 8 = = SYM cana-4527 183 9 π‘π‘˜2𝑑2 π‘π‘˜2𝑑2 PROPN cana-4527 183 10 2𝑆0 2𝑆0 NUM cana-4527 183 11 2 2 NUM cana-4527 183 12 ( ( PUNCT cana-4527 183 13 Ξ³(1 Ξ³(1 PROPN cana-4527 183 14 + + NUM cana-4527 183 15 ΞΌ ΞΌ NOUN cana-4527 183 16 ) ) PUNCT cana-4527 183 17 π‘Ÿ1+πœ‡ π‘Ÿ1+πœ‡ PROPN cana-4527 183 18 + + CCONJ cana-4527 183 19 Ξ³(1 Ξ³(1 PROPN cana-4527 183 20 + + NUM cana-4527 183 21 Ξ± Ξ± NOUN cana-4527 183 22 ) ) PUNCT cana-4527 183 23 π›Όπ‘Ÿ2+𝛼 π›Όπ‘Ÿ2+𝛼 NOUN cana-4527 183 24 ) ) PUNCT cana-4527 183 25 , , PUNCT cana-4527 183 26 πœ‘3(π‘Ÿ πœ‘3(π‘Ÿ NUM cana-4527 183 27 , , PUNCT cana-4527 183 28 𝑑 𝑑 NOUN cana-4527 183 29 ) ) PUNCT cana-4527 183 30 = = SYM cana-4527 184 1 π‘π‘˜3𝑑3 π‘π‘˜3𝑑3 PROPN cana-4527 184 2 6𝑆0 6𝑆0 NUM cana-4527 184 3 3 3 NUM cana-4527 184 4 ( ( PUNCT cana-4527 184 5 Ξ³(1 Ξ³(1 PROPN cana-4527 184 6 + + NUM cana-4527 184 7 2ΞΌ 2ΞΌ NOUN cana-4527 184 8 ) ) PUNCT cana-4527 184 9 π‘Ÿ1 π‘Ÿ1 NOUN cana-4527 184 10 + + NOUN cana-4527 184 11 2πœ‡ 2πœ‡ NOUN cana-4527 184 12 + + CCONJ cana-4527 184 13 Ξ³(1 Ξ³(1 PROPN cana-4527 184 14 + + NUM cana-4527 184 15 2Ξ± 2Ξ± NOUN cana-4527 184 16 ) ) PUNCT cana-4527 184 17 π›Όπ‘Ÿ2+𝛼 π›Όπ‘Ÿ2+𝛼 NOUN cana-4527 184 18 ) ) PUNCT cana-4527 184 19 , , PUNCT cana-4527 184 20 ( ( PUNCT cana-4527 184 21 4 4 X cana-4527 184 22 . . NOUN cana-4527 184 23 25 25 NUM cana-4527 184 24 ) ) PUNCT cana-4527 184 25 the the DET cana-4527 184 26 approximated approximate VERB cana-4527 184 27 solution solution NOUN cana-4527 184 28 to to ADP cana-4527 184 29 the the DET cana-4527 184 30 issue issue NOUN cana-4527 184 31 is be AUX cana-4527 184 32 given give VERB cana-4527 184 33 by by ADP cana-4527 184 34 the the DET cana-4527 184 35 following follow VERB cana-4527 184 36 expansion expansion NOUN cana-4527 184 37 , , PUNCT cana-4527 184 38 which which PRON cana-4527 184 39 is be AUX cana-4527 184 40 obtained obtain VERB cana-4527 184 41 by by ADP cana-4527 184 42 combining combine VERB cana-4527 184 43 equations equation NOUN cana-4527 184 44 ( ( PUNCT cana-4527 184 45 4.23 4.23 NUM cana-4527 184 46 ) ) PUNCT cana-4527 184 47 and and CCONJ cana-4527 184 48 ( ( PUNCT cana-4527 184 49 4.25 4.25 NUM cana-4527 184 50 ) ) PUNCT cana-4527 184 51 . . PUNCT cana-4527 185 1 πœ‘ πœ‘ NOUN cana-4527 185 2 ( ( PUNCT cana-4527 185 3 π‘Ÿ π‘Ÿ NOUN cana-4527 185 4 , , PUNCT cana-4527 185 5 𝑑 𝑑 X cana-4527 185 6 ) ) PUNCT cana-4527 185 7 = = SYM cana-4527 186 1 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-4527 187 1 + + NUM cana-4527 187 2 π‘π‘˜π‘‘ π‘π‘˜π‘‘ NOUN cana-4527 187 3 π‘Ÿπ‘†0 π‘Ÿπ‘†0 PROPN cana-4527 187 4 + + CCONJ cana-4527 187 5 π‘π‘˜2𝑑2 π‘π‘˜2𝑑2 PROPN cana-4527 187 6 2𝑆0 2𝑆0 NUM cana-4527 187 7 2 2 NUM cana-4527 187 8 ( ( PUNCT cana-4527 187 9 Ξ³ Ξ³ X cana-4527 187 10 ( ( PUNCT cana-4527 187 11 1 1 NUM cana-4527 187 12 + + NUM cana-4527 187 13 ΞΌ ΞΌ NUM cana-4527 187 14 ) ) PUNCT cana-4527 187 15 π‘Ÿ1+πœ‡ π‘Ÿ1+πœ‡ PROPN cana-4527 187 16 + + X cana-4527 187 17 Ξ³ Ξ³ X cana-4527 187 18 ( ( PUNCT cana-4527 187 19 1 1 NUM cana-4527 187 20 + + NUM cana-4527 187 21 Ξ± Ξ± NOUN cana-4527 187 22 ) ) PUNCT cana-4527 187 23 π›Όπ‘Ÿ2+𝛼 π›Όπ‘Ÿ2+𝛼 NOUN cana-4527 187 24 ) ) PUNCT cana-4527 188 1 + + CCONJ cana-4527 189 1 π‘π‘˜3𝑑3 π‘π‘˜3𝑑3 PROPN cana-4527 189 2 6𝑆0 6𝑆0 NUM cana-4527 189 3 3 3 NUM cana-4527 189 4 ( ( PUNCT cana-4527 189 5 Ξ³(1 Ξ³(1 PROPN cana-4527 189 6 + + NUM cana-4527 189 7 2ΞΌ 2ΞΌ NOUN cana-4527 189 8 ) ) PUNCT cana-4527 189 9 π‘Ÿ1 π‘Ÿ1 NOUN cana-4527 189 10 + + NOUN cana-4527 189 11 2πœ‡ 2πœ‡ NOUN cana-4527 189 12 + + CCONJ cana-4527 189 13 Ξ³(1 Ξ³(1 PROPN cana-4527 189 14 + + NUM cana-4527 189 15 2Ξ± 2Ξ± NOUN cana-4527 189 16 ) ) PUNCT cana-4527 189 17 π›Όπ‘Ÿ2 π›Όπ‘Ÿ2 PROPN cana-4527 189 18 + + PROPN cana-4527 189 19 2𝛼 2𝛼 NOUN cana-4527 189 20 ) ) PUNCT cana-4527 189 21 ( ( PUNCT cana-4527 189 22 4 4 NUM cana-4527 189 23 . . NOUN cana-4527 189 24 26 26 NUM cana-4527 189 25 ) ) PUNCT cana-4527 189 26 additionally additionally ADV cana-4527 189 27 , , PUNCT cana-4527 189 28 by by ADP cana-4527 189 29 using use VERB cana-4527 189 30 the the DET cana-4527 189 31 boundary boundary ADJ cana-4527 189 32 condition condition NOUN cana-4527 189 33 on on ADP cana-4527 189 34 ( ( PUNCT cana-4527 189 35 4.26 4.26 NUM cana-4527 189 36 ) ) PUNCT cana-4527 189 37 , , PUNCT cana-4527 189 38 we we PRON cana-4527 189 39 may may AUX cana-4527 189 40 determine determine VERB cana-4527 189 41 a a DET cana-4527 189 42 constant constant ADJ cana-4527 189 43 value value NOUN cana-4527 189 44 ( ( PUNCT cana-4527 189 45 a a NOUN cana-4527 189 46 ) ) PUNCT cana-4527 189 47 to to PART cana-4527 189 48 acquire acquire VERB cana-4527 189 49 . . PUNCT cana-4527 190 1 𝑝 𝑝 NOUN cana-4527 190 2 = = SYM cana-4527 190 3 𝑄 𝑄 PROPN cana-4527 190 4 2πœ‹ 2πœ‹ NOUN cana-4527 190 5 π‘Ÿπ‘π‘˜π‘‘π›Όπ‘“ π‘Ÿπ‘π‘˜π‘‘π›Όπ‘“ ADJ cana-4527 190 6 ( ( PUNCT cana-4527 190 7 π‘Ÿπ‘ π‘Ÿπ‘ PROPN cana-4527 190 8 , , PUNCT cana-4527 190 9 𝑑 𝑑 PROPN cana-4527 190 10 ) ) PUNCT cana-4527 190 11 ( ( PUNCT cana-4527 190 12 4 4 NUM cana-4527 190 13 . . NOUN cana-4527 190 14 27 27 NUM cana-4527 190 15 ) ) PUNCT cana-4527 190 16 where where SCONJ cana-4527 190 17 𝑓 𝑓 X cana-4527 190 18 ( ( PUNCT cana-4527 190 19 π‘Ÿπ‘ π‘Ÿπ‘ PROPN cana-4527 190 20 , , PUNCT cana-4527 190 21 𝑑 𝑑 PROPN cana-4527 190 22 ) ) PUNCT cana-4527 190 23 = = SYM cana-4527 190 24 1 1 NUM cana-4527 190 25 + + CCONJ cana-4527 190 26 π‘˜π‘‘Ξ³(1+∝ π‘˜π‘‘Ξ³(1+∝ X cana-4527 190 27 ) ) PUNCT cana-4527 190 28 𝑆0π‘Ÿπ‘ 𝑆0π‘Ÿπ‘ NOUN cana-4527 190 29 1+∝ 1+∝ NUM cana-4527 191 1 + + CCONJ cana-4527 191 2 π‘˜2𝑑2Ξ³(1 π‘˜2𝑑2Ξ³(1 PROPN cana-4527 191 3 + + PROPN cana-4527 191 4 ΞΌ ΞΌ PROPN cana-4527 191 5 + + CCONJ cana-4527 191 6 Ξ± Ξ± NOUN cana-4527 191 7 ) ) PUNCT cana-4527 191 8 2𝑆0 2𝑆0 NUM cana-4527 191 9 2π‘Ÿπ‘ 2π‘Ÿπ‘ ADJ cana-4527 191 10 1+𝛼+πœ‡ 1+𝛼+πœ‡ PUNCT cana-4527 192 1 + + CCONJ cana-4527 192 2 π‘˜2𝑑2Ξ³(2 π‘˜2𝑑2Ξ³(2 PROPN cana-4527 192 3 + + CCONJ cana-4527 192 4 2Ξ± 2Ξ± NOUN cana-4527 192 5 ) ) PUNCT cana-4527 192 6 2(1 2(1 NUM cana-4527 193 1 + + CCONJ cana-4527 193 2 𝛼)𝑆0 𝛼)𝑆0 ADJ cana-4527 193 3 2π‘Ÿπ‘ 2π‘Ÿπ‘ ADJ cana-4527 193 4 2 2 NUM cana-4527 193 5 + + NOUN cana-4527 193 6 2𝛼 2𝛼 NOUN cana-4527 193 7 + + CCONJ cana-4527 193 8 π‘˜3𝑑3 π‘˜3𝑑3 PROPN cana-4527 193 9 6𝑆0 6𝑆0 NUM cana-4527 193 10 3 3 NUM cana-4527 193 11 ( ( PUNCT cana-4527 193 12 Ξ³(1 Ξ³(1 PROPN cana-4527 193 13 + + NUM cana-4527 193 14 2ΞΌ 2ΞΌ NOUN cana-4527 193 15 + + CCONJ cana-4527 193 16 Ξ± Ξ± NOUN cana-4527 193 17 ) ) PUNCT cana-4527 193 18 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 193 19 1 1 NUM cana-4527 193 20 + + NOUN cana-4527 193 21 2πœ‡+𝛼 2πœ‡+𝛼 NUM cana-4527 193 22 + + CCONJ cana-4527 193 23 Ξ³(2 Ξ³(2 PROPN cana-4527 193 24 + + CCONJ cana-4527 193 25 3Ξ± 3Ξ± NOUN cana-4527 193 26 ) ) PUNCT cana-4527 193 27 ( ( PUNCT cana-4527 193 28 1 1 NUM cana-4527 193 29 + + NUM cana-4527 193 30 𝛼)(1 𝛼)(1 X cana-4527 193 31 + + X cana-4527 193 32 2𝛼)π‘Ÿπ‘ 2𝛼)π‘Ÿπ‘ NUM cana-4527 193 33 2 2 NUM cana-4527 193 34 + + NOUN cana-4527 193 35 3𝛼 3𝛼 X cana-4527 193 36 ) ) PUNCT cana-4527 193 37 ( ( PUNCT cana-4527 193 38 4 4 NUM cana-4527 193 39 . . NOUN cana-4527 193 40 28 28 NUM cana-4527 193 41 ) ) PUNCT cana-4527 193 42 communications communication NOUN cana-4527 193 43 on on ADP cana-4527 193 44 applied apply VERB cana-4527 193 45 nonlinear nonlinear ADJ cana-4527 193 46 analysis analysis NOUN cana-4527 193 47 issn issn NOUN cana-4527 193 48 : : PUNCT cana-4527 193 49 1074 1074 NUM cana-4527 193 50 - - PUNCT cana-4527 193 51 133x 133x NUM cana-4527 193 52 vol vol NOUN cana-4527 193 53 32 32 NUM cana-4527 193 54 no no NOUN cana-4527 193 55 . . PUNCT cana-4527 194 1 9s 9s NUM cana-4527 194 2 ( ( PUNCT cana-4527 194 3 2025 2025 NUM cana-4527 194 4 ) ) PUNCT cana-4527 194 5 2423 2423 NUM cana-4527 194 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 194 7 πœ‘(π‘Ÿ πœ‘(π‘Ÿ PROPN cana-4527 194 8 , , PUNCT cana-4527 194 9 𝑑 𝑑 NOUN cana-4527 194 10 ) ) PUNCT cana-4527 194 11 = = VERB cana-4527 194 12 π‘„π‘Ÿ2(1βˆ’ π‘„π‘Ÿ2(1βˆ’ VERB cana-4527 194 13 𝑛 𝑛 DET cana-4527 194 14 2⁄ 2⁄ NUM cana-4527 194 15 ) ) PUNCT cana-4527 194 16 4πœ‹ 4πœ‹ NOUN cana-4527 194 17 ( ( PUNCT cana-4527 194 18 𝑛 𝑛 PROPN cana-4527 194 19 2 2 X cana-4527 194 20 ) ) PUNCT cana-4527 194 21 π‘˜π‘‘1βˆ’π‘› π‘˜π‘‘1βˆ’π‘› PROPN cana-4527 194 22 Ξ³ Ξ³ PROPN cana-4527 194 23 [ [ PUNCT cana-4527 194 24 𝑛 𝑛 ADJ cana-4527 194 25 2 2 NUM cana-4527 194 26 βˆ’ βˆ’ NUM cana-4527 194 27 1 1 NUM cana-4527 194 28 , , PUNCT cana-4527 194 29 π‘Ÿ3𝑆0 π‘Ÿ3𝑆0 NUM cana-4527 194 30 4π‘˜π‘‘ 4π‘˜π‘‘ NOUN cana-4527 194 31 ] ] X cana-4527 194 32 ( ( PUNCT cana-4527 194 33 4 4 NUM cana-4527 194 34 . . NOUN cana-4527 194 35 29 29 NUM cana-4527 194 36 ) ) PUNCT cana-4527 194 37 gamma gamma NOUN cana-4527 194 38 function function NOUN cana-4527 194 39 is be AUX cana-4527 194 40 denoted denote VERB cana-4527 194 41 by by ADP cana-4527 194 42 Ξ³ Ξ³ PROPN cana-4527 194 43 . . PROPN cana-4527 194 44 Ξ· Ξ· PROPN cana-4527 194 45 : : PUNCT cana-4527 194 46 the the DET cana-4527 194 47 special special ADJ cana-4527 194 48 dimension dimension NOUN cana-4527 194 49 of of ADP cana-4527 194 50 the the DET cana-4527 194 51 flow flow NOUN cana-4527 194 52 , , PUNCT cana-4527 194 53 always always ADV cana-4527 194 54 an an DET cana-4527 194 55 integer integer NOUN cana-4527 194 56 n n CCONJ cana-4527 194 57 , , PUNCT cana-4527 194 58 is be AUX cana-4527 194 59 the the DET cana-4527 194 60 same same ADJ cana-4527 194 61 as as ADP cana-4527 194 62 the the DET cana-4527 194 63 special special ADJ cana-4527 194 64 dimension dimension NOUN cana-4527 194 65 . . PUNCT cana-4527 195 1 5 5 X cana-4527 195 2 . . X cana-4527 195 3 results result NOUN cana-4527 195 4 and and CCONJ cana-4527 195 5 discussion discussion NOUN cana-4527 195 6 the the DET cana-4527 195 7 revised revise VERB cana-4527 195 8 groundwater groundwater NOUN cana-4527 195 9 flow flow NOUN cana-4527 195 10 equation equation NOUN cana-4527 195 11 is be AUX cana-4527 195 12 precisely precisely ADV cana-4527 195 13 solved solve VERB cana-4527 195 14 in in ADP cana-4527 195 15 this this DET cana-4527 195 16 section section NOUN cana-4527 195 17 by by ADP cana-4527 195 18 utilizing utilize VERB cana-4527 195 19 the the DET cana-4527 195 20 suggested suggest VERB cana-4527 195 21 fixed fix VERB cana-4527 195 22 transform transform NOUN cana-4527 195 23 domain domain NOUN cana-4527 195 24 ( ( PUNCT cana-4527 195 25 ftd ftd NOUN cana-4527 195 26 ) ) PUNCT cana-4527 195 27 technique technique NOUN cana-4527 195 28 . . PUNCT cana-4527 196 1 we we PRON cana-4527 196 2 show show VERB cana-4527 196 3 the the DET cana-4527 196 4 suggested suggested ADJ cana-4527 196 5 approach approach NOUN cana-4527 196 6 along along ADP cana-4527 196 7 with with ADP cana-4527 196 8 its its PRON cana-4527 196 9 properties property NOUN cana-4527 196 10 for for ADP cana-4527 196 11 the the DET cana-4527 196 12 fractional fractional ADJ cana-4527 196 13 order order NOUN cana-4527 196 14 caputo caputo PROPN cana-4527 196 15 - - PUNCT cana-4527 196 16 fabrizio fabrizio PROPN cana-4527 196 17 derivative derivative NOUN cana-4527 196 18 . . PUNCT cana-4527 197 1 5.1 5.1 NUM cana-4527 197 2 numerical numerical ADJ cana-4527 197 3 simulations simulation NOUN cana-4527 197 4 table table NOUN cana-4527 197 5 1 1 NUM cana-4527 197 6 lists list VERB cana-4527 197 7 the the DET cana-4527 197 8 theoretical theoretical ADJ cana-4527 197 9 parameters parameter NOUN cana-4527 197 10 for for ADP cana-4527 197 11 the the DET cana-4527 197 12 aquifer aquifer NOUN cana-4527 197 13 's 's PART cana-4527 197 14 properties property NOUN cana-4527 197 15 and and CCONJ cana-4527 197 16 discharge discharge VERB cana-4527 197 17 rate rate NOUN cana-4527 197 18 used use VERB cana-4527 197 19 in in ADP cana-4527 197 20 the the DET cana-4527 197 21 numerical numerical PROPN cana-4527 197 22 simulation simulation PROPN cana-4527 197 23 . . PUNCT cana-4527 198 1 table table NOUN cana-4527 198 2 1 1 NUM cana-4527 198 3 . . PUNCT cana-4527 199 1 values value NOUN cana-4527 199 2 based base VERB cana-4527 199 3 on on ADP cana-4527 199 4 theory theory NOUN cana-4527 199 5 parameters parameter NOUN cana-4527 199 6 values value NOUN cana-4527 199 7 units unit NOUN cana-4527 199 8 𝑆0 𝑆0 VERB cana-4527 199 9 0.74 0.74 NUM cana-4527 199 10 π‘šβˆ’1 π‘šβˆ’1 PROPN cana-4527 199 11 q q PROPN cana-4527 199 12 10.7 10.7 NUM cana-4527 199 13 π‘š3π‘ βˆ’1 π‘š3π‘ βˆ’1 PROPN cana-4527 199 14 k k PROPN cana-4527 199 15 0.64 0.64 NUM cana-4527 199 16 π‘šπ‘ βˆ’1 π‘šπ‘ βˆ’1 PROPN cana-4527 199 17 d d PROPN cana-4527 199 18 39 39 NUM cana-4527 199 19 m m NOUN cana-4527 199 20 π‘Ÿπ‘ π‘Ÿπ‘ ADP cana-4527 199 21 0.14 0.14 NUM cana-4527 199 22 m m NUM cana-4527 199 23 πœ‘0(π‘Ÿ πœ‘0(π‘Ÿ NOUN cana-4527 199 24 , , PUNCT cana-4527 199 25 𝑑 𝑑 NOUN cana-4527 199 26 ) ) PUNCT cana-4527 199 27 0 0 NUM cana-4527 200 1 m m VERB cana-4527 200 2 ∝ ∝ NOUN cana-4527 200 3 0.004 0.004 NUM cana-4527 200 4 none none NOUN cana-4527 200 5 πœ‡ πœ‡ ADP cana-4527 200 6 1.004 1.004 NUM cana-4527 200 7 none none NOUN cana-4527 200 8 n n NOUN cana-4527 200 9 0.011 0.011 NUM cana-4527 200 10 none none NOUN cana-4527 200 11 this this DET cana-4527 200 12 section section NOUN cana-4527 200 13 contains contain VERB cana-4527 200 14 a a DET cana-4527 200 15 numerical numerical ADJ cana-4527 200 16 presentation presentation NOUN cana-4527 200 17 of of ADP cana-4527 200 18 the the DET cana-4527 200 19 new new ADJ cana-4527 200 20 groundwater groundwater NOUN cana-4527 200 21 model model NOUN cana-4527 200 22 's 's PART cana-4527 200 23 precise precise ADJ cana-4527 200 24 solution solution NOUN cana-4527 200 25 for for ADP cana-4527 200 26 various various ADJ cana-4527 200 27 alpha alpha ADJ cana-4527 200 28 values value NOUN cana-4527 200 29 . . PUNCT cana-4527 201 1 the the DET cana-4527 201 2 numerical numerical ADJ cana-4527 201 3 simulations simulation NOUN cana-4527 201 4 for for ADP cana-4527 201 5 various various ADJ cana-4527 201 6 alpha alpha ADJ cana-4527 201 7 values value NOUN cana-4527 201 8 are be AUX cana-4527 201 9 displayed display VERB cana-4527 201 10 in in ADP cana-4527 201 11 figures figure NOUN cana-4527 201 12 1 1 NUM cana-4527 201 13 , , PUNCT cana-4527 201 14 2 2 NUM cana-4527 201 15 , , PUNCT cana-4527 201 16 3 3 NUM cana-4527 201 17 , , PUNCT cana-4527 201 18 4 4 NUM cana-4527 201 19 , , PUNCT cana-4527 201 20 and and CCONJ cana-4527 201 21 5 5 NUM cana-4527 201 22 . . NOUN cana-4527 201 23 figure figure NOUN cana-4527 201 24 . . PUNCT cana-4527 202 1 1 1 X cana-4527 202 2 . . X cana-4527 202 3 precise precise ADJ cana-4527 202 4 solution solution NOUN cana-4527 202 5 simulation simulation NOUN cana-4527 202 6 for for ADP cana-4527 202 7 alpha alpha NOUN cana-4527 202 8 = = NOUN cana-4527 202 9 0.95 0.95 NUM cana-4527 202 10 as as ADP cana-4527 202 11 a a DET cana-4527 202 12 function function NOUN cana-4527 202 13 of of ADP cana-4527 202 14 space space NOUN cana-4527 202 15 and and CCONJ cana-4527 202 16 time time NOUN cana-4527 202 17 communications communication NOUN cana-4527 202 18 on on ADP cana-4527 202 19 applied apply VERB cana-4527 202 20 nonlinear nonlinear ADJ cana-4527 202 21 analysis analysis NOUN cana-4527 202 22 issn issn NOUN cana-4527 202 23 : : PUNCT cana-4527 202 24 1074 1074 NUM cana-4527 202 25 - - PUNCT cana-4527 202 26 133x 133x NUM cana-4527 202 27 vol vol NOUN cana-4527 202 28 32 32 NUM cana-4527 202 29 no no NOUN cana-4527 202 30 . . PUNCT cana-4527 203 1 9s 9s NUM cana-4527 203 2 ( ( PUNCT cana-4527 203 3 2025 2025 NUM cana-4527 203 4 ) ) PUNCT cana-4527 203 5 2424 2424 NUM cana-4527 204 1 https://internationalpubls.com https://internationalpubls.com X cana-4527 204 2 figure figure NOUN cana-4527 204 3 . . PUNCT cana-4527 205 1 2 2 NUM cana-4527 205 2 . . NOUN cana-4527 205 3 time time NOUN cana-4527 205 4 and and CCONJ cana-4527 205 5 space space NOUN cana-4527 205 6 - - PUNCT cana-4527 205 7 dependent dependent ADJ cana-4527 205 8 simulation simulation NOUN cana-4527 205 9 of of ADP cana-4527 205 10 the the DET cana-4527 205 11 precise precise ADJ cana-4527 205 12 solution solution NOUN cana-4527 205 13 for for ADP cana-4527 205 14 alpha alpha NOUN cana-4527 205 15 = = NOUN cana-4527 205 16 0.55 0.55 NUM cana-4527 205 17 figure figure NOUN cana-4527 205 18 . . PUNCT cana-4527 206 1 3 3 X cana-4527 206 2 . . X cana-4527 206 3 computation computation NOUN cana-4527 206 4 of of ADP cana-4527 206 5 the the DET cana-4527 206 6 precise precise ADJ cana-4527 206 7 solution solution NOUN cana-4527 206 8 for for ADP cana-4527 206 9 alpha alpha NOUN cana-4527 206 10 = = NOUN cana-4527 206 11 0.45 0.45 NUM cana-4527 206 12 as as ADP cana-4527 206 13 a a DET cana-4527 206 14 function function NOUN cana-4527 206 15 of of ADP cana-4527 206 16 time time NOUN cana-4527 206 17 and and CCONJ cana-4527 206 18 space space NOUN cana-4527 206 19 figure figure NOUN cana-4527 206 20 . . PUNCT cana-4527 207 1 4 4 X cana-4527 207 2 . . X cana-4527 207 3 simulate simulate VERB cana-4527 207 4 the the DET cana-4527 207 5 precise precise ADJ cana-4527 207 6 solution solution NOUN cana-4527 207 7 for for ADP cana-4527 207 8 alpha alpha NOUN cana-4527 207 9 in in ADP cana-4527 207 10 respect respect NOUN cana-4527 207 11 to to ADP cana-4527 207 12 time time NOUN cana-4527 207 13 and and CCONJ cana-4527 207 14 space space NOUN cana-4527 207 15 . . PUNCT cana-4527 208 1 figure figure NOUN cana-4527 208 2 . . PUNCT cana-4527 209 1 5 5 X cana-4527 209 2 . . X cana-4527 209 3 modelling modelling NOUN cana-4527 209 4 of of ADP cana-4527 209 5 the the DET cana-4527 209 6 precise precise ADJ cana-4527 209 7 response response NOUN cana-4527 209 8 as as ADP cana-4527 209 9 a a DET cana-4527 209 10 function function NOUN cana-4527 209 11 of of ADP cana-4527 209 12 space space NOUN cana-4527 209 13 for for ADP cana-4527 209 14 a a DET cana-4527 209 15 range range NOUN cana-4527 209 16 of of ADP cana-4527 209 17 alpha alpha ADJ cana-4527 209 18 values value NOUN cana-4527 209 19 the the DET cana-4527 209 20 above above ADV cana-4527 209 21 - - PUNCT cana-4527 209 22 mentioned mention VERB cana-4527 209 23 steady steady ADJ cana-4527 209 24 - - PUNCT cana-4527 209 25 state state NOUN cana-4527 209 26 model model NOUN cana-4527 209 27 is be AUX cana-4527 209 28 precisely precisely ADV cana-4527 209 29 solved solve VERB cana-4527 209 30 using use VERB cana-4527 209 31 the the DET cana-4527 209 32 newly newly ADV cana-4527 209 33 created create VERB cana-4527 209 34 fractional fractional ADJ cana-4527 209 35 derivative derivative NOUN cana-4527 209 36 . . PUNCT cana-4527 210 1 the the DET cana-4527 210 2 suggested suggest VERB cana-4527 210 3 method method NOUN cana-4527 210 4 's 's PART cana-4527 210 5 solution solution NOUN cana-4527 210 6 can can AUX cana-4527 210 7 be be AUX cana-4527 210 8 obtained obtain VERB cana-4527 210 9 when when SCONJ cana-4527 210 10 alpha alpha NOUN cana-4527 210 11 gets get VERB cana-4527 210 12 1 1 NUM cana-4527 210 13 . . PUNCT cana-4527 211 1 one one NUM cana-4527 211 2 additional additional ADJ cana-4527 211 3 feature feature NOUN cana-4527 211 4 of of ADP cana-4527 211 5 the the DET cana-4527 211 6 new new ADJ cana-4527 211 7 solution solution NOUN cana-4527 211 8 is be AUX cana-4527 211 9 that that SCONJ cana-4527 211 10 it it PRON cana-4527 211 11 may may AUX cana-4527 211 12 be be AUX cana-4527 211 13 applied apply VERB cana-4527 211 14 to to PART cana-4527 211 15 depict depict VERB cana-4527 211 16 critical critical ADJ cana-4527 211 17 real real ADJ cana-4527 211 18 - - PUNCT cana-4527 211 19 world world NOUN cana-4527 211 20 situations situation NOUN cana-4527 211 21 . . PUNCT cana-4527 212 1 diagrams diagram NOUN cana-4527 212 2 6 6 NUM cana-4527 212 3 , , PUNCT cana-4527 212 4 7 7 NUM cana-4527 212 5 , , PUNCT cana-4527 212 6 8 8 NUM cana-4527 212 7 , , PUNCT cana-4527 212 8 9 9 NUM cana-4527 212 9 , , PUNCT cana-4527 212 10 10 10 NUM cana-4527 212 11 , , PUNCT cana-4527 212 12 11 11 NUM cana-4527 212 13 , , PUNCT cana-4527 212 14 12 12 NUM cana-4527 212 15 , , PUNCT cana-4527 212 16 13 13 NUM cana-4527 212 17 , , PUNCT cana-4527 212 18 and and CCONJ cana-4527 212 19 14 14 NUM cana-4527 212 20 graphically graphically ADV cana-4527 212 21 depict depict VERB cana-4527 212 22 the the DET cana-4527 212 23 solutions solution NOUN cana-4527 212 24 of of ADP cana-4527 212 25 ( ( PUNCT cana-4527 212 26 4.9 4.9 NUM cana-4527 212 27 ) ) PUNCT cana-4527 212 28 and and CCONJ cana-4527 212 29 ( ( PUNCT cana-4527 212 30 4.19 4.19 NUM cana-4527 212 31 ) ) PUNCT cana-4527 212 32 for for ADP cana-4527 212 33 a a DET cana-4527 212 34 range range NOUN cana-4527 212 35 of of ADP cana-4527 212 36 parameter parameter NOUN cana-4527 212 37 values value NOUN cana-4527 212 38 . . PUNCT cana-4527 213 1 additionally additionally ADV cana-4527 213 2 , , PUNCT cana-4527 213 3 these these DET cana-4527 213 4 graphs graph NOUN cana-4527 213 5 demonstrate demonstrate VERB cana-4527 213 6 how how SCONJ cana-4527 213 7 the the DET cana-4527 213 8 drawdown drawdown NOUN cana-4527 213 9 behaved behave VERB cana-4527 213 10 during during ADP cana-4527 213 11 the the DET cana-4527 213 12 pumping pumping NOUN cana-4527 213 13 test test NOUN cana-4527 213 14 . . PUNCT cana-4527 214 1 communications communication NOUN cana-4527 214 2 on on ADP cana-4527 214 3 applied apply VERB cana-4527 214 4 nonlinear nonlinear ADJ cana-4527 214 5 analysis analysis NOUN cana-4527 214 6 issn issn NOUN cana-4527 214 7 : : PUNCT cana-4527 214 8 1074 1074 NUM cana-4527 214 9 - - PUNCT cana-4527 214 10 133x 133x NUM cana-4527 214 11 vol vol NOUN cana-4527 214 12 32 32 NUM cana-4527 214 13 no no NOUN cana-4527 214 14 . . PUNCT cana-4527 215 1 9s 9s NUM cana-4527 215 2 ( ( PUNCT cana-4527 215 3 2025 2025 NUM cana-4527 215 4 ) ) PUNCT cana-4527 215 5 2425 2425 NUM cana-4527 215 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 215 7 figure figure NOUN cana-4527 215 8 6 6 NUM cana-4527 215 9 . . PUNCT cana-4527 215 10 solution solution NOUN cana-4527 215 11 of of ADP cana-4527 215 12 ( ( PUNCT cana-4527 215 13 4.9 4.9 NUM cana-4527 215 14 ) ) PUNCT cana-4527 215 15 ( ( PUNCT cana-4527 215 16 drawdown drawdown NOUN cana-4527 215 17 ) ) PUNCT cana-4527 215 18 figure figure NOUN cana-4527 215 19 . . PUNCT cana-4527 216 1 7 7 X cana-4527 216 2 . . X cana-4527 216 3 solution solution NOUN cana-4527 216 4 of of ADP cana-4527 216 5 ( ( PUNCT cana-4527 216 6 4 4 NUM cana-4527 216 7 . . NOUN cana-4527 216 8 9 9 NUM cana-4527 216 9 ) ) PUNCT cana-4527 216 10 ( ( PUNCT cana-4527 216 11 piezometric piezometric ADJ cana-4527 216 12 head head NOUN cana-4527 216 13 ) ) PUNCT cana-4527 216 14 figure figure NOUN cana-4527 216 15 . . PUNCT cana-4527 217 1 8 8 NUM cana-4527 217 2 . . PUNCT cana-4527 217 3 solution solution NOUN cana-4527 217 4 of of ADP cana-4527 217 5 ( ( PUNCT cana-4527 217 6 4.19 4.19 NUM cana-4527 217 7 ) ) PUNCT cana-4527 217 8 ( ( PUNCT cana-4527 217 9 drawdown drawdown NOUN cana-4527 217 10 ) ) PUNCT cana-4527 217 11 figure figure NOUN cana-4527 217 12 . . PUNCT cana-4527 218 1 9 9 X cana-4527 218 2 . . PUNCT cana-4527 218 3 solution solution NOUN cana-4527 218 4 of of ADP cana-4527 218 5 ( ( PUNCT cana-4527 218 6 4.19 4.19 NUM cana-4527 218 7 ) ) PUNCT cana-4527 218 8 ( ( PUNCT cana-4527 218 9 piezometric piezometric ADJ cana-4527 218 10 head head NOUN cana-4527 218 11 ) ) PUNCT cana-4527 218 12 i. i. PROPN cana-4527 218 13 starting start VERB cana-4527 218 14 with with ADP cana-4527 218 15 the the DET cana-4527 218 16 well well NOUN cana-4527 218 17 as as ADP cana-4527 218 18 a a DET cana-4527 218 19 function function NOUN cana-4527 218 20 towards towards ADP cana-4527 218 21 the the DET cana-4527 218 22 perspective perspective ADJ cana-4527 218 23 point point NOUN cana-4527 218 24 of of ADP cana-4527 218 25 space space NOUN cana-4527 218 26 and and CCONJ cana-4527 218 27 time time NOUN cana-4527 218 28 . . PUNCT cana-4527 219 1 ii ii PROPN cana-4527 219 2 . . PUNCT cana-4527 220 1 considering consider VERB cana-4527 220 2 a a DET cana-4527 220 3 fixed fix VERB cana-4527 220 4 distance distance NOUN cana-4527 220 5 about about ADP cana-4527 220 6 the the DET cana-4527 220 7 well well NOUN cana-4527 220 8 as as ADP cana-4527 220 9 a a DET cana-4527 220 10 function function NOUN cana-4527 220 11 of of ADP cana-4527 220 12 time time NOUN cana-4527 220 13 . . PUNCT cana-4527 221 1 iii iii X cana-4527 221 2 . . PUNCT cana-4527 222 1 as as ADP cana-4527 222 2 a a DET cana-4527 222 3 function function NOUN cana-4527 222 4 of of ADP cana-4527 222 5 space space NOUN cana-4527 222 6 for for ADP cana-4527 222 7 a a DET cana-4527 222 8 predefined predefine VERB cana-4527 222 9 duration duration NOUN cana-4527 222 10 . . PUNCT cana-4527 223 1 the the DET cana-4527 223 2 following follow VERB cana-4527 223 3 barker barker PROPN cana-4527 223 4 solution solution NOUN cana-4527 223 5 for for ADP cana-4527 223 6 an an DET cana-4527 223 7 infinite infinite ADJ cana-4527 223 8 aquifer aquifer NOUN cana-4527 223 9 with with ADP cana-4527 223 10 a a DET cana-4527 223 11 line line NOUN cana-4527 223 12 source source NOUN cana-4527 223 13 is be AUX cana-4527 223 14 further far ADV cana-4527 223 15 contrasted contrast VERB cana-4527 223 16 with with ADP cana-4527 223 17 them they PRON cana-4527 223 18 . . PUNCT cana-4527 224 1 communications communication NOUN cana-4527 224 2 on on ADP cana-4527 224 3 applied apply VERB cana-4527 224 4 nonlinear nonlinear ADJ cana-4527 224 5 analysis analysis NOUN cana-4527 224 6 issn issn NOUN cana-4527 224 7 : : PUNCT cana-4527 224 8 1074 1074 NUM cana-4527 224 9 - - PUNCT cana-4527 224 10 133x 133x NUM cana-4527 224 11 vol vol NOUN cana-4527 224 12 32 32 NUM cana-4527 224 13 no no NOUN cana-4527 224 14 . . PUNCT cana-4527 225 1 9s 9s NUM cana-4527 225 2 ( ( PUNCT cana-4527 225 3 2025 2025 NUM cana-4527 225 4 ) ) PUNCT cana-4527 225 5 2426 2426 NUM cana-4527 225 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 225 7 figure figure NOUN cana-4527 225 8 . . PUNCT cana-4527 226 1 10 10 NUM cana-4527 226 2 . . PUNCT cana-4527 227 1 barker barker PROPN cana-4527 227 2 's 's PART cana-4527 227 3 solution solution NOUN cana-4527 227 4 ( ( PUNCT cana-4527 227 5 drawdown drawdown NOUN cana-4527 227 6 ) ) PUNCT cana-4527 227 7 figure figure NOUN cana-4527 227 8 . . PUNCT cana-4527 228 1 11 11 NUM cana-4527 228 2 . . PUNCT cana-4527 229 1 barker barker PROPN cana-4527 229 2 's 's PART cana-4527 229 3 solution solution NOUN cana-4527 229 4 ( ( PUNCT cana-4527 229 5 piezometric piezometric ADJ cana-4527 229 6 head head NOUN cana-4527 229 7 ) ) PUNCT cana-4527 229 8 figure figure NOUN cana-4527 229 9 . . PUNCT cana-4527 230 1 12 12 NUM cana-4527 230 2 . . PUNCT cana-4527 230 3 solution solution NOUN cana-4527 230 4 as as ADP cana-4527 230 5 a a DET cana-4527 230 6 function function NOUN cana-4527 230 7 of of ADP cana-4527 230 8 the the DET cana-4527 230 9 base base NOUN cana-4527 230 10 of of ADP cana-4527 230 11 ( ( PUNCT cana-4527 230 12 4.9 4.9 NUM cana-4527 230 13 ) ) PUNCT cana-4527 230 14 figure figure NOUN cana-4527 230 15 . . PUNCT cana-4527 231 1 13 13 NUM cana-4527 231 2 . . PUNCT cana-4527 231 3 solution solution NOUN cana-4527 231 4 as as ADP cana-4527 231 5 the the DET cana-4527 231 6 basis basis NOUN cana-4527 231 7 of of ADP cana-4527 231 8 ( ( PUNCT cana-4527 231 9 4.19 4.19 NUM cana-4527 231 10 ) ) PUNCT cana-4527 231 11 function function NOUN cana-4527 231 12 communications communication NOUN cana-4527 231 13 on on ADP cana-4527 231 14 applied apply VERB cana-4527 231 15 nonlinear nonlinear ADJ cana-4527 231 16 analysis analysis NOUN cana-4527 231 17 issn issn NOUN cana-4527 231 18 : : PUNCT cana-4527 231 19 1074 1074 NUM cana-4527 231 20 - - PUNCT cana-4527 231 21 133x 133x NUM cana-4527 231 22 vol vol NOUN cana-4527 231 23 32 32 NUM cana-4527 231 24 no no NOUN cana-4527 231 25 . . PUNCT cana-4527 232 1 9s 9s NUM cana-4527 232 2 ( ( PUNCT cana-4527 232 3 2025 2025 NUM cana-4527 232 4 ) ) PUNCT cana-4527 232 5 2427 2427 NUM cana-4527 232 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 232 7 figure figure NOUN cana-4527 232 8 . . PUNCT cana-4527 233 1 14 14 NUM cana-4527 233 2 . . PUNCT cana-4527 234 1 barker barker PROPN cana-4527 234 2 's 's PART cana-4527 234 3 solution solution NOUN cana-4527 234 4 as as ADP cana-4527 234 5 a a DET cana-4527 234 6 function function NOUN cana-4527 234 7 of of ADP cana-4527 234 8 space space NOUN cana-4527 234 9 problems problem NOUN cana-4527 234 10 with with ADP cana-4527 234 11 groundwater groundwater NOUN cana-4527 234 12 flow flow NOUN cana-4527 234 13 continue continue VERB cana-4527 234 14 to to PART cana-4527 234 15 grow grow VERB cana-4527 234 16 in in ADP cana-4527 234 17 importance importance NOUN cana-4527 234 18 as as ADP cana-4527 234 19 a a DET cana-4527 234 20 global global ADJ cana-4527 234 21 environmental environmental ADJ cana-4527 234 22 concern concern NOUN cana-4527 234 23 . . PUNCT cana-4527 235 1 this this PRON cana-4527 235 2 is be AUX cana-4527 235 3 one one NUM cana-4527 235 4 of of ADP cana-4527 235 5 the the DET cana-4527 235 6 main main ADJ cana-4527 235 7 reasons reason NOUN cana-4527 235 8 that that PRON cana-4527 235 9 numerous numerous ADJ cana-4527 235 10 scientists scientist NOUN cana-4527 235 11 from from ADP cana-4527 235 12 different different ADJ cana-4527 235 13 disciplines discipline NOUN cana-4527 235 14 have have AUX cana-4527 235 15 worked work VERB cana-4527 235 16 very very ADV cana-4527 235 17 hard hard ADV cana-4527 235 18 to to PART cana-4527 235 19 create create VERB cana-4527 235 20 new new ADJ cana-4527 235 21 mathematical mathematical ADJ cana-4527 235 22 formulas formula NOUN cana-4527 235 23 and and CCONJ cana-4527 235 24 models model NOUN cana-4527 235 25 that that PRON cana-4527 235 26 can can AUX cana-4527 235 27 be be AUX cana-4527 235 28 used use VERB cana-4527 235 29 to to PART cana-4527 235 30 illustrate illustrate VERB cana-4527 235 31 and and CCONJ cana-4527 235 32 comprehend comprehend VERB cana-4527 235 33 the the DET cana-4527 235 34 behaviour behaviour NOUN cana-4527 235 35 of of ADP cana-4527 235 36 groundwater groundwater NOUN cana-4527 235 37 flow flow NOUN cana-4527 235 38 over over ADP cana-4527 235 39 time time NOUN cana-4527 235 40 and and CCONJ cana-4527 235 41 space space NOUN cana-4527 235 42 . . PUNCT cana-4527 236 1 it it PRON cana-4527 236 2 is be AUX cana-4527 236 3 necessary necessary ADJ cana-4527 236 4 to to PART cana-4527 236 5 compare compare VERB cana-4527 236 6 the the DET cana-4527 236 7 outcomes outcome NOUN cana-4527 236 8 of of ADP cana-4527 236 9 the the DET cana-4527 236 10 suggested suggest VERB cana-4527 236 11 method method NOUN cana-4527 236 12 ( ( PUNCT cana-4527 236 13 ftd ftd NOUN cana-4527 236 14 ) ) PUNCT cana-4527 236 15 with with ADP cana-4527 236 16 barker barker PROPN cana-4527 236 17 's 's PART cana-4527 236 18 solution solution NOUN cana-4527 236 19 to to PART cana-4527 236 20 ascertain ascertain VERB cana-4527 236 21 whether whether SCONJ cana-4527 236 22 there there PRON cana-4527 236 23 is be VERB cana-4527 236 24 a a DET cana-4527 236 25 connection connection NOUN cana-4527 236 26 between between ADP cana-4527 236 27 the the DET cana-4527 236 28 parameter parameter NOUN cana-4527 236 29 fractal fractal NOUN cana-4527 236 30 that that PRON cana-4527 236 31 barker barker PROPN cana-4527 236 32 previously previously ADV cana-4527 236 33 described describe VERB cana-4527 236 34 and and CCONJ cana-4527 236 35 the the DET cana-4527 236 36 fractional fractional ADJ cana-4527 236 37 order order NOUN cana-4527 236 38 of of ADP cana-4527 236 39 the the DET cana-4527 236 40 derivative derivative NOUN cana-4527 236 41 . . PUNCT cana-4527 237 1 6 6 X cana-4527 237 2 . . X cana-4527 237 3 conclusion conclusion NOUN cana-4527 237 4 numerical numerical PROPN cana-4527 237 5 groundwater groundwater NOUN cana-4527 237 6 models model NOUN cana-4527 237 7 are be AUX cana-4527 237 8 constructed construct VERB cana-4527 237 9 by by ADP cana-4527 237 10 numerically numerically ADV cana-4527 237 11 approximating approximate VERB cana-4527 237 12 both both DET cana-4527 237 13 space space NOUN cana-4527 237 14 and and CCONJ cana-4527 237 15 time time NOUN cana-4527 237 16 . . PUNCT cana-4527 238 1 describing describe VERB cana-4527 238 2 underground underground ADJ cana-4527 238 3 water water NOUN cana-4527 238 4 passage passage NOUN cana-4527 238 5 across across ADP cana-4527 238 6 an an DET cana-4527 238 7 aquifer aquifer NOUN cana-4527 238 8 , , PUNCT cana-4527 238 9 a a DET cana-4527 238 10 diverse diverse ADJ cana-4527 238 11 geological geological ADJ cana-4527 238 12 structure structure NOUN cana-4527 238 13 is be AUX cana-4527 238 14 perhaps perhaps ADV cana-4527 238 15 the the DET cana-4527 238 16 most most ADV cana-4527 238 17 difficult difficult ADJ cana-4527 238 18 aspect aspect NOUN cana-4527 238 19 of of ADP cana-4527 238 20 studying study VERB cana-4527 238 21 groundwater groundwater NOUN cana-4527 238 22 . . PUNCT cana-4527 239 1 the the DET cana-4527 239 2 main main ADJ cana-4527 239 3 problem problem NOUN cana-4527 239 4 is be AUX cana-4527 239 5 that that SCONJ cana-4527 239 6 a a DET cana-4527 239 7 true true ADJ cana-4527 239 8 mathematical mathematical ADJ cana-4527 239 9 formula formula NOUN cana-4527 239 10 considering consider VERB cana-4527 239 11 the the DET cana-4527 239 12 aquifer aquifer NOUN cana-4527 239 13 's 's PART cana-4527 239 14 heterogeneity heterogeneity NOUN cana-4527 239 15 is be AUX cana-4527 239 16 difficult difficult ADJ cana-4527 239 17 to to PART cana-4527 239 18 develop develop VERB cana-4527 239 19 . . PUNCT cana-4527 240 1 recently recently ADV cana-4527 240 2 , , PUNCT cana-4527 240 3 a a DET cana-4527 240 4 new new ADJ cana-4527 240 5 fractional fractional ADJ cana-4527 240 6 order order NOUN cana-4527 240 7 derivative derivative NOUN cana-4527 240 8 has have AUX cana-4527 240 9 been be AUX cana-4527 240 10 introduced introduce VERB cana-4527 240 11 . . PUNCT cana-4527 241 1 this this DET cana-4527 241 2 derivative derivative NOUN cana-4527 241 3 is be AUX cana-4527 241 4 derived derive VERB cana-4527 241 5 from from ADP cana-4527 241 6 the the DET cana-4527 241 7 convolution convolution NOUN cana-4527 241 8 of of ADP cana-4527 241 9 an an DET cana-4527 241 10 exponential exponential ADJ cana-4527 241 11 function function NOUN cana-4527 241 12 with with ADP cana-4527 241 13 the the DET cana-4527 241 14 conventional conventional ADJ cana-4527 241 15 first first ADJ cana-4527 241 16 derivative derivative NOUN cana-4527 241 17 . . PUNCT cana-4527 242 1 the the DET cana-4527 242 2 derivative derivative NOUN cana-4527 242 3 can can AUX cana-4527 242 4 be be AUX cana-4527 242 5 employed employ VERB cana-4527 242 6 to to PART cana-4527 242 7 characterize characterize VERB cana-4527 242 8 material material NOUN cana-4527 242 9 diffusion diffusion NOUN cana-4527 242 10 on on ADP cana-4527 242 11 various various ADJ cana-4527 242 12 scales scale NOUN cana-4527 242 13 . . PUNCT cana-4527 243 1 we we PRON cana-4527 243 2 give give VERB cana-4527 243 3 a a DET cana-4527 243 4 way way NOUN cana-4527 243 5 to to PART cana-4527 243 6 demonstrate demonstrate VERB cana-4527 243 7 the the DET cana-4527 243 8 uniqueness uniqueness NOUN cana-4527 243 9 and and CCONJ cana-4527 243 10 existence existence NOUN cana-4527 243 11 of of ADP cana-4527 243 12 solutions solution NOUN cana-4527 243 13 to to ADP cana-4527 243 14 the the DET cana-4527 243 15 groundwater groundwater NOUN cana-4527 243 16 flow flow NOUN cana-4527 243 17 issue issue NOUN cana-4527 243 18 in in ADP cana-4527 243 19 a a DET cana-4527 243 20 confined confine VERB cana-4527 243 21 aquifer aquifer NOUN cana-4527 243 22 within within ADP cana-4527 243 23 the the DET cana-4527 243 24 context context NOUN cana-4527 243 25 of of ADP cana-4527 243 26 a a DET cana-4527 243 27 derivative derivative NOUN cana-4527 243 28 with with ADP cana-4527 243 29 fractional fractional ADJ cana-4527 243 30 order order NOUN cana-4527 243 31 . . PUNCT cana-4527 244 1 the the DET cana-4527 244 2 fixed fix VERB cana-4527 244 3 transform transform NOUN cana-4527 244 4 based base VERB cana-4527 244 5 domain domain NOUN cana-4527 244 6 ( ( PUNCT cana-4527 244 7 ftd ftd NOUN cana-4527 244 8 ) ) PUNCT cana-4527 244 9 technique technique NOUN cana-4527 244 10 confirmed confirm VERB cana-4527 244 11 the the DET cana-4527 244 12 exact exact ADJ cana-4527 244 13 solution solution NOUN cana-4527 244 14 . . PUNCT cana-4527 245 1 moreover moreover ADV cana-4527 245 2 , , PUNCT cana-4527 245 3 we we PRON cana-4527 245 4 can can AUX cana-4527 245 5 show show VERB cana-4527 245 6 only only ADV cana-4527 245 7 one one NUM cana-4527 245 8 solution solution NOUN cana-4527 245 9 to to ADP cana-4527 245 10 the the DET cana-4527 245 11 changed change VERB cana-4527 245 12 groundwater groundwater NOUN cana-4527 245 13 flow flow NOUN cana-4527 245 14 equation equation NOUN cana-4527 245 15 . . PUNCT cana-4527 246 1 the the DET cana-4527 246 2 local local ADJ cana-4527 246 3 temporal temporal ADJ cana-4527 246 4 and and CCONJ cana-4527 246 5 spatial spatial ADJ cana-4527 246 6 derivative derivative NOUN cana-4527 246 7 can can AUX cana-4527 246 8 be be AUX cana-4527 246 9 substituted substitute VERB cana-4527 246 10 with with ADP cana-4527 246 11 the the DET cana-4527 246 12 caputo caputo PROPN cana-4527 246 13 - - PUNCT cana-4527 246 14 fabrizio fabrizio PROPN cana-4527 246 15 derivative derivative NOUN cana-4527 246 16 to to PART cana-4527 246 17 create create VERB cana-4527 246 18 this this DET cana-4527 246 19 equation equation NOUN cana-4527 246 20 . . PUNCT cana-4527 247 1 given give VERB cana-4527 247 2 this this DET cana-4527 247 3 discovery discovery NOUN cana-4527 247 4 , , PUNCT cana-4527 247 5 numerical numerical ADJ cana-4527 247 6 simulations simulation NOUN cana-4527 247 7 and and CCONJ cana-4527 247 8 stability stability NOUN cana-4527 247 9 of of ADP cana-4527 247 10 solution solution NOUN cana-4527 247 11 analyses analysis NOUN cana-4527 247 12 were be AUX cana-4527 247 13 conducted conduct VERB cana-4527 247 14 for for ADP cana-4527 247 15 a a DET cana-4527 247 16 variety variety NOUN cana-4527 247 17 of of ADP cana-4527 247 18 Ξ± Ξ± PROPN cana-4527 247 19 values value NOUN cana-4527 247 20 to to PART cana-4527 247 21 illustrate illustrate VERB cana-4527 247 22 the the DET cana-4527 247 23 efficacy efficacy NOUN cana-4527 247 24 of of ADP cana-4527 247 25 this this DET cana-4527 247 26 innovative innovative ADJ cana-4527 247 27 model model NOUN cana-4527 247 28 . . PUNCT cana-4527 248 1 a a DET cana-4527 248 2 relationship relationship NOUN cana-4527 248 3 between between ADP cana-4527 248 4 the the DET cana-4527 248 5 flow flow NOUN cana-4527 248 6 's 's PART cana-4527 248 7 non non ADJ cana-4527 248 8 - - ADJ cana-4527 248 9 integral integral ADJ cana-4527 248 10 dimension dimension NOUN cana-4527 248 11 and and CCONJ cana-4527 248 12 the the DET cana-4527 248 13 derivative derivative NOUN cana-4527 248 14 's 's PART cana-4527 248 15 fractional fractional ADJ cana-4527 248 16 order order NOUN cana-4527 248 17 is be AUX cana-4527 248 18 demonstrated demonstrate VERB cana-4527 248 19 by by ADP cana-4527 248 20 comparing compare VERB cana-4527 248 21 the the DET cana-4527 248 22 results result NOUN cana-4527 248 23 of of ADP cana-4527 248 24 the the DET cana-4527 248 25 proposed propose VERB cana-4527 248 26 method method NOUN cana-4527 248 27 with with ADP cana-4527 248 28 those those PRON cana-4527 248 29 of of ADP cana-4527 248 30 barker barker PROPN cana-4527 248 31 's 's PART cana-4527 248 32 fractal fractal ADJ cana-4527 248 33 radial radial ADJ cana-4527 248 34 flow flow NOUN cana-4527 248 35 framework framework NOUN cana-4527 248 36 . . PUNCT cana-4527 249 1 additionally additionally ADV cana-4527 249 2 , , PUNCT cana-4527 249 3 the the DET cana-4527 249 4 proposed propose VERB cana-4527 249 5 method method NOUN cana-4527 249 6 is be AUX cana-4527 249 7 highly highly ADV cana-4527 249 8 effective effective ADJ cana-4527 249 9 and and CCONJ cana-4527 249 10 applicable applicable ADJ cana-4527 249 11 to to ADP cana-4527 249 12 similar similar ADJ cana-4527 249 13 problems problem NOUN cana-4527 249 14 . . PUNCT cana-4527 250 1 communications communication NOUN cana-4527 250 2 on on ADP cana-4527 250 3 applied apply VERB cana-4527 250 4 nonlinear nonlinear ADJ cana-4527 250 5 analysis analysis NOUN cana-4527 250 6 issn issn NOUN cana-4527 250 7 : : PUNCT cana-4527 250 8 1074 1074 NUM cana-4527 250 9 - - PUNCT cana-4527 250 10 133x 133x NUM cana-4527 250 11 vol vol NOUN cana-4527 250 12 32 32 NUM cana-4527 250 13 no no NOUN cana-4527 250 14 . . PUNCT cana-4527 251 1 9s 9s NUM cana-4527 251 2 ( ( PUNCT cana-4527 251 3 2025 2025 NUM cana-4527 251 4 ) ) PUNCT cana-4527 251 5 2428 2428 NUM cana-4527 251 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 251 7 refrences refrence VERB cana-4527 252 1 [ [ X cana-4527 252 2 1 1 NUM cana-4527 252 3 ] ] PUNCT cana-4527 252 4 . . PUNCT cana-4527 253 1 li li PROPN cana-4527 253 2 , , PUNCT cana-4527 253 3 changpin changpin NOUN cana-4527 253 4 , , PUNCT cana-4527 253 5 deliang deliang PROPN cana-4527 253 6 qian qian PROPN cana-4527 253 7 , , PUNCT cana-4527 253 8 and and CCONJ cana-4527 253 9 yangquan yangquan PROPN cana-4527 253 10 chen chen PROPN cana-4527 253 11 , , PUNCT cana-4527 253 12 on on ADP cana-4527 253 13 riemann‐liouville riemann‐liouville NOUN cana-4527 253 14 and and CCONJ cana-4527 253 15 caputo caputo PROPN cana-4527 253 16 derivatives derivative NOUN cana-4527 253 17 , , PUNCT cana-4527 253 18 discrete discrete ADJ cana-4527 253 19 dynamics dynamic NOUN cana-4527 253 20 in in ADP cana-4527 253 21 nature nature NOUN cana-4527 253 22 and and CCONJ cana-4527 253 23 society society NOUN cana-4527 253 24 2011 2011 NUM cana-4527 253 25 , , PUNCT cana-4527 253 26 no no INTJ cana-4527 253 27 . . NOUN cana-4527 253 28 1 1 NUM cana-4527 253 29 ( ( PUNCT cana-4527 253 30 2011 2011 NUM cana-4527 253 31 ) ) PUNCT cana-4527 253 32 . . PUNCT cana-4527 254 1 [ [ X cana-4527 254 2 2 2 NUM cana-4527 254 3 ] ] PUNCT cana-4527 254 4 . . PUNCT cana-4527 255 1 atangana atangana PROPN cana-4527 255 2 . . PUNCT cana-4527 256 1 a a DET cana-4527 256 2 , , PUNCT cana-4527 256 3 numerical numerical ADJ cana-4527 256 4 solution solution NOUN cana-4527 256 5 of of ADP cana-4527 256 6 space space NOUN cana-4527 256 7 - - PUNCT cana-4527 256 8 time time NOUN cana-4527 256 9 fractional fractional ADJ cana-4527 256 10 derivative derivative NOUN cana-4527 256 11 of of ADP cana-4527 256 12 groundwater groundwater NOUN cana-4527 256 13 flow flow NOUN cana-4527 256 14 equation equation NOUN cana-4527 256 15 , , PUNCT cana-4527 256 16 in in ADP cana-4527 256 17 proceedings proceeding NOUN cana-4527 256 18 of of ADP cana-4527 256 19 the the DET cana-4527 256 20 international international ADJ cana-4527 256 21 conference conference NOUN cana-4527 256 22 of of ADP cana-4527 256 23 algebra algebra NOUN cana-4527 256 24 and and CCONJ cana-4527 256 25 applied apply VERB cana-4527 256 26 analysis analysis NOUN cana-4527 256 27 , , PUNCT cana-4527 256 28 2 2 NUM cana-4527 256 29 ( ( PUNCT cana-4527 256 30 1 1 NUM cana-4527 256 31 ) ) PUNCT cana-4527 256 32 ( ( PUNCT cana-4527 256 33 2012 2012 NUM cana-4527 256 34 ) ) PUNCT cana-4527 256 35 , , PUNCT cana-4527 256 36 20 20 NUM cana-4527 256 37 . . PUNCT cana-4527 257 1 [ [ X cana-4527 257 2 3 3 NUM cana-4527 257 3 ] ] PUNCT cana-4527 257 4 . . PUNCT cana-4527 258 1 r. r. PROPN cana-4527 258 2 scherer scherer PROPN cana-4527 258 3 , , PUNCT cana-4527 258 4 s. s. PROPN cana-4527 258 5 l. l. PROPN cana-4527 258 6 kalla kalla PROPN cana-4527 258 7 , , PUNCT cana-4527 258 8 y. y. PROPN cana-4527 258 9 tang tang PROPN cana-4527 258 10 , , PUNCT cana-4527 258 11 j. j. PROPN cana-4527 258 12 huang huang PROPN cana-4527 258 13 , , PUNCT cana-4527 258 14 the the DET cana-4527 258 15 grunwald grunwald PROPN cana-4527 258 16 – – PUNCT cana-4527 258 17 letnikov letnikov NOUN cana-4527 258 18 method method NOUN cana-4527 258 19 for for ADP cana-4527 258 20 fractional fractional ADJ cana-4527 258 21 differential differential ADJ cana-4527 258 22 equations equation NOUN cana-4527 258 23 , , PUNCT cana-4527 258 24 computers computer NOUN cana-4527 258 25 & & CCONJ cana-4527 258 26 mathematics mathematic NOUN cana-4527 258 27 with with ADP cana-4527 258 28 applications application NOUN cana-4527 258 29 62 62 NUM cana-4527 258 30 ( ( PUNCT cana-4527 258 31 3 3 NUM cana-4527 258 32 ) ) PUNCT cana-4527 258 33 ( ( PUNCT cana-4527 258 34 2011 2011 NUM cana-4527 258 35 ) ) PUNCT cana-4527 258 36 , , PUNCT cana-4527 258 37 902–917 902–917 NUM cana-4527 258 38 . . PUNCT cana-4527 259 1 [ [ X cana-4527 259 2 4 4 NUM cana-4527 259 3 ] ] PUNCT cana-4527 259 4 . . PUNCT cana-4527 260 1 c. c. PROPN cana-4527 260 2 f. f. PROPN cana-4527 260 3 coimbra coimbra PROPN cana-4527 260 4 , , PUNCT cana-4527 260 5 mechanics mechanic NOUN cana-4527 260 6 with with ADP cana-4527 260 7 variable variable ADJ cana-4527 260 8 - - PUNCT cana-4527 260 9 order order NOUN cana-4527 260 10 differential differential NOUN cana-4527 260 11 operators operator NOUN cana-4527 260 12 , , PUNCT cana-4527 260 13 annalen annalen VERB cana-4527 260 14 der der NOUN cana-4527 260 15 physik physik PROPN cana-4527 260 16 12 12 NUM cana-4527 260 17 ( ( PUNCT cana-4527 260 18 11 11 NUM cana-4527 260 19 - - SYM cana-4527 260 20 12 12 NUM cana-4527 260 21 ) ) PUNCT cana-4527 260 22 ( ( PUNCT cana-4527 260 23 2003 2003 NUM cana-4527 260 24 ) ) PUNCT cana-4527 260 25 , , PUNCT cana-4527 260 26 692–703 692–703 NUM cana-4527 260 27 . . PUNCT cana-4527 261 1 [ [ X cana-4527 261 2 5 5 NUM cana-4527 261 3 ] ] PUNCT cana-4527 261 4 . . PUNCT cana-4527 262 1 g. g. PROPN cana-4527 262 2 jumarie jumarie PROPN cana-4527 262 3 , , PUNCT cana-4527 262 4 on on ADP cana-4527 262 5 the the DET cana-4527 262 6 solution solution NOUN cana-4527 262 7 of of ADP cana-4527 262 8 the the DET cana-4527 262 9 stochastic stochastic ADJ cana-4527 262 10 differential differential ADJ cana-4527 262 11 equation equation NOUN cana-4527 262 12 of of ADP cana-4527 262 13 exponential exponential ADJ cana-4527 262 14 growth growth NOUN cana-4527 262 15 driven drive VERB cana-4527 262 16 by by ADP cana-4527 262 17 fractional fractional ADJ cana-4527 262 18 brownian brownian PROPN cana-4527 262 19 motion motion NOUN cana-4527 262 20 , , PUNCT cana-4527 262 21 applied apply VERB cana-4527 262 22 mathematics mathematics NOUN cana-4527 262 23 letters letter NOUN cana-4527 262 24 18 18 NUM cana-4527 262 25 ( ( PUNCT cana-4527 262 26 7 7 NUM cana-4527 262 27 ) ) PUNCT cana-4527 262 28 ( ( PUNCT cana-4527 262 29 2005 2005 NUM cana-4527 262 30 ) ) PUNCT cana-4527 262 31 , , PUNCT cana-4527 262 32 817–826 817–826 NUM cana-4527 262 33 . . PUNCT cana-4527 263 1 [ [ X cana-4527 263 2 6 6 NUM cana-4527 263 3 ] ] PUNCT cana-4527 263 4 . . PUNCT cana-4527 264 1 n. n. PROPN cana-4527 264 2 a. a. PROPN cana-4527 264 3 sheikh sheikh PROPN cana-4527 264 4 , , PUNCT cana-4527 264 5 f. f. PROPN cana-4527 264 6 ali ali PROPN cana-4527 264 7 , , PUNCT cana-4527 264 8 m. m. NOUN cana-4527 264 9 saqib saqib PROPN cana-4527 264 10 , , PUNCT cana-4527 264 11 i. i. PROPN cana-4527 264 12 khan khan PROPN cana-4527 264 13 , , PUNCT cana-4527 264 14 s. s. PROPN cana-4527 264 15 a. a. PROPN cana-4527 264 16 a. a. PROPN cana-4527 264 17 jan jan PROPN cana-4527 264 18 , , PUNCT cana-4527 264 19 a. a. PROPN cana-4527 264 20 s. s. PROPN cana-4527 264 21 alshomrani alshomrani PROPN cana-4527 264 22 , , PUNCT cana-4527 264 23 m. m. NOUN cana-4527 264 24 s. s. PROPN cana-4527 264 25 alghamdi alghamdi PROPN cana-4527 264 26 , , PUNCT cana-4527 264 27 comparison comparison NOUN cana-4527 264 28 and and CCONJ cana-4527 264 29 analysis analysis NOUN cana-4527 264 30 of of ADP cana-4527 264 31 the the DET cana-4527 264 32 atangana atangana PROPN cana-4527 264 33 – – PUNCT cana-4527 264 34 baleanu baleanu PROPN cana-4527 264 35 and and CCONJ cana-4527 264 36 caputo caputo PROPN cana-4527 264 37 – – PUNCT cana-4527 264 38 fabrizio fabrizio ADJ cana-4527 264 39 fractional fractional ADJ cana-4527 264 40 derivatives derivative NOUN cana-4527 264 41 for for ADP cana-4527 264 42 generalized generalized ADJ cana-4527 264 43 casson casson PROPN cana-4527 264 44 fluid fluid NOUN cana-4527 264 45 model model NOUN cana-4527 264 46 with with ADP cana-4527 264 47 heat heat NOUN cana-4527 264 48 generation generation NOUN cana-4527 264 49 and and CCONJ cana-4527 264 50 chemical chemical NOUN cana-4527 264 51 reaction reaction NOUN cana-4527 264 52 , , PUNCT cana-4527 264 53 results result NOUN cana-4527 264 54 in in ADP cana-4527 264 55 physics physics NOUN cana-4527 264 56 7 7 NUM cana-4527 264 57 ( ( PUNCT cana-4527 264 58 2017 2017 NUM cana-4527 264 59 ) ) PUNCT cana-4527 264 60 , , PUNCT cana-4527 264 61 789–800 789–800 NUM cana-4527 264 62 . . PUNCT cana-4527 265 1 [ [ X cana-4527 265 2 7 7 NUM cana-4527 265 3 ] ] PUNCT cana-4527 265 4 . . PUNCT cana-4527 265 5 a. a. PROPN cana-4527 265 6 atangana atangana PROPN cana-4527 265 7 , , PUNCT cana-4527 265 8 b. b. PROPN cana-4527 265 9 s. s. PROPN cana-4527 265 10 t. t. PROPN cana-4527 265 11 alkahtani alkahtani PROPN cana-4527 265 12 , , PUNCT cana-4527 265 13 new new ADJ cana-4527 265 14 model model NOUN cana-4527 265 15 of of ADP cana-4527 265 16 groundwater groundwater NOUN cana-4527 265 17 flowing flow VERB cana-4527 265 18 within within ADP cana-4527 265 19 a a DET cana-4527 265 20 confine confine NOUN cana-4527 265 21 aquifer aquifer NOUN cana-4527 265 22 : : PUNCT cana-4527 265 23 application application NOUN cana-4527 265 24 of of ADP cana-4527 265 25 caputo caputo PROPN cana-4527 265 26 - - PUNCT cana-4527 265 27 fabrizio fabrizio PROPN cana-4527 265 28 derivative derivative NOUN cana-4527 265 29 , , PUNCT cana-4527 265 30 arabian arabian ADJ cana-4527 265 31 journal journal NOUN cana-4527 265 32 of of ADP cana-4527 265 33 geosciences geoscience NOUN cana-4527 265 34 9 9 NUM cana-4527 265 35 ( ( PUNCT cana-4527 265 36 1 1 NUM cana-4527 265 37 ) ) PUNCT cana-4527 265 38 ( ( PUNCT cana-4527 265 39 2016 2016 NUM cana-4527 265 40 ) ) PUNCT cana-4527 265 41 , , PUNCT cana-4527 265 42 1–6 1–6 X cana-4527 265 43 . . PUNCT cana-4527 266 1 [ [ X cana-4527 266 2 8 8 NUM cana-4527 266 3 ] ] PUNCT cana-4527 266 4 . . PUNCT cana-4527 267 1 t. t. PROPN cana-4527 267 2 akram akram PROPN cana-4527 267 3 , , PUNCT cana-4527 267 4 m. m. NOUN cana-4527 267 5 abbas abbas PROPN cana-4527 267 6 , , PUNCT cana-4527 267 7 a. a. PROPN cana-4527 267 8 ali ali PROPN cana-4527 267 9 , , PUNCT cana-4527 267 10 a. a. PROPN cana-4527 267 11 iqbal iqbal PROPN cana-4527 267 12 , , PUNCT cana-4527 267 13 d. d. PROPN cana-4527 267 14 baleanu baleanu PROPN cana-4527 267 15 , , PUNCT cana-4527 267 16 a a DET cana-4527 267 17 numerical numerical ADJ cana-4527 267 18 approach approach NOUN cana-4527 267 19 of of ADP cana-4527 267 20 a a DET cana-4527 267 21 time time NOUN cana-4527 267 22 - - PUNCT cana-4527 267 23 fractional fractional ADJ cana-4527 267 24 reaction reaction NOUN cana-4527 267 25 - - PUNCT cana-4527 267 26 diffusion diffusion NOUN cana-4527 267 27 model model NOUN cana-4527 267 28 with with ADP cana-4527 267 29 a a DET cana-4527 267 30 non non ADJ cana-4527 267 31 - - ADJ cana-4527 267 32 singular singular ADJ cana-4527 267 33 kernel kernel NOUN cana-4527 267 34 , , PUNCT cana-4527 267 35 symmetry symmetry NOUN cana-4527 267 36 12 12 NUM cana-4527 267 37 ( ( PUNCT cana-4527 267 38 10 10 NUM cana-4527 267 39 ) ) PUNCT cana-4527 267 40 ( ( PUNCT cana-4527 267 41 2020 2020 NUM cana-4527 267 42 ) ) PUNCT cana-4527 267 43 , , PUNCT cana-4527 267 44 1653 1653 NUM cana-4527 267 45 . . PUNCT cana-4527 268 1 [ [ X cana-4527 268 2 9 9 NUM cana-4527 268 3 ] ] PUNCT cana-4527 268 4 . . PUNCT cana-4527 268 5 m. m. PROPN cana-4527 268 6 shafiq shafiq PROPN cana-4527 268 7 , , PUNCT cana-4527 268 8 m. m. NOUN cana-4527 268 9 abbas abbas PROPN cana-4527 268 10 , , PUNCT cana-4527 268 11 k. k. PROPN cana-4527 268 12 m. m. PROPN cana-4527 268 13 abualnaja abualnaja PROPN cana-4527 268 14 , , PUNCT cana-4527 268 15 m. m. NOUN cana-4527 268 16 huntul huntul PROPN cana-4527 268 17 , , PUNCT cana-4527 268 18 a. a. NOUN cana-4527 268 19 majeed majeed PROPN cana-4527 268 20 , , PUNCT cana-4527 268 21 t. t. PROPN cana-4527 268 22 nazir nazir PROPN cana-4527 268 23 , , PUNCT cana-4527 268 24 an an DET cana-4527 268 25 efficient efficient ADJ cana-4527 268 26 technique technique NOUN cana-4527 268 27 based base VERB cana-4527 268 28 on on ADP cana-4527 268 29 cubic cubic ADJ cana-4527 268 30 b b PROPN cana-4527 268 31 - - PUNCT cana-4527 268 32 spline spline NOUN cana-4527 268 33 functions function NOUN cana-4527 268 34 for for ADP cana-4527 268 35 solving solve VERB cana-4527 268 36 time time NOUN cana-4527 268 37 - - PUNCT cana-4527 268 38 fractional fractional ADJ cana-4527 268 39 advection advection NOUN cana-4527 268 40 - - PUNCT cana-4527 268 41 diffusion diffusion NOUN cana-4527 268 42 equation equation NOUN cana-4527 268 43 involving involve VERB cana-4527 268 44 atangana atangana PROPN cana-4527 268 45 – – PUNCT cana-4527 268 46 baleanu baleanu ADJ cana-4527 268 47 derivative derivative NOUN cana-4527 268 48 , , PUNCT cana-4527 268 49 engineering engineering NOUN cana-4527 268 50 with with ADP cana-4527 268 51 computers computer NOUN cana-4527 268 52 38 38 NUM cana-4527 268 53 ( ( PUNCT cana-4527 268 54 1 1 NUM cana-4527 268 55 ) ) PUNCT cana-4527 268 56 ( ( PUNCT cana-4527 268 57 2021 2021 NUM cana-4527 268 58 ) ) PUNCT cana-4527 268 59 901–917 901–917 NUM cana-4527 268 60 . . PUNCT cana-4527 269 1 [ [ X cana-4527 269 2 10 10 NUM cana-4527 269 3 ] ] PUNCT cana-4527 269 4 . . PUNCT cana-4527 269 5 a. a. PROPN cana-4527 269 6 h. h. PROPN cana-4527 269 7 bhrawy bhrawy PROPN cana-4527 269 8 , , PUNCT cana-4527 269 9 t. t. PROPN cana-4527 269 10 m. m. PROPN cana-4527 269 11 taha taha PROPN cana-4527 269 12 , , PUNCT cana-4527 269 13 j. j. PROPN cana-4527 269 14 a. a. PROPN cana-4527 269 15 t. t. PROPN cana-4527 269 16 machado machado PROPN cana-4527 269 17 , , PUNCT cana-4527 269 18 a a DET cana-4527 269 19 review review NOUN cana-4527 269 20 of of ADP cana-4527 269 21 operational operational ADJ cana-4527 269 22 matrices matrix NOUN cana-4527 269 23 and and CCONJ cana-4527 269 24 spectral spectral ADJ cana-4527 269 25 techniques technique NOUN cana-4527 269 26 for for ADP cana-4527 269 27 fractional fractional ADJ cana-4527 269 28 calculus calculus NOUN cana-4527 269 29 , , PUNCT cana-4527 269 30 non non ADJ cana-4527 269 31 - - ADJ cana-4527 269 32 linear linear ADJ cana-4527 269 33 dynamics dynamic NOUN cana-4527 269 34 81 81 NUM cana-4527 269 35 ( ( PUNCT cana-4527 269 36 3 3 NUM cana-4527 269 37 ) ) PUNCT cana-4527 269 38 ( ( PUNCT cana-4527 269 39 2015 2015 NUM cana-4527 269 40 ) ) PUNCT cana-4527 269 41 , , PUNCT cana-4527 269 42 1023–1052 1023–1052 NUM cana-4527 269 43 . . PUNCT cana-4527 270 1 [ [ X cana-4527 270 2 11 11 NUM cana-4527 270 3 ] ] PUNCT cana-4527 270 4 . . PUNCT cana-4527 271 1 h. h. PROPN cana-4527 271 2 sun sun PROPN cana-4527 271 3 , , PUNCT cana-4527 271 4 w. w. PROPN cana-4527 271 5 chen chen PROPN cana-4527 271 6 , , PUNCT cana-4527 271 7 y. y. PROPN cana-4527 271 8 chen chen PROPN cana-4527 271 9 , , PUNCT cana-4527 271 10 variable variable ADJ cana-4527 271 11 - - PUNCT cana-4527 271 12 order order NOUN cana-4527 271 13 fractional fractional ADJ cana-4527 271 14 differential differential NOUN cana-4527 271 15 operators operator NOUN cana-4527 271 16 in in ADP cana-4527 271 17 anomalous anomalous ADJ cana-4527 271 18 diffusion diffusion NOUN cana-4527 271 19 modeling modeling NOUN cana-4527 271 20 , , PUNCT cana-4527 271 21 physica physica NOUN cana-4527 271 22 a a DET cana-4527 271 23 : : PUNCT cana-4527 271 24 statistical statistical ADJ cana-4527 271 25 mechanics mechanic NOUN cana-4527 271 26 and and CCONJ cana-4527 271 27 its its PRON cana-4527 271 28 applications application NOUN cana-4527 271 29 388 388 NUM cana-4527 271 30 ( ( PUNCT cana-4527 271 31 21 21 NUM cana-4527 271 32 ) ) PUNCT cana-4527 271 33 ( ( PUNCT cana-4527 271 34 2009 2009 NUM cana-4527 271 35 ) ) PUNCT cana-4527 271 36 , , PUNCT cana-4527 271 37 4586–4592 4586–4592 NUM cana-4527 271 38 . . PUNCT cana-4527 272 1 [ [ X cana-4527 272 2 12 12 NUM cana-4527 272 3 ] ] PUNCT cana-4527 272 4 . . PUNCT cana-4527 273 1 l. l. PROPN cana-4527 273 2 vzquez vzquez PROPN cana-4527 273 3 , , PUNCT cana-4527 273 4 from from ADP cana-4527 273 5 newtons newton NOUN cana-4527 273 6 equation equation NOUN cana-4527 273 7 to to ADP cana-4527 273 8 fractional fractional ADJ cana-4527 273 9 diffusion diffusion NOUN cana-4527 273 10 and and CCONJ cana-4527 273 11 wave wave NOUN cana-4527 273 12 equations equation NOUN cana-4527 273 13 , , PUNCT cana-4527 273 14 adv adv PROPN cana-4527 273 15 . . PUNCT cana-4527 273 16 difference difference NOUN cana-4527 273 17 equ equ PROPN cana-4527 273 18 ( ( PUNCT cana-4527 273 19 2011),1–13 2011),1–13 NUM cana-4527 273 20 . . PUNCT cana-4527 274 1 [ [ X cana-4527 274 2 13 13 NUM cana-4527 274 3 ] ] PUNCT cana-4527 274 4 . . PUNCT cana-4527 275 1 i. i. PROPN cana-4527 275 2 podlubny podlubny PROPN cana-4527 275 3 , , PUNCT cana-4527 275 4 a. a. NOUN cana-4527 275 5 chechkin chechkin NOUN cana-4527 275 6 , , PUNCT cana-4527 275 7 t. t. PROPN cana-4527 275 8 skovranek skovranek NOUN cana-4527 275 9 , , PUNCT cana-4527 275 10 y. y. PROPN cana-4527 275 11 chen chen PROPN cana-4527 275 12 , , PUNCT cana-4527 275 13 b. b. PROPN cana-4527 275 14 m. m. PROPN cana-4527 275 15 vinagre vinagre PROPN cana-4527 275 16 jara jara PROPN cana-4527 275 17 , , PUNCT cana-4527 275 18 matrix matrix NOUN cana-4527 275 19 approach approach NOUN cana-4527 275 20 to to PART cana-4527 275 21 discrete discrete VERB cana-4527 275 22 fractional fractional ADJ cana-4527 275 23 calculus calculus NOUN cana-4527 275 24 ii ii PROPN cana-4527 275 25 : : PUNCT cana-4527 275 26 partial partial ADJ cana-4527 275 27 fractional fractional ADJ cana-4527 275 28 differential differential NOUN cana-4527 275 29 equations equation NOUN cana-4527 275 30 , , PUNCT cana-4527 275 31 journal journal NOUN cana-4527 275 32 of of ADP cana-4527 275 33 computational computational ADJ cana-4527 275 34 physics physic NOUN cana-4527 275 35 228 228 NUM cana-4527 275 36 ( ( PUNCT cana-4527 275 37 8) 8) NUM cana-4527 275 38 ( ( PUNCT cana-4527 275 39 2009 2009 NUM cana-4527 275 40 ) ) PUNCT cana-4527 275 41 , , PUNCT cana-4527 275 42 3137–3153 3137–3153 NUM cana-4527 275 43 . . PUNCT cana-4527 276 1 [ [ X cana-4527 276 2 14 14 NUM cana-4527 276 3 ] ] PUNCT cana-4527 276 4 . . PUNCT cana-4527 277 1 a. a. PROPN cana-4527 277 2 atangana atangana PROPN cana-4527 277 3 , , PUNCT cana-4527 277 4 r. r. PROPN cana-4527 277 5 t. t. PROPN cana-4527 277 6 alqahtani alqahtani PROPN cana-4527 277 7 , , PUNCT cana-4527 277 8 numerical numerical ADJ cana-4527 277 9 approximation approximation NOUN cana-4527 277 10 of of ADP cana-4527 277 11 the the DET cana-4527 277 12 space space NOUN cana-4527 277 13 - - PUNCT cana-4527 277 14 time time NOUN cana-4527 277 15 caputo caputo PROPN cana-4527 277 16 - - PUNCT cana-4527 277 17 fabrizio fabrizio PROPN cana-4527 277 18 fractional fractional ADJ cana-4527 277 19 derivative derivative NOUN cana-4527 277 20 and and CCONJ cana-4527 277 21 application application NOUN cana-4527 277 22 to to PART cana-4527 277 23 groundwater groundwater VERB cana-4527 277 24 pollution pollution NOUN cana-4527 277 25 equation equation NOUN cana-4527 277 26 , , PUNCT cana-4527 277 27 advances advance NOUN cana-4527 277 28 in in ADP cana-4527 277 29 difference difference NOUN cana-4527 277 30 equations equation NOUN cana-4527 277 31 2016 2016 NUM cana-4527 277 32 ( ( PUNCT cana-4527 277 33 1 1 NUM cana-4527 277 34 ) ) PUNCT cana-4527 277 35 ( ( PUNCT cana-4527 277 36 2016 2016 NUM cana-4527 277 37 ) ) PUNCT cana-4527 277 38 , , PUNCT cana-4527 277 39 1–13 1–13 NOUN cana-4527 277 40 . . PUNCT cana-4527 278 1 communications communication NOUN cana-4527 278 2 on on ADP cana-4527 278 3 applied apply VERB cana-4527 278 4 nonlinear nonlinear ADJ cana-4527 278 5 analysis analysis NOUN cana-4527 278 6 issn issn NOUN cana-4527 278 7 : : PUNCT cana-4527 278 8 1074 1074 NUM cana-4527 278 9 - - PUNCT cana-4527 278 10 133x 133x NUM cana-4527 278 11 vol vol NOUN cana-4527 278 12 32 32 NUM cana-4527 278 13 no no NOUN cana-4527 278 14 . . PUNCT cana-4527 279 1 9s 9s NUM cana-4527 279 2 ( ( PUNCT cana-4527 279 3 2025 2025 NUM cana-4527 279 4 ) ) PUNCT cana-4527 279 5 2429 2429 NUM cana-4527 279 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 280 1 [ [ X cana-4527 280 2 15 15 NUM cana-4527 280 3 ] ] PUNCT cana-4527 280 4 . . PUNCT cana-4527 281 1 s. s. PROPN cana-4527 281 2 momani momani PROPN cana-4527 281 3 , , PUNCT cana-4527 281 4 z. z. PROPN cana-4527 281 5 odibat odibat PROPN cana-4527 281 6 , , PUNCT cana-4527 281 7 numerical numerical ADJ cana-4527 281 8 approach approach NOUN cana-4527 281 9 to to ADP cana-4527 281 10 differential differential ADJ cana-4527 281 11 equations equation NOUN cana-4527 281 12 of of ADP cana-4527 281 13 fractional fractional ADJ cana-4527 281 14 order order NOUN cana-4527 281 15 , , PUNCT cana-4527 281 16 journal journal NOUN cana-4527 281 17 of of ADP cana-4527 281 18 computational computational ADJ cana-4527 281 19 and and CCONJ cana-4527 281 20 applied apply VERB cana-4527 281 21 mathematics mathematic NOUN cana-4527 281 22 207 207 NUM cana-4527 281 23 ( ( PUNCT cana-4527 281 24 1 1 NUM cana-4527 281 25 ) ) PUNCT cana-4527 281 26 ( ( PUNCT cana-4527 281 27 2007 2007 NUM cana-4527 281 28 ) ) PUNCT cana-4527 281 29 , , PUNCT cana-4527 281 30 96–110 96–110 PROPN cana-4527 281 31 . . PUNCT cana-4527 282 1 [ [ X cana-4527 282 2 16 16 NUM cana-4527 282 3 ] ] PUNCT cana-4527 282 4 . . PUNCT cana-4527 283 1 s. s. PROPN cana-4527 283 2 jafari jafari PROPN cana-4527 283 3 , , PUNCT cana-4527 283 4 j. j. PROPN cana-4527 283 5 sprott sprott PROPN cana-4527 283 6 , , PUNCT cana-4527 283 7 simple simple ADJ cana-4527 283 8 chaotic chaotic ADJ cana-4527 283 9 flows flow NOUN cana-4527 283 10 with with ADP cana-4527 283 11 a a DET cana-4527 283 12 line line NOUN cana-4527 283 13 equilibrium equilibrium NOUN cana-4527 283 14 , , PUNCT cana-4527 283 15 chaos chaos NOUN cana-4527 283 16 , , PUNCT cana-4527 283 17 solitons soliton NOUN cana-4527 283 18 & & CCONJ cana-4527 283 19 fractals fractal NOUN cana-4527 283 20 57 57 NUM cana-4527 283 21 ( ( PUNCT cana-4527 283 22 2013 2013 NUM cana-4527 283 23 ) ) PUNCT cana-4527 283 24 , , PUNCT cana-4527 283 25 79–84 79–84 NOUN cana-4527 283 26 . . PUNCT cana-4527 284 1 [ [ X cana-4527 284 2 17 17 NUM cana-4527 284 3 ] ] PUNCT cana-4527 284 4 . . PUNCT cana-4527 285 1 k. k. PROPN cana-4527 285 2 m. m. NOUN cana-4527 285 3 owolabi owolabi NOUN cana-4527 285 4 , , PUNCT cana-4527 285 5 mathematical mathematical ADJ cana-4527 285 6 analysis analysis NOUN cana-4527 285 7 and and CCONJ cana-4527 285 8 numerical numerical PROPN cana-4527 285 9 simulation simulation NOUN cana-4527 285 10 of of ADP cana-4527 285 11 patterns pattern NOUN cana-4527 285 12 in in ADP cana-4527 285 13 fractional fractional ADJ cana-4527 285 14 and and CCONJ cana-4527 285 15 classical classical ADJ cana-4527 285 16 reaction reaction NOUN cana-4527 285 17 - - PUNCT cana-4527 285 18 diffusion diffusion NOUN cana-4527 285 19 systems system NOUN cana-4527 285 20 , , PUNCT cana-4527 285 21 chaos chaos NOUN cana-4527 285 22 , , PUNCT cana-4527 285 23 solitons soliton NOUN cana-4527 285 24 & & CCONJ cana-4527 285 25 fractals fractal NOUN cana-4527 285 26 93 93 NUM cana-4527 285 27 ( ( PUNCT cana-4527 285 28 2016 2016 NUM cana-4527 285 29 ) ) PUNCT cana-4527 285 30 , , PUNCT cana-4527 285 31 89–98 89–98 NUM cana-4527 285 32 . . PUNCT cana-4527 286 1 [ [ X cana-4527 286 2 18 18 NUM cana-4527 286 3 ] ] PUNCT cana-4527 286 4 . . PUNCT cana-4527 287 1 k. k. PROPN cana-4527 287 2 m. m. PROPN cana-4527 287 3 owolabi owolabi NOUN cana-4527 287 4 , , PUNCT cana-4527 287 5 mathematical mathematical ADJ cana-4527 287 6 study study NOUN cana-4527 287 7 of of ADP cana-4527 287 8 multispecies multispecie NOUN cana-4527 287 9 dynamics dynamic NOUN cana-4527 287 10 modelling model VERB cana-4527 287 11 predator predator NOUN cana-4527 287 12 - - PUNCT cana-4527 287 13 prey prey NOUN cana-4527 287 14 spatial spatial ADJ cana-4527 287 15 interactions interaction NOUN cana-4527 287 16 , , PUNCT cana-4527 287 17 journal journal NOUN cana-4527 287 18 of of ADP cana-4527 287 19 numerical numerical ADJ cana-4527 287 20 mathematics mathematic NOUN cana-4527 287 21 25 25 NUM cana-4527 287 22 ( ( PUNCT cana-4527 287 23 1 1 NUM cana-4527 287 24 ) ) PUNCT cana-4527 287 25 ( ( PUNCT cana-4527 287 26 2017 2017 NUM cana-4527 287 27 ) ) PUNCT cana-4527 287 28 , , PUNCT cana-4527 287 29 1–16 1–16 PROPN cana-4527 287 30 . . PUNCT cana-4527 288 1 [ [ X cana-4527 288 2 19 19 NUM cana-4527 288 3 ] ] PUNCT cana-4527 288 4 . . PUNCT cana-4527 288 5 m. m. PROPN cana-4527 288 6 m. m. PROPN cana-4527 288 7 meerschaert meerschaert PROPN cana-4527 288 8 , , PUNCT cana-4527 288 9 c. c. PROPN cana-4527 288 10 tadjeran tadjeran NOUN cana-4527 288 11 , , PUNCT cana-4527 288 12 finite finite ADJ cana-4527 288 13 difference difference NOUN cana-4527 288 14 approximations approximation NOUN cana-4527 288 15 for for ADP cana-4527 288 16 two two NUM cana-4527 288 17 - - PUNCT cana-4527 288 18 sided sided ADJ cana-4527 288 19 spacefractional spacefractional ADJ cana-4527 288 20 partial partial ADJ cana-4527 288 21 differential differential NOUN cana-4527 288 22 equations equation NOUN cana-4527 288 23 , , PUNCT cana-4527 288 24 applied apply VERB cana-4527 288 25 numerical numerical ADJ cana-4527 288 26 mathematics mathematic NOUN cana-4527 288 27 56 56 NUM cana-4527 288 28 ( ( PUNCT cana-4527 288 29 1 1 NUM cana-4527 288 30 ) ) PUNCT cana-4527 288 31 ( ( PUNCT cana-4527 288 32 2006 2006 NUM cana-4527 288 33 ) ) PUNCT cana-4527 288 34 , , PUNCT cana-4527 288 35 80–90 80–90 NUM cana-4527 288 36 . . PUNCT cana-4527 289 1 [ [ X cana-4527 289 2 20 20 NUM cana-4527 289 3 ] ] PUNCT cana-4527 289 4 . . PUNCT cana-4527 290 1 k. k. PROPN cana-4527 290 2 a. a. PROPN cana-4527 290 3 rybakov rybakov PROPN cana-4527 290 4 , , PUNCT cana-4527 290 5 spectral spectral ADJ cana-4527 290 6 method method NOUN cana-4527 290 7 of of ADP cana-4527 290 8 analysis analysis NOUN cana-4527 290 9 and and CCONJ cana-4527 290 10 optimal optimal ADJ cana-4527 290 11 estimation estimation NOUN cana-4527 290 12 in in ADP cana-4527 290 13 linear linear PROPN cana-4527 290 14 stochastic stochastic NOUN cana-4527 290 15 systems system NOUN cana-4527 290 16 , , PUNCT cana-4527 290 17 international international ADJ cana-4527 290 18 journal journal NOUN cana-4527 290 19 of of ADP cana-4527 290 20 modeling modeling NOUN cana-4527 290 21 , , PUNCT cana-4527 290 22 simulation simulation NOUN cana-4527 290 23 , , PUNCT cana-4527 290 24 and and CCONJ cana-4527 290 25 scientific scientific ADJ cana-4527 290 26 computing compute VERB cana-4527 290 27 11 11 NUM cana-4527 290 28 ( ( PUNCT cana-4527 290 29 03 03 NUM cana-4527 290 30 ) ) PUNCT cana-4527 290 31 ( ( PUNCT cana-4527 290 32 2020 2020 NUM cana-4527 290 33 ) ) PUNCT cana-4527 290 34 . . PUNCT cana-4527 291 1 [ [ X cana-4527 291 2 21 21 NUM cana-4527 291 3 ] ] PUNCT cana-4527 291 4 . . PUNCT cana-4527 292 1 he he PRON cana-4527 292 2 , , PUNCT cana-4527 292 3 ji ji PROPN cana-4527 292 4 - - PROPN cana-4527 292 5 huan huan PROPN cana-4527 292 6 , , PUNCT cana-4527 292 7 and and CCONJ cana-4527 292 8 habibolla habibolla NOUN cana-4527 292 9 latifizadeh latifizadeh PROPN cana-4527 292 10 , , PUNCT cana-4527 292 11 a a DET cana-4527 292 12 general general ADJ cana-4527 292 13 numerical numerical ADJ cana-4527 292 14 algorithm algorithm NOUN cana-4527 292 15 for for ADP cana-4527 292 16 nonlinear nonlinear ADJ cana-4527 292 17 differential differential ADJ cana-4527 292 18 equations equation NOUN cana-4527 292 19 by by ADP cana-4527 292 20 the the DET cana-4527 292 21 variational variational ADJ cana-4527 292 22 iteration iteration NOUN cana-4527 292 23 method method NOUN cana-4527 292 24 , , PUNCT cana-4527 292 25 international international ADJ cana-4527 292 26 journal journal NOUN cana-4527 292 27 of of ADP cana-4527 292 28 numerical numerical ADJ cana-4527 292 29 methods method NOUN cana-4527 292 30 for for ADP cana-4527 292 31 heat heat NOUN cana-4527 292 32 & & CCONJ cana-4527 292 33 fluid fluid ADJ cana-4527 292 34 flow flow NOUN cana-4527 292 35 30 30 NUM cana-4527 292 36 , , PUNCT cana-4527 292 37 no no INTJ cana-4527 292 38 . . NOUN cana-4527 292 39 11 11 NUM cana-4527 292 40 ( ( PUNCT cana-4527 292 41 2020 2020 NUM cana-4527 292 42 ) ) PUNCT cana-4527 292 43 , , PUNCT cana-4527 292 44 4797 4797 NUM cana-4527 292 45 - - SYM cana-4527 292 46 4810 4810 NUM cana-4527 292 47 . . PUNCT cana-4527 293 1 [ [ X cana-4527 293 2 22 22 NUM cana-4527 293 3 ] ] PUNCT cana-4527 293 4 . . PUNCT cana-4527 294 1 s. s. PROPN cana-4527 294 2 s. s. PROPN cana-4527 294 3 ray ray PROPN cana-4527 294 4 , , PUNCT cana-4527 294 5 analytical analytical ADJ cana-4527 294 6 solution solution NOUN cana-4527 294 7 for for ADP cana-4527 294 8 the the DET cana-4527 294 9 space space NOUN cana-4527 294 10 fractional fractional ADJ cana-4527 294 11 diffusion diffusion NOUN cana-4527 294 12 equation equation NOUN cana-4527 294 13 by by ADP cana-4527 294 14 two two NUM cana-4527 294 15 - - PUNCT cana-4527 294 16 step step NOUN cana-4527 294 17 adomian adomian NOUN cana-4527 294 18 decomposition decomposition NOUN cana-4527 294 19 method method NOUN cana-4527 294 20 , , PUNCT cana-4527 294 21 communications communication NOUN cana-4527 294 22 in in ADP cana-4527 294 23 non non ADJ cana-4527 294 24 - - ADJ cana-4527 294 25 linear linear ADJ cana-4527 294 26 science science NOUN cana-4527 294 27 and and CCONJ cana-4527 294 28 numerical numerical PROPN cana-4527 294 29 simulation simulation PROPN cana-4527 294 30 14 14 NUM cana-4527 294 31 ( ( PUNCT cana-4527 294 32 4 4 NUM cana-4527 294 33 ) ) PUNCT cana-4527 294 34 ( ( PUNCT cana-4527 294 35 2009 2009 NUM cana-4527 294 36 ) ) PUNCT cana-4527 294 37 , , PUNCT cana-4527 294 38 1295–1306 1295–1306 NUM cana-4527 294 39 . . PUNCT cana-4527 295 1 [ [ X cana-4527 295 2 23 23 NUM cana-4527 295 3 ] ] PUNCT cana-4527 295 4 . . PUNCT cana-4527 295 5 q. q. PROPN cana-4527 295 6 yu yu PROPN cana-4527 295 7 , , PUNCT cana-4527 295 8 f. f. PROPN cana-4527 295 9 liu liu PROPN cana-4527 295 10 , , PUNCT cana-4527 295 11 v. v. PROPN cana-4527 295 12 anh anh PROPN cana-4527 295 13 , , PUNCT cana-4527 295 14 i. i. PROPN cana-4527 295 15 turner turner PROPN cana-4527 295 16 , , PUNCT cana-4527 295 17 solving solve VERB cana-4527 295 18 linear linear NOUN cana-4527 295 19 and and CCONJ cana-4527 295 20 non non ADJ cana-4527 295 21 - - ADJ cana-4527 295 22 linear linear ADJ cana-4527 295 23 space space NOUN cana-4527 295 24 - - PUNCT cana-4527 295 25 time time NOUN cana-4527 295 26 fractional fractional ADJ cana-4527 295 27 reactiondiffusion reactiondiffusion NOUN cana-4527 295 28 equations equation NOUN cana-4527 295 29 by by ADP cana-4527 295 30 the the DET cana-4527 295 31 adomian adomian NOUN cana-4527 295 32 decomposition decomposition NOUN cana-4527 295 33 method method NOUN cana-4527 295 34 , , PUNCT cana-4527 295 35 international international ADJ cana-4527 295 36 journal journal NOUN cana-4527 295 37 for for ADP cana-4527 295 38 numerical numerical ADJ cana-4527 295 39 methods method NOUN cana-4527 295 40 in in ADP cana-4527 295 41 engineering engineer VERB cana-4527 295 42 74 74 NUM cana-4527 295 43 ( ( PUNCT cana-4527 295 44 1 1 NUM cana-4527 295 45 ) ) PUNCT cana-4527 295 46 ( ( PUNCT cana-4527 295 47 2008 2008 NUM cana-4527 295 48 ) ) PUNCT cana-4527 295 49 , , PUNCT cana-4527 295 50 138–158 138–158 NUM cana-4527 295 51 . . PUNCT cana-4527 296 1 [ [ X cana-4527 296 2 24 24 NUM cana-4527 296 3 ] ] PUNCT cana-4527 296 4 . . PUNCT cana-4527 297 1 y. y. PROPN cana-4527 297 2 zhao zhao PROPN cana-4527 297 3 , , PUNCT cana-4527 297 4 s. s. PROPN cana-4527 297 5 sun sun PROPN cana-4527 297 6 , , PUNCT cana-4527 297 7 z. z. PROPN cana-4527 297 8 han han PROPN cana-4527 297 9 , , PUNCT cana-4527 297 10 q. q. PROPN cana-4527 297 11 li li PROPN cana-4527 297 12 , , PUNCT cana-4527 297 13 theory theory NOUN cana-4527 297 14 of of ADP cana-4527 297 15 fractional fractional ADJ cana-4527 297 16 hybrid hybrid ADJ cana-4527 297 17 differential differential NOUN cana-4527 297 18 equations equation NOUN cana-4527 297 19 , , PUNCT cana-4527 297 20 comput comput NOUN cana-4527 297 21 . . PUNCT cana-4527 298 1 math.appl math.appl PROPN cana-4527 298 2 . . PROPN cana-4527 299 1 , , PUNCT cana-4527 299 2 62 62 NUM cana-4527 299 3 ( ( PUNCT cana-4527 299 4 2011 2011 NUM cana-4527 299 5 ) ) PUNCT cana-4527 299 6 , , PUNCT cana-4527 299 7 1312–1324 1312–1324 NUM cana-4527 299 8 . . PUNCT cana-4527 300 1 http://dx.doi.org/10.1016/j.camwa.2011.03.041 http://dx.doi.org/10.1016/j.camwa.2011.03.041 NOUN cana-4527 301 1 [ [ X cana-4527 301 2 25 25 NUM cana-4527 301 3 ] ] PUNCT cana-4527 301 4 . . PUNCT cana-4527 302 1 s. s. PROPN cana-4527 302 2 sitho sitho PROPN cana-4527 302 3 , , PUNCT cana-4527 302 4 s. s. PROPN cana-4527 302 5 ntouyas ntouyas PROPN cana-4527 302 6 , , PUNCT cana-4527 302 7 j. j. PROPN cana-4527 302 8 tariboon tariboon PROPN cana-4527 302 9 , , PUNCT cana-4527 302 10 existence existence NOUN cana-4527 302 11 results result VERB cana-4527 302 12 for for ADP cana-4527 302 13 hybrid hybrid ADJ cana-4527 302 14 fractional fractional ADJ cana-4527 302 15 integro integro ADJ cana-4527 302 16 - - PUNCT cana-4527 302 17 differential differential NOUN cana-4527 302 18 equations equation NOUN cana-4527 302 19 , , PUNCT cana-4527 302 20 bound bind VERB cana-4527 302 21 . . PUNCT cana-4527 303 1 value value PROPN cana-4527 303 2 probl probl PROPN cana-4527 303 3 . . PUNCT cana-4527 303 4 , , PUNCT cana-4527 303 5 2015 2015 NUM cana-4527 303 6 ( ( PUNCT cana-4527 303 7 2015 2015 NUM cana-4527 303 8 ) ) PUNCT cana-4527 303 9 , , PUNCT cana-4527 303 10 113 113 NUM cana-4527 303 11 . . PUNCT cana-4527 304 1 http://dx.doi.org/10.1186/s13661-015-03767 http://dx.doi.org/10.1186/s13661-015-03767 PUNCT cana-4527 305 1 [ [ X cana-4527 305 2 26 26 NUM cana-4527 305 3 ] ] PUNCT cana-4527 305 4 . . PUNCT cana-4527 306 1 m. m. NOUN cana-4527 306 2 awadalla awadalla PROPN cana-4527 306 3 , , PUNCT cana-4527 306 4 k. k. PROPN cana-4527 306 5 abuasbeh abuasbeh PROPN cana-4527 306 6 , , PUNCT cana-4527 306 7 on on ADP cana-4527 306 8 the the DET cana-4527 306 9 system system NOUN cana-4527 306 10 of of ADP cana-4527 306 11 non non ADJ cana-4527 306 12 - - ADJ cana-4527 306 13 linear linear ADJ cana-4527 306 14 sequential sequential ADJ cana-4527 306 15 hybrid hybrid ADJ cana-4527 306 16 fractional fractional ADJ cana-4527 306 17 differential differential NOUN cana-4527 306 18 equations equation NOUN cana-4527 306 19 , , PUNCT cana-4527 306 20 math math NOUN cana-4527 306 21 . . PUNCT cana-4527 307 1 probl probl PROPN cana-4527 307 2 . . PUNCT cana-4527 308 1 eng eng PROPN cana-4527 308 2 . . PROPN cana-4527 308 3 , , PUNCT cana-4527 308 4 2022 2022 NUM cana-4527 308 5 ( ( PUNCT cana-4527 308 6 2022 2022 NUM cana-4527 308 7 ) ) PUNCT cana-4527 308 8 , , PUNCT cana-4527 308 9 8556578 8556578 NUM cana-4527 308 10 . . PUNCT cana-4527 309 1 http://dx.doi.org/10.1155/2022/8556578 http://dx.doi.org/10.1155/2022/8556578 X cana-4527 310 1 [ [ X cana-4527 310 2 27 27 NUM cana-4527 310 3 ] ] PUNCT cana-4527 310 4 . . PUNCT cana-4527 311 1 s. s. PROPN cana-4527 311 2 gul gul PROPN cana-4527 311 3 , , PUNCT cana-4527 311 4 r. r. PROPN cana-4527 311 5 khan khan PROPN cana-4527 311 6 , , PUNCT cana-4527 311 7 h. h. PROPN cana-4527 311 8 khan khan PROPN cana-4527 311 9 , , PUNCT cana-4527 311 10 r. r. PROPN cana-4527 311 11 george george PROPN cana-4527 311 12 , , PUNCT cana-4527 311 13 s. s. PROPN cana-4527 311 14 etemad etemad PROPN cana-4527 311 15 , , PUNCT cana-4527 311 16 s. s. PROPN cana-4527 311 17 rezapour rezapour PROPN cana-4527 311 18 , , PUNCT cana-4527 311 19 analysis analysis NOUN cana-4527 311 20 on on ADP cana-4527 311 21 a a DET cana-4527 311 22 coupled couple VERB cana-4527 311 23 system system NOUN cana-4527 311 24 of of ADP cana-4527 311 25 two two NUM cana-4527 311 26 sequential sequential ADJ cana-4527 311 27 hybrid hybrid ADJ cana-4527 311 28 bvps bvps NOUN cana-4527 311 29 with with ADP cana-4527 311 30 numerical numerical ADJ cana-4527 311 31 simulations simulation NOUN cana-4527 311 32 to to ADP cana-4527 311 33 a a DET cana-4527 311 34 model model NOUN cana-4527 311 35 of of ADP cana-4527 311 36 typhoid typhoid NOUN cana-4527 311 37 treatment treatment NOUN cana-4527 311 38 , , PUNCT cana-4527 311 39 alex.eng.j alex.eng.j PROPN cana-4527 311 40 . . PROPN cana-4527 311 41 , , PUNCT cana-4527 311 42 61 61 NUM cana-4527 311 43 ( ( PUNCT cana-4527 311 44 2022 2022 NUM cana-4527 311 45 ) ) PUNCT cana-4527 311 46 , , PUNCT cana-4527 311 47 10085–10098 10085–10098 NUM cana-4527 311 48 . . PUNCT cana-4527 312 1 http://dx.doi.org/10.1016/j.aej.2022.03.020 http://dx.doi.org/10.1016/j.aej.2022.03.020 VERB cana-4527 313 1 [ [ X cana-4527 313 2 28 28 NUM cana-4527 313 3 ] ] PUNCT cana-4527 313 4 . . PUNCT cana-4527 314 1 a. a. PROPN cana-4527 314 2 khan khan PROPN cana-4527 314 3 , , PUNCT cana-4527 314 4 z. z. PROPN cana-4527 314 5 khan khan PROPN cana-4527 314 6 , , PUNCT cana-4527 314 7 t. t. PROPN cana-4527 314 8 abdeljawad abdeljawad PROPN cana-4527 314 9 , , PUNCT cana-4527 314 10 h. h. PROPN cana-4527 314 11 khan khan PROPN cana-4527 314 12 , , PUNCT cana-4527 314 13 analytical analytical ADJ cana-4527 314 14 analysis analysis NOUN cana-4527 314 15 of of ADP cana-4527 314 16 the the DET cana-4527 314 17 fractional fractional ADJ cana-4527 314 18 - - PUNCT cana-4527 314 19 order order NOUN cana-4527 314 20 sequential sequential ADJ cana-4527 314 21 hybrid hybrid ADJ cana-4527 314 22 system system NOUN cana-4527 314 23 with with ADP cana-4527 314 24 the the DET cana-4527 314 25 numerical numerical ADJ cana-4527 314 26 application application NOUN cana-4527 314 27 , , PUNCT cana-4527 314 28 adv adv PROPN cana-4527 314 29 . . PUNCT cana-4527 314 30 cont cont PROPN cana-4527 314 31 . . PUNCT cana-4527 315 1 discr discr PROPN cana-4527 315 2 . . PUNCT cana-4527 316 1 mod mod PROPN cana-4527 316 2 . . PROPN cana-4527 316 3 , , PUNCT cana-4527 316 4 2022 2022 NUM cana-4527 316 5 ( ( PUNCT cana-4527 316 6 2022 2022 NUM cana-4527 316 7 ) ) PUNCT cana-4527 316 8 , , PUNCT cana-4527 316 9 12.http://dx.doi.org/10.1186/s13662-022-03685-w 12.http://dx.doi.org/10.1186/s13662-022-03685-w NUM cana-4527 316 10 . . PUNCT cana-4527 316 11 http://dx.doi.org/10.1186/s13661-015-0376-7 http://dx.doi.org/10.1186/s13661-015-0376-7 PROPN cana-4527 317 1 http://dx.doi.org/10.1186/s13661-015-0376-7 http://dx.doi.org/10.1186/s13661-015-0376-7 PROPN cana-4527 317 2 communications communication NOUN cana-4527 317 3 on on ADP cana-4527 317 4 applied apply VERB cana-4527 317 5 nonlinear nonlinear ADJ cana-4527 317 6 analysis analysis NOUN cana-4527 317 7 issn issn NOUN cana-4527 317 8 : : PUNCT cana-4527 317 9 1074 1074 NUM cana-4527 317 10 - - PUNCT cana-4527 317 11 133x 133x NUM cana-4527 317 12 vol vol NOUN cana-4527 317 13 32 32 NUM cana-4527 317 14 no no NOUN cana-4527 317 15 . . PUNCT cana-4527 318 1 9s 9s NUM cana-4527 318 2 ( ( PUNCT cana-4527 318 3 2025 2025 NUM cana-4527 318 4 ) ) PUNCT cana-4527 318 5 2430 2430 NUM cana-4527 318 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 319 1 [ [ X cana-4527 319 2 29 29 NUM cana-4527 319 3 ] ] PUNCT cana-4527 319 4 . . PUNCT cana-4527 320 1 j. j. PROPN cana-4527 320 2 losada losada PROPN cana-4527 320 3 , , PUNCT cana-4527 320 4 j. j. PROPN cana-4527 320 5 nieto nieto PROPN cana-4527 320 6 , , PUNCT cana-4527 320 7 properties property NOUN cana-4527 320 8 of of ADP cana-4527 320 9 a a DET cana-4527 320 10 new new ADJ cana-4527 320 11 fractional fractional ADJ cana-4527 320 12 derivative derivative NOUN cana-4527 320 13 without without ADP cana-4527 320 14 singular singular ADJ cana-4527 320 15 kernel kernel NOUN cana-4527 320 16 , , PUNCT cana-4527 320 17 progr progr NOUN cana-4527 320 18 . . PUNCT cana-4527 321 1 fract.differ fract.differ NOUN cana-4527 321 2 . . PUNCT cana-4527 321 3 appl appl PROPN cana-4527 321 4 . . PROPN cana-4527 321 5 , , PUNCT cana-4527 321 6 1 1 NUM cana-4527 321 7 ( ( PUNCT cana-4527 321 8 2015 2015 NUM cana-4527 321 9 ) ) PUNCT cana-4527 321 10 , , PUNCT cana-4527 321 11 87–92 87–92 NUM cana-4527 321 12 . . PUNCT cana-4527 322 1 http://dx.doi.org/10.12785/pfda/010202 http://dx.doi.org/10.12785/pfda/010202 PROPN cana-4527 322 2 . . PUNCT cana-4527 323 1 [ [ X cana-4527 323 2 30 30 NUM cana-4527 323 3 ] ] PUNCT cana-4527 323 4 . . PUNCT cana-4527 323 5 m. m. PROPN cana-4527 323 6 caputo caputo PROPN cana-4527 323 7 , , PUNCT cana-4527 323 8 m. m. PROPN cana-4527 323 9 fabrizio fabrizio PROPN cana-4527 323 10 , , PUNCT cana-4527 323 11 a a DET cana-4527 323 12 new new ADJ cana-4527 323 13 definition definition NOUN cana-4527 323 14 of of ADP cana-4527 323 15 fractional fractional ADJ cana-4527 323 16 derivative derivative NOUN cana-4527 323 17 without without ADP cana-4527 323 18 singular singular ADJ cana-4527 323 19 kernel kernel NOUN cana-4527 323 20 , , PUNCT cana-4527 323 21 progr.fract progr.fract NUM cana-4527 323 22 . . PUNCT cana-4527 323 23 differ differ VERB cana-4527 323 24 . . PUNCT cana-4527 324 1 appl appl PROPN cana-4527 324 2 . . PROPN cana-4527 324 3 , , PUNCT cana-4527 324 4 1 1 NUM cana-4527 324 5 ( ( PUNCT cana-4527 324 6 2015 2015 NUM cana-4527 324 7 ) ) PUNCT cana-4527 324 8 , , PUNCT cana-4527 324 9 73–85 73–85 PROPN cana-4527 324 10 . . PUNCT cana-4527 325 1 http://dx.doi.org/10.12785/pfda/01020 http://dx.doi.org/10.12785/pfda/01020 PROPN cana-4527 325 2 . . PUNCT cana-4527 326 1 [ [ X cana-4527 326 2 31 31 NUM cana-4527 326 3 ] ] PUNCT cana-4527 326 4 . . PUNCT cana-4527 327 1 a. a. PROPN cana-4527 327 2 atangana atangana PROPN cana-4527 327 3 , , PUNCT cana-4527 327 4 d. d. PROPN cana-4527 327 5 baleanu baleanu PROPN cana-4527 327 6 , , PUNCT cana-4527 327 7 new new ADJ cana-4527 327 8 fractional fractional ADJ cana-4527 327 9 derivatives derivative NOUN cana-4527 327 10 with with ADP cana-4527 327 11 nonlocal nonlocal ADJ cana-4527 327 12 and and CCONJ cana-4527 327 13 non non ADJ cana-4527 327 14 - - ADJ cana-4527 327 15 singular singular ADJ cana-4527 327 16 kernel kernel NOUN cana-4527 327 17 : : PUNCT cana-4527 327 18 theory theory NOUN cana-4527 327 19 and and CCONJ cana-4527 327 20 application application NOUN cana-4527 327 21 to to PART cana-4527 327 22 heat heat NOUN cana-4527 327 23 transfer transfer NOUN cana-4527 327 24 model model NOUN cana-4527 327 25 , , PUNCT cana-4527 327 26 therm therm PROPN cana-4527 327 27 . . PUNCT cana-4527 328 1 sci sci PROPN cana-4527 328 2 . . PROPN cana-4527 328 3 , , PUNCT cana-4527 328 4 20 20 NUM cana-4527 328 5 ( ( PUNCT cana-4527 328 6 2016 2016 NUM cana-4527 328 7 ) ) PUNCT cana-4527 328 8 , , PUNCT cana-4527 328 9 763–769 763–769 NUM cana-4527 328 10 . . PUNCT cana-4527 329 1 http://dx.doi.org/10.2298/tsci160111018a http://dx.doi.org/10.2298/tsci160111018a NOUN cana-4527 329 2 . . PUNCT cana-4527 330 1 [ [ X cana-4527 330 2 32 32 NUM cana-4527 330 3 ] ] PUNCT cana-4527 330 4 . . PUNCT cana-4527 331 1 m. m. PROPN cana-4527 331 2 al al PROPN cana-4527 331 3 - - PUNCT cana-4527 331 4 refai refai PROPN cana-4527 331 5 , , PUNCT cana-4527 331 6 d. d. PROPN cana-4527 331 7 baleanu baleanu PROPN cana-4527 331 8 , , PUNCT cana-4527 331 9 on on ADP cana-4527 331 10 an an DET cana-4527 331 11 extension extension NOUN cana-4527 331 12 of of ADP cana-4527 331 13 the the DET cana-4527 331 14 operator operator NOUN cana-4527 331 15 with with ADP cana-4527 331 16 mittag mittag ADJ cana-4527 331 17 - - PUNCT cana-4527 331 18 leffler leffler NOUN cana-4527 331 19 kernel kernel NOUN cana-4527 331 20 , , PUNCT cana-4527 331 21 fractals fractal NOUN cana-4527 331 22 , , PUNCT cana-4527 331 23 30(2022 30(2022 NUM cana-4527 331 24 ) ) PUNCT cana-4527 331 25 , , PUNCT cana-4527 331 26 2240129 2240129 NUM cana-4527 331 27 . . PUNCT cana-4527 332 1 http://dx.doi.org/10.1142/s0218348x22401296 http://dx.doi.org/10.1142/s0218348x22401296 AUX cana-4527 332 2 . . PUNCT cana-4527 333 1 [ [ X cana-4527 333 2 33 33 NUM cana-4527 333 3 ] ] PUNCT cana-4527 333 4 . . PUNCT cana-4527 334 1 b. b. PROPN cana-4527 334 2 dhage dhage PROPN cana-4527 334 3 , , PUNCT cana-4527 334 4 g. g. PROPN cana-4527 334 5 khurape khurape PROPN cana-4527 334 6 , , PUNCT cana-4527 334 7 a. a. PROPN cana-4527 334 8 shete shete PROPN cana-4527 334 9 , , PUNCT cana-4527 334 10 j. j. PROPN cana-4527 334 11 salunkhe salunkhe PROPN cana-4527 334 12 , , PUNCT cana-4527 334 13 existence existence NOUN cana-4527 334 14 and and CCONJ cana-4527 334 15 approximate approximate ADJ cana-4527 334 16 solutions solution NOUN cana-4527 334 17 for for ADP cana-4527 334 18 nonlinear nonlinear ADJ cana-4527 334 19 hybrid hybrid ADJ cana-4527 334 20 fractional fractional ADJ cana-4527 334 21 integro integro ADJ cana-4527 334 22 - - PUNCT cana-4527 334 23 differential differential NOUN cana-4527 334 24 equations equation NOUN cana-4527 334 25 , , PUNCT cana-4527 334 26 int int NOUN cana-4527 334 27 . . PUNCT cana-4527 335 1 j. j. PROPN cana-4527 335 2 anal anal PROPN cana-4527 335 3 . . PUNCT cana-4527 336 1 appl appl PROPN cana-4527 336 2 . . PROPN cana-4527 336 3 , , PUNCT cana-4527 336 4 11 11 NUM cana-4527 336 5 ( ( PUNCT cana-4527 336 6 2016 2016 NUM cana-4527 336 7 ) ) PUNCT cana-4527 336 8 , , PUNCT cana-4527 336 9 157–167 157–167 NUM cana-4527 336 10 . . PUNCT cana-4527 337 1 [ [ X cana-4527 337 2 34 34 NUM cana-4527 337 3 ] ] PUNCT cana-4527 337 4 . . PUNCT cana-4527 338 1 b. b. PROPN cana-4527 338 2 dhage dhage PROPN cana-4527 338 3 , , PUNCT cana-4527 338 4 existence existence NOUN cana-4527 338 5 and and CCONJ cana-4527 338 6 attractivity attractivity NOUN cana-4527 338 7 theorems theorem NOUN cana-4527 338 8 for for ADP cana-4527 338 9 non non ADJ cana-4527 338 10 - - ADJ cana-4527 338 11 linear linear ADJ cana-4527 338 12 hybrid hybrid ADJ cana-4527 338 13 fractional fractional ADJ cana-4527 338 14 differential differential ADJ cana-4527 338 15 equations equation NOUN cana-4527 338 16 with with ADP cana-4527 338 17 anticipation anticipation NOUN cana-4527 338 18 and and CCONJ cana-4527 338 19 retardation retardation NOUN cana-4527 338 20 , , PUNCT cana-4527 338 21 j. j. PROPN cana-4527 338 22 non non ADJ cana-4527 338 23 - - ADJ cana-4527 338 24 linear linear ADJ cana-4527 338 25 funct funct NOUN cana-4527 338 26 . . PUNCT cana-4527 339 1 anal anal PROPN cana-4527 339 2 . . PROPN cana-4527 339 3 , , PUNCT cana-4527 339 4 2020 2020 NUM cana-4527 339 5 ( ( PUNCT cana-4527 339 6 2020 2020 NUM cana-4527 339 7 ) ) PUNCT cana-4527 339 8 , , PUNCT cana-4527 339 9 47 47 NUM cana-4527 339 10 . . PUNCT cana-4527 340 1 http://dx.doi.org/10.23952/jnfa.2020.47 http://dx.doi.org/10.23952/jnfa.2020.47 X cana-4527 340 2 . . PUNCT cana-4527 341 1 [ [ X cana-4527 341 2 35 35 NUM cana-4527 341 3 ] ] PUNCT cana-4527 341 4 . . PUNCT cana-4527 342 1 m. m. PROPN cana-4527 342 2 al al PROPN cana-4527 342 3 - - PUNCT cana-4527 342 4 refai refai PROPN cana-4527 342 5 , , PUNCT cana-4527 342 6 proper proper ADJ cana-4527 342 7 inverse inverse NOUN cana-4527 342 8 operators operator NOUN cana-4527 342 9 of of ADP cana-4527 342 10 fractional fractional ADJ cana-4527 342 11 derivatives derivative NOUN cana-4527 342 12 with with ADP cana-4527 342 13 non non ADJ cana-4527 342 14 - - ADJ cana-4527 342 15 singular singular ADJ cana-4527 342 16 kernels kernel NOUN cana-4527 342 17 , , PUNCT cana-4527 342 18 rend.circ rend.circ PROPN cana-4527 342 19 . . PUNCT cana-4527 343 1 mat mat NOUN cana-4527 343 2 . . PUNCT cana-4527 343 3 palermo palermo PROPN cana-4527 343 4 . . PUNCT cana-4527 343 5 ii ii PROPN cana-4527 343 6 ser ser PROPN cana-4527 343 7 . . PROPN cana-4527 343 8 , , PUNCT cana-4527 343 9 71 71 NUM cana-4527 343 10 ( ( PUNCT cana-4527 343 11 2022 2022 NUM cana-4527 343 12 ) ) PUNCT cana-4527 343 13 , , PUNCT cana-4527 343 14 525–535 525–535 NUM cana-4527 343 15 . . PUNCT cana-4527 343 16 http://dx.doi.org/10.1007/s12215-02100638-2 http://dx.doi.org/10.1007/s12215-02100638-2 NOUN cana-4527 343 17 . . PUNCT cana-4527 344 1 [ [ X cana-4527 344 2 36 36 NUM cana-4527 344 3 ] ] PUNCT cana-4527 344 4 . . PUNCT cana-4527 345 1 z. z. PROPN cana-4527 345 2 khan khan PROPN cana-4527 345 3 , , PUNCT cana-4527 345 4 a. a. PROPN cana-4527 345 5 khan khan PROPN cana-4527 345 6 , , PUNCT cana-4527 345 7 t. t. PROPN cana-4527 345 8 abdeljawad abdeljawad PROPN cana-4527 345 9 , , PUNCT cana-4527 345 10 h. h. PROPN cana-4527 345 11 khan khan PROPN cana-4527 345 12 , , PUNCT cana-4527 345 13 computational computational ADJ cana-4527 345 14 analysis analysis NOUN cana-4527 345 15 of of ADP cana-4527 345 16 fractional fractional ADJ cana-4527 345 17 order order NOUN cana-4527 345 18 imperfect imperfect ADJ cana-4527 345 19 testing testing NOUN cana-4527 345 20 infection infection NOUN cana-4527 345 21 disease disease NOUN cana-4527 345 22 model model NOUN cana-4527 345 23 , , PUNCT cana-4527 345 24 fractals fractal NOUN cana-4527 345 25 , , PUNCT cana-4527 345 26 30 30 NUM cana-4527 345 27 ( ( PUNCT cana-4527 345 28 2022 2022 NUM cana-4527 345 29 ) ) PUNCT cana-4527 345 30 , , PUNCT cana-4527 345 31 2240169.http://dx.doi.org/10.1142/s0218348x22401697 2240169.http://dx.doi.org/10.1142/s0218348x22401697 X cana-4527 345 32 . . PUNCT cana-4527 346 1 [ [ X cana-4527 346 2 37 37 NUM cana-4527 346 3 ] ] PUNCT cana-4527 346 4 . . PUNCT cana-4527 347 1 r. r. PROPN cana-4527 347 2 shi shi PROPN cana-4527 347 3 , , PUNCT cana-4527 347 4 y. y. PROPN cana-4527 347 5 cui cui PROPN cana-4527 347 6 , , PUNCT cana-4527 347 7 global global ADJ cana-4527 347 8 analysis analysis NOUN cana-4527 347 9 of of ADP cana-4527 347 10 a a DET cana-4527 347 11 mathematical mathematical ADJ cana-4527 347 12 model model NOUN cana-4527 347 13 for for ADP cana-4527 347 14 hepatitis hepatitis PROPN cana-4527 347 15 c c PROPN cana-4527 347 16 virus virus NOUN cana-4527 347 17 transmissions transmission NOUN cana-4527 347 18 , , PUNCT cana-4527 347 19 virus virus NOUN cana-4527 347 20 res re NOUN cana-4527 347 21 . . PROPN cana-4527 347 22 , , PUNCT cana-4527 347 23 217 217 NUM cana-4527 347 24 ( ( PUNCT cana-4527 347 25 2016 2016 NUM cana-4527 347 26 ) ) PUNCT cana-4527 347 27 , , PUNCT cana-4527 347 28 8–17 8–17 PROPN cana-4527 347 29 . . PUNCT cana-4527 347 30 http://dx.doi.org/10.1016/j.virusres.2016.02.006 http://dx.doi.org/10.1016/j.virusres.2016.02.006 PROPN cana-4527 348 1 [ [ X cana-4527 348 2 38 38 NUM cana-4527 348 3 ] ] PUNCT cana-4527 348 4 . . PUNCT cana-4527 349 1 m. m. PROPN cana-4527 349 2 subramanian subramanian PROPN cana-4527 349 3 , , PUNCT cana-4527 349 4 m. m. NOUN cana-4527 349 5 manigandan manigandan PROPN cana-4527 349 6 , , PUNCT cana-4527 349 7 c. c. PROPN cana-4527 349 8 tunΓ§ tunΓ§ PROPN cana-4527 349 9 , , PUNCT cana-4527 349 10 t. t. PROPN cana-4527 349 11 gopal gopal PROPN cana-4527 349 12 , , PUNCT cana-4527 349 13 j. j. PROPN cana-4527 349 14 alzabut alzabut PROPN cana-4527 349 15 , , PUNCT cana-4527 349 16 on on ADP cana-4527 349 17 the the DET cana-4527 349 18 system system NOUN cana-4527 349 19 of of ADP cana-4527 349 20 non non ADJ cana-4527 349 21 - - ADJ cana-4527 349 22 linear linear ADJ cana-4527 349 23 coupled couple VERB cana-4527 349 24 differential differential ADJ cana-4527 349 25 equations equation NOUN cana-4527 349 26 and and CCONJ cana-4527 349 27 inclusions inclusion NOUN cana-4527 349 28 involving involve VERB cana-4527 349 29 caputo caputo NOUN cana-4527 349 30 - - PUNCT cana-4527 349 31 type type NOUN cana-4527 349 32 sequential sequential ADJ cana-4527 349 33 derivatives derivative NOUN cana-4527 349 34 of of ADP cana-4527 349 35 fractional fractional ADJ cana-4527 349 36 order order NOUN cana-4527 349 37 , , PUNCT cana-4527 349 38 j. j. PROPN cana-4527 349 39 taibah taibah PROPN cana-4527 350 1 uni uni PROPN cana-4527 350 2 . . PUNCT cana-4527 351 1 sci sci PROPN cana-4527 351 2 . . PROPN cana-4527 351 3 , , PUNCT cana-4527 351 4 16 16 NUM cana-4527 351 5 ( ( PUNCT cana-4527 351 6 2022 2022 NUM cana-4527 351 7 ) ) PUNCT cana-4527 351 8 , , PUNCT cana-4527 351 9 1–23 1–23 PROPN cana-4527 351 10 . . PUNCT cana-4527 352 1 http://dx.doi.org/10.1080/16583655.2021.2010984 http://dx.doi.org/10.1080/16583655.2021.2010984 PROPN cana-4527 353 1 [ [ X cana-4527 353 2 39 39 NUM cana-4527 353 3 ] ] PUNCT cana-4527 353 4 . . PUNCT cana-4527 354 1 m. m. PROPN cana-4527 354 2 subramanian subramanian PROPN cana-4527 354 3 , , PUNCT cana-4527 354 4 j. j. PROPN cana-4527 354 5 alzabut alzabut PROPN cana-4527 354 6 , , PUNCT cana-4527 354 7 m. m. NOUN cana-4527 354 8 abbas abbas PROPN cana-4527 354 9 , , PUNCT cana-4527 354 10 c. c. PROPN cana-4527 354 11 thaiprayoon thaiprayoon PROPN cana-4527 354 12 , , PUNCT cana-4527 354 13 w. w. PROPN cana-4527 354 14 sudsutad sudsutad PROPN cana-4527 354 15 , , PUNCT cana-4527 354 16 existence existence NOUN cana-4527 354 17 of of ADP cana-4527 354 18 solutions solution NOUN cana-4527 354 19 for for ADP cana-4527 354 20 coupled couple VERB cana-4527 354 21 higher high ADJ cana-4527 354 22 - - PUNCT cana-4527 354 23 order order NOUN cana-4527 354 24 fractional fractional ADJ cana-4527 354 25 integro integro ADJ cana-4527 354 26 - - PUNCT cana-4527 354 27 differential differential NOUN cana-4527 354 28 equations equation NOUN cana-4527 354 29 with with ADP cana-4527 354 30 nonlocal nonlocal ADJ cana-4527 354 31 integral integral ADJ cana-4527 354 32 and and CCONJ cana-4527 354 33 multi multi ADJ cana-4527 354 34 - - ADJ cana-4527 354 35 point point ADJ cana-4527 354 36 boundary boundary ADJ cana-4527 354 37 conditions condition NOUN cana-4527 354 38 depending depend VERB cana-4527 354 39 on on ADP cana-4527 354 40 lower low ADJ cana-4527 354 41 - - PUNCT cana-4527 354 42 order order NOUN cana-4527 354 43 fractional fractional ADJ cana-4527 354 44 derivatives derivative NOUN cana-4527 354 45 and and CCONJ cana-4527 354 46 integrals integral NOUN cana-4527 354 47 , , PUNCT cana-4527 354 48 mathematics,10 mathematics,10 NOUN cana-4527 354 49 ( ( PUNCT cana-4527 354 50 2022 2022 NUM cana-4527 354 51 ) ) PUNCT cana-4527 354 52 , , PUNCT cana-4527 354 53 1823 1823 NUM cana-4527 354 54 . . PUNCT cana-4527 355 1 http://dx.doi.org/10.3390/math10111823 http://dx.doi.org/10.3390/math10111823 PROPN cana-4527 356 1 [ [ X cana-4527 356 2 40 40 NUM cana-4527 356 3 ] ] PUNCT cana-4527 356 4 . . PUNCT cana-4527 357 1 a. a. PROPN cana-4527 357 2 zada zada PROPN cana-4527 357 3 , , PUNCT cana-4527 357 4 j. j. PROPN cana-4527 357 5 alzabut alzabut PROPN cana-4527 357 6 , , PUNCT cana-4527 357 7 h. h. PROPN cana-4527 357 8 waheed waheed PROPN cana-4527 357 9 , , PUNCT cana-4527 357 10 i. i. PROPN cana-4527 357 11 popa popa PROPN cana-4527 357 12 , , PUNCT cana-4527 357 13 ulam ulam NOUN cana-4527 357 14 - - PUNCT cana-4527 357 15 hyers hyer NOUN cana-4527 357 16 stability stability NOUN cana-4527 357 17 of of ADP cana-4527 357 18 impulsive impulsive ADJ cana-4527 357 19 integro integro ADJ cana-4527 357 20 differential differential ADJ cana-4527 357 21 equations equation NOUN cana-4527 357 22 with with ADP cana-4527 357 23 riemann riemann PROPN cana-4527 357 24 - - PUNCT cana-4527 357 25 liouville liouville VERB cana-4527 357 26 boundary boundary ADJ cana-4527 357 27 conditions condition NOUN cana-4527 357 28 , , PUNCT cana-4527 357 29 adv adv PROPN cana-4527 357 30 . . PUNCT cana-4527 357 31 differ differ VERB cana-4527 357 32 . . PUNCT cana-4527 358 1 equ equ PROPN cana-4527 358 2 . . PROPN cana-4527 358 3 , , PUNCT cana-4527 358 4 2020 2020 NUM cana-4527 358 5 ( ( PUNCT cana-4527 358 6 2020 2020 NUM cana-4527 358 7 ) ) PUNCT cana-4527 358 8 , , PUNCT cana-4527 358 9 64.http://dx.doi.org/10.1186/s13662-020-2534-1 64.http://dx.doi.org/10.1186/s13662-020-2534-1 PUNCT cana-4527 359 1 [ [ X cana-4527 359 2 41 41 NUM cana-4527 359 3 ] ] PUNCT cana-4527 359 4 . . PUNCT cana-4527 360 1 belgacem belgacem NOUN cana-4527 360 2 , , PUNCT cana-4527 360 3 fethi fethi PROPN cana-4527 360 4 bin bin NOUN cana-4527 360 5 muhammed muhamme VERB cana-4527 360 6 , , PUNCT cana-4527 360 7 introducing introduce VERB cana-4527 360 8 and and CCONJ cana-4527 360 9 analyzing analyze VERB cana-4527 360 10 deeper deep ADJ cana-4527 360 11 sumudu sumudu NOUN cana-4527 360 12 properties property NOUN cana-4527 360 13 , , PUNCT cana-4527 360 14 nonlinear nonlinear ADJ cana-4527 360 15 studies study NOUN cana-4527 360 16 13 13 NUM cana-4527 360 17 , , PUNCT cana-4527 360 18 no no INTJ cana-4527 360 19 . . NOUN cana-4527 360 20 1 1 NUM cana-4527 360 21 ( ( PUNCT cana-4527 360 22 2006 2006 NUM cana-4527 360 23 ): ): PUNCT cana-4527 360 24 23 23 NUM cana-4527 360 25 . . PUNCT cana-4527 361 1 http://dx.doi.org/10.3390/math10111823 http://dx.doi.org/10.3390/math10111823 PROPN cana-4527 361 2 communications communication NOUN cana-4527 361 3 on on ADP cana-4527 361 4 applied apply VERB cana-4527 361 5 nonlinear nonlinear ADJ cana-4527 361 6 analysis analysis NOUN cana-4527 361 7 issn issn NOUN cana-4527 361 8 : : PUNCT cana-4527 361 9 1074 1074 NUM cana-4527 361 10 - - PUNCT cana-4527 361 11 133x 133x NUM cana-4527 361 12 vol vol NOUN cana-4527 361 13 32 32 NUM cana-4527 361 14 no no NOUN cana-4527 361 15 . . PUNCT cana-4527 362 1 9s 9s NUM cana-4527 362 2 ( ( PUNCT cana-4527 362 3 2025 2025 NUM cana-4527 362 4 ) ) PUNCT cana-4527 362 5 2431 2431 NUM cana-4527 362 6 https://internationalpubls.com https://internationalpubls.com X cana-4527 363 1 [ [ X cana-4527 363 2 42 42 NUM cana-4527 363 3 ] ] PUNCT cana-4527 363 4 . . PUNCT cana-4527 364 1 van van PROPN cana-4527 364 2 tonder tonder PROPN cana-4527 364 3 , , PUNCT cana-4527 364 4 g. g. PROPN cana-4527 364 5 j. j. PROPN cana-4527 364 6 , , PUNCT cana-4527 364 7 j. j. PROPN cana-4527 364 8 f. f. PROPN cana-4527 364 9 botha botha PROPN cana-4527 364 10 , , PUNCT cana-4527 364 11 w w PROPN cana-4527 364 12 - - PUNCT cana-4527 364 13 h. h. PROPN cana-4527 364 14 chiang chiang PROPN cana-4527 364 15 , , PUNCT cana-4527 364 16 harald harald PROPN cana-4527 364 17 kunstmann kunstmann PROPN cana-4527 364 18 , , PUNCT cana-4527 364 19 and and CCONJ cana-4527 364 20 y. y. PROPN cana-4527 364 21 xu xu PROPN cana-4527 364 22 , , PUNCT cana-4527 364 23 estimation estimation NOUN cana-4527 364 24 of of ADP cana-4527 364 25 the the DET cana-4527 364 26 sustainable sustainable ADJ cana-4527 364 27 yields yield NOUN cana-4527 364 28 of of ADP cana-4527 364 29 boreholes borehole NOUN cana-4527 364 30 in in ADP cana-4527 364 31 fractured fractured ADJ cana-4527 364 32 rock rock NOUN cana-4527 364 33 formations formation NOUN cana-4527 364 34 , , PUNCT cana-4527 364 35 journal journal NOUN cana-4527 364 36 of of ADP cana-4527 364 37 hydrology hydrology NOUN cana-4527 364 38 241 241 NUM cana-4527 364 39 , , PUNCT cana-4527 364 40 no no INTJ cana-4527 364 41 . . NOUN cana-4527 364 42 1 1 NUM cana-4527 364 43 - - SYM cana-4527 364 44 2 2 NUM cana-4527 364 45 ( ( PUNCT cana-4527 364 46 2001 2001 NUM cana-4527 364 47 ): ): PUNCT cana-4527 364 48 70 70 NUM cana-4527 364 49 - - SYM cana-4527 364 50 90 90 NUM cana-4527 364 51 , , PUNCT cana-4527 364 52 doi.org/10.1016/s0022-1694(00)00369-3 doi.org/10.1016/s0022-1694(00)00369-3 PUNCT cana-4527 365 1 [ [ X cana-4527 365 2 43 43 NUM cana-4527 365 3 ] ] PUNCT cana-4527 365 4 . . PUNCT cana-4527 365 5 cloot cloot PROPN cana-4527 365 6 , , PUNCT cana-4527 365 7 a. a. NOUN cana-4527 365 8 , , PUNCT cana-4527 365 9 and and CCONJ cana-4527 365 10 j. j. PROPN cana-4527 365 11 f. f. PROPN cana-4527 365 12 botha botha PROPN cana-4527 365 13 , , PUNCT cana-4527 365 14 a a DET cana-4527 365 15 generalized generalized ADJ cana-4527 365 16 groundwater groundwater NOUN cana-4527 365 17 flow flow NOUN cana-4527 365 18 equation equation NOUN cana-4527 365 19 using use VERB cana-4527 365 20 the the DET cana-4527 365 21 concept concept NOUN cana-4527 365 22 of of ADP cana-4527 365 23 noninteger noninteger NOUN cana-4527 365 24 order order NOUN cana-4527 365 25 derivatives derivative NOUN cana-4527 365 26 , , PUNCT cana-4527 365 27 water water NOUN cana-4527 365 28 sa sa NOUN cana-4527 365 29 32 32 NUM cana-4527 365 30 , , PUNCT cana-4527 365 31 no no INTJ cana-4527 365 32 . . NOUN cana-4527 365 33 1 1 NUM cana-4527 365 34 ( ( PUNCT cana-4527 365 35 2006 2006 NUM cana-4527 365 36 ): ): PUNCT cana-4527 365 37 1 1 NUM cana-4527 365 38 - - SYM cana-4527 365 39 7 7 NUM cana-4527 365 40 , , PUNCT cana-4527 365 41 https://doi.org/10.4314/wsa.v32i1.5225 https://doi.org/10.4314/wsa.v32i1.5225 PUNCT cana-4527 365 42 . . PUNCT cana-4527 366 1 https://doi.org/10.1016/s0022-1694(00)00369-3 https://doi.org/10.1016/s0022-1694(00)00369-3 PROPN cana-4527 366 2 https://doi.org/10.4314/wsa.v32i1.5225 https://doi.org/10.4314/wsa.v32i1.5225 PUNCT