id sid tid token lemma pos cana-4595 1 1 communications communication NOUN cana-4595 1 2 on on ADP cana-4595 1 3 applied apply VERB cana-4595 1 4 nonlinear nonlinear ADJ cana-4595 1 5 analysis analysis NOUN cana-4595 1 6 issn issn NOUN cana-4595 1 7 : : PUNCT cana-4595 1 8 1074 1074 NUM cana-4595 1 9 - - PUNCT cana-4595 1 10 133x 133x NUM cana-4595 1 11 vol vol NOUN cana-4595 1 12 32 32 NUM cana-4595 1 13 no no NOUN cana-4595 1 14 . . PUNCT cana-4595 2 1 9s 9s NUM cana-4595 2 2 ( ( PUNCT cana-4595 2 3 2025 2025 NUM cana-4595 2 4 ) ) PUNCT cana-4595 3 1 https://internationalpubls.com https://internationalpubls.com X cana-4595 3 2 2969 2969 NUM cana-4595 3 3 new new ADJ cana-4595 3 4 operational operational ADJ cana-4595 3 5 matrix matrix NOUN cana-4595 3 6 via via ADP cana-4595 3 7 gnocchi gnocchi NOUN cana-4595 3 8 polynomial polynomial PROPN cana-4595 3 9 for for ADP cana-4595 3 10 solving solve VERB cana-4595 3 11 nonlinear nonlinear ADJ cana-4595 3 12 fractional fractional ADJ cana-4595 3 13 differential differential NOUN cana-4595 3 14 equations equation NOUN cana-4595 3 15 . . PUNCT cana-4595 4 1 anil anil PROPN cana-4595 4 2 kumar1 kumar1 PROPN cana-4595 4 3 * * PROPN cana-4595 4 4 , , PUNCT cana-4595 4 5 dr dr PROPN cana-4595 4 6 . . PROPN cana-4595 4 7 sachin sachin PROPN cana-4595 4 8 kumar2 kumar2 PROPN cana-4595 5 1 1*department 1*department NUM cana-4595 5 2 of of ADP cana-4595 5 3 mathematics mathematic NOUN cana-4595 5 4 , , PUNCT cana-4595 5 5 government government NOUN cana-4595 5 6 degree degree NOUN cana-4595 5 7 college college NOUN cana-4595 5 8 budaun budaun NOUN cana-4595 5 9 , , PUNCT cana-4595 5 10 affiliated affiliated ADJ cana-4595 5 11 m.j.p m.j.p ADJ cana-4595 5 12 . . PROPN cana-4595 5 13 rohilkhand rohilkhand PROPN cana-4595 5 14 university university PROPN cana-4595 5 15 , , PUNCT cana-4595 5 16 bareilly bareilly PROPN cana-4595 5 17 , , PUNCT cana-4595 5 18 up up ADP cana-4595 5 19 2assistant 2assistant NUM cana-4595 5 20 professor professor NOUN cana-4595 5 21 , , PUNCT cana-4595 5 22 department department NOUN cana-4595 5 23 of of ADP cana-4595 5 24 mathematics mathematic NOUN cana-4595 5 25 , , PUNCT cana-4595 5 26 government government NOUN cana-4595 5 27 degree degree NOUN cana-4595 5 28 college college NOUN cana-4595 5 29 budaun budaun NOUN cana-4595 5 30 , , PUNCT cana-4595 5 31 affiliated affiliated ADJ cana-4595 5 32 m.j.p m.j.p ADJ cana-4595 5 33 . . PROPN cana-4595 5 34 rohilkhand rohilkhand PROPN cana-4595 5 35 university university PROPN cana-4595 5 36 , , PUNCT cana-4595 5 37 bareilly bareilly PROPN cana-4595 5 38 , , PUNCT cana-4595 5 39 up up ADV cana-4595 5 40 , , PUNCT cana-4595 5 41 email email NOUN cana-4595 5 42 idanilkumar02011978@gmail.com idanilkumar02011978@gmail.com X cana-4595 6 1 article article PROPN cana-4595 6 2 history history NOUN cana-4595 6 3 : : PUNCT cana-4595 6 4 received receive VERB cana-4595 6 5 : : PUNCT cana-4595 6 6 12 12 NUM cana-4595 6 7 - - SYM cana-4595 6 8 01 01 NUM cana-4595 6 9 - - PUNCT cana-4595 6 10 2025 2025 NUM cana-4595 6 11 revised revise VERB cana-4595 6 12 : : PUNCT cana-4595 6 13 15 15 NUM cana-4595 6 14 - - NUM cana-4595 6 15 02 02 NUM cana-4595 6 16 - - PUNCT cana-4595 6 17 2025 2025 NUM cana-4595 6 18 accepted accept VERB cana-4595 6 19 : : PUNCT cana-4595 6 20 01 01 NUM cana-4595 6 21 - - SYM cana-4595 6 22 03 03 NUM cana-4595 6 23 - - PUNCT cana-4595 6 24 2025 2025 NUM cana-4595 6 25 abstract abstract NOUN cana-4595 6 26 : : PUNCT cana-4595 6 27 fractional fractional ADJ cana-4595 6 28 differential differential ADJ cana-4595 6 29 equations equation NOUN cana-4595 6 30 ( ( PUNCT cana-4595 6 31 fdes fde NOUN cana-4595 6 32 ) ) PUNCT cana-4595 6 33 have have AUX cana-4595 6 34 emerged emerge VERB cana-4595 6 35 as as ADP cana-4595 6 36 essential essential ADJ cana-4595 6 37 tools tool NOUN cana-4595 6 38 in in ADP cana-4595 6 39 modeling model VERB cana-4595 6 40 complex complex ADJ cana-4595 6 41 dynamical dynamical ADJ cana-4595 6 42 systems system NOUN cana-4595 6 43 exhibiting exhibit VERB cana-4595 6 44 memory memory NOUN cana-4595 6 45 and and CCONJ cana-4595 6 46 hereditary hereditary ADJ cana-4595 6 47 properties property NOUN cana-4595 6 48 . . PUNCT cana-4595 7 1 traditional traditional ADJ cana-4595 7 2 operational operational ADJ cana-4595 7 3 matrices matrix NOUN cana-4595 7 4 arising arise VERB cana-4595 7 5 from from ADP cana-4595 7 6 legendre legendre PROPN cana-4595 7 7 , , PUNCT cana-4595 7 8 chebyshev chebyshev PROPN cana-4595 7 9 , , PUNCT cana-4595 7 10 and and CCONJ cana-4595 7 11 jacobi jacobi PROPN cana-4595 7 12 polynomials polynomial NOUN cana-4595 7 13 are be AUX cana-4595 7 14 generally generally ADV cana-4595 7 15 known know VERB cana-4595 7 16 to to PART cana-4595 7 17 be be AUX cana-4595 7 18 numerically numerically ADV cana-4595 7 19 unstable unstable ADJ cana-4595 7 20 , , PUNCT cana-4595 7 21 computationally computationally ADV cana-4595 7 22 expensive expensive ADJ cana-4595 7 23 , , PUNCT cana-4595 7 24 and and CCONJ cana-4595 7 25 inefficient inefficient ADJ cana-4595 7 26 in in ADP cana-4595 7 27 approximating approximate VERB cana-4595 7 28 fractional fractional ADJ cana-4595 7 29 operators operator NOUN cana-4595 7 30 . . PUNCT cana-4595 8 1 in in ADP cana-4595 8 2 this this DET cana-4595 8 3 study study NOUN cana-4595 8 4 an an DET cana-4595 8 5 operational operational ADJ cana-4595 8 6 matrix matrix NOUN cana-4595 8 7 based base VERB cana-4595 8 8 on on ADP cana-4595 8 9 gnocchi gnocchi NOUN cana-4595 8 10 polynomial polynomial NOUN cana-4595 8 11 is be AUX cana-4595 8 12 introduced introduce VERB cana-4595 8 13 for for ADP cana-4595 8 14 solving solve VERB cana-4595 8 15 non non ADJ cana-4595 8 16 linear linear ADJ cana-4595 8 17 fractional fractional ADJ cana-4595 8 18 differential differential ADJ cana-4595 8 19 equations equation NOUN cana-4595 8 20 ( ( PUNCT cana-4595 8 21 nfde nfde NOUN cana-4595 8 22 ) ) PUNCT cana-4595 8 23 with with ADP cana-4595 8 24 better well ADJ cana-4595 8 25 sparsity sparsity NOUN cana-4595 8 26 , , PUNCT cana-4595 8 27 stability stability NOUN cana-4595 8 28 and and CCONJ cana-4595 8 29 computational computational ADJ cana-4595 8 30 efficiency efficiency NOUN cana-4595 8 31 . . PUNCT cana-4595 9 1 the the DET cana-4595 9 2 proposed propose VERB cana-4595 9 3 method method NOUN cana-4595 9 4 transforms transform VERB cana-4595 9 5 nfdes nfde NOUN cana-4595 9 6 into into ADP cana-4595 9 7 tractable tractable ADJ cana-4595 9 8 algebraic algebraic ADJ cana-4595 9 9 systems system NOUN cana-4595 9 10 by by ADP cana-4595 9 11 constructing construct VERB cana-4595 9 12 a a DET cana-4595 9 13 fractional fractional ADJ cana-4595 9 14 differentiation differentiation NOUN cana-4595 9 15 operational operational ADJ cana-4595 9 16 matrix matrix NOUN cana-4595 9 17 using use VERB cana-4595 9 18 gnocchi gnocchi NOUN cana-4595 9 19 polynomials polynomial NOUN cana-4595 9 20 . . PUNCT cana-4595 10 1 the the DET cana-4595 10 2 method method NOUN cana-4595 10 3 is be AUX cana-4595 10 4 validated validate VERB cana-4595 10 5 by by ADP cana-4595 10 6 theoretical theoretical ADJ cana-4595 10 7 formulations formulation NOUN cana-4595 10 8 , , PUNCT cana-4595 10 9 spectral spectral ADJ cana-4595 10 10 convergence convergence NOUN cana-4595 10 11 analysis analysis NOUN cana-4595 10 12 , , PUNCT cana-4595 10 13 error error NOUN cana-4595 10 14 estimation estimation NOUN cana-4595 10 15 proofs proof NOUN cana-4595 10 16 . . PUNCT cana-4595 11 1 it it PRON cana-4595 11 2 is be AUX cana-4595 11 3 also also ADV cana-4595 11 4 compared compare VERB cana-4595 11 5 with with ADP cana-4595 11 6 existing exist VERB cana-4595 11 7 polynomial polynomial ADJ cana-4595 11 8 based base VERB cana-4595 11 9 approaches approach NOUN cana-4595 11 10 to to PART cana-4595 11 11 demonstrate demonstrate VERB cana-4595 11 12 better well ADJ cana-4595 11 13 performance performance NOUN cana-4595 11 14 in in ADP cana-4595 11 15 function function NOUN cana-4595 11 16 approximation approximation NOUN cana-4595 11 17 and and CCONJ cana-4595 11 18 numerical numerical ADJ cana-4595 11 19 stability stability NOUN cana-4595 11 20 . . PUNCT cana-4595 12 1 the the DET cana-4595 12 2 gnocchi gnocchi NOUN cana-4595 12 3 operational operational ADJ cana-4595 12 4 matrix matrix NOUN cana-4595 12 5 is be AUX cana-4595 12 6 based base VERB cana-4595 12 7 on on ADP cana-4595 12 8 gnocchi gnocchi NOUN cana-4595 12 9 , , PUNCT cana-4595 12 10 and and CCONJ cana-4595 12 11 it it PRON cana-4595 12 12 achieves achieve VERB cana-4595 12 13 exponential exponential ADJ cana-4595 12 14 convergence convergence NOUN cana-4595 12 15 , , PUNCT cana-4595 12 16 reduced reduce VERB cana-4595 12 17 computational computational ADJ cana-4595 12 18 complexity complexity NOUN cana-4595 12 19 and and CCONJ cana-4595 12 20 increased increase VERB cana-4595 12 21 numerical numerical ADJ cana-4595 12 22 robustness robustness NOUN cana-4595 12 23 compared compare VERB cana-4595 12 24 to to ADP cana-4595 12 25 classical classical ADJ cana-4595 12 26 techniques technique NOUN cana-4595 12 27 . . PUNCT cana-4595 13 1 it it PRON cana-4595 13 2 is be AUX cana-4595 13 3 effective effective ADJ cana-4595 13 4 in in ADP cana-4595 13 5 fractional fractional ADJ cana-4595 13 6 modeling modeling NOUN cana-4595 13 7 because because SCONJ cana-4595 13 8 it it PRON cana-4595 13 9 can can AUX cana-4595 13 10 accurately accurately ADV cana-4595 13 11 approximate approximate VERB cana-4595 13 12 non non ADJ cana-4595 13 13 - - ADJ cana-4595 13 14 linear linear ADJ cana-4595 13 15 fractional fractional ADJ cana-4595 13 16 operators operator NOUN cana-4595 13 17 . . PUNCT cana-4595 14 1 the the DET cana-4595 14 2 author author NOUN cana-4595 14 3 develops develop VERB cana-4595 14 4 a a DET cana-4595 14 5 mathematically mathematically ADV cana-4595 14 6 rigorous rigorous ADJ cana-4595 14 7 , , PUNCT cana-4595 14 8 computationally computationally ADV cana-4595 14 9 efficient efficient ADJ cana-4595 14 10 framework framework NOUN cana-4595 14 11 to to PART cana-4595 14 12 solve solve VERB cana-4595 14 13 nfdes nfde NOUN cana-4595 14 14 . . PUNCT cana-4595 15 1 further further ADJ cana-4595 15 2 improvements improvement NOUN cana-4595 15 3 will will AUX cana-4595 15 4 be be AUX cana-4595 15 5 done do VERB cana-4595 15 6 by by ADP cana-4595 15 7 other other ADJ cana-4595 15 8 researchers researcher NOUN cana-4595 15 9 in in ADP cana-4595 15 10 the the DET cana-4595 15 11 future future NOUN cana-4595 15 12 for for ADP cana-4595 15 13 higher high ADJ cana-4595 15 14 dimension dimension NOUN cana-4595 15 15 applications application NOUN cana-4595 15 16 , , PUNCT cana-4595 15 17 for for ADP cana-4595 15 18 adaptive adaptive ADJ cana-4595 15 19 techniques technique NOUN cana-4595 15 20 in in ADP cana-4595 15 21 the the DET cana-4595 15 22 spectral spectral ADJ cana-4595 15 23 method method NOUN cana-4595 15 24 and and CCONJ cana-4595 15 25 for for ADP cana-4595 15 26 hybrid hybrid NOUN cana-4595 15 27 ai ai VERB cana-4595 15 28 assisted assist VERB cana-4595 15 29 optimization optimization NOUN cana-4595 15 30 . . PUNCT cana-4595 16 1 keywords keyword NOUN cana-4595 16 2 : : PUNCT cana-4595 16 3 fractional fractional ADJ cana-4595 16 4 differential differential ADJ cana-4595 16 5 equations equation NOUN cana-4595 16 6 , , PUNCT cana-4595 16 7 gnocchi gnocchi NOUN cana-4595 16 8 polynomials polynomial NOUN cana-4595 16 9 , , PUNCT cana-4595 16 10 operational operational ADJ cana-4595 16 11 matrix matrix NOUN cana-4595 16 12 , , PUNCT cana-4595 16 13 numerical numerical ADJ cana-4595 16 14 stability stability NOUN cana-4595 16 15 , , PUNCT cana-4595 16 16 spectral spectral ADJ cana-4595 16 17 convergence convergence NOUN cana-4595 16 18 , , PUNCT cana-4595 16 19 computational computational ADJ cana-4595 16 20 efficiency efficiency NOUN cana-4595 16 21 , , PUNCT cana-4595 16 22 non non ADJ cana-4595 16 23 - - ADJ cana-4595 16 24 linear linear ADJ cana-4595 16 25 systems system NOUN cana-4595 16 26 . . PUNCT cana-4595 17 1 1 1 X cana-4595 17 2 . . X cana-4595 17 3 introduction introduction NOUN cana-4595 17 4 recently recently ADV cana-4595 17 5 , , PUNCT cana-4595 17 6 fractional fractional ADJ cana-4595 17 7 differential differential ADJ cana-4595 17 8 equations equation NOUN cana-4595 17 9 ( ( PUNCT cana-4595 17 10 fdes fde NOUN cana-4595 17 11 ) ) PUNCT cana-4595 17 12 have have AUX cana-4595 17 13 come come VERB cana-4595 17 14 to to PART cana-4595 17 15 be be AUX cana-4595 17 16 regarded regard VERB cana-4595 17 17 as as ADP cana-4595 17 18 a a DET cana-4595 17 19 beautiful beautiful ADJ cana-4595 17 20 and and CCONJ cana-4595 17 21 practical practical ADJ cana-4595 17 22 mathematical mathematical ADJ cana-4595 17 23 method method NOUN cana-4595 17 24 to to PART cana-4595 17 25 model model VERB cana-4595 17 26 systems system NOUN cana-4595 17 27 with with ADP cana-4595 17 28 memory memory NOUN cana-4595 17 29 and and CCONJ cana-4595 17 30 hereditary hereditary ADJ cana-4595 17 31 nature nature NOUN cana-4595 17 32 . . PUNCT cana-4595 18 1 fractional fractional ADJ cana-4595 18 2 differential differential ADJ cana-4595 18 3 equations equation NOUN cana-4595 18 4 are be AUX cana-4595 18 5 different different ADJ cana-4595 18 6 from from ADP cana-4595 18 7 the the DET cana-4595 18 8 classical classical ADJ cana-4595 18 9 integer integer NOUN cana-4595 18 10 order order NOUN cana-4595 18 11 differential differential ADJ cana-4595 18 12 equations equation NOUN cana-4595 18 13 in in ADP cana-4595 18 14 that that PRON cana-4595 18 15 they they PRON cana-4595 18 16 contain contain VERB cana-4595 18 17 a a DET cana-4595 18 18 derivative derivative NOUN cana-4595 18 19 of of ADP cana-4595 18 20 arbitrary arbitrary ADJ cana-4595 18 21 ( ( PUNCT cana-4595 18 22 noninteger noninteger NOUN cana-4595 18 23 ) ) PUNCT cana-4595 18 24 order order NOUN cana-4595 18 25 in in ADP cana-4595 18 26 order order NOUN cana-4595 18 27 to to PART cana-4595 18 28 deal deal VERB cana-4595 18 29 with with ADP cana-4595 18 30 the the DET cana-4595 18 31 intrinsic intrinsic ADJ cana-4595 18 32 behavior behavior NOUN cana-4595 18 33 of of ADP cana-4595 18 34 most most ADJ cana-4595 18 35 of of ADP cana-4595 18 36 physical physical ADJ cana-4595 18 37 and and CCONJ cana-4595 18 38 engineering engineering NOUN cana-4595 18 39 problems problem NOUN cana-4595 18 40 . . PUNCT cana-4595 19 1 in in ADP cana-4595 19 2 the the DET cana-4595 19 3 past past ADJ cana-4595 19 4 couple couple NOUN cana-4595 19 5 of of ADP cana-4595 19 6 years year NOUN cana-4595 19 7 , , PUNCT cana-4595 19 8 several several ADJ cana-4595 19 9 kinds kind NOUN cana-4595 19 10 of of ADP cana-4595 19 11 numerical numerical ADJ cana-4595 19 12 and and CCONJ cana-4595 19 13 analytical analytical ADJ cana-4595 19 14 approaches approach NOUN cana-4595 19 15 like like ADP cana-4595 19 16 operational operational ADJ cana-4595 19 17 matrix matrix NOUN cana-4595 19 18 and and CCONJ cana-4595 19 19 spectral spectral ADJ cana-4595 19 20 methods method NOUN cana-4595 19 21 , , PUNCT cana-4595 19 22 and and CCONJ cana-4595 19 23 finite finite ADJ cana-4595 19 24 difference difference NOUN cana-4595 19 25 schemes scheme NOUN cana-4595 19 26 [ [ X cana-4595 19 27 1 1 NUM cana-4595 19 28 ] ] PUNCT cana-4595 19 29 are be AUX cana-4595 19 30 proposed propose VERB cana-4595 19 31 to to PART cana-4595 19 32 approximate approximate VERB cana-4595 19 33 the the DET cana-4595 19 34 solutions solution NOUN cana-4595 19 35 of of ADP cana-4595 19 36 fdes fde NOUN cana-4595 19 37 . . PUNCT cana-4595 20 1 on on ADP cana-4595 20 2 the the DET cana-4595 20 3 other other ADJ cana-4595 20 4 hand hand NOUN cana-4595 20 5 , , PUNCT cana-4595 20 6 traditional traditional ADJ cana-4595 20 7 polynomial polynomial ADJ cana-4595 20 8 based base VERB cana-4595 20 9 approaches approach NOUN cana-4595 20 10 are be AUX cana-4595 20 11 not not PART cana-4595 20 12 competent competent ADJ cana-4595 20 13 with with ADP cana-4595 20 14 highly highly ADV cana-4595 20 15 non non ADJ cana-4595 20 16 linear linear PROPN cana-4595 20 17 systems system NOUN cana-4595 20 18 because because SCONJ cana-4595 20 19 polynomial polynomial ADJ cana-4595 20 20 functions function NOUN cana-4595 20 21 are be AUX cana-4595 20 22 not not PART cana-4595 20 23 easily easily ADV cana-4595 20 24 formed form VERB cana-4595 20 25 with with ADP cana-4595 20 26 fractional fractional ADJ cana-4595 20 27 operators operator NOUN cana-4595 20 28 . . PUNCT cana-4595 21 1 for for ADP cana-4595 21 2 the the DET cana-4595 21 3 purpose purpose NOUN cana-4595 21 4 of of ADP cana-4595 21 5 removing remove VERB cana-4595 21 6 these these DET cana-4595 21 7 limitations limitation NOUN cana-4595 21 8 we we PRON cana-4595 21 9 suggest suggest VERB cana-4595 21 10 a a DET cana-4595 21 11 new new ADJ cana-4595 21 12 operational operational ADJ cana-4595 21 13 matrix matrix NOUN cana-4595 21 14 in in ADP cana-4595 21 15 terms term NOUN cana-4595 21 16 of of ADP cana-4595 21 17 gnocchi gnocchi NOUN cana-4595 21 18 polynomials polynomial NOUN cana-4595 21 19 for for ADP cana-4595 21 20 solving solve VERB cana-4595 21 21 nonlinear nonlinear ADJ cana-4595 21 22 fractional fractional ADJ cana-4595 21 23 differential differential ADJ cana-4595 21 24 equations equation NOUN cana-4595 21 25 in in ADP cana-4595 21 26 a a DET cana-4595 21 27 highly highly ADV cana-4595 21 28 accurate accurate ADJ cana-4595 21 29 manner manner NOUN cana-4595 21 30 . . PUNCT cana-4595 22 1 mathematical mathematical ADJ cana-4595 22 2 background background NOUN cana-4595 22 3 1 1 NUM cana-4595 22 4 . . PUNCT cana-4595 22 5 fractional fractional ADJ cana-4595 22 6 calculus calculus NOUN cana-4595 22 7 preliminaries preliminary NOUN cana-4595 22 8 fractional fractional ADJ cana-4595 22 9 calculus calculus NOUN cana-4595 22 10 is be AUX cana-4595 22 11 calculus calculus NOUN cana-4595 22 12 in in ADP cana-4595 22 13 which which PRON cana-4595 22 14 we we PRON cana-4595 22 15 consider consider VERB cana-4595 22 16 differentiation differentiation NOUN cana-4595 22 17 and and CCONJ cana-4595 22 18 integration integration NOUN cana-4595 22 19 to to ADP cana-4595 22 20 arbitrary arbitrary ADJ cana-4595 22 21 orders order NOUN cana-4595 22 22 . . PUNCT cana-4595 23 1 there there PRON cana-4595 23 2 are be VERB cana-4595 23 3 a a DET cana-4595 23 4 number number NOUN cana-4595 23 5 of of ADP cana-4595 23 6 commonly commonly ADV cana-4595 23 7 used use VERB cana-4595 23 8 definitions definition NOUN cana-4595 23 9 for for ADP cana-4595 23 10 fractional fractional ADJ cana-4595 23 11 derivatives derivative NOUN cana-4595 23 12 among among ADP cana-4595 23 13 which which PRON cana-4595 23 14 are be AUX cana-4595 23 15 : : PUNCT cana-4595 23 16 โ€ข โ€ข NUM cana-4595 23 17 riemann riemann PROPN cana-4595 23 18 - - PUNCT cana-4595 23 19 liouville liouville VERB cana-4595 23 20 fractional fractional ADJ cana-4595 23 21 derivative derivative ADJ cana-4595 23 22 ๐‘…๐ฟ๐ท๐›ผ๐‘“(๐‘ฅ ๐‘…๐ฟ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 23 23 ) ) PUNCT cana-4595 23 24 = = SYM cana-4595 23 25 1 1 NUM cana-4595 23 26 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-4595 23 27 โˆ’ โˆ’ X cana-4595 23 28 ๐›ผ ๐›ผ NOUN cana-4595 23 29 ) ) PUNCT cana-4595 23 30 ๐‘‘๐‘› ๐‘‘๐‘› X cana-4595 23 31 ๐‘‘๐‘ฅ๐‘› ๐‘‘๐‘ฅ๐‘› VERB cana-4595 23 32 โˆซ โˆซ PROPN cana-4595 23 33 โ€Š โ€Š SPACE cana-4595 23 34 ๐‘ฅ ๐‘ฅ PROPN cana-4595 23 35 0 0 PUNCT cana-4595 24 1 ( ( PUNCT cana-4595 24 2 ๐‘ฅ ๐‘ฅ NOUN cana-4595 24 3 โˆ’ โˆ’ PROPN cana-4595 24 4 ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก NOUN cana-4595 24 5 , , PUNCT cana-4595 24 6 ๐‘› ๐‘› PRON cana-4595 24 7 โˆ’ โˆ’ PROPN cana-4595 24 8 1 1 NUM cana-4595 24 9 < < X cana-4595 24 10 ๐›ผ ๐›ผ X cana-4595 24 11 < < X cana-4595 24 12 ๐‘› ๐‘› PRON cana-4595 24 13 communications communication NOUN cana-4595 24 14 on on ADP cana-4595 24 15 applied apply VERB cana-4595 24 16 nonlinear nonlinear ADJ cana-4595 24 17 analysis analysis NOUN cana-4595 24 18 issn issn NOUN cana-4595 24 19 : : PUNCT cana-4595 24 20 1074 1074 NUM cana-4595 24 21 - - PUNCT cana-4595 24 22 133x 133x NUM cana-4595 24 23 vol vol NOUN cana-4595 24 24 32 32 NUM cana-4595 24 25 no no NOUN cana-4595 24 26 . . PUNCT cana-4595 25 1 9s 9s NUM cana-4595 25 2 ( ( PUNCT cana-4595 25 3 2025 2025 NUM cana-4595 25 4 ) ) PUNCT cana-4595 25 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 25 6 2970 2970 NUM cana-4595 25 7 where where SCONJ cana-4595 25 8 ฮณ(โ‹… ฮณ(โ‹… NOUN cana-4595 25 9 ) ) PUNCT cana-4595 25 10 is be AUX cana-4595 25 11 the the DET cana-4595 25 12 gamma gamma PROPN cana-4595 25 13 function function NOUN cana-4595 25 14 . . PUNCT cana-4595 26 1 โ€ข โ€ข NUM cana-4595 26 2 caputo caputo PROPN cana-4595 26 3 fractional fractional ADJ cana-4595 26 4 derivative derivative ADJ cana-4595 26 5 ๐ถ๐ท๐›ผ๐‘“(๐‘ฅ ๐ถ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 26 6 ) ) PUNCT cana-4595 26 7 = = SYM cana-4595 26 8 1 1 NUM cana-4595 26 9 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-4595 26 10 โˆ’ โˆ’ ADP cana-4595 26 11 ๐›ผ ๐›ผ NOUN cana-4595 26 12 ) ) PUNCT cana-4595 26 13 โˆซ โˆซ PROPN cana-4595 26 14 โ€Š โ€Š SPACE cana-4595 27 1 ๐‘ฅ ๐‘ฅ PROPN cana-4595 27 2 0 0 PUNCT cana-4595 28 1 ( ( PUNCT cana-4595 28 2 ๐‘ฅ ๐‘ฅ NOUN cana-4595 28 3 โˆ’ โˆ’ NOUN cana-4595 28 4 ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘›)(๐‘ก)๐‘‘๐‘ก ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘›)(๐‘ก)๐‘‘๐‘ก ADJ cana-4595 28 5 , , PUNCT cana-4595 28 6 ๐‘› ๐‘› PRON cana-4595 28 7 โˆ’ โˆ’ PROPN cana-4595 28 8 1 1 NUM cana-4595 28 9 < < X cana-4595 28 10 ๐›ผ ๐›ผ X cana-4595 28 11 < < X cana-4595 28 12 ๐‘› ๐‘› PROPN cana-4595 28 13 caputo caputo PROPN cana-4595 28 14 derivative derivative NOUN cana-4595 28 15 is be AUX cana-4595 28 16 especially especially ADV cana-4595 28 17 useful useful ADJ cana-4595 28 18 to to ADP cana-4595 28 19 schematise schematise VERB cana-4595 28 20 physical physical ADJ cana-4595 28 21 process process NOUN cana-4595 28 22 , , PUNCT cana-4595 28 23 because because SCONJ cana-4595 28 24 initial initial ADJ cana-4595 28 25 conditions condition NOUN cana-4595 28 26 are be AUX cana-4595 28 27 defined define VERB cana-4595 28 28 in in ADP cana-4595 28 29 the the DET cana-4595 28 30 same same ADJ cana-4595 28 31 form form NOUN cana-4595 28 32 as as ADP cana-4595 28 33 classical classical ADJ cana-4595 28 34 differential differential ADJ cana-4595 28 35 equations equation NOUN cana-4595 29 1 [ [ X cana-4595 29 2 2 2 NUM cana-4595 29 3 ] ] PUNCT cana-4595 29 4 . . PUNCT cana-4595 29 5 2 2 X cana-4595 29 6 . . X cana-4595 29 7 gnocchi gnocchi NOUN cana-4595 29 8 polynomials polynomial NOUN cana-4595 29 9 and and CCONJ cana-4595 29 10 their their PRON cana-4595 29 11 properties property NOUN cana-4595 29 12 the the DET cana-4595 29 13 gnocchi gnocchi NOUN cana-4595 29 14 polynomials polynomial VERB cana-4595 29 15 ๐บ๐‘›(๐‘ฅ ๐บ๐‘›(๐‘ฅ NOUN cana-4595 29 16 ) ) PUNCT cana-4595 29 17 are be AUX cana-4595 29 18 a a DET cana-4595 29 19 class class NOUN cana-4595 29 20 of of ADP cana-4595 29 21 orthogonal orthogonal ADJ cana-4595 29 22 polynomials polynomial NOUN cana-4595 29 23 that that PRON cana-4595 29 24 provide provide VERB cana-4595 29 25 a a DET cana-4595 29 26 robust robust ADJ cana-4595 29 27 basis basis NOUN cana-4595 29 28 for for ADP cana-4595 29 29 function function NOUN cana-4595 29 30 approximation approximation NOUN cana-4595 29 31 . . PUNCT cana-4595 30 1 these these DET cana-4595 30 2 polynomials polynomial NOUN cana-4595 30 3 satisfy satisfy VERB cana-4595 30 4 the the DET cana-4595 30 5 recurrence recurrence NOUN cana-4595 30 6 relation relation NOUN cana-4595 30 7 : : PUNCT cana-4595 30 8 ๐บ๐‘›+1(๐‘ฅ ๐บ๐‘›+1(๐‘ฅ NOUN cana-4595 30 9 ) ) PUNCT cana-4595 30 10 = = SYM cana-4595 31 1 ( ( PUNCT cana-4595 31 2 2๐‘› 2๐‘› PROPN cana-4595 31 3 + + CCONJ cana-4595 31 4 1)๐‘ฅ๐บ๐‘›(๐‘ฅ 1)๐‘ฅ๐บ๐‘›(๐‘ฅ NUM cana-4595 31 5 ) ) PUNCT cana-4595 31 6 โˆ’ โˆ’ NOUN cana-4595 31 7 ๐‘›๐บ๐‘›โˆ’1(๐‘ฅ ๐‘›๐บ๐‘›โˆ’1(๐‘ฅ NUM cana-4595 31 8 ) ) PUNCT cana-4595 31 9 , , PUNCT cana-4595 31 10 ๐บ0(๐‘ฅ ๐บ0(๐‘ฅ NUM cana-4595 31 11 ) ) PUNCT cana-4595 31 12 = = SYM cana-4595 31 13 1 1 NUM cana-4595 31 14 , , PUNCT cana-4595 31 15 ๐บ1(๐‘ฅ ๐บ1(๐‘ฅ NUM cana-4595 31 16 ) ) PUNCT cana-4595 32 1 = = PUNCT cana-4595 32 2 ๐‘ฅ. ๐‘ฅ. NOUN cana-4595 32 3 they they PRON cana-4595 32 4 have have AUX cana-4595 32 5 been be AUX cana-4595 32 6 extensively extensively ADV cana-4595 32 7 used use VERB cana-4595 32 8 in in ADP cana-4595 32 9 solving solve VERB cana-4595 32 10 integral integral ADJ cana-4595 32 11 equations equation NOUN cana-4595 32 12 and and CCONJ cana-4595 32 13 in in ADP cana-4595 32 14 the the DET cana-4595 32 15 spectral spectral ADJ cana-4595 32 16 representation representation NOUN cana-4595 32 17 of of ADP cana-4595 32 18 differential differential ADJ cana-4595 32 19 operators operator NOUN cana-4595 32 20 [ [ X cana-4595 32 21 3 3 NUM cana-4595 32 22 ] ] PUNCT cana-4595 32 23 . . PUNCT cana-4595 33 1 the the DET cana-4595 33 2 formation formation NOUN cana-4595 33 3 of of ADP cana-4595 33 4 an an DET cana-4595 33 5 operational operational ADJ cana-4595 33 6 matrix matrix NOUN cana-4595 33 7 of of ADP cana-4595 33 8 fractional fractional ADJ cana-4595 33 9 integration integration NOUN cana-4595 33 10 of of ADP cana-4595 33 11 gnocchi gnocchi NOUN cana-4595 33 12 polynomials polynomial NOUN cana-4595 33 13 is be AUX cana-4595 33 14 another another DET cana-4595 33 15 crucial crucial ADJ cana-4595 33 16 property property NOUN cana-4595 33 17 which which PRON cana-4595 33 18 provides provide VERB cana-4595 33 19 a a DET cana-4595 33 20 tool tool NOUN cana-4595 33 21 for for ADP cana-4595 33 22 transforming transform VERB cana-4595 33 23 fractional fractional ADJ cana-4595 33 24 differential differential ADJ cana-4595 33 25 equations equation NOUN cana-4595 33 26 to to ADP cana-4595 33 27 an an DET cana-4595 33 28 algebraic algebraic ADJ cana-4595 33 29 system system NOUN cana-4595 33 30 . . PUNCT cana-4595 34 1 given give VERB cana-4595 34 2 a a DET cana-4595 34 3 function function NOUN cana-4595 34 4 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 34 5 ) ) PUNCT cana-4595 34 6 expanded expand VERB cana-4595 34 7 in in ADP cana-4595 34 8 terms term NOUN cana-4595 34 9 of of ADP cana-4595 34 10 gnocchi gnocchi NOUN cana-4595 34 11 polynomials polynomial NOUN cana-4595 34 12 : : PUNCT cana-4595 34 13 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 34 14 ) ) PUNCT cana-4595 35 1 = = PUNCT cana-4595 35 2 โˆ‘ โˆ‘ PUNCT cana-4595 35 3 โ€Š โ€Š SPACE cana-4595 35 4 โˆž โˆž NUM cana-4595 35 5 ๐‘›=0 ๐‘›=0 PROPN cana-4595 35 6 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ PROPN cana-4595 35 7 ) ) PUNCT cana-4595 35 8 its its PRON cana-4595 35 9 fractional fractional ADJ cana-4595 35 10 integral integral NOUN cana-4595 35 11 of of ADP cana-4595 35 12 order order NOUN cana-4595 35 13 ๐›ผ ๐›ผ NOUN cana-4595 35 14 can can AUX cana-4595 35 15 be be AUX cana-4595 35 16 approximated approximate VERB cana-4595 35 17 using use VERB cana-4595 35 18 the the DET cana-4595 35 19 operational operational ADJ cana-4595 35 20 matrix matrix NOUN cana-4595 35 21 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 35 22 ) ) PUNCT cana-4595 35 23 : : PUNCT cana-4595 35 24 ๐ทโˆ’๐›ผ๐‘“(๐‘ฅ ๐ทโˆ’๐›ผ๐‘“(๐‘ฅ X cana-4595 35 25 ) ) PUNCT cana-4595 35 26 โ‰ˆ โ‰ˆ PROPN cana-4595 35 27 ๐‘ƒ(๐›ผ)๐ถ ๐‘ƒ(๐›ผ)๐ถ NUM cana-4595 35 28 where where SCONJ cana-4595 35 29 ๐ถ ๐ถ PROPN cana-4595 35 30 = = PUNCT cana-4595 36 1 [ [ X cana-4595 36 2 ๐‘0 ๐‘0 NOUN cana-4595 36 3 , , PUNCT cana-4595 36 4 ๐‘1 ๐‘1 NOUN cana-4595 36 5 , , PUNCT cana-4595 36 6 โ€ฆ โ€ฆ PUNCT cana-4595 36 7 , , PUNCT cana-4595 36 8 ๐‘๐‘›]๐‘‡ ๐‘๐‘›]๐‘‡ NOUN cana-4595 36 9 represents represent VERB cana-4595 36 10 the the DET cana-4595 36 11 vector vector NOUN cana-4595 36 12 of of ADP cana-4595 36 13 coefficients coefficient NOUN cana-4595 36 14 in in ADP cana-4595 36 15 the the DET cana-4595 36 16 polynomial polynomial ADJ cana-4595 36 17 expansion expansion NOUN cana-4595 36 18 . . PUNCT cana-4595 37 1 3 3 X cana-4595 37 2 . . X cana-4595 37 3 the the DET cana-4595 37 4 need need NOUN cana-4595 37 5 for for ADP cana-4595 37 6 a a DET cana-4595 37 7 new new ADJ cana-4595 37 8 operational operational ADJ cana-4595 37 9 matrix matrix NOUN cana-4595 37 10 existing exist VERB cana-4595 37 11 numerical numerical ADJ cana-4595 37 12 methods method NOUN cana-4595 37 13 , , PUNCT cana-4595 37 14 such such ADJ cana-4595 37 15 as as ADP cana-4595 37 16 legendre legendre PROPN cana-4595 37 17 , , PUNCT cana-4595 37 18 chebyshev chebyshev PROPN cana-4595 37 19 , , PUNCT cana-4595 37 20 and and CCONJ cana-4595 37 21 laguerre laguerre NOUN cana-4595 37 22 polynomial polynomial ADJ cana-4595 37 23 - - PUNCT cana-4595 37 24 based base VERB cana-4595 37 25 approaches approach NOUN cana-4595 37 26 , , PUNCT cana-4595 37 27 often often ADV cana-4595 37 28 struggle struggle VERB cana-4595 37 29 with with ADP cana-4595 37 30 high high ADJ cana-4595 37 31 computational computational ADJ cana-4595 37 32 complexity complexity NOUN cana-4595 37 33 and and CCONJ cana-4595 37 34 numerical numerical ADJ cana-4595 37 35 instability instability NOUN cana-4595 37 36 in in ADP cana-4595 37 37 solving solve VERB cana-4595 37 38 fdes fde NOUN cana-4595 37 39 [ [ X cana-4595 37 40 4 4 NUM cana-4595 37 41 ] ] PUNCT cana-4595 37 42 . . PUNCT cana-4595 38 1 to to PART cana-4595 38 2 address address VERB cana-4595 38 3 these these DET cana-4595 38 4 challenges challenge NOUN cana-4595 38 5 , , PUNCT cana-4595 38 6 we we PRON cana-4595 38 7 propose propose VERB cana-4595 38 8 a a DET cana-4595 38 9 new new ADJ cana-4595 38 10 gnocchi gnocchi NOUN cana-4595 38 11 polynomial polynomial ADJ cana-4595 38 12 - - PUNCT cana-4595 38 13 based base VERB cana-4595 38 14 operational operational ADJ cana-4595 38 15 matrix matrix NOUN cana-4595 38 16 that that PRON cana-4595 38 17 : : PUNCT cana-4595 38 18 โ€ข โ€ข NOUN cana-4595 38 19 efficiently efficiently ADV cana-4595 38 20 approximates approximate VERB cana-4595 38 21 fractional fractional ADJ cana-4595 38 22 derivatives derivative NOUN cana-4595 38 23 and and CCONJ cana-4595 38 24 integrals integral NOUN cana-4595 38 25 . . PUNCT cana-4595 39 1 โ€ข โ€ข NUM cana-4595 39 2 reduces reduce VERB cana-4595 39 3 computational computational ADJ cana-4595 39 4 overhead overhead NOUN cana-4595 39 5 compared compare VERB cana-4595 39 6 to to ADP cana-4595 39 7 classical classical ADJ cana-4595 39 8 polynomial polynomial ADJ cana-4595 39 9 methods method NOUN cana-4595 39 10 . . PUNCT cana-4595 40 1 โ€ข โ€ข NUM cana-4595 40 2 provides provide VERB cana-4595 40 3 better well ADJ cana-4595 40 4 convergence convergence NOUN cana-4595 40 5 for for ADP cana-4595 40 6 non non ADJ cana-4595 40 7 - - ADJ cana-4595 40 8 linear linear ADJ cana-4595 40 9 fractional fractional ADJ cana-4595 40 10 differential differential ADJ cana-4595 40 11 equations equation NOUN cana-4595 40 12 . . PUNCT cana-4595 41 1 the the DET cana-4595 41 2 operational operational ADJ cana-4595 41 3 matrix matrix NOUN cana-4595 41 4 developed develop VERB cana-4595 41 5 in in ADP cana-4595 41 6 this this DET cana-4595 41 7 work work NOUN cana-4595 41 8 provides provide VERB cana-4595 41 9 a a DET cana-4595 41 10 novel novel ADJ cana-4595 41 11 framework framework NOUN cana-4595 41 12 for for ADP cana-4595 41 13 approximating approximate VERB cana-4595 41 14 fractional fractional ADJ cana-4595 41 15 differential differential ADJ cana-4595 41 16 operators operator NOUN cana-4595 41 17 and and CCONJ cana-4595 41 18 transforming transform VERB cana-4595 41 19 non non ADJ cana-4595 41 20 - - ADJ cana-4595 41 21 linear linear ADJ cana-4595 41 22 fdes fde NOUN cana-4595 41 23 into into ADP cana-4595 41 24 solvable solvable ADJ cana-4595 41 25 algebraic algebraic ADJ cana-4595 41 26 systems system NOUN cana-4595 41 27 . . PUNCT cana-4595 42 1 2 2 X cana-4595 42 2 . . X cana-4595 42 3 mathematical mathematical ADJ cana-4595 42 4 preliminaries preliminary NOUN cana-4595 42 5 2.1 2.1 NUM cana-4595 42 6 fractional fractional ADJ cana-4595 42 7 calculus calculus NOUN cana-4595 42 8 basics basic NOUN cana-4595 42 9 fractional fractional ADJ cana-4595 42 10 calculus calculus NOUN cana-4595 42 11 ( ( PUNCT cana-4595 42 12 or or CCONJ cana-4595 42 13 sometimes sometimes ADV cana-4595 42 14 fractal fractal ADJ cana-4595 42 15 calculus calculus NOUN cana-4595 42 16 ) ) PUNCT cana-4595 42 17 is be AUX cana-4595 42 18 a a DET cana-4595 42 19 branch branch NOUN cana-4595 42 20 of of ADP cana-4595 42 21 mathematical mathematical ADJ cana-4595 42 22 analysis analysis NOUN cana-4595 42 23 that that PRON cana-4595 42 24 studies study VERB cana-4595 42 25 the the DET cana-4595 42 26 possibility possibility NOUN cana-4595 42 27 of of ADP cana-4595 42 28 taking take VERB cana-4595 42 29 an an DET cana-4595 42 30 arbitrary arbitrary ADJ cana-4595 42 31 ( ( PUNCT cana-4595 42 32 non non X cana-4595 42 33 integer integer NOUN cana-4595 42 34 ) ) PUNCT cana-4595 42 35 order order NOUN cana-4595 42 36 of of ADP cana-4595 42 37 differentiation differentiation NOUN cana-4595 42 38 or or CCONJ cana-4595 42 39 integration integration NOUN cana-4595 42 40 . . PUNCT cana-4595 43 1 in in ADP cana-4595 43 2 contrast contrast NOUN cana-4595 43 3 to to ADP cana-4595 43 4 integer integer NOUN cana-4595 43 5 order order NOUN cana-4595 43 6 derivatives derivative NOUN cana-4595 43 7 , , PUNCT cana-4595 43 8 fractional fractional ADJ cana-4595 43 9 derivatives derivative NOUN cana-4595 43 10 are be AUX cana-4595 43 11 a a DET cana-4595 43 12 more more ADV cana-4595 43 13 accurate accurate ADJ cana-4595 43 14 description description NOUN cana-4595 43 15 of of ADP cana-4595 43 16 memory memory NOUN cana-4595 43 17 and and CCONJ cana-4595 43 18 hereditary hereditary ADJ cana-4595 43 19 effects effect NOUN cana-4595 43 20 of of ADP cana-4595 43 21 complex complex ADJ cana-4595 43 22 systems system NOUN cana-4595 43 23 . . PUNCT cana-4595 44 1 fractional fractional ADJ cana-4595 44 2 integrals integral NOUN cana-4595 44 3 and and CCONJ cana-4595 44 4 derivatives derivative NOUN cana-4595 44 5 are be AUX cana-4595 44 6 defined define VERB cana-4595 44 7 formally formally ADV cana-4595 44 8 using use VERB cana-4595 44 9 integral integral ADJ cana-4595 44 10 transforms transform NOUN cana-4595 44 11 , , PUNCT cana-4595 44 12 convolution convolution NOUN cana-4595 44 13 operations operation NOUN cana-4595 44 14 and and CCONJ cana-4595 44 15 series series NOUN cana-4595 44 16 expansions expansion NOUN cana-4595 44 17 [ [ X cana-4595 44 18 5 5 NUM cana-4595 44 19 ] ] PUNCT cana-4595 44 20 . . PUNCT cana-4595 45 1 the the DET cana-4595 45 2 most most ADV cana-4595 45 3 commonly commonly ADV cana-4595 45 4 used use VERB cana-4595 45 5 definitions definition NOUN cana-4595 45 6 in in ADP cana-4595 45 7 fractional fractional ADJ cana-4595 45 8 calculus calculus NOUN cana-4595 45 9 are be AUX cana-4595 45 10 riemann riemann NOUN cana-4595 45 11 - - PUNCT cana-4595 45 12 liouville liouville PROPN cana-4595 45 13 , , PUNCT cana-4595 45 14 caputo caputo PROPN cana-4595 45 15 and and CCONJ cana-4595 45 16 grรผnwald grรผnwald NOUN cana-4595 45 17 - - PUNCT cana-4595 45 18 letnikov letnikov NOUN cana-4595 45 19 derivatives derivative NOUN cana-4595 45 20 . . PUNCT cana-4595 46 1 2.1.1 2.1.1 NUM cana-4595 46 2 definition definition NOUN cana-4595 46 3 of of ADP cana-4595 46 4 fractional fractional ADJ cana-4595 46 5 integrals integral NOUN cana-4595 46 6 the the DET cana-4595 46 7 fractional fractional ADJ cana-4595 46 8 integral integral NOUN cana-4595 46 9 of of ADP cana-4595 46 10 order order NOUN cana-4595 46 11 ๐›ผ ๐›ผ VERB cana-4595 46 12 > > X cana-4595 46 13 0 0 NUM cana-4595 46 14 of of ADP cana-4595 46 15 a a DET cana-4595 46 16 function function NOUN cana-4595 46 17 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 46 18 ) ) PUNCT cana-4595 46 19 is be AUX cana-4595 46 20 given give VERB cana-4595 46 21 by by ADP cana-4595 46 22 the the DET cana-4595 46 23 riemann riemann PROPN cana-4595 46 24 - - PUNCT cana-4595 46 25 liouville liouville VERB cana-4595 46 26 fractional fractional ADJ cana-4595 46 27 integral integral ADJ cana-4595 46 28 : : PUNCT cana-4595 46 29 ๐ผ๐›ผ๐‘“(๐‘ฅ ๐ผ๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 46 30 ) ) PUNCT cana-4595 46 31 = = SYM cana-4595 46 32 1 1 NUM cana-4595 46 33 ฮณ(๐›ผ ฮณ(๐›ผ NUM cana-4595 46 34 ) ) PUNCT cana-4595 46 35 โˆซ โˆซ PROPN cana-4595 46 36 โ€Š โ€Š SPACE cana-4595 47 1 ๐‘ฅ ๐‘ฅ PROPN cana-4595 47 2 0 0 PUNCT cana-4595 48 1 ( ( PUNCT cana-4595 48 2 ๐‘ฅ ๐‘ฅ PRON cana-4595 48 3 โˆ’ โˆ’ NOUN cana-4595 48 4 ๐‘ก)๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก ๐‘ก)๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก NOUN cana-4595 48 5 , , PUNCT cana-4595 48 6 ๐‘ฅ ๐‘ฅ X cana-4595 48 7 > > X cana-4595 48 8 0 0 NUM cana-4595 48 9 where where SCONJ cana-4595 48 10 ฮณ(๐›ผ ฮณ(๐›ผ NUM cana-4595 48 11 ) ) PUNCT cana-4595 48 12 is be AUX cana-4595 48 13 the the DET cana-4595 48 14 gamma gamma PROPN cana-4595 48 15 function function NOUN cana-4595 48 16 , , PUNCT cana-4595 48 17 defined define VERB cana-4595 48 18 as as ADP cana-4595 48 19 : : PUNCT cana-4595 48 20 communications communication NOUN cana-4595 48 21 on on ADP cana-4595 48 22 applied apply VERB cana-4595 48 23 nonlinear nonlinear ADJ cana-4595 48 24 analysis analysis NOUN cana-4595 48 25 issn issn NOUN cana-4595 48 26 : : PUNCT cana-4595 48 27 1074 1074 NUM cana-4595 48 28 - - PUNCT cana-4595 48 29 133x 133x NUM cana-4595 48 30 vol vol NOUN cana-4595 48 31 32 32 NUM cana-4595 48 32 no no NOUN cana-4595 48 33 . . PUNCT cana-4595 49 1 9s 9s NUM cana-4595 49 2 ( ( PUNCT cana-4595 49 3 2025 2025 NUM cana-4595 49 4 ) ) PUNCT cana-4595 49 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 49 6 2971 2971 NUM cana-4595 49 7 ฮณ(๐›ผ ฮณ(๐›ผ NUM cana-4595 49 8 ) ) PUNCT cana-4595 49 9 = = SYM cana-4595 50 1 โˆซ โˆซ AUX cana-4595 50 2 โ€Š โ€Š SPACE cana-4595 50 3 โˆž โˆž PROPN cana-4595 50 4 0 0 NUM cana-4595 50 5 ๐‘ก๐›ผโˆ’1๐‘’โˆ’๐‘ก๐‘‘๐‘ก ๐‘ก๐›ผโˆ’1๐‘’โˆ’๐‘ก๐‘‘๐‘ก NOUN cana-4595 50 6 fractional fractional ADJ cana-4595 50 7 integrals integral NOUN cana-4595 50 8 serve serve VERB cana-4595 50 9 as as ADP cana-4595 50 10 a a DET cana-4595 50 11 foundation foundation NOUN cana-4595 50 12 for for ADP cana-4595 50 13 defining define VERB cana-4595 50 14 fractional fractional ADJ cana-4595 50 15 derivatives derivative NOUN cana-4595 50 16 , , PUNCT cana-4595 50 17 extending extend VERB cana-4595 50 18 traditional traditional ADJ cana-4595 50 19 calculus calculus NOUN cana-4595 50 20 operators operator NOUN cana-4595 50 21 to to ADP cana-4595 50 22 non non ADJ cana-4595 50 23 - - ADJ cana-4595 50 24 integer integer ADJ cana-4595 50 25 orders order NOUN cana-4595 50 26 [ [ X cana-4595 50 27 6 6 NUM cana-4595 50 28 ] ] PUNCT cana-4595 50 29 . . PUNCT cana-4595 51 1 2.1.2 2.1.2 NUM cana-4595 51 2 fractional fractional ADJ cana-4595 51 3 derivatives derivative NOUN cana-4595 51 4 the the DET cana-4595 51 5 three three NUM cana-4595 51 6 primary primary ADJ cana-4595 51 7 definitions definition NOUN cana-4595 51 8 of of ADP cana-4595 51 9 fractional fractional ADJ cana-4595 51 10 differentiation differentiation NOUN cana-4595 51 11 are be AUX cana-4595 51 12 : : PUNCT cana-4595 51 13 1 1 X cana-4595 51 14 . . X cana-4595 51 15 riemann riemann PROPN cana-4595 51 16 - - PUNCT cana-4595 51 17 liouville liouville VERB cana-4595 51 18 fractional fractional ADJ cana-4595 51 19 derivative derivative NOUN cana-4595 51 20 the the DET cana-4595 51 21 riemann riemann PROPN cana-4595 51 22 - - PUNCT cana-4595 51 23 liouville liouville VERB cana-4595 51 24 fractional fractional ADJ cana-4595 51 25 derivative derivative NOUN cana-4595 51 26 of of ADP cana-4595 51 27 order order NOUN cana-4595 51 28 ๐›ผ ๐›ผ NOUN cana-4595 51 29 is be AUX cana-4595 51 30 given give VERB cana-4595 51 31 by by ADP cana-4595 51 32 : : PUNCT cana-4595 51 33 ๐ท๐›ผ๐‘“(๐‘ฅ ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 51 34 ) ) PUNCT cana-4595 52 1 = = PRON cana-4595 52 2 ๐‘‘๐‘› ๐‘‘๐‘› AUX cana-4595 52 3 ๐‘‘๐‘ฅ๐‘› ๐‘‘๐‘ฅ๐‘› VERB cana-4595 52 4 [ [ PUNCT cana-4595 52 5 1 1 NUM cana-4595 52 6 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-4595 52 7 โˆ’ โˆ’ NUM cana-4595 52 8 ๐›ผ ๐›ผ NOUN cana-4595 52 9 ) ) PUNCT cana-4595 52 10 โˆซ โˆซ PROPN cana-4595 52 11 โ€Š โ€Š SPACE cana-4595 53 1 ๐‘ฅ ๐‘ฅ PROPN cana-4595 53 2 0 0 NUM cana-4595 53 3 โ€Š โ€Š SPACE cana-4595 53 4 ( ( PUNCT cana-4595 53 5 ๐‘ฅ ๐‘ฅ NOUN cana-4595 53 6 โˆ’ โˆ’ NOUN cana-4595 53 7 ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘ก)๐‘‘๐‘ก NOUN cana-4595 53 8 ] ] PUNCT cana-4595 53 9 where where SCONJ cana-4595 53 10 ๐‘› ๐‘› ADJ cana-4595 53 11 = = PRON cana-4595 53 12 โŒˆ๐›ผโŒ‰ โŒˆ๐›ผโŒ‰ NOUN cana-4595 53 13 is be AUX cana-4595 53 14 the the DET cana-4595 53 15 ceiling ceiling NOUN cana-4595 53 16 function function NOUN cana-4595 53 17 of of ADP cana-4595 53 18 ๐›ผ. ๐›ผ. NOUN cana-4595 53 19 this this DET cana-4595 53 20 definition definition NOUN cana-4595 53 21 is be AUX cana-4595 53 22 widely widely ADV cana-4595 53 23 used use VERB cana-4595 53 24 in in ADP cana-4595 53 25 theoretical theoretical ADJ cana-4595 53 26 analysis analysis NOUN cana-4595 53 27 but but CCONJ cana-4595 53 28 is be AUX cana-4595 53 29 less less ADV cana-4595 53 30 practical practical ADJ cana-4595 53 31 for for ADP cana-4595 53 32 initial initial ADJ cana-4595 53 33 - - PUNCT cana-4595 53 34 value value NOUN cana-4595 53 35 problems problem NOUN cana-4595 53 36 since since SCONJ cana-4595 53 37 it it PRON cana-4595 53 38 does do AUX cana-4595 53 39 not not PART cana-4595 53 40 accommodate accommodate VERB cana-4595 53 41 standard standard ADJ cana-4595 53 42 boundary boundary ADJ cana-4595 53 43 conditions condition NOUN cana-4595 54 1 [ [ X cana-4595 54 2 7 7 NUM cana-4595 54 3 ] ] PUNCT cana-4595 54 4 . . PUNCT cana-4595 55 1 2 2 X cana-4595 55 2 . . X cana-4595 55 3 caputo caputo PROPN cana-4595 55 4 fractional fractional PROPN cana-4595 55 5 derivative derivative PROPN cana-4595 55 6 the the DET cana-4595 55 7 caputo caputo PROPN cana-4595 55 8 derivative derivative PROPN cana-4595 55 9 modifies modify VERB cana-4595 55 10 the the DET cana-4595 55 11 riemann riemann PROPN cana-4595 55 12 - - PUNCT cana-4595 55 13 liouville liouville VERB cana-4595 55 14 derivative derivative NOUN cana-4595 55 15 by by ADP cana-4595 55 16 shifting shift VERB cana-4595 55 17 the the DET cana-4595 55 18 differentiation differentiation NOUN cana-4595 55 19 inside inside ADP cana-4595 55 20 the the DET cana-4595 55 21 integral integral ADJ cana-4595 55 22 : : PUNCT cana-4595 55 23 ๐ถ๐ท๐›ผ๐‘“(๐‘ฅ ๐ถ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 55 24 ) ) PUNCT cana-4595 55 25 = = SYM cana-4595 55 26 1 1 NUM cana-4595 55 27 ฮณ(๐‘› ฮณ(๐‘› PROPN cana-4595 55 28 โˆ’ โˆ’ ADP cana-4595 55 29 ๐›ผ ๐›ผ NOUN cana-4595 55 30 ) ) PUNCT cana-4595 55 31 โˆซ โˆซ PROPN cana-4595 55 32 โ€Š โ€Š SPACE cana-4595 56 1 ๐‘ฅ ๐‘ฅ PROPN cana-4595 56 2 0 0 PUNCT cana-4595 57 1 ( ( PUNCT cana-4595 57 2 ๐‘ฅ ๐‘ฅ NOUN cana-4595 57 3 โˆ’ โˆ’ NOUN cana-4595 57 4 ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘›)(๐‘ก)๐‘‘๐‘ก ๐‘ก)๐‘›โˆ’๐›ผโˆ’1๐‘“(๐‘›)(๐‘ก)๐‘‘๐‘ก PROPN cana-4595 57 5 the the DET cana-4595 57 6 caputo caputo PROPN cana-4595 57 7 derivative derivative NOUN cana-4595 57 8 is be AUX cana-4595 57 9 often often ADV cana-4595 57 10 preferred prefer VERB cana-4595 57 11 in in ADP cana-4595 57 12 physical physical ADJ cana-4595 57 13 applications application NOUN cana-4595 57 14 because because SCONJ cana-4595 57 15 it it PRON cana-4595 57 16 allows allow VERB cana-4595 57 17 for for ADP cana-4595 57 18 the the DET cana-4595 57 19 use use NOUN cana-4595 57 20 of of ADP cana-4595 57 21 classical classical ADJ cana-4595 57 22 initial initial ADJ cana-4595 57 23 conditions condition NOUN cana-4595 57 24 , , PUNCT cana-4595 57 25 unlike unlike ADP cana-4595 57 26 the the DET cana-4595 57 27 riemann riemann PROPN cana-4595 57 28 - - PUNCT cana-4595 57 29 liouville liouville VERB cana-4595 57 30 approach approach NOUN cana-4595 57 31 [ [ X cana-4595 57 32 8 8 NUM cana-4595 57 33 ] ] PUNCT cana-4595 57 34 . . PUNCT cana-4595 58 1 3 3 X cana-4595 58 2 . . X cana-4595 58 3 grรผnwald grรผnwald NOUN cana-4595 58 4 - - PUNCT cana-4595 58 5 letnikov letnikov NOUN cana-4595 58 6 fractional fractional ADJ cana-4595 58 7 derivative derivative NOUN cana-4595 58 8 the the DET cana-4595 58 9 grรผnwald grรผnwald NOUN cana-4595 58 10 - - PUNCT cana-4595 58 11 letnikov letnikov NOUN cana-4595 58 12 derivative derivative NOUN cana-4595 58 13 is be AUX cana-4595 58 14 a a DET cana-4595 58 15 discrete discrete ADJ cana-4595 58 16 approximation approximation NOUN cana-4595 58 17 of of ADP cana-4595 58 18 fractional fractional ADJ cana-4595 58 19 differentiation differentiation NOUN cana-4595 58 20 : : PUNCT cana-4595 58 21 ๐ท๐›ผ๐‘“(๐‘ฅ ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 58 22 ) ) PUNCT cana-4595 59 1 = = SYM cana-4595 59 2 lim lim PROPN cana-4595 59 3 โ„Žโ†’0 โ„Žโ†’0 PUNCT cana-4595 59 4 โ€Š โ€Š SPACE cana-4595 59 5 1 1 NUM cana-4595 59 6 โ„Ž๐›ผ โ„Ž๐›ผ INTJ cana-4595 59 7 โˆ‘ โˆ‘ DET cana-4595 59 8 โ€Š โ€Š SPACE cana-4595 59 9 โˆž โˆž PROPN cana-4595 59 10 ๐‘˜=0 ๐‘˜=0 PROPN cana-4595 59 11 ( ( PUNCT cana-4595 59 12 โˆ’1)๐‘˜ โˆ’1)๐‘˜ X cana-4595 59 13 ( ( PUNCT cana-4595 59 14 ๐›ผ ๐›ผ NOUN cana-4595 59 15 ๐‘˜ ๐‘˜ NOUN cana-4595 59 16 ) ) PUNCT cana-4595 59 17 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 59 18 โˆ’ โˆ’ NUM cana-4595 59 19 ๐‘˜โ„Ž ๐‘˜โ„Ž NOUN cana-4595 59 20 ) ) PUNCT cana-4595 59 21 this this DET cana-4595 59 22 definition definition NOUN cana-4595 59 23 is be AUX cana-4595 59 24 widely widely ADV cana-4595 59 25 used use VERB cana-4595 59 26 in in ADP cana-4595 59 27 numerical numerical ADJ cana-4595 59 28 analysis analysis NOUN cana-4595 59 29 , , PUNCT cana-4595 59 30 particularly particularly ADV cana-4595 59 31 in in ADP cana-4595 59 32 fractional fractional ADJ cana-4595 59 33 difference difference NOUN cana-4595 59 34 equations equation NOUN cana-4595 59 35 and and CCONJ cana-4595 59 36 computational computational ADJ cana-4595 59 37 methods method NOUN cana-4595 59 38 [ [ X cana-4595 59 39 9 9 NUM cana-4595 59 40 ] ] PUNCT cana-4595 59 41 . . PUNCT cana-4595 60 1 2.1.3 2.1.3 NUM cana-4595 60 2 properties property NOUN cana-4595 60 3 of of ADP cana-4595 60 4 fractional fractional ADJ cana-4595 60 5 operators operator NOUN cana-4595 60 6 fractional fractional ADJ cana-4595 60 7 differentiation differentiation NOUN cana-4595 60 8 exhibits exhibit VERB cana-4595 60 9 several several ADJ cana-4595 60 10 distinct distinct ADJ cana-4595 60 11 properties property NOUN cana-4595 60 12 that that PRON cana-4595 60 13 differ differ VERB cana-4595 60 14 from from ADP cana-4595 60 15 classical classical ADJ cana-4595 60 16 integer integer NOUN cana-4595 60 17 - - PUNCT cana-4595 60 18 order order NOUN cana-4595 60 19 derivatives derivative NOUN cana-4595 60 20 : : PUNCT cana-4595 60 21 โ€ข โ€ข NUM cana-4595 60 22 linearity linearity NOUN cana-4595 60 23 : : PUNCT cana-4595 60 24 if if SCONJ cana-4595 60 25 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 60 26 ) ) PUNCT cana-4595 60 27 and and CCONJ cana-4595 60 28 ๐‘”(๐‘ฅ ๐‘”(๐‘ฅ NUM cana-4595 60 29 ) ) PUNCT cana-4595 60 30 are be AUX cana-4595 60 31 functions function NOUN cana-4595 60 32 , , PUNCT cana-4595 60 33 then then ADV cana-4595 60 34 : : PUNCT cana-4595 60 35 ๐ท๐›ผ(๐‘Ž๐‘“(๐‘ฅ ๐ท๐›ผ(๐‘Ž๐‘“(๐‘ฅ NOUN cana-4595 60 36 ) ) PUNCT cana-4595 61 1 + + NUM cana-4595 61 2 ๐‘๐‘”(๐‘ฅ ๐‘๐‘”(๐‘ฅ NUM cana-4595 61 3 ) ) PUNCT cana-4595 61 4 ) ) PUNCT cana-4595 62 1 = = SYM cana-4595 62 2 ๐‘Ž๐ท๐›ผ๐‘“(๐‘ฅ ๐‘Ž๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 62 3 ) ) PUNCT cana-4595 62 4 + + CCONJ cana-4595 62 5 ๐‘๐ท๐›ผ๐‘”(๐‘ฅ ๐‘๐ท๐›ผ๐‘”(๐‘ฅ ADJ cana-4595 62 6 ) ) PUNCT cana-4595 62 7 โ€ข โ€ข NUM cana-4595 62 8 semigroup semigroup NOUN cana-4595 62 9 property property NOUN cana-4595 62 10 : : PUNCT cana-4595 62 11 for for ADP cana-4595 62 12 any any DET cana-4595 62 13 two two NUM cana-4595 62 14 orders order NOUN cana-4595 62 15 ๐›ผ ๐›ผ NOUN cana-4595 62 16 , , PUNCT cana-4595 62 17 ๐›ฝ ๐›ฝ NOUN cana-4595 62 18 > > X cana-4595 62 19 0 0 NUM cana-4595 62 20 : : PUNCT cana-4595 62 21 ๐ท๐›ผ๐ท๐›ฝ๐‘“(๐‘ฅ ๐ท๐›ผ๐ท๐›ฝ๐‘“(๐‘ฅ X cana-4595 62 22 ) ) PUNCT cana-4595 62 23 = = SYM cana-4595 62 24 ๐ท๐›ผ+๐›ฝ๐‘“(๐‘ฅ ๐ท๐›ผ+๐›ฝ๐‘“(๐‘ฅ NOUN cana-4595 62 25 ) ) PUNCT cana-4595 62 26 โ€ข โ€ข NUM cana-4595 62 27 fractional fractional ADJ cana-4595 62 28 derivative derivative NOUN cana-4595 62 29 of of ADP cana-4595 62 30 a a DET cana-4595 62 31 power power NOUN cana-4595 62 32 function function NOUN cana-4595 62 33 : : PUNCT cana-4595 62 34 ๐ท๐›ผ๐‘ฅ๐‘ ๐ท๐›ผ๐‘ฅ๐‘ PROPN cana-4595 62 35 = = PUNCT cana-4595 62 36 ฮณ(๐‘ ฮณ(๐‘ NOUN cana-4595 62 37 + + CCONJ cana-4595 62 38 1 1 X cana-4595 62 39 ) ) PUNCT cana-4595 62 40 ฮณ(๐‘ ฮณ(๐‘ NOUN cana-4595 62 41 + + CCONJ cana-4595 62 42 1 1 NUM cana-4595 62 43 โˆ’ โˆ’ NUM cana-4595 62 44 ๐›ผ ๐›ผ NOUN cana-4595 62 45 ) ) PUNCT cana-4595 62 46 ๐‘ฅ๐‘โˆ’๐›ผ ๐‘ฅ๐‘โˆ’๐›ผ VERB cana-4595 62 47 these these DET cana-4595 62 48 properties property NOUN cana-4595 62 49 form form VERB cana-4595 62 50 the the DET cana-4595 62 51 basis basis NOUN cana-4595 62 52 for for ADP cana-4595 62 53 constructing construct VERB cana-4595 62 54 operational operational ADJ cana-4595 62 55 matrices matrix NOUN cana-4595 62 56 , , PUNCT cana-4595 62 57 which which PRON cana-4595 62 58 facilitate facilitate VERB cana-4595 62 59 numerical numerical ADJ cana-4595 62 60 approximations approximation NOUN cana-4595 62 61 of of ADP cana-4595 62 62 fractional fractional ADJ cana-4595 62 63 derivatives derivative NOUN cana-4595 62 64 . . PUNCT cana-4595 63 1 2.2 2.2 NUM cana-4595 63 2 gnocchi gnocchi NOUN cana-4595 63 3 polynomials polynomial NOUN cana-4595 63 4 : : PUNCT cana-4595 63 5 definition definition NOUN cana-4595 63 6 and and CCONJ cana-4595 63 7 properties property NOUN cana-4595 63 8 2.2.1 2.2.1 NUM cana-4595 63 9 definition definition NOUN cana-4595 63 10 of of ADP cana-4595 63 11 gnocchi gnocchi NOUN cana-4595 63 12 polynomials polynomial VERB cana-4595 63 13 the the DET cana-4595 63 14 gnocchi gnocchi NOUN cana-4595 63 15 polynomials polynomial NOUN cana-4595 63 16 , , PUNCT cana-4595 63 17 denoted denote VERB cana-4595 63 18 by by ADP cana-4595 63 19 ๐บ๐‘›(๐‘ฅ ๐บ๐‘›(๐‘ฅ NOUN cana-4595 63 20 ) ) PUNCT cana-4595 63 21 , , PUNCT cana-4595 63 22 are be AUX cana-4595 63 23 a a DET cana-4595 63 24 sequence sequence NOUN cana-4595 63 25 of of ADP cana-4595 63 26 orthogonal orthogonal ADJ cana-4595 63 27 polynomials polynomial NOUN cana-4595 63 28 that that PRON cana-4595 63 29 satisfy satisfy VERB cana-4595 63 30 the the DET cana-4595 63 31 recurrence recurrence NOUN cana-4595 63 32 relation relation NOUN cana-4595 63 33 : : PUNCT cana-4595 63 34 communications communication NOUN cana-4595 63 35 on on ADP cana-4595 63 36 applied apply VERB cana-4595 63 37 nonlinear nonlinear ADJ cana-4595 63 38 analysis analysis NOUN cana-4595 63 39 issn issn NOUN cana-4595 63 40 : : PUNCT cana-4595 63 41 1074 1074 NUM cana-4595 63 42 - - PUNCT cana-4595 63 43 133x 133x NUM cana-4595 63 44 vol vol NOUN cana-4595 63 45 32 32 NUM cana-4595 63 46 no no NOUN cana-4595 63 47 . . PUNCT cana-4595 64 1 9s 9s NUM cana-4595 64 2 ( ( PUNCT cana-4595 64 3 2025 2025 NUM cana-4595 64 4 ) ) PUNCT cana-4595 64 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 64 6 2972 2972 NUM cana-4595 64 7 ๐บ๐‘›+1(๐‘ฅ ๐บ๐‘›+1(๐‘ฅ NOUN cana-4595 64 8 ) ) PUNCT cana-4595 64 9 = = PUNCT cana-4595 65 1 ( ( PUNCT cana-4595 65 2 2๐‘› 2๐‘› PROPN cana-4595 65 3 + + CCONJ cana-4595 65 4 1)๐‘ฅ๐บ๐‘›(๐‘ฅ 1)๐‘ฅ๐บ๐‘›(๐‘ฅ NUM cana-4595 65 5 ) ) PUNCT cana-4595 65 6 โˆ’ โˆ’ NOUN cana-4595 65 7 ๐‘›๐บ๐‘›โˆ’1(๐‘ฅ ๐‘›๐บ๐‘›โˆ’1(๐‘ฅ NUM cana-4595 65 8 ) ) PUNCT cana-4595 65 9 with with ADP cana-4595 65 10 initial initial ADJ cana-4595 65 11 conditions condition NOUN cana-4595 65 12 : : PUNCT cana-4595 65 13 ๐บ0(๐‘ฅ ๐บ0(๐‘ฅ NUM cana-4595 65 14 ) ) PUNCT cana-4595 65 15 = = SYM cana-4595 65 16 1 1 NUM cana-4595 65 17 , , PUNCT cana-4595 65 18 ๐บ1(๐‘ฅ ๐บ1(๐‘ฅ NUM cana-4595 65 19 ) ) PUNCT cana-4595 66 1 = = PUNCT cana-4595 66 2 ๐‘ฅ ๐‘ฅ ADP cana-4595 66 3 these these DET cana-4595 66 4 polynomials polynomial NOUN cana-4595 66 5 play play VERB cana-4595 66 6 a a DET cana-4595 66 7 crucial crucial ADJ cana-4595 66 8 role role NOUN cana-4595 66 9 in in ADP cana-4595 66 10 approximating approximate VERB cana-4595 66 11 functions function NOUN cana-4595 66 12 and and CCONJ cana-4595 66 13 constructing construct VERB cana-4595 66 14 spectral spectral ADJ cana-4595 66 15 methods method NOUN cana-4595 66 16 for for ADP cana-4595 66 17 solving solve VERB cana-4595 66 18 differential differential ADJ cana-4595 66 19 equations equation NOUN cana-4595 66 20 [ [ X cana-4595 66 21 10 10 NUM cana-4595 66 22 ] ] PUNCT cana-4595 66 23 . . PUNCT cana-4595 67 1 2.2.2 2.2.2 NUM cana-4595 67 2 orthogonality orthogonality NOUN cana-4595 67 3 and and CCONJ cana-4595 67 4 function function NOUN cana-4595 67 5 space space NOUN cana-4595 67 6 expansion expansion NOUN cana-4595 67 7 gnocchi gnocchi NOUN cana-4595 67 8 polynomials polynomial NOUN cana-4595 67 9 form form VERB cana-4595 67 10 an an DET cana-4595 67 11 orthogonal orthogonal ADJ cana-4595 67 12 basis basis NOUN cana-4595 67 13 over over ADP cana-4595 67 14 the the DET cana-4595 67 15 interval interval NOUN cana-4595 67 16 [ [ X cana-4595 67 17 โˆ’1,1 โˆ’1,1 X cana-4595 67 18 ] ] PUNCT cana-4595 67 19 with with ADP cana-4595 67 20 respect respect NOUN cana-4595 67 21 to to ADP cana-4595 67 22 a a DET cana-4595 67 23 weight weight NOUN cana-4595 67 24 function function NOUN cana-4595 67 25 ๐‘ค(๐‘ฅ ๐‘ค(๐‘ฅ NOUN cana-4595 67 26 ) ) PUNCT cana-4595 67 27 , , PUNCT cana-4595 67 28 meaning mean VERB cana-4595 67 29 : : PUNCT cana-4595 67 30 โˆซ โˆซ PROPN cana-4595 67 31 โ€Š โ€Š SPACE cana-4595 67 32 1 1 NUM cana-4595 67 33 โˆ’1 โˆ’1 NOUN cana-4595 67 34 ๐บ๐‘š(๐‘ฅ)๐บ๐‘›(๐‘ฅ)๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ ๐บ๐‘š(๐‘ฅ)๐บ๐‘›(๐‘ฅ)๐‘ค(๐‘ฅ)๐‘‘๐‘ฅ PUNCT cana-4595 67 35 = = SYM cana-4595 67 36 0,๐‘š 0,๐‘š PROPN cana-4595 67 37 โ‰  โ‰  PROPN cana-4595 67 38 ๐‘› ๐‘› ADP cana-4595 67 39 any any DET cana-4595 67 40 sufficiently sufficiently ADV cana-4595 67 41 smooth smooth ADJ cana-4595 67 42 function function NOUN cana-4595 67 43 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 67 44 ) ) PUNCT cana-4595 67 45 can can AUX cana-4595 67 46 be be AUX cana-4595 67 47 expressed express VERB cana-4595 67 48 in in ADP cana-4595 67 49 terms term NOUN cana-4595 67 50 of of ADP cana-4595 67 51 gnocchi gnocchi NOUN cana-4595 67 52 polynomials polynomial NOUN cana-4595 67 53 as as ADP cana-4595 67 54 : : PUNCT cana-4595 67 55 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 67 56 ) ) PUNCT cana-4595 67 57 = = PUNCT cana-4595 67 58 โˆ‘ โˆ‘ PUNCT cana-4595 67 59 โ€Š โ€Š SPACE cana-4595 67 60 โˆž โˆž NUM cana-4595 67 61 ๐‘›=0 ๐‘›=0 PROPN cana-4595 67 62 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ PROPN cana-4595 67 63 ) ) PUNCT cana-4595 67 64 where where SCONJ cana-4595 67 65 ๐‘๐‘› ๐‘๐‘› PRON cana-4595 67 66 are be AUX cana-4595 67 67 expansion expansion NOUN cana-4595 67 68 coefficients coefficient NOUN cana-4595 67 69 determined determine VERB cana-4595 67 70 using use VERB cana-4595 67 71 inner inner ADJ cana-4595 67 72 product product NOUN cana-4595 67 73 projection projection NOUN cana-4595 67 74 . . PUNCT cana-4595 68 1 2.3 2.3 NUM cana-4595 68 2 introduction introduction NOUN cana-4595 68 3 to to ADP cana-4595 68 4 operational operational ADJ cana-4595 68 5 matrices matrix NOUN cana-4595 68 6 2.3.1 2.3.1 NUM cana-4595 68 7 concept concept NOUN cana-4595 68 8 of of ADP cana-4595 68 9 an an DET cana-4595 68 10 operational operational ADJ cana-4595 68 11 matrix matrix NOUN cana-4595 68 12 for for ADP cana-4595 68 13 function function NOUN cana-4595 68 14 approximation approximation NOUN cana-4595 68 15 an an DET cana-4595 68 16 operational operational ADJ cana-4595 68 17 matrix matrix NOUN cana-4595 68 18 changes change VERB cana-4595 68 19 a a DET cana-4595 68 20 function function NOUN cana-4595 68 21 expansion expansion NOUN cana-4595 68 22 into into ADP cana-4595 68 23 a a DET cana-4595 68 24 matrix matrix NOUN cana-4595 68 25 , , PUNCT cana-4595 68 26 which which PRON cana-4595 68 27 leads lead VERB cana-4595 68 28 to to ADP cana-4595 68 29 easy easy ADJ cana-4595 68 30 computation computation NOUN cana-4595 68 31 of of ADP cana-4595 68 32 derivatives derivative NOUN cana-4595 68 33 and and CCONJ cana-4595 68 34 integrals integral NOUN cana-4595 68 35 . . PUNCT cana-4595 69 1 if if SCONJ cana-4595 69 2 a a DET cana-4595 69 3 function function NOUN cana-4595 69 4 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 69 5 ) ) PUNCT cana-4595 69 6 is be AUX cana-4595 69 7 approximated approximate VERB cana-4595 69 8 by by ADP cana-4595 69 9 gnocchi gnocchi NOUN cana-4595 69 10 polynomials polynomial NOUN cana-4595 69 11 : : PUNCT cana-4595 69 12 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 69 13 ) ) PUNCT cana-4595 70 1 โ‰ˆ โ‰ˆ PROPN cana-4595 70 2 ๐ถ๐‘‡๐บ(๐‘ฅ ๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 70 3 ) ) PUNCT cana-4595 70 4 then then ADV cana-4595 70 5 its its PRON cana-4595 70 6 fractional fractional ADJ cana-4595 70 7 derivative derivative NOUN cana-4595 70 8 can can AUX cana-4595 70 9 be be AUX cana-4595 70 10 expressed express VERB cana-4595 70 11 as as ADP cana-4595 70 12 : : PUNCT cana-4595 70 13 ๐ท๐›ผ๐‘“(๐‘ฅ ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 70 14 ) ) PUNCT cana-4595 71 1 โ‰ˆ โ‰ˆ PROPN cana-4595 71 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 71 3 ) ) PUNCT cana-4595 71 4 where where SCONJ cana-4595 71 5 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 71 6 ) ) PUNCT cana-4595 71 7 is be AUX cana-4595 71 8 the the DET cana-4595 71 9 fractional fractional ADJ cana-4595 71 10 derivative derivative ADJ cana-4595 71 11 operational operational ADJ cana-4595 71 12 matrix matrix NOUN cana-4595 72 1 [ [ X cana-4595 72 2 11 11 NUM cana-4595 72 3 ] ] PUNCT cana-4595 72 4 . . PUNCT cana-4595 73 1 2.3.2 2.3.2 NUM cana-4595 73 2 review review NOUN cana-4595 73 3 of of ADP cana-4595 73 4 existing exist VERB cana-4595 73 5 polynomial polynomial ADJ cana-4595 73 6 - - PUNCT cana-4595 73 7 based base VERB cana-4595 73 8 operational operational ADJ cana-4595 73 9 matrices matrix NOUN cana-4595 73 10 several several ADJ cana-4595 73 11 polynomial polynomial ADJ cana-4595 73 12 families family NOUN cana-4595 73 13 have have AUX cana-4595 73 14 been be AUX cana-4595 73 15 used use VERB cana-4595 73 16 to to PART cana-4595 73 17 construct construct VERB cana-4595 73 18 operational operational ADJ cana-4595 73 19 matrices matrix NOUN cana-4595 73 20 : : PUNCT cana-4595 73 21 โ€ข โ€ข NUM cana-4595 73 22 legendre legendre NOUN cana-4595 73 23 polynomials polynomial NOUN cana-4595 73 24 : : PUNCT cana-4595 73 25 used use VERB cana-4595 73 26 in in ADP cana-4595 73 27 spectral spectral ADJ cana-4595 73 28 methods method NOUN cana-4595 73 29 due due ADP cana-4595 73 30 to to ADP cana-4595 73 31 their their PRON cana-4595 73 32 simplicity simplicity NOUN cana-4595 73 33 . . PUNCT cana-4595 74 1 โ€ข โ€ข NUM cana-4595 74 2 chebyshev chebyshev NOUN cana-4595 74 3 polynomials polynomial NOUN cana-4595 74 4 : : PUNCT cana-4595 74 5 common common ADJ cana-4595 74 6 in in ADP cana-4595 74 7 numerical numerical ADJ cana-4595 74 8 approximations approximation NOUN cana-4595 74 9 for for ADP cana-4595 74 10 their their PRON cana-4595 74 11 minimal minimal ADJ cana-4595 74 12 error error NOUN cana-4595 74 13 properties property NOUN cana-4595 74 14 . . PUNCT cana-4595 75 1 โ€ข โ€ข NOUN cana-4595 75 2 jacobi jacobi PROPN cana-4595 75 3 polynomials polynomial NOUN cana-4595 75 4 : : PUNCT cana-4595 75 5 useful useful ADJ cana-4595 75 6 in in ADP cana-4595 75 7 weight weight NOUN cana-4595 75 8 - - PUNCT cana-4595 75 9 function function NOUN cana-4595 75 10 - - PUNCT cana-4595 75 11 based base VERB cana-4595 75 12 approximations approximation NOUN cana-4595 75 13 . . PUNCT cana-4595 76 1 though though SCONJ cana-4595 76 2 these these DET cana-4595 76 3 approaches approach NOUN cana-4595 76 4 work work VERB cana-4595 76 5 well well ADV cana-4595 76 6 , , PUNCT cana-4595 76 7 they they PRON cana-4595 76 8 are be AUX cana-4595 76 9 numerically numerically ADV cana-4595 76 10 unstable unstable ADJ cana-4595 76 11 when when SCONJ cana-4595 76 12 applied apply VERB cana-4595 76 13 to to ADP cana-4595 76 14 fractional fractional ADJ cana-4595 76 15 differential differential ADJ cana-4595 76 16 equations equation NOUN cana-4595 76 17 . . PUNCT cana-4595 77 1 this this DET cana-4595 77 2 research research NOUN cana-4595 77 3 introduces introduce VERB cana-4595 77 4 the the DET cana-4595 77 5 gnocchi gnocchi NOUN cana-4595 77 6 polynomial polynomial PROPN cana-4595 77 7 based base VERB cana-4595 77 8 operational operational ADJ cana-4595 77 9 matrix matrix NOUN cana-4595 77 10 which which PRON cana-4595 77 11 is be AUX cana-4595 77 12 more more ADV cana-4595 77 13 stable stable ADJ cana-4595 77 14 and and CCONJ cana-4595 77 15 computationally computationally ADV cana-4595 77 16 efficient efficient ADJ cana-4595 77 17 than than ADP cana-4595 77 18 the the DET cana-4595 77 19 techniques technique NOUN cana-4595 77 20 mentioned mention VERB cana-4595 77 21 above above ADV cana-4595 77 22 . . PUNCT cana-4595 78 1 3 3 X cana-4595 78 2 . . X cana-4595 78 3 construction construction NOUN cana-4595 78 4 of of ADP cana-4595 78 5 the the DET cana-4595 78 6 new new ADJ cana-4595 78 7 operational operational ADJ cana-4595 78 8 matrix matrix NOUN cana-4595 78 9 using use VERB cana-4595 78 10 gnocchi gnocchi NOUN cana-4595 78 11 polynomials polynomial VERB cana-4595 78 12 3.1 3.1 NUM cana-4595 78 13 definition definition NOUN cana-4595 78 14 and and CCONJ cana-4595 78 15 construction construction NOUN cana-4595 78 16 3.1.1 3.1.1 NUM cana-4595 78 17 formulation formulation NOUN cana-4595 78 18 of of ADP cana-4595 78 19 the the DET cana-4595 78 20 new new ADJ cana-4595 78 21 operational operational ADJ cana-4595 78 22 matrix matrix NOUN cana-4595 78 23 for for ADP cana-4595 78 24 fractional fractional ADJ cana-4595 78 25 differentiation differentiation NOUN cana-4595 78 26 in in ADP cana-4595 78 27 this this DET cana-4595 78 28 section section NOUN cana-4595 78 29 , , PUNCT cana-4595 78 30 the the DET cana-4595 78 31 main main ADJ cana-4595 78 32 aim aim NOUN cana-4595 78 33 is be AUX cana-4595 78 34 to to PART cana-4595 78 35 develop develop VERB cana-4595 78 36 an an DET cana-4595 78 37 operational operational ADJ cana-4595 78 38 matrix matrix NOUN cana-4595 78 39 of of ADP cana-4595 78 40 fractional fractional ADJ cana-4595 78 41 differentiation differentiation NOUN cana-4595 78 42 by by ADP cana-4595 78 43 means mean NOUN cana-4595 78 44 of of ADP cana-4595 78 45 gnocchi gnocchi NOUN cana-4595 78 46 polynomials polynomial NOUN cana-4595 78 47 . . PUNCT cana-4595 79 1 given give VERB cana-4595 79 2 that that PRON cana-4595 79 3 gnocchi gnocchi NOUN cana-4595 79 4 polynomials polynomial NOUN cana-4595 79 5 { { PUNCT cana-4595 79 6 ๐บ๐‘›(๐‘ฅ ๐บ๐‘›(๐‘ฅ NOUN cana-4595 79 7 ) ) PUNCT cana-4595 79 8 } } PUNCT cana-4595 79 9 form form VERB cana-4595 79 10 an an DET cana-4595 79 11 orthogonal orthogonal ADJ cana-4595 79 12 basis basis NOUN cana-4595 79 13 , , PUNCT cana-4595 79 14 any any DET cana-4595 79 15 smooth smooth ADJ cana-4595 79 16 function function NOUN cana-4595 79 17 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 79 18 ) ) PUNCT cana-4595 79 19 can can AUX cana-4595 79 20 be be AUX cana-4595 79 21 approximated approximate VERB cana-4595 79 22 as as ADP cana-4595 79 23 : : PUNCT cana-4595 79 24 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 79 25 ) ) PUNCT cana-4595 80 1 โ‰ˆ โ‰ˆ PROPN cana-4595 80 2 โˆ‘ โˆ‘ PUNCT cana-4595 80 3 โ€Š โ€Š SPACE cana-4595 80 4 โˆž โˆž NUM cana-4595 80 5 ๐‘›=0 ๐‘›=0 PROPN cana-4595 80 6 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ PROPN cana-4595 80 7 ) ) PUNCT cana-4595 80 8 where where SCONJ cana-4595 80 9 ๐‘๐‘› ๐‘๐‘› PRON cana-4595 80 10 are be AUX cana-4595 80 11 the the DET cana-4595 80 12 expansion expansion NOUN cana-4595 80 13 coefficients coefficient NOUN cana-4595 80 14 determined determine VERB cana-4595 80 15 using use VERB cana-4595 80 16 inner inner ADJ cana-4595 80 17 product product NOUN cana-4595 80 18 projection projection NOUN cana-4595 80 19 . . PUNCT cana-4595 81 1 now now ADV cana-4595 81 2 , , PUNCT cana-4595 81 3 applying apply VERB cana-4595 81 4 the the DET cana-4595 81 5 fractional fractional ADJ cana-4595 81 6 derivative derivative ADJ cana-4595 81 7 operator operator NOUN cana-4595 81 8 ๐ท๐›ผ ๐ท๐›ผ NOUN cana-4595 81 9 to to ADP cana-4595 81 10 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 81 11 ) ) PUNCT cana-4595 81 12 , , PUNCT cana-4595 81 13 we we PRON cana-4595 81 14 obtain obtain VERB cana-4595 81 15 : : PUNCT cana-4595 81 16 ๐ท๐›ผ๐‘“(๐‘ฅ ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 81 17 ) ) PUNCT cana-4595 81 18 = = SYM cana-4595 82 1 ๐ท๐›ผ ๐ท๐›ผ NOUN cana-4595 82 2 โˆ‘ โˆ‘ NOUN cana-4595 82 3 โ€Š โ€Š SPACE cana-4595 82 4 โˆž โˆž NUM cana-4595 82 5 ๐‘›=0 ๐‘›=0 PROPN cana-4595 82 6 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ ADJ cana-4595 82 7 ) ) PUNCT cana-4595 82 8 communications communication NOUN cana-4595 82 9 on on ADP cana-4595 82 10 applied apply VERB cana-4595 82 11 nonlinear nonlinear ADJ cana-4595 82 12 analysis analysis NOUN cana-4595 82 13 issn issn NOUN cana-4595 82 14 : : PUNCT cana-4595 82 15 1074 1074 NUM cana-4595 82 16 - - PUNCT cana-4595 82 17 133x 133x NUM cana-4595 82 18 vol vol NOUN cana-4595 82 19 32 32 NUM cana-4595 82 20 no no NOUN cana-4595 82 21 . . PUNCT cana-4595 83 1 9s 9s NUM cana-4595 83 2 ( ( PUNCT cana-4595 83 3 2025 2025 NUM cana-4595 83 4 ) ) PUNCT cana-4595 83 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 83 6 2973 2973 NUM cana-4595 83 7 to to PART cana-4595 83 8 efficiently efficiently ADV cana-4595 83 9 compute compute VERB cana-4595 83 10 this this DET cana-4595 83 11 transformation transformation NOUN cana-4595 83 12 , , PUNCT cana-4595 83 13 we we PRON cana-4595 83 14 define define VERB cana-4595 83 15 the the DET cana-4595 83 16 fractional fractional ADJ cana-4595 83 17 differentiation differentiation NOUN cana-4595 83 18 operational operational ADJ cana-4595 83 19 matrix matrix NOUN cana-4595 83 20 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 83 21 ) ) PUNCT cana-4595 83 22 , , PUNCT cana-4595 83 23 such such ADJ cana-4595 83 24 that that SCONJ cana-4595 83 25 : : PUNCT cana-4595 83 26 ๐ท๐›ผ๐บ(๐‘ฅ ๐ท๐›ผ๐บ(๐‘ฅ X cana-4595 83 27 ) ) PUNCT cana-4595 83 28 = = SYM cana-4595 83 29 ๐‘ƒ(๐›ผ)๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐บ(๐‘ฅ NOUN cana-4595 83 30 ) ) PUNCT cana-4595 83 31 thus thus ADV cana-4595 83 32 , , PUNCT cana-4595 83 33 applying apply VERB cana-4595 83 34 ๐ท๐›ผ ๐ท๐›ผ NOUN cana-4595 83 35 to to ADP cana-4595 83 36 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 83 37 ) ) PUNCT cana-4595 83 38 , , PUNCT cana-4595 83 39 we we PRON cana-4595 83 40 get get VERB cana-4595 83 41 : : PUNCT cana-4595 83 42 ๐ท๐›ผ๐‘“(๐‘ฅ ๐ท๐›ผ๐‘“(๐‘ฅ NOUN cana-4595 83 43 ) ) PUNCT cana-4595 84 1 โ‰ˆ โ‰ˆ PROPN cana-4595 84 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 84 3 ) ) PUNCT cana-4595 84 4 where where SCONJ cana-4595 84 5 ๐ถ ๐ถ PROPN cana-4595 84 6 = = PUNCT cana-4595 85 1 [ [ X cana-4595 85 2 ๐‘0 ๐‘0 NOUN cana-4595 85 3 , , PUNCT cana-4595 85 4 ๐‘1 ๐‘1 NOUN cana-4595 85 5 , , PUNCT cana-4595 85 6 โ€ฆ โ€ฆ PUNCT cana-4595 85 7 , , PUNCT cana-4595 85 8 ๐‘๐‘›]๐‘‡ ๐‘๐‘›]๐‘‡ NOUN cana-4595 85 9 is be AUX cana-4595 85 10 the the DET cana-4595 85 11 vector vector NOUN cana-4595 85 12 of of ADP cana-4595 85 13 expansion expansion NOUN cana-4595 85 14 coefficients coefficient NOUN cana-4595 85 15 , , PUNCT cana-4595 85 16 and and CCONJ cana-4595 85 17 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ PRON cana-4595 85 18 ) ) PUNCT cana-4595 85 19 is be AUX cana-4595 85 20 the the DET cana-4595 85 21 matrix matrix NOUN cana-4595 85 22 representation representation NOUN cana-4595 85 23 of of ADP cana-4595 85 24 the the DET cana-4595 85 25 fractional fractional ADJ cana-4595 85 26 differentiation differentiation NOUN cana-4595 85 27 operator operator NOUN cana-4595 85 28 in in ADP cana-4595 85 29 the the DET cana-4595 85 30 gnocchi gnocchi NOUN cana-4595 85 31 polynomial polynomial ADJ cana-4595 85 32 basis basis NOUN cana-4595 85 33 . . PUNCT cana-4595 86 1 3.1.2 3.1.2 NUM cana-4595 86 2 step step NOUN cana-4595 86 3 - - PUNCT cana-4595 86 4 by by ADP cana-4595 86 5 - - PUNCT cana-4595 86 6 step step NOUN cana-4595 86 7 derivation derivation NOUN cana-4595 86 8 using use VERB cana-4595 86 9 gnocchi gnocchi NOUN cana-4595 86 10 polynomials polynomial NOUN cana-4595 86 11 to to PART cana-4595 86 12 derive derive VERB cana-4595 86 13 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 86 14 ) ) PUNCT cana-4595 86 15 , , PUNCT cana-4595 86 16 we we PRON cana-4595 86 17 use use VERB cana-4595 86 18 the the DET cana-4595 86 19 property property NOUN cana-4595 86 20 of of ADP cana-4595 86 21 fractional fractional ADJ cana-4595 86 22 differentiation differentiation NOUN cana-4595 86 23 on on ADP cana-4595 86 24 gnocchi gnocchi NOUN cana-4595 86 25 polynomials polynomial NOUN cana-4595 86 26 . . PUNCT cana-4595 87 1 the the DET cana-4595 87 2 fractional fractional ADJ cana-4595 87 3 derivative derivative NOUN cana-4595 87 4 of of ADP cana-4595 87 5 a a DET cana-4595 87 6 gnocchi gnocchi NOUN cana-4595 87 7 polynomial polynomial NOUN cana-4595 87 8 follows follow VERB cana-4595 87 9 the the DET cana-4595 87 10 general general ADJ cana-4595 87 11 rule rule NOUN cana-4595 87 12 : : PUNCT cana-4595 87 13 ๐ท๐›ผ๐บ๐‘›(๐‘ฅ ๐ท๐›ผ๐บ๐‘›(๐‘ฅ PROPN cana-4595 87 14 ) ) PUNCT cana-4595 88 1 = = PUNCT cana-4595 88 2 โˆ‘ โˆ‘ PUNCT cana-4595 88 3 โ€Š โ€Š SPACE cana-4595 88 4 ๐‘› ๐‘› PRON cana-4595 88 5 ๐‘š=0 ๐‘š=0 PUNCT cana-4595 88 6 ๐‘ƒ๐‘›,๐‘š ๐‘ƒ๐‘›,๐‘š NOUN cana-4595 88 7 ( ( PUNCT cana-4595 88 8 ๐›ผ ๐›ผ NOUN cana-4595 88 9 ) ) PUNCT cana-4595 88 10 ๐บ๐‘š(๐‘ฅ ๐บ๐‘š(๐‘ฅ NOUN cana-4595 88 11 ) ) PUNCT cana-4595 88 12 by by ADP cana-4595 88 13 defining define VERB cana-4595 88 14 the the DET cana-4595 88 15 matrix matrix NOUN cana-4595 88 16 elements element NOUN cana-4595 88 17 ๐‘ƒ๐‘›,๐‘š ๐‘ƒ๐‘›,๐‘š NOUN cana-4595 88 18 ( ( PUNCT cana-4595 88 19 ๐›ผ ๐›ผ NOUN cana-4595 88 20 ) ) PUNCT cana-4595 88 21 , , PUNCT cana-4595 88 22 the the DET cana-4595 88 23 operational operational ADJ cana-4595 88 24 matrix matrix NOUN cana-4595 88 25 takes take VERB cana-4595 88 26 the the DET cana-4595 88 27 form form NOUN cana-4595 88 28 : : PUNCT cana-4595 88 29 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 88 30 ) ) PUNCT cana-4595 88 31 = = SYM cana-4595 88 32 [ [ PUNCT cana-4595 88 33 ๐‘0,0 ๐‘0,0 NOUN cana-4595 88 34 ( ( PUNCT cana-4595 88 35 ๐›ผ ๐›ผ NOUN cana-4595 88 36 ) ) PUNCT cana-4595 89 1 ๐‘0,1 ๐‘0,1 NOUN cana-4595 89 2 ( ( PUNCT cana-4595 89 3 ๐›ผ ๐›ผ NOUN cana-4595 89 4 ) ) PUNCT cana-4595 89 5 ๐‘0,2 ๐‘0,2 NOUN cana-4595 89 6 ( ( PUNCT cana-4595 89 7 ๐›ผ ๐›ผ NOUN cana-4595 89 8 ) ) PUNCT cana-4595 89 9 โ€ฆ โ€ฆ PUNCT cana-4595 90 1 ๐‘1,0 ๐‘1,0 PROPN cana-4595 90 2 ( ( PUNCT cana-4595 90 3 ๐›ผ ๐›ผ NOUN cana-4595 90 4 ) ) PUNCT cana-4595 90 5 ๐‘1,1 ๐‘1,1 NOUN cana-4595 90 6 ( ( PUNCT cana-4595 90 7 ๐›ผ ๐›ผ NOUN cana-4595 90 8 ) ) PUNCT cana-4595 90 9 ๐‘1,2 ๐‘1,2 NOUN cana-4595 90 10 ( ( PUNCT cana-4595 90 11 ๐›ผ ๐›ผ NOUN cana-4595 90 12 ) ) PUNCT cana-4595 90 13 โ€ฆ โ€ฆ PUNCT cana-4595 90 14 ๐‘2,0 ๐‘2,0 PROPN cana-4595 90 15 ( ( PUNCT cana-4595 90 16 ๐›ผ ๐›ผ NOUN cana-4595 90 17 ) ) PUNCT cana-4595 90 18 ๐‘2,1 ๐‘2,1 PROPN cana-4595 90 19 ( ( PUNCT cana-4595 90 20 ๐›ผ ๐›ผ NOUN cana-4595 90 21 ) ) PUNCT cana-4595 90 22 ๐‘2,2 ๐‘2,2 PROPN cana-4595 90 23 ( ( PUNCT cana-4595 90 24 ๐›ผ ๐›ผ NOUN cana-4595 90 25 ) ) PUNCT cana-4595 90 26 โ€ฆ โ€ฆ PUNCT cana-4595 90 27 โ‹ฎ โ‹ฎ NOUN cana-4595 90 28 โ‹ฎ โ‹ฎ ADJ cana-4595 90 29 โ‹ฎ โ‹ฎ NOUN cana-4595 90 30 โ‹ฑ โ‹ฑ X cana-4595 90 31 ] ] PUNCT cana-4595 90 32 each each DET cana-4595 90 33 element element NOUN cana-4595 90 34 ๐‘๐‘›,๐‘š ๐‘๐‘›,๐‘š PUNCT cana-4595 90 35 ( ( PUNCT cana-4595 90 36 ๐›ผ ๐›ผ X cana-4595 90 37 ) ) PUNCT cana-4595 90 38 is be AUX cana-4595 90 39 determined determine VERB cana-4595 90 40 by by ADP cana-4595 90 41 fractional fractional ADJ cana-4595 90 42 integration integration NOUN cana-4595 90 43 formulas formula NOUN cana-4595 90 44 and and CCONJ cana-4595 90 45 the the DET cana-4595 90 46 recurrence recurrence NOUN cana-4595 90 47 relations relation NOUN cana-4595 90 48 of of ADP cana-4595 90 49 gnocchi gnocchi NOUN cana-4595 90 50 polynomials polynomial NOUN cana-4595 90 51 [ [ X cana-4595 90 52 1 1 NUM cana-4595 90 53 ] ] PUNCT cana-4595 90 54 . . PUNCT cana-4595 91 1 this this DET cana-4595 91 2 operational operational ADJ cana-4595 91 3 matrix matrix NOUN cana-4595 91 4 is be AUX cana-4595 91 5 used use VERB cana-4595 91 6 for for ADP cana-4595 91 7 the the DET cana-4595 91 8 computation computation NOUN cana-4595 91 9 of of ADP cana-4595 91 10 fractional fractional ADJ cana-4595 91 11 derivatives derivative NOUN cana-4595 91 12 without without ADP cana-4595 91 13 numerical numerical ADJ cana-4595 91 14 differentiation differentiation NOUN cana-4595 91 15 . . PUNCT cana-4595 92 1 3.2 3.2 NUM cana-4595 92 2 properties property NOUN cana-4595 92 3 of of ADP cana-4595 92 4 the the DET cana-4595 92 5 new new ADJ cana-4595 92 6 operational operational ADJ cana-4595 92 7 matrix matrix NOUN cana-4595 92 8 3.2.1 3.2.1 NUM cana-4595 92 9 sparsity sparsity NOUN cana-4595 92 10 and and CCONJ cana-4595 92 11 structure structure NOUN cana-4595 92 12 analysis analysis NOUN cana-4595 92 13 a a DET cana-4595 92 14 crucial crucial ADJ cana-4595 92 15 property property NOUN cana-4595 92 16 of of ADP cana-4595 92 17 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ PUNCT cana-4595 92 18 ) ) PUNCT cana-4595 92 19 is be AUX cana-4595 92 20 banded band VERB cana-4595 92 21 sparsity sparsity NOUN cana-4595 92 22 , , PUNCT cana-4595 92 23 meaning mean VERB cana-4595 92 24 that that SCONJ cana-4595 92 25 most most ADJ cana-4595 92 26 elements element NOUN cana-4595 92 27 in in ADP cana-4595 92 28 the the DET cana-4595 92 29 matrix matrix NOUN cana-4595 92 30 are be AUX cana-4595 92 31 zero zero NUM cana-4595 92 32 except except SCONJ cana-4595 92 33 for for ADP cana-4595 92 34 a a DET cana-4595 92 35 few few ADJ cana-4595 92 36 dominant dominant ADJ cana-4595 92 37 terms term NOUN cana-4595 92 38 . . PUNCT cana-4595 93 1 reduced reduce VERB cana-4595 93 2 computational computational ADJ cana-4595 93 3 complexity complexity NOUN cana-4595 93 4 when when SCONJ cana-4595 93 5 solving solve VERB cana-4595 93 6 non non ADJ cana-4595 93 7 linear linear ADJ cana-4595 93 8 fractional fractional ADJ cana-4595 93 9 differential differential NOUN cana-4595 93 10 equations equation NOUN cana-4595 93 11 is be AUX cana-4595 93 12 possible possible ADJ cana-4595 93 13 due due ADP cana-4595 93 14 to to ADP cana-4595 93 15 this this DET cana-4595 93 16 sparsity sparsity NOUN cana-4595 93 17 structure structure NOUN cana-4595 93 18 . . PUNCT cana-4595 94 1 formally formally ADV cana-4595 94 2 , , PUNCT cana-4595 94 3 the the DET cana-4595 94 4 bandwidth bandwidth ADJ cana-4595 94 5 ๐ต ๐ต NOUN cana-4595 94 6 of of ADP cana-4595 94 7 the the DET cana-4595 94 8 operational operational ADJ cana-4595 94 9 matrix matrix NOUN cana-4595 94 10 satisfies satisfie NOUN cana-4595 94 11 : : PUNCT cana-4595 94 12 ๐ต ๐ต NOUN cana-4595 94 13 = = SYM cana-4595 94 14 ๐’ช(๐›ผ๐‘› ๐’ช(๐›ผ๐‘› PROPN cana-4595 94 15 ) ) PUNCT cana-4595 94 16 where where SCONJ cana-4595 94 17 ๐‘› ๐‘› PROPN cana-4595 94 18 is be AUX cana-4595 94 19 the the DET cana-4595 94 20 polynomial polynomial ADJ cana-4595 94 21 order order NOUN cana-4595 94 22 . . PUNCT cana-4595 95 1 the the DET cana-4595 95 2 sparsity sparsity NOUN cana-4595 95 3 improves improve VERB cana-4595 95 4 as as ADP cana-4595 95 5 ๐›ผ ๐›ผ NOUN cana-4595 95 6 decreases decrease NOUN cana-4595 95 7 , , PUNCT cana-4595 95 8 leading lead VERB cana-4595 95 9 to to ADP cana-4595 95 10 a a DET cana-4595 95 11 more more ADV cana-4595 95 12 efficient efficient ADJ cana-4595 95 13 representation representation NOUN cana-4595 95 14 of of ADP cana-4595 95 15 fractional fractional ADJ cana-4595 95 16 operators operator NOUN cana-4595 95 17 [ [ X cana-4595 95 18 2 2 NUM cana-4595 95 19 ] ] PUNCT cana-4595 95 20 . . PUNCT cana-4595 96 1 3.2.2 3.2.2 NUM cana-4595 96 2 convergence convergence NOUN cana-4595 96 3 and and CCONJ cana-4595 96 4 stability stability NOUN cana-4595 96 5 of of ADP cana-4595 96 6 the the DET cana-4595 96 7 matrix matrix NOUN cana-4595 96 8 - - PUNCT cana-4595 96 9 based base VERB cana-4595 96 10 approximation approximation NOUN cana-4595 96 11 spectral spectral ADJ cana-4595 96 12 convergence convergence NOUN cana-4595 96 13 techniques technique NOUN cana-4595 96 14 are be AUX cana-4595 96 15 used use VERB cana-4595 96 16 to to PART cana-4595 96 17 analyze analyze VERB cana-4595 96 18 the the DET cana-4595 96 19 accuracy accuracy NOUN cana-4595 96 20 of of ADP cana-4595 96 21 the the DET cana-4595 96 22 operational operational ADJ cana-4595 96 23 matrix matrix NOUN cana-4595 96 24 approach approach NOUN cana-4595 96 25 . . PUNCT cana-4595 97 1 given give VERB cana-4595 97 2 a a DET cana-4595 97 3 function function NOUN cana-4595 97 4 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 97 5 ) ) PUNCT cana-4595 97 6 with with ADP cana-4595 97 7 an an DET cana-4595 97 8 exact exact ADJ cana-4595 97 9 solution solution NOUN cana-4595 97 10 ๐‘“exact ๐‘“exact NOUN cana-4595 97 11 ( ( PUNCT cana-4595 97 12 ๐‘ฅ ๐‘ฅ NOUN cana-4595 97 13 ) ) PUNCT cana-4595 97 14 , , PUNCT cana-4595 97 15 the the DET cana-4595 97 16 error error NOUN cana-4595 97 17 function function NOUN cana-4595 97 18 is be AUX cana-4595 97 19 defined define VERB cana-4595 97 20 as as ADP cana-4595 97 21 : : PUNCT cana-4595 97 22 ๐ธ(๐‘ฅ ๐ธ(๐‘ฅ NOUN cana-4595 97 23 ) ) PUNCT cana-4595 97 24 = = SYM cana-4595 97 25 โ€–๐‘“(๐‘ฅ โ€–๐‘“(๐‘ฅ NOUN cana-4595 97 26 ) ) PUNCT cana-4595 97 27 โˆ’ โˆ’ PROPN cana-4595 97 28 ๐‘“exact ๐‘“exact NOUN cana-4595 97 29 ( ( PUNCT cana-4595 97 30 ๐‘ฅ)โ€–โˆž ๐‘ฅ)โ€–โˆž ADJ cana-4595 97 31 the the DET cana-4595 97 32 gnocchi gnocchi NOUN cana-4595 97 33 polynomial polynomial ADJ cana-4595 97 34 expansion expansion NOUN cana-4595 97 35 ensures ensure VERB cana-4595 97 36 that that SCONJ cana-4595 97 37 the the DET cana-4595 97 38 approximation approximation NOUN cana-4595 97 39 error error NOUN cana-4595 97 40 decreases decrease VERB cana-4595 97 41 exponentially exponentially ADV cana-4595 97 42 as as ADP cana-4595 97 43 the the DET cana-4595 97 44 polynomial polynomial ADJ cana-4595 97 45 order order NOUN cana-4595 97 46 ๐‘› ๐‘› DET cana-4595 97 47 increases increase NOUN cana-4595 97 48 : : PUNCT cana-4595 97 49 โ€–๐ธ(๐‘ฅ)โ€–โˆž โ€–๐ธ(๐‘ฅ)โ€–โˆž NOUN cana-4595 97 50 = = SYM cana-4595 97 51 ๐’ช(๐‘’โˆ’๐‘› ๐’ช(๐‘’โˆ’๐‘› PROPN cana-4595 97 52 ) ) PUNCT cana-4595 97 53 thus thus ADV cana-4595 97 54 , , PUNCT cana-4595 97 55 it it PRON cana-4595 97 56 confirms confirm VERB cana-4595 97 57 that that SCONJ cana-4595 97 58 the the DET cana-4595 97 59 proposed propose VERB cana-4595 97 60 operational operational ADJ cana-4595 97 61 matrix matrix NOUN cana-4595 97 62 results result NOUN cana-4595 97 63 in in ADP cana-4595 97 64 exponential exponential ADJ cana-4595 97 65 convergence convergence NOUN cana-4595 97 66 when when SCONJ cana-4595 97 67 smooth smooth ADJ cana-4595 97 68 function function NOUN cana-4595 97 69 approximations approximation NOUN cana-4595 97 70 are be AUX cana-4595 97 71 considered consider VERB cana-4595 97 72 [ [ PUNCT cana-4595 97 73 3 3 NUM cana-4595 97 74 ] ] PUNCT cana-4595 97 75 . . PUNCT cana-4595 98 1 3.2.3 3.2.3 NUM cana-4595 98 2 comparison comparison NOUN cana-4595 98 3 with with ADP cana-4595 98 4 existing exist VERB cana-4595 98 5 polynomial polynomial ADJ cana-4595 98 6 - - PUNCT cana-4595 98 7 based base VERB cana-4595 98 8 operational operational ADJ cana-4595 98 9 matrices matrix NOUN cana-4595 98 10 the the DET cana-4595 98 11 newly newly ADV cana-4595 98 12 proposed propose VERB cana-4595 98 13 gnocchi gnocchi NOUN cana-4595 98 14 polynomial polynomial ADJ cana-4595 98 15 - - PUNCT cana-4595 98 16 based base VERB cana-4595 98 17 operational operational ADJ cana-4595 98 18 matrix matrix NOUN cana-4595 98 19 is be AUX cana-4595 98 20 compared compare VERB cana-4595 98 21 against against ADP cana-4595 98 22 legendre legendre PROPN cana-4595 98 23 , , PUNCT cana-4595 98 24 chebyshev chebyshev PROPN cana-4595 98 25 , , PUNCT cana-4595 98 26 and and CCONJ cana-4595 98 27 jacobi jacobi PROPN cana-4595 98 28 polynomial polynomial PROPN cana-4595 98 29 - - PUNCT cana-4595 98 30 based base VERB cana-4595 98 31 matrices matrix NOUN cana-4595 98 32 . . PUNCT cana-4595 99 1 a a DET cana-4595 99 2 comparative comparative ADJ cana-4595 99 3 analysis analysis NOUN cana-4595 99 4 shows show VERB cana-4595 99 5 that that SCONJ cana-4595 99 6 : : PUNCT cana-4595 99 7 1 1 X cana-4595 99 8 . . X cana-4595 99 9 runge runge NOUN cana-4595 99 10 's 's PART cana-4595 99 11 phenomenon phenomenon NOUN cana-4595 99 12 leads lead VERB cana-4595 99 13 to to ADP cana-4595 99 14 numerical numerical ADJ cana-4595 99 15 instability instability NOUN cana-4595 99 16 of of ADP cana-4595 99 17 chebyshev chebyshev PROPN cana-4595 99 18 based base VERB cana-4595 99 19 matrices matrix NOUN cana-4595 99 20 for for ADP cana-4595 99 21 large large ADJ cana-4595 99 22 fractional fractional ADJ cana-4595 99 23 orders order NOUN cana-4595 99 24 . . PUNCT cana-4595 100 1 communications communication NOUN cana-4595 100 2 on on ADP cana-4595 100 3 applied apply VERB cana-4595 100 4 nonlinear nonlinear ADJ cana-4595 100 5 analysis analysis NOUN cana-4595 100 6 issn issn NOUN cana-4595 100 7 : : PUNCT cana-4595 100 8 1074 1074 NUM cana-4595 100 9 - - PUNCT cana-4595 100 10 133x 133x NUM cana-4595 100 11 vol vol NOUN cana-4595 100 12 32 32 NUM cana-4595 100 13 no no NOUN cana-4595 100 14 . . PUNCT cana-4595 101 1 9s 9s NUM cana-4595 101 2 ( ( PUNCT cana-4595 101 3 2025 2025 NUM cana-4595 101 4 ) ) PUNCT cana-4595 102 1 https://internationalpubls.com https://internationalpubls.com X cana-4595 102 2 2974 2974 NUM cana-4595 102 3 2 2 NUM cana-4595 102 4 . . PUNCT cana-4595 102 5 additional additional ADJ cana-4595 102 6 weight weight NOUN cana-4595 102 7 functions function NOUN cana-4595 102 8 are be AUX cana-4595 102 9 required require VERB cana-4595 102 10 for for ADP cana-4595 102 11 accurate accurate ADJ cana-4595 102 12 approximations approximation NOUN cana-4595 102 13 of of ADP cana-4595 102 14 legendre legendre PROPN cana-4595 102 15 - - PUNCT cana-4595 102 16 based base VERB cana-4595 102 17 matrices matrix NOUN cana-4595 102 18 . . PUNCT cana-4595 103 1 3 3 X cana-4595 103 2 . . X cana-4595 103 3 while while SCONJ cana-4595 103 4 being be AUX cana-4595 103 5 more more ADV cana-4595 103 6 accurate accurate ADJ cana-4595 103 7 , , PUNCT cana-4595 103 8 jacobi jacobi PROPN cana-4595 103 9 - - PUNCT cana-4595 103 10 based base VERB cana-4595 103 11 matrices matrix NOUN cana-4595 103 12 have have VERB cana-4595 103 13 a a DET cana-4595 103 14 computational computational ADJ cana-4595 103 15 overhead overhead NOUN cana-4595 103 16 . . PUNCT cana-4595 104 1 4 4 X cana-4595 104 2 . . NUM cana-4595 104 3 matrices matrix NOUN cana-4595 104 4 based base VERB cana-4595 104 5 on on ADP cana-4595 104 6 gnocchi gnocchi NOUN cana-4595 104 7 are be AUX cana-4595 104 8 optimal optimal ADJ cana-4595 104 9 in in ADP cana-4595 104 10 terms term NOUN cana-4595 104 11 of of ADP cana-4595 104 12 sparsity sparsity NOUN cana-4595 104 13 , , PUNCT cana-4595 104 14 lower low ADJ cana-4595 104 15 computational computational ADJ cana-4595 104 16 cost cost NOUN cana-4595 104 17 and and CCONJ cana-4595 104 18 higher high ADJ cana-4595 104 19 numerical numerical ADJ cana-4595 104 20 stability stability NOUN cana-4595 104 21 , , PUNCT cana-4595 104 22 and and CCONJ cana-4595 104 23 hence hence ADV cana-4595 104 24 , , PUNCT cana-4595 104 25 suited suit VERB cana-4595 104 26 for for ADP cana-4595 104 27 solving solve VERB cana-4595 104 28 non non ADJ cana-4595 104 29 - - ADJ cana-4595 104 30 linear linear ADJ cana-4595 104 31 fractional fractional ADJ cana-4595 104 32 differential differential ADJ cana-4595 104 33 equations equation NOUN cana-4595 104 34 [ [ X cana-4595 104 35 4 4 NUM cana-4595 104 36 ] ] PUNCT cana-4595 104 37 . . PUNCT cana-4595 105 1 3.3 3.3 NUM cana-4595 105 2 analytical analytical ADJ cana-4595 105 3 proofs proof NOUN cana-4595 105 4 and and CCONJ cana-4595 105 5 theorems theorem NOUN cana-4595 105 6 3.3.1 3.3.1 NUM cana-4595 105 7 proof proof NOUN cana-4595 105 8 of of ADP cana-4595 105 9 existence existence NOUN cana-4595 105 10 and and CCONJ cana-4595 105 11 uniqueness uniqueness NOUN cana-4595 105 12 of of ADP cana-4595 105 13 the the DET cana-4595 105 14 proposed propose VERB cana-4595 105 15 method method NOUN cana-4595 105 16 we we PRON cana-4595 105 17 establish establish VERB cana-4595 105 18 the the DET cana-4595 105 19 existence existence NOUN cana-4595 105 20 and and CCONJ cana-4595 105 21 uniqueness uniqueness NOUN cana-4595 105 22 of of ADP cana-4595 105 23 the the DET cana-4595 105 24 proposed propose VERB cana-4595 105 25 method method NOUN cana-4595 105 26 using use VERB cana-4595 105 27 spectral spectral ADJ cana-4595 105 28 theory theory NOUN cana-4595 105 29 . . PUNCT cana-4595 106 1 theorem theorem ADJ cana-4595 106 2 1 1 NUM cana-4595 106 3 ( ( PUNCT cana-4595 106 4 existence existence NOUN cana-4595 106 5 and and CCONJ cana-4595 106 6 uniqueness uniqueness NOUN cana-4595 106 7 of of ADP cana-4595 106 8 the the DET cana-4595 106 9 gnocchi gnocchi NOUN cana-4595 106 10 polynomial polynomial ADJ cana-4595 106 11 expansion expansion NOUN cana-4595 106 12 ): ): PUNCT cana-4595 106 13 let let VERB cana-4595 106 14 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 106 15 ) ) PUNCT cana-4595 106 16 be be AUX cana-4595 106 17 an an DET cana-4595 106 18 analytic analytic ADJ cana-4595 106 19 function function NOUN cana-4595 106 20 on on ADP cana-4595 106 21 [ [ X cana-4595 106 22 โˆ’1,1 โˆ’1,1 NOUN cana-4595 106 23 ] ] PUNCT cana-4595 106 24 . . PUNCT cana-4595 107 1 then then ADV cana-4595 107 2 , , PUNCT cana-4595 107 3 there there PRON cana-4595 107 4 exists exist VERB cana-4595 107 5 a a DET cana-4595 107 6 unique unique ADJ cana-4595 107 7 set set NOUN cana-4595 107 8 of of ADP cana-4595 107 9 coefficients coefficient NOUN cana-4595 107 10 ๐‘๐‘› ๐‘๐‘› ADP cana-4595 107 11 such such ADJ cana-4595 107 12 that that SCONJ cana-4595 107 13 : : PUNCT cana-4595 107 14 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 107 15 ) ) PUNCT cana-4595 107 16 = = PUNCT cana-4595 107 17 โˆ‘ โˆ‘ PUNCT cana-4595 107 18 โ€Š โ€Š SPACE cana-4595 107 19 โˆž โˆž NUM cana-4595 108 1 ๐‘›=0 ๐‘›=0 PROPN cana-4595 108 2 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ ADJ cana-4595 108 3 ) ) PUNCT cana-4595 108 4 proof proof NOUN cana-4595 108 5 : : PUNCT cana-4595 108 6 any any DET cana-4595 108 7 continuous continuous ADJ cana-4595 108 8 function function NOUN cana-4595 108 9 on on ADP cana-4595 108 10 a a DET cana-4595 108 11 closed closed ADJ cana-4595 108 12 interval interval NOUN cana-4595 108 13 can can AUX cana-4595 108 14 be be AUX cana-4595 108 15 approximated approximate VERB cana-4595 108 16 by by ADP cana-4595 108 17 a a DET cana-4595 108 18 sequence sequence NOUN cana-4595 108 19 of of ADP cana-4595 108 20 polynomials polynomial NOUN cana-4595 108 21 by by ADP cana-4595 108 22 the the DET cana-4595 108 23 weierstrass weierstrass NOUN cana-4595 108 24 approximation approximation NOUN cana-4595 108 25 theorem theorem NOUN cana-4595 108 26 . . PUNCT cana-4595 109 1 since since SCONJ cana-4595 109 2 the the DET cana-4595 109 3 gnocchi gnocchi NOUN cana-4595 109 4 polynomials polynomial NOUN cana-4595 109 5 are be AUX cana-4595 109 6 complete complete ADJ cana-4595 109 7 orthonormal orthonormal ADJ cana-4595 109 8 , , PUNCT cana-4595 109 9 the the DET cana-4595 109 10 expansion expansion NOUN cana-4595 109 11 : : PUNCT cana-4595 109 12 ๐‘๐‘› ๐‘๐‘› PROPN cana-4595 110 1 = = SYM cana-4595 110 2 โˆซ โˆซ PROPN cana-4595 110 3 โ€Š โ€Š SPACE cana-4595 110 4 1 1 NUM cana-4595 110 5 โˆ’1 โˆ’1 NOUN cana-4595 110 6 โ€Š โ€Š SPACE cana-4595 110 7 ๐‘“(๐‘ฅ)๐บ๐‘›(๐‘ฅ)๐‘‘๐‘ฅ ๐‘“(๐‘ฅ)๐บ๐‘›(๐‘ฅ)๐‘‘๐‘ฅ PROPN cana-4595 110 8 โˆซ โˆซ PROPN cana-4595 110 9 โ€Š โ€Š SPACE cana-4595 110 10 1 1 NUM cana-4595 110 11 โˆ’1 โˆ’1 NOUN cana-4595 110 12 โ€Š โ€Š SPACE cana-4595 110 13 ๐บ๐‘› ๐บ๐‘› PROPN cana-4595 110 14 2(๐‘ฅ)๐‘‘๐‘ฅ 2(๐‘ฅ)๐‘‘๐‘ฅ NUM cana-4595 110 15 is be AUX cana-4595 110 16 unique unique ADJ cana-4595 110 17 , , PUNCT cana-4595 110 18 proving prove VERB cana-4595 110 19 the the DET cana-4595 110 20 theorem theorem NOUN cana-4595 110 21 . . PROPN cana-4595 111 1 3.3.2 3.3.2 NUM cana-4595 111 2 mathematical mathematical ADJ cana-4595 111 3 justifications justification NOUN cana-4595 111 4 for for ADP cana-4595 111 5 accuracy accuracy NOUN cana-4595 111 6 the the DET cana-4595 111 7 accuracy accuracy NOUN cana-4595 111 8 of of ADP cana-4595 111 9 the the DET cana-4595 111 10 gnocchi gnocchi NOUN cana-4595 111 11 - - PUNCT cana-4595 111 12 based base VERB cana-4595 111 13 operational operational ADJ cana-4595 111 14 matrix matrix NOUN cana-4595 111 15 is be AUX cana-4595 111 16 established establish VERB cana-4595 111 17 using use VERB cana-4595 111 18 an an DET cana-4595 111 19 error error NOUN cana-4595 111 20 bound bind VERB cana-4595 111 21 theorem theorem NOUN cana-4595 111 22 : : PUNCT cana-4595 111 23 theorem theorem ADJ cana-4595 111 24 2 2 NUM cana-4595 111 25 ( ( PUNCT cana-4595 111 26 error error NOUN cana-4595 111 27 bound bind VERB cana-4595 111 28 for for ADP cana-4595 111 29 gnocchi gnocchi NOUN cana-4595 111 30 polynomial polynomial ADJ cana-4595 111 31 approximation approximation NOUN cana-4595 111 32 ): ): PUNCT cana-4595 111 33 if if SCONJ cana-4595 111 34 ๐‘“(๐‘ฅ ๐‘“(๐‘ฅ NOUN cana-4595 111 35 ) ) PUNCT cana-4595 111 36 is be AUX cana-4595 111 37 ๐‘˜-times ๐‘˜-time NOUN cana-4595 111 38 continuously continuously ADV cana-4595 111 39 differentiable differentiable ADJ cana-4595 111 40 , , PUNCT cana-4595 111 41 then then ADV cana-4595 111 42 the the DET cana-4595 111 43 approximation approximation NOUN cana-4595 111 44 error error NOUN cana-4595 111 45 satisfies satisfie NOUN cana-4595 111 46 : : PUNCT cana-4595 111 47 โ€–๐‘“(๐‘ฅ โ€–๐‘“(๐‘ฅ NOUN cana-4595 111 48 ) ) PUNCT cana-4595 111 49 โˆ’ โˆ’ NOUN cana-4595 111 50 ๐‘“๐‘(๐‘ฅ)โ€–โˆž ๐‘“๐‘(๐‘ฅ)โ€–โˆž ADJ cana-4595 111 51 โ‰ค โ‰ค ADJ cana-4595 111 52 ๐‘€ ๐‘€ PROPN cana-4595 111 53 ( ( PUNCT cana-4595 111 54 ๐‘ ๐‘ PROPN cana-4595 111 55 + + NOUN cana-4595 111 56 1)๐‘˜ 1)๐‘˜ NUM cana-4595 111 57 where where SCONJ cana-4595 111 58 ๐‘€ ๐‘€ PROPN cana-4595 111 59 is be AUX cana-4595 111 60 a a DET cana-4595 111 61 constant constant ADJ cana-4595 111 62 and and CCONJ cana-4595 111 63 ๐‘ ๐‘ PROPN cana-4595 111 64 is be AUX cana-4595 111 65 the the DET cana-4595 111 66 polynomial polynomial ADJ cana-4595 111 67 order order NOUN cana-4595 111 68 . . PUNCT cana-4595 112 1 this this PRON cana-4595 112 2 confirms confirm VERB cana-4595 112 3 superpolynomial superpolynomial ADJ cana-4595 112 4 convergence convergence NOUN cana-4595 112 5 for for ADP cana-4595 112 6 sufficiently sufficiently ADV cana-4595 112 7 smooth smooth ADJ cana-4595 112 8 functions function NOUN cana-4595 112 9 [ [ X cana-4595 112 10 5 5 NUM cana-4595 112 11 ] ] PUNCT cana-4595 112 12 . . PUNCT cana-4595 113 1 4 4 X cana-4595 113 2 . . X cana-4595 113 3 application application NOUN cana-4595 113 4 to to ADP cana-4595 113 5 non non ADJ cana-4595 113 6 - - ADJ cana-4595 113 7 linear linear ADJ cana-4595 113 8 fractional fractional ADJ cana-4595 113 9 differential differential ADJ cana-4595 113 10 equations equation NOUN cana-4595 113 11 4.1 4.1 NUM cana-4595 113 12 theoretical theoretical ADJ cana-4595 113 13 formulation formulation NOUN cana-4595 113 14 fractional fractional ADJ cana-4595 113 15 differential differential NOUN cana-4595 113 16 equations equation NOUN cana-4595 113 17 ( ( PUNCT cana-4595 113 18 fdes fde NOUN cana-4595 113 19 ) ) PUNCT cana-4595 113 20 enlarge enlarge NOUN cana-4595 113 21 classical classical ADJ cana-4595 113 22 differential differential NOUN cana-4595 113 23 equations equation NOUN cana-4595 113 24 by by ADP cana-4595 113 25 fdes fde NOUN cana-4595 113 26 introduce introduce VERB cana-4595 113 27 noninteger noninteger NOUN cana-4595 113 28 order order NOUN cana-4595 113 29 derivatives derivative NOUN cana-4595 113 30 which which PRON cana-4595 113 31 can can AUX cana-4595 113 32 be be AUX cana-4595 113 33 used use VERB cana-4595 113 34 to to PART cana-4595 113 35 better well ADV cana-4595 113 36 describe describe VERB cana-4595 113 37 memory memory NOUN cana-4595 113 38 and and CCONJ cana-4595 113 39 heredity heredity NOUN cana-4595 113 40 properties property NOUN cana-4595 113 41 of of ADP cana-4595 113 42 such such ADJ cana-4595 113 43 systems system NOUN cana-4595 113 44 . . PUNCT cana-4595 114 1 a a DET cana-4595 114 2 non non ADJ cana-4595 114 3 - - ADJ cana-4595 114 4 linear linear ADJ cana-4595 114 5 fractional fractional ADJ cana-4595 114 6 differential differential NOUN cana-4595 114 7 equation equation NOUN cana-4595 114 8 ( ( PUNCT cana-4595 114 9 nfde nfde NOUN cana-4595 114 10 ) ) PUNCT cana-4595 114 11 has have VERB cana-4595 114 12 the the DET cana-4595 114 13 general general ADJ cana-4595 114 14 form form NOUN cana-4595 114 15 : : PUNCT cana-4595 114 16 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ NOUN cana-4595 114 17 ) ) PUNCT cana-4595 114 18 + + CCONJ cana-4595 114 19 ๐’ฉ(๐‘ฆ(๐‘ฅ ๐’ฉ(๐‘ฆ(๐‘ฅ NOUN cana-4595 114 20 ) ) PUNCT cana-4595 114 21 ) ) PUNCT cana-4595 115 1 = = SYM cana-4595 115 2 ๐‘”(๐‘ฅ ๐‘”(๐‘ฅ PROPN cana-4595 115 3 ) ) PUNCT cana-4595 115 4 , , PUNCT cana-4595 115 5 0 0 PUNCT cana-4595 115 6 < < X cana-4595 115 7 ๐›ผ ๐›ผ X cana-4595 115 8 < < X cana-4595 115 9 1 1 NUM cana-4595 115 10 , , PUNCT cana-4595 115 11 where where SCONJ cana-4595 115 12 ๐ท๐›ผ ๐ท๐›ผ NOUN cana-4595 115 13 represents represent VERB cana-4595 115 14 the the DET cana-4595 115 15 fractional fractional ADJ cana-4595 115 16 derivative derivative ADJ cana-4595 115 17 operator operator NOUN cana-4595 115 18 ( ( PUNCT cana-4595 115 19 caputo caputo PROPN cana-4595 115 20 or or CCONJ cana-4595 115 21 riemann riemann PROPN cana-4595 115 22 - - PUNCT cana-4595 115 23 liouville liouville NOUN cana-4595 115 24 ) ) PUNCT cana-4595 115 25 , , PUNCT cana-4595 115 26 ๐’ฉ(๐‘ฆ(๐‘ฅ ๐’ฉ(๐‘ฆ(๐‘ฅ NOUN cana-4595 115 27 ) ) PUNCT cana-4595 115 28 ) ) PUNCT cana-4595 115 29 is be AUX cana-4595 115 30 a a DET cana-4595 115 31 non non ADJ cana-4595 115 32 - - ADJ cana-4595 115 33 linear linear ADJ cana-4595 115 34 function function NOUN cana-4595 115 35 of of ADP cana-4595 115 36 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NOUN cana-4595 115 37 ) ) PUNCT cana-4595 115 38 , , PUNCT cana-4595 115 39 and and CCONJ cana-4595 115 40 ๐‘”(๐‘ฅ ๐‘”(๐‘ฅ NUM cana-4595 115 41 ) ) PUNCT cana-4595 115 42 is be AUX cana-4595 115 43 a a DET cana-4595 115 44 given give VERB cana-4595 115 45 forcing force VERB cana-4595 115 46 function function NOUN cana-4595 115 47 . . PUNCT cana-4595 116 1 to to PART cana-4595 116 2 efficiently efficiently ADV cana-4595 116 3 solve solve VERB cana-4595 116 4 this this DET cana-4595 116 5 equation equation NOUN cana-4595 116 6 , , PUNCT cana-4595 116 7 we we PRON cana-4595 116 8 approximate approximate VERB cana-4595 116 9 the the DET cana-4595 116 10 function function NOUN cana-4595 116 11 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NOUN cana-4595 116 12 ) ) PUNCT cana-4595 116 13 using use VERB cana-4595 116 14 gnocchi gnocchi NOUN cana-4595 116 15 polynomials polynomial NOUN cana-4595 116 16 : : PUNCT cana-4595 116 17 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ X cana-4595 116 18 ) ) PUNCT cana-4595 117 1 โ‰ˆ โ‰ˆ NUM cana-4595 117 2 โˆ‘ โˆ‘ PUNCT cana-4595 117 3 โ€Š โ€Š SPACE cana-4595 117 4 โˆž โˆž NUM cana-4595 118 1 ๐‘›=0 ๐‘›=0 PROPN cana-4595 118 2 ๐‘๐‘›๐บ๐‘›(๐‘ฅ ๐‘๐‘›๐บ๐‘›(๐‘ฅ PROPN cana-4595 118 3 ) ) PUNCT cana-4595 118 4 applying apply VERB cana-4595 118 5 the the DET cana-4595 118 6 newly newly ADV cana-4595 118 7 developed develop VERB cana-4595 118 8 gnocchi gnocchi NOUN cana-4595 118 9 polynomial polynomial ADJ cana-4595 118 10 - - PUNCT cana-4595 118 11 based base VERB cana-4595 118 12 operational operational ADJ cana-4595 118 13 matrix matrix NOUN cana-4595 118 14 , , PUNCT cana-4595 118 15 the the DET cana-4595 118 16 fractional fractional ADJ cana-4595 118 17 derivative derivative NOUN cana-4595 118 18 of of ADP cana-4595 118 19 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ PROPN cana-4595 118 20 ) ) PUNCT cana-4595 118 21 can can AUX cana-4595 118 22 be be AUX cana-4595 118 23 expressed express VERB cana-4595 118 24 as as ADP cana-4595 118 25 : : PUNCT cana-4595 118 26 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ NOUN cana-4595 118 27 ) ) PUNCT cana-4595 119 1 โ‰ˆ โ‰ˆ PROPN cana-4595 119 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 119 3 ) ) PUNCT cana-4595 119 4 where where SCONJ cana-4595 119 5 : : PUNCT cana-4595 119 6 โ€ข โ€ข NUM cana-4595 119 7 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 119 8 ) ) PUNCT cana-4595 119 9 is be AUX cana-4595 119 10 the the DET cana-4595 119 11 fractional fractional ADJ cana-4595 119 12 differentiation differentiation NOUN cana-4595 119 13 operational operational ADJ cana-4595 119 14 matrix matrix NOUN cana-4595 119 15 . . PUNCT cana-4595 120 1 โ€ข โ€ข NUM cana-4595 120 2 ๐ถ ๐ถ PROPN cana-4595 120 3 = = PUNCT cana-4595 121 1 [ [ X cana-4595 121 2 ๐‘0 ๐‘0 NOUN cana-4595 121 3 , , PUNCT cana-4595 121 4 ๐‘1 ๐‘1 NOUN cana-4595 121 5 , , PUNCT cana-4595 121 6 โ€ฆ โ€ฆ PUNCT cana-4595 121 7 , , PUNCT cana-4595 121 8 ๐‘๐‘›]๐‘‡ ๐‘๐‘›]๐‘‡ NOUN cana-4595 121 9 is be AUX cana-4595 121 10 the the DET cana-4595 121 11 coefficient coefficient NOUN cana-4595 121 12 vector vector NOUN cana-4595 121 13 . . PUNCT cana-4595 122 1 communications communication NOUN cana-4595 122 2 on on ADP cana-4595 122 3 applied apply VERB cana-4595 122 4 nonlinear nonlinear ADJ cana-4595 122 5 analysis analysis NOUN cana-4595 122 6 issn issn NOUN cana-4595 122 7 : : PUNCT cana-4595 122 8 1074 1074 NUM cana-4595 122 9 - - PUNCT cana-4595 122 10 133x 133x NUM cana-4595 122 11 vol vol NOUN cana-4595 122 12 32 32 NUM cana-4595 122 13 no no NOUN cana-4595 122 14 . . PUNCT cana-4595 123 1 9s 9s NUM cana-4595 123 2 ( ( PUNCT cana-4595 123 3 2025 2025 NUM cana-4595 123 4 ) ) PUNCT cana-4595 123 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 123 6 2975 2975 NUM cana-4595 123 7 โ€ข โ€ข NUM cana-4595 123 8 ๐บ(๐‘ฅ ๐บ(๐‘ฅ NOUN cana-4595 123 9 ) ) PUNCT cana-4595 123 10 represents represent VERB cana-4595 123 11 the the DET cana-4595 123 12 basis basis NOUN cana-4595 123 13 expansion expansion NOUN cana-4595 123 14 using use VERB cana-4595 123 15 gnocchi gnocchi NOUN cana-4595 123 16 polynomials polynomial NOUN cana-4595 123 17 . . PUNCT cana-4595 124 1 substituting substitute VERB cana-4595 124 2 this this DET cana-4595 124 3 approximation approximation NOUN cana-4595 124 4 into into ADP cana-4595 124 5 the the DET cana-4595 124 6 given give VERB cana-4595 124 7 nfde nfde NOUN cana-4595 124 8 results result NOUN cana-4595 124 9 in in ADP cana-4595 124 10 : : PUNCT cana-4595 124 11 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 124 12 ) ) PUNCT cana-4595 124 13 + + SYM cana-4595 124 14 ๐’ฉ(๐ถ๐‘‡๐บ(๐‘ฅ ๐’ฉ(๐ถ๐‘‡๐บ(๐‘ฅ NOUN cana-4595 124 15 ) ) PUNCT cana-4595 124 16 ) ) PUNCT cana-4595 125 1 = = SYM cana-4595 125 2 ๐‘”(๐‘ฅ ๐‘”(๐‘ฅ PROPN cana-4595 125 3 ) ) PUNCT cana-4595 125 4 this this DET cana-4595 125 5 formulation formulation NOUN cana-4595 125 6 transforms transform VERB cana-4595 125 7 the the DET cana-4595 125 8 original original ADJ cana-4595 125 9 differential differential NOUN cana-4595 125 10 equation equation NOUN cana-4595 125 11 into into ADP cana-4595 125 12 an an DET cana-4595 125 13 algebraic algebraic ADJ cana-4595 125 14 system system NOUN cana-4595 125 15 that that PRON cana-4595 125 16 can can AUX cana-4595 125 17 be be AUX cana-4595 125 18 solved solve VERB cana-4595 125 19 for for ADP cana-4595 125 20 the the DET cana-4595 125 21 unknown unknown ADJ cana-4595 125 22 coefficients coefficient NOUN cana-4595 125 23 ๐ถ. ๐ถ. PROPN cana-4595 125 24 4.2 4.2 NUM cana-4595 125 25 analytical analytical ADJ cana-4595 125 26 solution solution NOUN cana-4595 125 27 approximation approximation NOUN cana-4595 125 28 to to PART cana-4595 125 29 obtain obtain VERB cana-4595 125 30 an an DET cana-4595 125 31 explicit explicit ADJ cana-4595 125 32 solution solution NOUN cana-4595 125 33 , , PUNCT cana-4595 125 34 we we PRON cana-4595 125 35 express express VERB cana-4595 125 36 the the DET cana-4595 125 37 non non ADJ cana-4595 125 38 - - ADJ cana-4595 125 39 linear linear ADJ cana-4595 125 40 term term NOUN cana-4595 125 41 using use VERB cana-4595 125 42 a a DET cana-4595 125 43 power power NOUN cana-4595 125 44 series series NOUN cana-4595 125 45 expansion expansion NOUN cana-4595 125 46 : : PUNCT cana-4595 125 47 ๐’ฉ(๐‘ฆ(๐‘ฅ ๐’ฉ(๐‘ฆ(๐‘ฅ NOUN cana-4595 125 48 ) ) PUNCT cana-4595 125 49 ) ) PUNCT cana-4595 126 1 = = PUNCT cana-4595 126 2 โˆ‘ โˆ‘ PUNCT cana-4595 126 3 โ€Š โ€Š SPACE cana-4595 126 4 โˆž โˆž NUM cana-4595 126 5 ๐‘š=0 ๐‘š=0 PUNCT cana-4595 126 6 ๐‘Ž๐‘š(๐‘ฆ(๐‘ฅ))๐‘š. ๐‘Ž๐‘š(๐‘ฆ(๐‘ฅ))๐‘š. NOUN cana-4595 126 7 using use VERB cana-4595 126 8 the the DET cana-4595 126 9 gnocchi gnocchi NOUN cana-4595 126 10 expansion expansion NOUN cana-4595 126 11 : : PUNCT cana-4595 126 12 ( ( PUNCT cana-4595 126 13 ๐‘ฆ(๐‘ฅ))๐‘š ๐‘ฆ(๐‘ฅ))๐‘š PROPN cana-4595 126 14 โ‰ˆ โ‰ˆ PROPN cana-4595 126 15 โˆ‘ โˆ‘ PUNCT cana-4595 126 16 โ€Š โ€Š SPACE cana-4595 126 17 โˆž โˆž PROPN cana-4595 126 18 ๐‘˜=0 ๐‘˜=0 PROPN cana-4595 126 19 ๐‘‘๐‘˜๐บ๐‘˜(๐‘ฅ ๐‘‘๐‘˜๐บ๐‘˜(๐‘ฅ ADJ cana-4595 126 20 ) ) PUNCT cana-4595 126 21 we we PRON cana-4595 126 22 rewrite rewrite VERB cana-4595 126 23 the the DET cana-4595 126 24 transformed transform VERB cana-4595 126 25 nfde nfde NOUN cana-4595 126 26 as as ADP cana-4595 126 27 : : PUNCT cana-4595 126 28 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 126 29 ) ) PUNCT cana-4595 126 30 + + CCONJ cana-4595 126 31 โˆ‘ โˆ‘ PUNCT cana-4595 126 32 โ€Š โ€Š SPACE cana-4595 126 33 โˆž โˆž NUM cana-4595 126 34 ๐‘š=0 ๐‘š=0 SYM cana-4595 126 35 ๐‘Ž๐‘š ๐‘Ž๐‘š ADV cana-4595 126 36 โˆ‘ โˆ‘ ADP cana-4595 126 37 โ€Š โ€Š SPACE cana-4595 126 38 โˆž โˆž PROPN cana-4595 126 39 ๐‘˜=0 ๐‘˜=0 PROPN cana-4595 126 40 ๐‘‘๐‘˜๐บ๐‘˜(๐‘ฅ ๐‘‘๐‘˜๐บ๐‘˜(๐‘ฅ ADJ cana-4595 126 41 ) ) PUNCT cana-4595 126 42 = = PUNCT cana-4595 126 43 ๐‘”(๐‘ฅ ๐‘”(๐‘ฅ PROPN cana-4595 126 44 ) ) PUNCT cana-4595 126 45 . . PUNCT cana-4595 127 1 by by ADP cana-4595 127 2 taking take VERB cana-4595 127 3 the the DET cana-4595 127 4 inner inner ADJ cana-4595 127 5 product product NOUN cana-4595 127 6 with with ADP cana-4595 127 7 ๐บ๐‘›(๐‘ฅ ๐บ๐‘›(๐‘ฅ NOUN cana-4595 127 8 ) ) PUNCT cana-4595 127 9 and and CCONJ cana-4595 127 10 utilizing utilize VERB cana-4595 127 11 orthogonality orthogonality NOUN cana-4595 127 12 properties property NOUN cana-4595 127 13 , , PUNCT cana-4595 127 14 we we PRON cana-4595 127 15 obtain obtain VERB cana-4595 127 16 a a DET cana-4595 127 17 non non ADJ cana-4595 127 18 - - ADJ cana-4595 127 19 linear linear ADJ cana-4595 127 20 algebraic algebraic ADJ cana-4595 127 21 system system NOUN cana-4595 127 22 : : PUNCT cana-4595 127 23 ๐‘ƒ(๐›ผ)๐ถ ๐‘ƒ(๐›ผ)๐ถ X cana-4595 128 1 + + CCONJ cana-4595 128 2 โˆ‘ โˆ‘ PROPN cana-4595 128 3 โ€Š โ€Š SPACE cana-4595 128 4 โˆž โˆž NUM cana-4595 128 5 ๐‘š=0 ๐‘š=0 PUNCT cana-4595 128 6 ๐‘Ž๐‘š๐ท(๐‘š)๐ถ ๐‘Ž๐‘š๐ท(๐‘š)๐ถ ADV cana-4595 128 7 = = X cana-4595 128 8 ๐บ ๐บ NOUN cana-4595 128 9 where where SCONJ cana-4595 128 10 ๐ท(๐‘š ๐ท(๐‘š X cana-4595 128 11 ) ) PUNCT cana-4595 128 12 represents represent VERB cana-4595 128 13 the the DET cana-4595 128 14 coefficient coefficient NOUN cana-4595 128 15 transformation transformation NOUN cana-4595 128 16 matrix matrix NOUN cana-4595 128 17 for for ADP cana-4595 128 18 the the DET cana-4595 128 19 non non ADJ cana-4595 128 20 - - ADJ cana-4595 128 21 linear linear ADJ cana-4595 128 22 term term NOUN cana-4595 128 23 . . PUNCT cana-4595 129 1 this this DET cana-4595 129 2 system system NOUN cana-4595 129 3 can can AUX cana-4595 129 4 be be AUX cana-4595 129 5 solved solve VERB cana-4595 129 6 iteratively iteratively ADV cana-4595 129 7 using use VERB cana-4595 129 8 newton newton PROPN cana-4595 129 9 's 's PART cana-4595 129 10 method method NOUN cana-4595 129 11 or or CCONJ cana-4595 129 12 other other ADJ cana-4595 129 13 numerical numerical ADJ cana-4595 129 14 techniques technique NOUN cana-4595 129 15 . . PUNCT cana-4595 130 1 4.3 4.3 NUM cana-4595 130 2 error error NOUN cana-4595 130 3 analysis analysis NOUN cana-4595 130 4 and and CCONJ cana-4595 130 5 convergence convergence NOUN cana-4595 130 6 to to PART cana-4595 130 7 ensure ensure VERB cana-4595 130 8 the the DET cana-4595 130 9 validity validity NOUN cana-4595 130 10 of of ADP cana-4595 130 11 our our PRON cana-4595 130 12 method method NOUN cana-4595 130 13 , , PUNCT cana-4595 130 14 we we PRON cana-4595 130 15 analyze analyze VERB cana-4595 130 16 the the DET cana-4595 130 17 theoretical theoretical ADJ cana-4595 130 18 error error NOUN cana-4595 130 19 bounds bound NOUN cana-4595 130 20 . . PUNCT cana-4595 131 1 4.3.1 4.3.1 NUM cana-4595 131 2 error error NOUN cana-4595 131 3 bound bind VERB cana-4595 131 4 for for ADP cana-4595 131 5 gnocchi gnocchi NOUN cana-4595 131 6 polynomial polynomial ADJ cana-4595 131 7 approximation approximation NOUN cana-4595 131 8 for for ADP cana-4595 131 9 a a DET cana-4595 131 10 function function NOUN cana-4595 131 11 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NOUN cana-4595 131 12 ) ) PUNCT cana-4595 131 13 that that PRON cana-4595 131 14 is be AUX cana-4595 131 15 ๐‘˜-times ๐‘˜-time NOUN cana-4595 131 16 differentiable differentiable ADJ cana-4595 131 17 , , PUNCT cana-4595 131 18 the the DET cana-4595 131 19 error error NOUN cana-4595 131 20 in in ADP cana-4595 131 21 the the DET cana-4595 131 22 gnocchi gnocchi NOUN cana-4595 131 23 polynomial polynomial ADJ cana-4595 131 24 approximation approximation NOUN cana-4595 131 25 satisfies satisfie NOUN cana-4595 131 26 : : PUNCT cana-4595 131 27 โ€–๐‘ฆ(๐‘ฅ โ€–๐‘ฆ(๐‘ฅ X cana-4595 131 28 ) ) PUNCT cana-4595 131 29 โˆ’ โˆ’ PROPN cana-4595 131 30 ๐‘ฆ๐‘(๐‘ฅ)โ€–โˆž ๐‘ฆ๐‘(๐‘ฅ)โ€–โˆž NOUN cana-4595 131 31 โ‰ค โ‰ค PUNCT cana-4595 131 32 ๐‘€ ๐‘€ PROPN cana-4595 131 33 ( ( PUNCT cana-4595 131 34 ๐‘ ๐‘ PROPN cana-4595 131 35 + + NOUN cana-4595 131 36 1)๐‘˜ 1)๐‘˜ NUM cana-4595 131 37 where where SCONJ cana-4595 131 38 ๐‘€ ๐‘€ PROPN cana-4595 131 39 is be AUX cana-4595 131 40 a a DET cana-4595 131 41 constant constant ADJ cana-4595 131 42 dependent dependent NOUN cana-4595 131 43 on on ADP cana-4595 131 44 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NOUN cana-4595 131 45 ) ) PUNCT cana-4595 131 46 and and CCONJ cana-4595 131 47 ๐‘ ๐‘ PROPN cana-4595 131 48 is be AUX cana-4595 131 49 the the DET cana-4595 131 50 truncation truncation NOUN cana-4595 131 51 order order NOUN cana-4595 131 52 . . PUNCT cana-4595 132 1 this this DET cana-4595 132 2 result result NOUN cana-4595 132 3 guarantees guarantee VERB cana-4595 132 4 superpolynomial superpolynomial ADJ cana-4595 132 5 convergence convergence NOUN cana-4595 132 6 . . PUNCT cana-4595 133 1 4.3.2 4.3.2 NUM cana-4595 133 2 convergence convergence NOUN cana-4595 133 3 theorem theorem NOUN cana-4595 133 4 and and CCONJ cana-4595 133 5 proof proof NOUN cana-4595 133 6 theorem theorem VERB cana-4595 133 7 1 1 NUM cana-4595 133 8 ( ( PUNCT cana-4595 133 9 convergence convergence NOUN cana-4595 133 10 of of ADP cana-4595 133 11 gnocchi gnocchi NOUN cana-4595 133 12 - - PUNCT cana-4595 133 13 based base VERB cana-4595 133 14 operational operational ADJ cana-4595 133 15 matrix matrix NOUN cana-4595 133 16 ): ): PUNCT cana-4595 133 17 let let VERB cana-4595 133 18 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ X cana-4595 133 19 ) ) PUNCT cana-4595 133 20 be be AUX cana-4595 133 21 a a DET cana-4595 133 22 sufficiently sufficiently ADV cana-4595 133 23 smooth smooth ADJ cana-4595 133 24 function function NOUN cana-4595 133 25 , , PUNCT cana-4595 133 26 and and CCONJ cana-4595 133 27 let let VERB cana-4595 133 28 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ VERB cana-4595 133 29 ) ) PUNCT cana-4595 133 30 be be AUX cana-4595 133 31 the the DET cana-4595 133 32 corresponding corresponding ADJ cana-4595 133 33 operational operational ADJ cana-4595 133 34 matrix matrix NOUN cana-4595 133 35 . . PUNCT cana-4595 134 1 then then ADV cana-4595 134 2 , , PUNCT cana-4595 134 3 the the DET cana-4595 134 4 approximation approximation NOUN cana-4595 134 5 : : PUNCT cana-4595 134 6 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ NOUN cana-4595 134 7 ) ) PUNCT cana-4595 135 1 โ‰ˆ โ‰ˆ PROPN cana-4595 135 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 135 3 ) ) PUNCT cana-4595 135 4 converges converge VERB cana-4595 135 5 in in ADP cana-4595 135 6 the the DET cana-4595 135 7 ๐ฟ2 ๐ฟ2 NOUN cana-4595 135 8 - - PUNCT cana-4595 135 9 norm norm NOUN cana-4595 135 10 , , PUNCT cana-4595 135 11 with with ADP cana-4595 135 12 the the DET cana-4595 135 13 error error NOUN cana-4595 135 14 bound bind VERB cana-4595 135 15 : : PUNCT cana-4595 135 16 โ€–๐ท๐›ผ๐‘ฆ(๐‘ฅ โ€–๐ท๐›ผ๐‘ฆ(๐‘ฅ NUM cana-4595 135 17 ) ) PUNCT cana-4595 135 18 โˆ’ โˆ’ ADP cana-4595 135 19 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ)โ€– ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ)โ€– ADP cana-4595 135 20 2 2 NUM cana-4595 135 21 โ‰ค โ‰ค NUM cana-4595 135 22 ๐’ช(๐‘’โˆ’๐‘ ๐’ช(๐‘’โˆ’๐‘ PROPN cana-4595 135 23 ) ) PUNCT cana-4595 135 24 . . PUNCT cana-4595 136 1 proof proof NOUN cana-4595 136 2 : : PUNCT cana-4595 136 3 using use VERB cana-4595 136 4 the the DET cana-4595 136 5 orthogonality orthogonality NOUN cana-4595 136 6 property property NOUN cana-4595 136 7 of of ADP cana-4595 136 8 gnocchi gnocchi NOUN cana-4595 136 9 polynomials polynomial NOUN cana-4595 136 10 , , PUNCT cana-4595 136 11 we we PRON cana-4595 136 12 express express VERB cana-4595 136 13 the the DET cana-4595 136 14 residual residual ADJ cana-4595 136 15 error error NOUN cana-4595 136 16 : : PUNCT cana-4595 136 17 ๐ธ(๐‘ฅ ๐ธ(๐‘ฅ NOUN cana-4595 136 18 ) ) PUNCT cana-4595 136 19 = = PUNCT cana-4595 136 20 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ PROPN cana-4595 136 21 ) ) PUNCT cana-4595 136 22 โˆ’ โˆ’ ADP cana-4595 136 23 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 136 24 ) ) PUNCT cana-4595 136 25 taking take VERB cana-4595 136 26 the the DET cana-4595 136 27 ๐ฟ2 ๐ฟ2 NOUN cana-4595 136 28 - - PUNCT cana-4595 136 29 norm norm NOUN cana-4595 136 30 and and CCONJ cana-4595 136 31 applying apply VERB cana-4595 136 32 spectral spectral ADJ cana-4595 136 33 approximation approximation NOUN cana-4595 136 34 results result NOUN cana-4595 136 35 , , PUNCT cana-4595 136 36 we we PRON cana-4595 136 37 obtain obtain VERB cana-4595 136 38 : : PUNCT cana-4595 136 39 โ€–๐ธ(๐‘ฅ)โ€–2 โ€–๐ธ(๐‘ฅ)โ€–2 NUM cana-4595 136 40 โ‰ค โ‰ค NUM cana-4595 136 41 ๐’ช(๐‘’โˆ’๐‘ ๐’ช(๐‘’โˆ’๐‘ X cana-4595 136 42 ) ) PUNCT cana-4595 136 43 communications communication NOUN cana-4595 136 44 on on ADP cana-4595 136 45 applied apply VERB cana-4595 136 46 nonlinear nonlinear ADJ cana-4595 136 47 analysis analysis NOUN cana-4595 136 48 issn issn NOUN cana-4595 136 49 : : PUNCT cana-4595 136 50 1074 1074 NUM cana-4595 136 51 - - PUNCT cana-4595 136 52 133x 133x NUM cana-4595 136 53 vol vol NOUN cana-4595 136 54 32 32 NUM cana-4595 136 55 no no NOUN cana-4595 136 56 . . PUNCT cana-4595 137 1 9s 9s NUM cana-4595 137 2 ( ( PUNCT cana-4595 137 3 2025 2025 NUM cana-4595 137 4 ) ) PUNCT cana-4595 137 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 137 6 2976 2976 NUM cana-4595 137 7 which which PRON cana-4595 137 8 confirms confirm VERB cana-4595 137 9 exponential exponential ADJ cana-4595 137 10 convergence convergence NOUN cana-4595 137 11 . . PUNCT cana-4595 138 1 4.3.3 4.3.3 ADJ cana-4595 138 2 stability stability NOUN cana-4595 138 3 of of ADP cana-4595 138 4 the the DET cana-4595 138 5 proposed propose VERB cana-4595 138 6 operational operational ADJ cana-4595 138 7 matrix matrix NOUN cana-4595 138 8 approach approach VERB cana-4595 138 9 the the DET cana-4595 138 10 stability stability NOUN cana-4595 138 11 of of ADP cana-4595 138 12 the the DET cana-4595 138 13 proposed propose VERB cana-4595 138 14 gnocchi gnocchi NOUN cana-4595 138 15 - - PUNCT cana-4595 138 16 based base VERB cana-4595 138 17 operational operational ADJ cana-4595 138 18 matrix matrix NOUN cana-4595 138 19 is be AUX cana-4595 138 20 analyzed analyze VERB cana-4595 138 21 using use VERB cana-4595 138 22 spectral spectral ADJ cana-4595 138 23 condition condition NOUN cana-4595 138 24 numbers number NOUN cana-4595 138 25 . . PUNCT cana-4595 139 1 the the DET cana-4595 139 2 spectral spectral ADJ cana-4595 139 3 radius radius NOUN cana-4595 139 4 ๐œŒ(๐‘ƒ(๐›ผ ๐œŒ(๐‘ƒ(๐›ผ NUM cana-4595 139 5 ) ) PUNCT cana-4595 139 6 ) ) PUNCT cana-4595 139 7 satisfies satisfie NOUN cana-4595 139 8 : : PUNCT cana-4595 139 9 ๐œŒ(๐‘ƒ(๐›ผ ๐œŒ(๐‘ƒ(๐›ผ NUM cana-4595 139 10 ) ) PUNCT cana-4595 139 11 ) ) PUNCT cana-4595 140 1 < < X cana-4595 140 2 โˆž โˆž NUM cana-4595 140 3 implying implying ADJ cana-4595 140 4 numerical numerical ADJ cana-4595 140 5 stability stability NOUN cana-4595 140 6 for for ADP cana-4595 140 7 well well ADV cana-4595 140 8 - - PUNCT cana-4595 140 9 conditioned condition VERB cana-4595 140 10 problems problem NOUN cana-4595 140 11 . . PUNCT cana-4595 141 1 5 5 X cana-4595 141 2 . . X cana-4595 141 3 comparative comparative ADJ cana-4595 141 4 analysis analysis NOUN cana-4595 141 5 with with ADP cana-4595 141 6 existing exist VERB cana-4595 141 7 theoretical theoretical ADJ cana-4595 141 8 approaches approach NOUN cana-4595 141 9 5.1 5.1 NUM cana-4595 141 10 mathematical mathematical ADJ cana-4595 141 11 differences difference NOUN cana-4595 141 12 from from ADP cana-4595 141 13 other other ADJ cana-4595 141 14 operational operational ADJ cana-4595 141 15 matrices matrix NOUN cana-4595 141 16 the the DET cana-4595 141 17 construction construction NOUN cana-4595 141 18 of of ADP cana-4595 141 19 operational operational ADJ cana-4595 141 20 matrices matrix NOUN cana-4595 141 21 for for ADP cana-4595 141 22 fractional fractional ADJ cana-4595 141 23 differentiation differentiation NOUN cana-4595 141 24 has have AUX cana-4595 141 25 been be AUX cana-4595 141 26 widely widely ADV cana-4595 141 27 studied study VERB cana-4595 141 28 using use VERB cana-4595 141 29 various various ADJ cana-4595 141 30 polynomial polynomial ADJ cana-4595 141 31 bases basis NOUN cana-4595 141 32 , , PUNCT cana-4595 141 33 including include VERB cana-4595 141 34 legendre legendre PROPN cana-4595 141 35 , , PUNCT cana-4595 141 36 chebyshev chebyshev PROPN cana-4595 141 37 , , PUNCT cana-4595 141 38 and and CCONJ cana-4595 141 39 jacobi jacobi PROPN cana-4595 141 40 polynomials polynomial NOUN cana-4595 141 41 . . PUNCT cana-4595 142 1 however however ADV cana-4595 142 2 , , PUNCT cana-4595 142 3 the the DET cana-4595 142 4 newly newly ADV cana-4595 142 5 developed develop VERB cana-4595 142 6 gnocchi gnocchi NOUN cana-4595 142 7 polynomial polynomial ADJ cana-4595 142 8 - - PUNCT cana-4595 142 9 based base VERB cana-4595 142 10 operational operational ADJ cana-4595 142 11 matrix matrix NOUN cana-4595 142 12 introduces introduce VERB cana-4595 142 13 several several ADJ cana-4595 142 14 key key ADJ cana-4595 142 15 differences difference NOUN cana-4595 142 16 that that PRON cana-4595 142 17 improve improve VERB cana-4595 142 18 computational computational ADJ cana-4595 142 19 efficiency efficiency NOUN cana-4595 142 20 and and CCONJ cana-4595 142 21 numerical numerical ADJ cana-4595 142 22 stability stability NOUN cana-4595 142 23 . . PUNCT cana-4595 143 1 5.1.1 5.1.1 NUM cana-4595 143 2 comparison comparison NOUN cana-4595 143 3 with with ADP cana-4595 143 4 legendre legendre PROPN cana-4595 143 5 polynomial polynomial PROPN cana-4595 143 6 - - PUNCT cana-4595 143 7 based base VERB cana-4595 143 8 operational operational ADJ cana-4595 143 9 matrices matrix NOUN cana-4595 143 10 legendre legendre PROPN cana-4595 143 11 polynomials polynomials PROPN cana-4595 143 12 ๐‘ƒ๐‘›(๐‘ฅ ๐‘ƒ๐‘›(๐‘ฅ NOUN cana-4595 143 13 ) ) PUNCT cana-4595 143 14 are be AUX cana-4595 143 15 widely widely ADV cana-4595 143 16 used use VERB cana-4595 143 17 in in ADP cana-4595 143 18 spectral spectral ADJ cana-4595 143 19 approximations approximation NOUN cana-4595 143 20 due due ADP cana-4595 143 21 to to ADP cana-4595 143 22 their their PRON cana-4595 143 23 orthogonality orthogonality NOUN cana-4595 143 24 on on ADP cana-4595 143 25 the the DET cana-4595 143 26 interval interval NOUN cana-4595 144 1 [ [ X cana-4595 144 2 โˆ’1,1 โˆ’1,1 X cana-4595 144 3 ] ] X cana-4595 144 4 . . PUNCT cana-4595 145 1 the the DET cana-4595 145 2 operational operational ADJ cana-4595 145 3 matrix matrix NOUN cana-4595 145 4 of of ADP cana-4595 145 5 fractional fractional ADJ cana-4595 145 6 differentiation differentiation NOUN cana-4595 145 7 constructed construct VERB cana-4595 145 8 using use VERB cana-4595 145 9 legendre legendre PROPN cana-4595 145 10 polynomials polynomial NOUN cana-4595 145 11 follows follow VERB cana-4595 145 12 the the DET cana-4595 145 13 form form NOUN cana-4595 145 14 : : PUNCT cana-4595 145 15 ๐ท๐›ผ๐‘ƒ๐‘›(๐‘ฅ ๐ท๐›ผ๐‘ƒ๐‘›(๐‘ฅ PROPN cana-4595 145 16 ) ) PUNCT cana-4595 146 1 = = PUNCT cana-4595 146 2 โˆ‘ โˆ‘ PUNCT cana-4595 146 3 โ€Š โ€Š SPACE cana-4595 146 4 ๐‘› ๐‘› PRON cana-4595 146 5 ๐‘š=0 ๐‘š=0 PUNCT cana-4595 146 6 ๐ฟ๐‘›,๐‘š ๐ฟ๐‘›,๐‘š NOUN cana-4595 146 7 ( ( PUNCT cana-4595 146 8 ๐›ผ ๐›ผ NOUN cana-4595 146 9 ) ) PUNCT cana-4595 146 10 ๐‘ƒ๐‘š(๐‘ฅ ๐‘ƒ๐‘š(๐‘ฅ NOUN cana-4595 146 11 ) ) PUNCT cana-4595 146 12 where where SCONJ cana-4595 146 13 ๐ฟ๐‘›,๐‘š ๐ฟ๐‘›,๐‘š NOUN cana-4595 146 14 ( ( PUNCT cana-4595 146 15 ๐›ผ ๐›ผ X cana-4595 146 16 ) ) PUNCT cana-4595 146 17 are be AUX cana-4595 146 18 the the DET cana-4595 146 19 transformation transformation NOUN cana-4595 146 20 coefficients coefficient NOUN cana-4595 146 21 . . PUNCT cana-4595 147 1 however however ADV cana-4595 147 2 , , PUNCT cana-4595 147 3 the the DET cana-4595 147 4 legendre legendre NOUN cana-4595 147 5 - - PUNCT cana-4595 147 6 based base VERB cana-4595 147 7 approach approach NOUN cana-4595 147 8 suffers suffer VERB cana-4595 147 9 from from ADP cana-4595 147 10 the the DET cana-4595 147 11 following following ADJ cana-4595 147 12 issues issue NOUN cana-4595 147 13 : : PUNCT cana-4595 147 14 โ€ข โ€ข NUM cana-4595 147 15 requires require VERB cana-4595 147 16 additional additional ADJ cana-4595 147 17 weighting weighting NOUN cana-4595 147 18 functions function NOUN cana-4595 147 19 to to PART cana-4595 147 20 approximate approximate VERB cana-4595 147 21 fractional fractional ADJ cana-4595 147 22 derivatives derivative NOUN cana-4595 147 23 accurately accurately ADV cana-4595 147 24 . . PUNCT cana-4595 148 1 โ€ข โ€ข ADP cana-4595 148 2 the the DET cana-4595 148 3 coefficients coefficient NOUN cana-4595 148 4 ๐ฟ๐‘›,๐‘š ๐ฟ๐‘›,๐‘š NOUN cana-4595 148 5 ( ( PUNCT cana-4595 148 6 ๐›ผ ๐›ผ X cana-4595 148 7 ) ) PUNCT cana-4595 148 8 do do AUX cana-4595 148 9 not not PART cana-4595 148 10 exhibit exhibit VERB cana-4595 148 11 sparsity sparsity NOUN cana-4595 148 12 , , PUNCT cana-4595 148 13 leading lead VERB cana-4595 148 14 to to ADP cana-4595 148 15 higher high ADJ cana-4595 148 16 computational computational ADJ cana-4595 148 17 cost cost NOUN cana-4595 148 18 . . PUNCT cana-4595 149 1 โ€ข โ€ข NOUN cana-4595 149 2 prone prone ADJ cana-4595 149 3 to to ADP cana-4595 149 4 ill ill ADJ cana-4595 149 5 - - PUNCT cana-4595 149 6 conditioning conditioning NOUN cana-4595 149 7 in in ADP cana-4595 149 8 higher high ADJ cana-4595 149 9 - - PUNCT cana-4595 149 10 order order NOUN cana-4595 149 11 approximations approximation NOUN cana-4595 149 12 . . PUNCT cana-4595 150 1 in in ADP cana-4595 150 2 contrast contrast NOUN cana-4595 150 3 , , PUNCT cana-4595 150 4 the the DET cana-4595 150 5 gnocchi gnocchi NOUN cana-4595 150 6 - - PUNCT cana-4595 150 7 based base VERB cana-4595 150 8 operational operational ADJ cana-4595 150 9 matrix matrix NOUN cana-4595 150 10 offers offer VERB cana-4595 150 11 an an DET cana-4595 150 12 inherently inherently ADV cana-4595 150 13 sparse sparse ADJ cana-4595 150 14 structure structure NOUN cana-4595 150 15 , , PUNCT cana-4595 150 16 reducing reduce VERB cana-4595 150 17 computational computational ADJ cana-4595 150 18 overhead overhead NOUN cana-4595 150 19 and and CCONJ cana-4595 150 20 improving improve VERB cana-4595 150 21 numerical numerical ADJ cana-4595 150 22 efficiency efficiency NOUN cana-4595 150 23 . . PUNCT cana-4595 151 1 5.1.2 5.1.2 NUM cana-4595 151 2 comparison comparison NOUN cana-4595 151 3 with with ADP cana-4595 151 4 chebyshev chebyshev NOUN cana-4595 151 5 polynomial polynomial ADJ cana-4595 151 6 - - PUNCT cana-4595 151 7 based base VERB cana-4595 151 8 operational operational ADJ cana-4595 151 9 matrices matrix NOUN cana-4595 151 10 chebyshev chebyshev NOUN cana-4595 151 11 polynomials polynomial NOUN cana-4595 151 12 ๐‘‡๐‘›(๐‘ฅ ๐‘‡๐‘›(๐‘ฅ NOUN cana-4595 151 13 ) ) PUNCT cana-4595 151 14 are be AUX cana-4595 151 15 another another DET cana-4595 151 16 popular popular ADJ cana-4595 151 17 choice choice NOUN cana-4595 151 18 due due ADP cana-4595 151 19 to to ADP cana-4595 151 20 their their PRON cana-4595 151 21 minimization minimization NOUN cana-4595 151 22 of of ADP cana-4595 151 23 runge runge NOUN cana-4595 151 24 's 's PART cana-4595 151 25 phenomenon phenomenon NOUN cana-4595 151 26 . . PUNCT cana-4595 152 1 the the DET cana-4595 152 2 fractional fractional ADJ cana-4595 152 3 differentiation differentiation NOUN cana-4595 152 4 operational operational ADJ cana-4595 152 5 matrix matrix NOUN cana-4595 152 6 using use VERB cana-4595 152 7 chebyshev chebyshev NOUN cana-4595 152 8 polynomials polynomial NOUN cana-4595 152 9 is be AUX cana-4595 152 10 defined define VERB cana-4595 152 11 as as ADP cana-4595 152 12 : : PUNCT cana-4595 152 13 ๐ท๐›ผ๐‘‡๐‘›(๐‘ฅ ๐ท๐›ผ๐‘‡๐‘›(๐‘ฅ PROPN cana-4595 152 14 ) ) PUNCT cana-4595 153 1 = = PUNCT cana-4595 153 2 โˆ‘ โˆ‘ PUNCT cana-4595 153 3 โ€Š โ€Š SPACE cana-4595 153 4 ๐‘› ๐‘› DET cana-4595 153 5 ๐‘š=0 ๐‘š=0 X cana-4595 153 6 ๐ถ๐‘›,๐‘š ๐ถ๐‘›,๐‘š PROPN cana-4595 153 7 ( ( PUNCT cana-4595 153 8 ๐›ผ ๐›ผ NOUN cana-4595 153 9 ) ) PUNCT cana-4595 153 10 ๐‘‡๐‘š(๐‘ฅ ๐‘‡๐‘š(๐‘ฅ NOUN cana-4595 153 11 ) ) PUNCT cana-4595 153 12 however however ADV cana-4595 153 13 , , PUNCT cana-4595 153 14 chebyshev chebyshev PROPN cana-4595 153 15 polynomial polynomial ADJ cana-4595 153 16 - - PUNCT cana-4595 153 17 based base VERB cana-4595 153 18 approaches approach NOUN cana-4595 153 19 exhibit exhibit VERB cana-4595 153 20 : : PUNCT cana-4595 153 21 โ€ข โ€ข NUM cana-4595 153 22 oscillatory oscillatory ADJ cana-4595 153 23 behavior behavior NOUN cana-4595 153 24 near near ADP cana-4595 153 25 the the DET cana-4595 153 26 boundaries boundary NOUN cana-4595 153 27 , , PUNCT cana-4595 153 28 leading lead VERB cana-4595 153 29 to to ADP cana-4595 153 30 accuracy accuracy NOUN cana-4595 153 31 loss loss NOUN cana-4595 153 32 . . PUNCT cana-4595 154 1 โ€ข โ€ข NUM cana-4595 154 2 poor poor ADJ cana-4595 154 3 performance performance NOUN cana-4595 154 4 in in ADP cana-4595 154 5 approximating approximate VERB cana-4595 154 6 non non ADJ cana-4595 154 7 - - ADJ cana-4595 154 8 smooth smooth ADJ cana-4595 154 9 functions function NOUN cana-4595 154 10 . . PUNCT cana-4595 155 1 โ€ข โ€ข NUM cana-4595 155 2 high high ADJ cana-4595 155 3 sensitivity sensitivity NOUN cana-4595 155 4 to to PART cana-4595 155 5 round round VERB cana-4595 155 6 - - PUNCT cana-4595 155 7 off off ADP cana-4595 155 8 errors error NOUN cana-4595 155 9 . . PUNCT cana-4595 156 1 the the DET cana-4595 156 2 gnocchi gnocchi NOUN cana-4595 156 3 - - PUNCT cana-4595 156 4 based base VERB cana-4595 156 5 method method NOUN cana-4595 156 6 mitigates mitigate VERB cana-4595 156 7 these these DET cana-4595 156 8 challenges challenge NOUN cana-4595 156 9 by by ADP cana-4595 156 10 ensuring ensure VERB cana-4595 156 11 smooth smooth ADJ cana-4595 156 12 function function NOUN cana-4595 156 13 approximation approximation NOUN cana-4595 156 14 while while SCONJ cana-4595 156 15 maintaining maintain VERB cana-4595 156 16 computational computational ADJ cana-4595 156 17 efficiency efficiency NOUN cana-4595 156 18 . . PUNCT cana-4595 157 1 5.1.3 5.1.3 NUM cana-4595 157 2 comparison comparison NOUN cana-4595 157 3 with with ADP cana-4595 157 4 jacobi jacobi PROPN cana-4595 157 5 polynomial polynomial PROPN cana-4595 157 6 - - PUNCT cana-4595 157 7 based base VERB cana-4595 157 8 operational operational ADJ cana-4595 157 9 matrices matrix NOUN cana-4595 157 10 jacobi jacobi NOUN cana-4595 157 11 polynomials polynomials PROPN cana-4595 158 1 ๐ฝ๐‘› ๐ฝ๐‘› PROPN cana-4595 158 2 ( ( PUNCT cana-4595 158 3 ๐‘Ž,๐‘ ๐‘Ž,๐‘ NOUN cana-4595 158 4 ) ) PUNCT cana-4595 158 5 ( ( PUNCT cana-4595 158 6 ๐‘ฅ ๐‘ฅ X cana-4595 158 7 ) ) PUNCT cana-4595 158 8 generalize generalize VERB cana-4595 158 9 legendre legendre PROPN cana-4595 158 10 and and CCONJ cana-4595 158 11 chebyshev chebyshev NOUN cana-4595 158 12 polynomials polynomial NOUN cana-4595 158 13 with with ADP cana-4595 158 14 additional additional ADJ cana-4595 158 15 shape shape NOUN cana-4595 158 16 parameters parameter NOUN cana-4595 158 17 ( ( PUNCT cana-4595 158 18 ๐‘Ž ๐‘Ž X cana-4595 158 19 , , PUNCT cana-4595 158 20 ๐‘ ๐‘ NOUN cana-4595 158 21 ) ) PUNCT cana-4595 158 22 . . PUNCT cana-4595 159 1 the the DET cana-4595 159 2 fractional fractional ADJ cana-4595 159 3 differentiation differentiation NOUN cana-4595 159 4 matrix matrix NOUN cana-4595 159 5 is be AUX cana-4595 159 6 given give VERB cana-4595 159 7 by by ADP cana-4595 159 8 : : PUNCT cana-4595 159 9 communications communication NOUN cana-4595 159 10 on on ADP cana-4595 159 11 applied apply VERB cana-4595 159 12 nonlinear nonlinear ADJ cana-4595 159 13 analysis analysis NOUN cana-4595 159 14 issn issn NOUN cana-4595 159 15 : : PUNCT cana-4595 159 16 1074 1074 NUM cana-4595 159 17 - - PUNCT cana-4595 159 18 133x 133x NUM cana-4595 159 19 vol vol NOUN cana-4595 159 20 32 32 NUM cana-4595 159 21 no no NOUN cana-4595 159 22 . . PUNCT cana-4595 160 1 9s 9s NUM cana-4595 160 2 ( ( PUNCT cana-4595 160 3 2025 2025 NUM cana-4595 160 4 ) ) PUNCT cana-4595 161 1 https://internationalpubls.com https://internationalpubls.com X cana-4595 161 2 2977 2977 NUM cana-4595 161 3 ๐ท๐›ผ๐ฝ๐‘› ๐ท๐›ผ๐ฝ๐‘› PROPN cana-4595 161 4 ( ( PUNCT cana-4595 161 5 ๐‘Ž,๐‘ ๐‘Ž,๐‘ NOUN cana-4595 161 6 ) ) PUNCT cana-4595 161 7 ( ( PUNCT cana-4595 161 8 ๐‘ฅ ๐‘ฅ NOUN cana-4595 161 9 ) ) PUNCT cana-4595 161 10 = = PUNCT cana-4595 161 11 โˆ‘ โˆ‘ PUNCT cana-4595 161 12 โ€Š โ€Š SPACE cana-4595 161 13 ๐‘› ๐‘› PRON cana-4595 161 14 ๐‘š=0 ๐‘š=0 PUNCT cana-4595 161 15 ๐ฝ๐‘›,๐‘š ๐ฝ๐‘›,๐‘š NOUN cana-4595 161 16 ( ( PUNCT cana-4595 161 17 ๐›ผ ๐›ผ X cana-4595 161 18 ) ) PUNCT cana-4595 161 19 ๐ฝ๐‘š ๐ฝ๐‘š PROPN cana-4595 161 20 ( ( PUNCT cana-4595 161 21 ๐‘Ž,๐‘ ๐‘Ž,๐‘ NOUN cana-4595 161 22 ) ) PUNCT cana-4595 161 23 ( ( PUNCT cana-4595 161 24 ๐‘ฅ ๐‘ฅ NOUN cana-4595 161 25 ) ) PUNCT cana-4595 161 26 despite despite SCONJ cana-4595 161 27 their their PRON cana-4595 161 28 flexibility flexibility NOUN cana-4595 161 29 , , PUNCT cana-4595 161 30 jacobi jacobi PROPN cana-4595 161 31 polynomial polynomial PROPN cana-4595 161 32 - - PUNCT cana-4595 161 33 based base VERB cana-4595 161 34 operational operational ADJ cana-4595 161 35 matrices matrix NOUN cana-4595 161 36 : : PUNCT cana-4595 161 37 โ€ข โ€ข NUM cana-4595 161 38 require require VERB cana-4595 161 39 tuning tuning NOUN cana-4595 161 40 of of ADP cana-4595 161 41 ๐‘Ž ๐‘Ž NOUN cana-4595 161 42 , , PUNCT cana-4595 161 43 ๐‘ ๐‘ DET cana-4595 161 44 parameters parameter NOUN cana-4595 161 45 for for ADP cana-4595 161 46 optimal optimal ADJ cana-4595 161 47 performance performance NOUN cana-4595 161 48 . . PUNCT cana-4595 162 1 โ€ข โ€ข NUM cana-4595 162 2 are be AUX cana-4595 162 3 computationally computationally ADV cana-4595 162 4 expensive expensive ADJ cana-4595 162 5 due due ADP cana-4595 162 6 to to ADP cana-4595 162 7 the the DET cana-4595 162 8 added add VERB cana-4595 162 9 complexity complexity NOUN cana-4595 162 10 of of ADP cana-4595 162 11 basis basis NOUN cana-4595 162 12 functions function NOUN cana-4595 162 13 . . PUNCT cana-4595 163 1 โ€ข โ€ข NUM cana-4595 163 2 exhibit exhibit NOUN cana-4595 163 3 instability instability NOUN cana-4595 163 4 when when SCONJ cana-4595 163 5 fractional fractional ADJ cana-4595 163 6 orders order NOUN cana-4595 163 7 are be AUX cana-4595 163 8 non non ADJ cana-4595 163 9 - - ADJ cana-4595 163 10 uniform uniform ADJ cana-4595 163 11 . . PUNCT cana-4595 164 1 the the DET cana-4595 164 2 gnocchi gnocchi NOUN cana-4595 164 3 - - PUNCT cana-4595 164 4 based base VERB cana-4595 164 5 matrix matrix NOUN cana-4595 164 6 eliminates eliminate VERB cana-4595 164 7 the the DET cana-4595 164 8 need need NOUN cana-4595 164 9 for for ADP cana-4595 164 10 parameter parameter NOUN cana-4595 164 11 tuning tuning NOUN cana-4595 164 12 while while SCONJ cana-4595 164 13 preserving preserve VERB cana-4595 164 14 computational computational ADJ cana-4595 164 15 feasibility feasibility NOUN cana-4595 164 16 and and CCONJ cana-4595 164 17 stability stability NOUN cana-4595 164 18 . . PUNCT cana-4595 165 1 5.2 5.2 NUM cana-4595 165 2 advantages advantage NOUN cana-4595 165 3 of of ADP cana-4595 165 4 the the DET cana-4595 165 5 new new ADJ cana-4595 165 6 matrix matrix NOUN cana-4595 165 7 the the DET cana-4595 165 8 gnocchi gnocchi NOUN cana-4595 165 9 polynomial polynomial ADV cana-4595 165 10 - - PUNCT cana-4595 165 11 based base VERB cana-4595 165 12 operational operational ADJ cana-4595 165 13 matrix matrix NOUN cana-4595 165 14 introduces introduce VERB cana-4595 165 15 several several ADJ cana-4595 165 16 theoretical theoretical ADJ cana-4595 165 17 advantages advantage NOUN cana-4595 165 18 over over ADP cana-4595 165 19 existing exist VERB cana-4595 165 20 polynomial polynomial ADJ cana-4595 165 21 - - PUNCT cana-4595 165 22 based base VERB cana-4595 165 23 approaches approach NOUN cana-4595 165 24 : : PUNCT cana-4595 165 25 5.2.1 5.2.1 NUM cana-4595 165 26 better well ADJ cana-4595 165 27 representation representation NOUN cana-4595 165 28 of of ADP cana-4595 165 29 fractional fractional ADJ cana-4595 165 30 operators operator NOUN cana-4595 165 31 unlike unlike ADP cana-4595 165 32 legendre legendre PROPN cana-4595 165 33 and and CCONJ cana-4595 165 34 chebyshev chebyshev PROPN cana-4595 165 35 matrices matrix NOUN cana-4595 165 36 , , PUNCT cana-4595 165 37 which which PRON cana-4595 165 38 require require VERB cana-4595 165 39 additional additional ADJ cana-4595 165 40 weight weight NOUN cana-4595 165 41 functions function NOUN cana-4595 165 42 to to PART cana-4595 165 43 approximate approximate VERB cana-4595 165 44 fractional fractional ADJ cana-4595 165 45 derivatives derivative NOUN cana-4595 165 46 , , PUNCT cana-4595 165 47 the the DET cana-4595 165 48 gnocchi gnocchi NOUN cana-4595 165 49 - - PUNCT cana-4595 165 50 based base VERB cana-4595 165 51 matrix matrix NOUN cana-4595 165 52 directly directly ADV cana-4595 165 53 encodes encode VERB cana-4595 165 54 fractional fractional ADJ cana-4595 165 55 operations operation NOUN cana-4595 165 56 within within ADP cana-4595 165 57 its its PRON cana-4595 165 58 structure structure NOUN cana-4595 165 59 . . PUNCT cana-4595 166 1 this this PRON cana-4595 166 2 ensures ensure VERB cana-4595 166 3 : : PUNCT cana-4595 166 4 โ€ข โ€ข NUM cana-4595 166 5 higher high ADJ cana-4595 166 6 accuracy accuracy NOUN cana-4595 166 7 in in ADP cana-4595 166 8 fractional fractional ADJ cana-4595 166 9 derivative derivative ADJ cana-4595 166 10 approximations approximation NOUN cana-4595 166 11 . . PUNCT cana-4595 167 1 โ€ข โ€ข NUM cana-4595 167 2 improved improve VERB cana-4595 167 3 adaptability adaptability NOUN cana-4595 167 4 to to ADP cana-4595 167 5 non non ADJ cana-4595 167 6 - - ADJ cana-4595 167 7 linear linear ADJ cana-4595 167 8 differential differential ADJ cana-4595 167 9 equations equation NOUN cana-4595 167 10 . . PUNCT cana-4595 168 1 5.2.2 5.2.2 NUM cana-4595 168 2 improved improve VERB cana-4595 168 3 sparsity sparsity NOUN cana-4595 168 4 and and CCONJ cana-4595 168 5 computational computational ADJ cana-4595 168 6 feasibility feasibility NOUN cana-4595 168 7 the the DET cana-4595 168 8 operational operational ADJ cana-4595 168 9 matrix matrix NOUN cana-4595 168 10 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ VERB cana-4595 168 11 ) ) PUNCT cana-4595 168 12 derived derive VERB cana-4595 168 13 using use VERB cana-4595 168 14 gnocchi gnocchi NOUN cana-4595 168 15 polynomials polynomial NOUN cana-4595 168 16 exhibits exhibit VERB cana-4595 168 17 a a DET cana-4595 168 18 sparse sparse ADJ cana-4595 168 19 banded band VERB cana-4595 168 20 structure structure NOUN cana-4595 168 21 , , PUNCT cana-4595 168 22 reducing reduce VERB cana-4595 168 23 computational computational ADJ cana-4595 168 24 complexity complexity NOUN cana-4595 168 25 : : PUNCT cana-4595 168 26 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ X cana-4595 168 27 ) ) PUNCT cana-4595 168 28 = = SYM cana-4595 168 29 [ [ PUNCT cana-4595 168 30 ๐‘0,0 ๐‘0,0 NOUN cana-4595 168 31 ( ( PUNCT cana-4595 168 32 ๐›ผ ๐›ผ NOUN cana-4595 168 33 ) ) PUNCT cana-4595 168 34 ๐‘0,1 ๐‘0,1 NOUN cana-4595 168 35 ( ( PUNCT cana-4595 168 36 ๐›ผ ๐›ผ NOUN cana-4595 168 37 ) ) PUNCT cana-4595 168 38 0 0 NUM cana-4595 168 39 0 0 NUM cana-4595 169 1 โ€ฆ โ€ฆ PUNCT cana-4595 169 2 ๐‘1,0 ๐‘1,0 PROPN cana-4595 169 3 ( ( PUNCT cana-4595 169 4 ๐›ผ ๐›ผ NOUN cana-4595 169 5 ) ) PUNCT cana-4595 169 6 ๐‘1,1 ๐‘1,1 NOUN cana-4595 169 7 ( ( PUNCT cana-4595 169 8 ๐›ผ ๐›ผ NOUN cana-4595 169 9 ) ) PUNCT cana-4595 169 10 ๐‘1,2 ๐‘1,2 NOUN cana-4595 169 11 ( ( PUNCT cana-4595 169 12 ๐›ผ ๐›ผ NOUN cana-4595 169 13 ) ) PUNCT cana-4595 169 14 0 0 NUM cana-4595 169 15 โ€ฆ โ€ฆ SYM cana-4595 169 16 0 0 NUM cana-4595 169 17 ๐‘2,1 ๐‘2,1 PROPN cana-4595 169 18 ( ( PUNCT cana-4595 169 19 ๐›ผ ๐›ผ NOUN cana-4595 169 20 ) ) PUNCT cana-4595 169 21 ๐‘2,2 ๐‘2,2 PROPN cana-4595 169 22 ( ( PUNCT cana-4595 169 23 ๐›ผ ๐›ผ NOUN cana-4595 169 24 ) ) PUNCT cana-4595 169 25 ๐‘2,3 ๐‘2,3 NOUN cana-4595 169 26 ( ( PUNCT cana-4595 169 27 ๐›ผ ๐›ผ NOUN cana-4595 169 28 ) ) PUNCT cana-4595 169 29 โ€ฆ โ€ฆ PUNCT cana-4595 169 30 โ‹ฎ โ‹ฎ NOUN cana-4595 169 31 โ‹ฎ โ‹ฎ ADJ cana-4595 169 32 โ‹ฎ โ‹ฎ ADJ cana-4595 169 33 โ‹ฎ โ‹ฎ PROPN cana-4595 169 34 โ‹ฑ โ‹ฑ X cana-4595 169 35 ] ] PUNCT cana-4595 169 36 this this DET cana-4595 169 37 sparsity sparsity NOUN cana-4595 169 38 reduces reduce VERB cana-4595 169 39 matrix matrix NOUN cana-4595 169 40 - - PUNCT cana-4595 169 41 vector vector NOUN cana-4595 169 42 multiplication multiplication NOUN cana-4595 169 43 costs cost NOUN cana-4595 169 44 from from ADP cana-4595 169 45 ๐’ช(๐‘›2 ๐’ช(๐‘›2 PROPN cana-4595 169 46 ) ) PUNCT cana-4595 169 47 to to ADP cana-4595 169 48 ๐’ช(๐‘› ๐’ช(๐‘› NOUN cana-4595 169 49 ) ) PUNCT cana-4595 169 50 , , PUNCT cana-4595 169 51 significantly significantly ADV cana-4595 169 52 enhancing enhance VERB cana-4595 169 53 computational computational ADJ cana-4595 169 54 efficiency efficiency NOUN cana-4595 169 55 . . PUNCT cana-4595 170 1 5.2.3 5.2.3 NUM cana-4595 170 2 theoretical theoretical ADJ cana-4595 170 3 advantages advantage NOUN cana-4595 170 4 over over ADP cana-4595 170 5 existing exist VERB cana-4595 170 6 techniques technique NOUN cana-4595 170 7 key key ADJ cana-4595 170 8 theoretical theoretical ADJ cana-4595 170 9 benefits benefit NOUN cana-4595 170 10 of of ADP cana-4595 170 11 the the DET cana-4595 170 12 gnocchi gnocchi NOUN cana-4595 170 13 polynomial polynomial ADJ cana-4595 170 14 - - PUNCT cana-4595 170 15 based base VERB cana-4595 170 16 operational operational ADJ cana-4595 170 17 matrix matrix NOUN cana-4595 170 18 include include VERB cana-4595 170 19 : : PUNCT cana-4595 170 20 โ€ข โ€ข NUM cana-4595 170 21 exponential exponential ADJ cana-4595 170 22 convergence convergence NOUN cana-4595 170 23 in in ADP cana-4595 170 24 function function NOUN cana-4595 170 25 approximation approximation NOUN cana-4595 170 26 . . PUNCT cana-4595 171 1 โ€ข โ€ข NUM cana-4595 171 2 reduced reduce VERB cana-4595 171 3 error error NOUN cana-4595 171 4 propagation propagation NOUN cana-4595 171 5 due due ADP cana-4595 171 6 to to ADP cana-4595 171 7 well well ADV cana-4595 171 8 - - PUNCT cana-4595 171 9 conditioned condition VERB cana-4595 171 10 basis basis NOUN cana-4595 171 11 functions function NOUN cana-4595 171 12 . . PUNCT cana-4595 172 1 โ€ข โ€ข NUM cana-4595 172 2 robustness robustness NOUN cana-4595 172 3 in in ADP cana-4595 172 4 handling handle VERB cana-4595 172 5 non non ADJ cana-4595 172 6 - - ADJ cana-4595 172 7 linear linear ADJ cana-4595 172 8 terms term NOUN cana-4595 172 9 without without ADP cana-4595 172 10 additional additional ADJ cana-4595 172 11 transformations transformation NOUN cana-4595 172 12 . . PUNCT cana-4595 173 1 these these DET cana-4595 173 2 advantages advantage NOUN cana-4595 173 3 make make VERB cana-4595 173 4 it it PRON cana-4595 173 5 an an DET cana-4595 173 6 optimal optimal ADJ cana-4595 173 7 choice choice NOUN cana-4595 173 8 for for ADP cana-4595 173 9 solving solve VERB cana-4595 173 10 non non ADJ cana-4595 173 11 - - ADJ cana-4595 173 12 linear linear ADJ cana-4595 173 13 fractional fractional ADJ cana-4595 173 14 differential differential NOUN cana-4595 173 15 equations equation NOUN cana-4595 173 16 . . PUNCT cana-4595 174 1 5.3 5.3 NUM cana-4595 174 2 limitations limitation NOUN cana-4595 174 3 and and CCONJ cana-4595 174 4 challenges challenge VERB cana-4595 174 5 the the DET cana-4595 174 6 proposed propose VERB cana-4595 174 7 gnocchi gnocchi NOUN cana-4595 174 8 polynomial polynomial PROPN cana-4595 174 9 based base VERB cana-4595 174 10 operational operational ADJ cana-4595 174 11 matrix matrix NOUN cana-4595 174 12 has have VERB cana-4595 174 13 some some DET cana-4595 174 14 limitations limitation NOUN cana-4595 174 15 which which PRON cana-4595 174 16 must must AUX cana-4595 174 17 be be AUX cana-4595 174 18 recognized recognize VERB cana-4595 174 19 . . PUNCT cana-4595 175 1 the the DET cana-4595 175 2 most most ADV cana-4595 175 3 important important ADJ cana-4595 175 4 challenge challenge NOUN cana-4595 175 5 is be AUX cana-4595 175 6 in in ADP cana-4595 175 7 the the DET cana-4595 175 8 case case NOUN cana-4595 175 9 of of ADP cana-4595 175 10 highly highly ADV cana-4595 175 11 oscillatory oscillatory ADJ cana-4595 175 12 functions function NOUN cana-4595 175 13 , , PUNCT cana-4595 175 14 since since SCONJ cana-4595 175 15 rapid rapid ADJ cana-4595 175 16 oscillations oscillation NOUN cana-4595 175 17 of of ADP cana-4595 175 18 the the DET cana-4595 175 19 solution solution NOUN cana-4595 175 20 require require VERB cana-4595 175 21 high high ADJ cana-4595 175 22 order order NOUN cana-4595 175 23 approximations approximation NOUN cana-4595 175 24 that that PRON cana-4595 175 25 result result VERB cana-4595 175 26 in in ADP cana-4595 175 27 very very ADV cana-4595 175 28 high high ADJ cana-4595 175 29 computational computational ADJ cana-4595 175 30 effort effort NOUN cana-4595 175 31 . . PUNCT cana-4595 176 1 the the DET cana-4595 176 2 method method NOUN cana-4595 176 3 also also ADV cana-4595 176 4 assumes assume VERB cana-4595 176 5 smooth smooth ADJ cana-4595 176 6 solutions solution NOUN cana-4595 176 7 , , PUNCT cana-4595 176 8 which which PRON cana-4595 176 9 means mean VERB cana-4595 176 10 that that SCONJ cana-4595 176 11 it it PRON cana-4595 176 12 is be AUX cana-4595 176 13 not not PART cana-4595 176 14 very very ADV cana-4595 176 15 efficient efficient ADJ cana-4595 176 16 when when SCONJ cana-4595 176 17 it it PRON cana-4595 176 18 comes come VERB cana-4595 176 19 to to ADP cana-4595 176 20 solving solve VERB cana-4595 176 21 singular singular ADJ cana-4595 176 22 solutions solution NOUN cana-4595 176 23 , , PUNCT cana-4595 176 24 i.e. i.e. X cana-4595 176 25 solutions solution NOUN cana-4595 176 26 with with ADP cana-4595 176 27 discontinuities discontinuity NOUN cana-4595 176 28 or or CCONJ cana-4595 176 29 non non ADJ cana-4595 176 30 - - ADJ cana-4595 176 31 smooth smooth ADJ cana-4595 176 32 behavior behavior NOUN cana-4595 176 33 . . PUNCT cana-4595 177 1 two two NUM cana-4595 177 2 problems problem NOUN cana-4595 177 3 , , PUNCT cana-4595 177 4 i.e. i.e. X cana-4595 177 5 , , PUNCT cana-4595 177 6 boundary boundary ADJ cana-4595 177 7 layer layer NOUN cana-4595 177 8 problem problem NOUN cana-4595 177 9 in in ADP cana-4595 177 10 fractional fractional ADJ cana-4595 177 11 partial partial ADJ cana-4595 177 12 differential differential NOUN cana-4595 177 13 equations equation NOUN cana-4595 177 14 ( ( PUNCT cana-4595 177 15 pdes pde NOUN cana-4595 177 16 ) ) PUNCT cana-4595 177 17 that that PRON cana-4595 177 18 have have VERB cana-4595 177 19 steep steep ADJ cana-4595 177 20 gradient gradient NOUN cana-4595 177 21 near near ADP cana-4595 177 22 boundary boundary NOUN cana-4595 177 23 , , PUNCT cana-4595 177 24 have have VERB cana-4595 177 25 limitations limitation NOUN cana-4595 177 26 . . PUNCT cana-4595 178 1 in in ADP cana-4595 178 2 such such ADJ cana-4595 178 3 cases case NOUN cana-4595 178 4 , , PUNCT cana-4595 178 5 a a DET cana-4595 178 6 standard standard ADJ cana-4595 178 7 representation representation NOUN cana-4595 178 8 based base VERB cana-4595 178 9 on on ADP cana-4595 178 10 gnocchi gnocchi NOUN cana-4595 178 11 may may AUX cana-4595 178 12 need need VERB cana-4595 178 13 adaptive adaptive ADJ cana-4595 178 14 basis basis NOUN cana-4595 178 15 functions function NOUN cana-4595 178 16 to to PART cana-4595 178 17 keep keep VERB cana-4595 178 18 accuracy accuracy NOUN cana-4595 178 19 and and CCONJ cana-4595 178 20 efficiency efficiency NOUN cana-4595 178 21 . . PUNCT cana-4595 179 1 some some PRON cana-4595 179 2 of of ADP cana-4595 179 3 the the DET cana-4595 179 4 things thing NOUN cana-4595 179 5 future future ADJ cana-4595 179 6 research research NOUN cana-4595 179 7 should should AUX cana-4595 179 8 focus focus VERB cana-4595 179 9 on on ADP cana-4595 179 10 to to PART cana-4595 179 11 overcome overcome VERB cana-4595 179 12 these these DET cana-4595 179 13 challenges challenge NOUN cana-4595 179 14 are be AUX cana-4595 179 15 as as SCONJ cana-4595 179 16 follows follow VERB cana-4595 179 17 . . PUNCT cana-4595 180 1 this this PRON cana-4595 180 2 leads lead VERB cana-4595 180 3 to to ADP cana-4595 180 4 the the DET cana-4595 180 5 development development NOUN cana-4595 180 6 of of ADP cana-4595 180 7 adaptive adaptive ADJ cana-4595 180 8 gnocchi gnocchi NOUN cana-4595 180 9 polynomial polynomial ADJ cana-4595 180 10 methods method NOUN cana-4595 180 11 , , PUNCT cana-4595 180 12 as as SCONJ cana-4595 180 13 one one NUM cana-4595 180 14 promising promising ADJ cana-4595 180 15 direction direction NOUN cana-4595 180 16 for for ADP cana-4595 180 17 communications communication NOUN cana-4595 180 18 on on ADP cana-4595 180 19 applied apply VERB cana-4595 180 20 nonlinear nonlinear ADJ cana-4595 180 21 analysis analysis NOUN cana-4595 180 22 issn issn NOUN cana-4595 180 23 : : PUNCT cana-4595 180 24 1074 1074 NUM cana-4595 180 25 - - PUNCT cana-4595 180 26 133x 133x NUM cana-4595 180 27 vol vol NOUN cana-4595 180 28 32 32 NUM cana-4595 181 1 no no NOUN cana-4595 181 2 . . PUNCT cana-4595 182 1 9s 9s NUM cana-4595 182 2 ( ( PUNCT cana-4595 182 3 2025 2025 NUM cana-4595 182 4 ) ) PUNCT cana-4595 183 1 https://internationalpubls.com https://internationalpubls.com X cana-4595 183 2 2978 2978 NUM cana-4595 183 3 basis basis NOUN cana-4595 183 4 selection selection NOUN cana-4595 183 5 dynamically dynamically ADV cana-4595 183 6 and and CCONJ cana-4595 183 7 thereby thereby ADV cana-4595 183 8 improve improve VERB cana-4595 183 9 singularities singularity NOUN cana-4595 183 10 as as ADV cana-4595 183 11 well well ADV cana-4595 183 12 as as ADP cana-4595 183 13 boundary boundary ADJ cana-4595 183 14 - - PUNCT cana-4595 183 15 layer layer NOUN cana-4595 183 16 handling handling NOUN cana-4595 183 17 . . PUNCT cana-4595 184 1 this this DET cana-4595 184 2 approach approach NOUN cana-4595 184 3 can can AUX cana-4595 184 4 be be AUX cana-4595 184 5 extended extend VERB cana-4595 184 6 to to ADP cana-4595 184 7 higher high ADJ cana-4595 184 8 dimensional dimensional ADJ cana-4595 184 9 fractional fractional ADJ cana-4595 184 10 pdes pde NOUN cana-4595 184 11 , , PUNCT cana-4595 184 12 and and CCONJ cana-4595 184 13 hence hence ADV cana-4595 184 14 the the DET cana-4595 184 15 applicability applicability NOUN cana-4595 184 16 of of ADP cana-4595 184 17 this this DET cana-4595 184 18 approach approach NOUN cana-4595 184 19 can can AUX cana-4595 184 20 be be AUX cana-4595 184 21 extended extend VERB cana-4595 184 22 to to ADP cana-4595 184 23 complex complex ADJ cana-4595 184 24 multi multi ADJ cana-4595 184 25 - - ADJ cana-4595 184 26 dimensional dimensional ADJ cana-4595 184 27 systems system NOUN cana-4595 184 28 . . PUNCT cana-4595 185 1 additionally additionally ADV cana-4595 185 2 , , PUNCT cana-4595 185 3 it it PRON cana-4595 185 4 would would AUX cana-4595 185 5 be be AUX cana-4595 185 6 advantageous advantageous ADJ cana-4595 185 7 to to PART cana-4595 185 8 develop develop VERB cana-4595 185 9 hybrid hybrid ADJ cana-4595 185 10 techniques technique NOUN cana-4595 185 11 by by ADP cana-4595 185 12 combining combine VERB cana-4595 185 13 gnocchi gnocchi NOUN cana-4595 185 14 based base VERB cana-4595 185 15 matrices matrix NOUN cana-4595 185 16 with with ADP cana-4595 185 17 a a DET cana-4595 185 18 combination combination NOUN cana-4595 185 19 of of ADP cana-4595 185 20 deep deep ADJ cana-4595 185 21 learning learning NOUN cana-4595 185 22 models model NOUN cana-4595 185 23 to to PART cana-4595 185 24 improve improve VERB cana-4595 185 25 the the DET cana-4595 185 26 precision precision NOUN cana-4595 185 27 and and CCONJ cana-4595 185 28 computational computational ADJ cana-4595 185 29 complexity complexity NOUN cana-4595 185 30 of of ADP cana-4595 185 31 the the DET cana-4595 185 32 proposed propose VERB cana-4595 185 33 procedure procedure NOUN cana-4595 185 34 for for ADP cana-4595 185 35 problems problem NOUN cana-4595 185 36 that that PRON cana-4595 185 37 are be AUX cana-4595 185 38 highly highly ADV cana-4595 185 39 non non ADJ cana-4595 185 40 - - ADJ cana-4595 185 41 linear linear ADJ cana-4595 185 42 and and CCONJ cana-4595 185 43 data datum NOUN cana-4595 185 44 driven drive VERB cana-4595 185 45 . . PUNCT cana-4595 186 1 the the DET cana-4595 186 2 possible possible ADJ cana-4595 186 3 directions direction NOUN cana-4595 186 4 of of ADP cana-4595 186 5 research research NOUN cana-4595 186 6 mentioned mention VERB cana-4595 186 7 above above ADP cana-4595 186 8 themselves themselves PRON cana-4595 186 9 provide provide VERB cana-4595 186 10 promising promising ADJ cana-4595 186 11 possibilities possibility NOUN cana-4595 186 12 for for ADP cana-4595 186 13 further further ADJ cana-4595 186 14 refining refining NOUN cana-4595 186 15 and and CCONJ cana-4595 186 16 enlarging enlarge VERB cana-4595 186 17 the the DET cana-4595 186 18 capabilities capability NOUN cana-4595 186 19 of of ADP cana-4595 186 20 the the DET cana-4595 186 21 introduced introduce VERB cana-4595 186 22 framework framework NOUN cana-4595 186 23 in in ADP cana-4595 186 24 solving solve VERB cana-4595 186 25 the the DET cana-4595 186 26 nonlinear nonlinear ADJ cana-4595 186 27 fractional fractional ADJ cana-4595 186 28 differential differential NOUN cana-4595 186 29 equations equation NOUN cana-4595 186 30 . . PUNCT cana-4595 187 1 6 6 NUM cana-4595 187 2 . . X cana-4595 187 3 numerical numerical PROPN cana-4595 187 4 validation validation NOUN cana-4595 187 5 and and CCONJ cana-4595 187 6 conclusion conclusion NOUN cana-4595 187 7 6.1 6.1 NUM cana-4595 187 8 numerical numerical ADJ cana-4595 187 9 validation validation NOUN cana-4595 187 10 to to PART cana-4595 187 11 validate validate VERB cana-4595 187 12 the the DET cana-4595 187 13 proposed propose VERB cana-4595 187 14 gnocchi gnocchi NOUN cana-4595 187 15 polynomial polynomial ADJ cana-4595 187 16 - - PUNCT cana-4595 187 17 based base VERB cana-4595 187 18 operational operational ADJ cana-4595 187 19 matrix matrix NOUN cana-4595 187 20 , , PUNCT cana-4595 187 21 we we PRON cana-4595 187 22 apply apply VERB cana-4595 187 23 it it PRON cana-4595 187 24 to to ADP cana-4595 187 25 a a DET cana-4595 187 26 benchmark benchmark NOUN cana-4595 187 27 function function NOUN cana-4595 187 28 and and CCONJ cana-4595 187 29 analyze analyze VERB cana-4595 187 30 its its PRON cana-4595 187 31 accuracy accuracy NOUN cana-4595 187 32 . . PUNCT cana-4595 188 1 consider consider VERB cana-4595 188 2 the the DET cana-4595 188 3 function function NOUN cana-4595 188 4 : : PUNCT cana-4595 188 5 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NUM cana-4595 188 6 ) ) PUNCT cana-4595 189 1 = = SYM cana-4595 189 2 ๐‘’โˆ’๐‘ฅ ๐‘’โˆ’๐‘ฅ NOUN cana-4595 189 3 which which PRON cana-4595 189 4 is be AUX cana-4595 189 5 commonly commonly ADV cana-4595 189 6 used use VERB cana-4595 189 7 in in ADP cana-4595 189 8 fractional fractional ADJ cana-4595 189 9 differential differential ADJ cana-4595 189 10 equation equation NOUN cana-4595 189 11 approximations approximation NOUN cana-4595 189 12 due due ADP cana-4595 189 13 to to ADP cana-4595 189 14 its its PRON cana-4595 189 15 smooth smooth ADJ cana-4595 189 16 nature nature NOUN cana-4595 189 17 . . PUNCT cana-4595 190 1 the the DET cana-4595 190 2 fractional fractional ADJ cana-4595 190 3 derivative derivative NOUN cana-4595 190 4 of of ADP cana-4595 190 5 ๐‘ฆ(๐‘ฅ ๐‘ฆ(๐‘ฅ NOUN cana-4595 190 6 ) ) PUNCT cana-4595 190 7 using use VERB cana-4595 190 8 the the DET cana-4595 190 9 gnocchi gnocchi NOUN cana-4595 190 10 polynomial polynomial ADJ cana-4595 190 11 expansion expansion NOUN cana-4595 190 12 is be AUX cana-4595 190 13 approximated approximate VERB cana-4595 190 14 as as ADP cana-4595 190 15 : : PUNCT cana-4595 190 16 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ NOUN cana-4595 190 17 ) ) PUNCT cana-4595 191 1 โ‰ˆ โ‰ˆ PROPN cana-4595 191 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 191 3 ) ) PUNCT cana-4595 191 4 where where SCONJ cana-4595 191 5 : : PUNCT cana-4595 191 6 โ€ข โ€ข NUM cana-4595 191 7 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 191 8 ) ) PUNCT cana-4595 191 9 is be AUX cana-4595 191 10 the the DET cana-4595 191 11 fractional fractional ADJ cana-4595 191 12 differentiation differentiation NOUN cana-4595 191 13 matrix matrix NOUN cana-4595 191 14 derived derive VERB cana-4595 191 15 using use VERB cana-4595 191 16 gnocchi gnocchi NOUN cana-4595 191 17 polynomials polynomial NOUN cana-4595 191 18 . . PUNCT cana-4595 192 1 โ€ข โ€ข NUM cana-4595 192 2 ๐ถ ๐ถ PROPN cana-4595 192 3 is be AUX cana-4595 192 4 the the DET cana-4595 192 5 coefficient coefficient NOUN cana-4595 192 6 vector vector NOUN cana-4595 192 7 in in ADP cana-4595 192 8 the the DET cana-4595 192 9 polynomial polynomial ADJ cana-4595 192 10 expansion expansion NOUN cana-4595 192 11 . . PUNCT cana-4595 193 1 โ€ข โ€ข NUM cana-4595 193 2 ๐บ(๐‘ฅ ๐บ(๐‘ฅ NOUN cana-4595 193 3 ) ) PUNCT cana-4595 193 4 represents represent VERB cana-4595 193 5 the the DET cana-4595 193 6 basis basis NOUN cana-4595 193 7 expansion expansion NOUN cana-4595 193 8 using use VERB cana-4595 193 9 gnocchi gnocchi NOUN cana-4595 193 10 polynomials polynomial NOUN cana-4595 193 11 . . PUNCT cana-4595 194 1 6.1.1 6.1.1 NUM cana-4595 194 2 error error NOUN cana-4595 194 3 analysis analysis NOUN cana-4595 194 4 and and CCONJ cana-4595 194 5 convergence convergence NOUN cana-4595 194 6 to to PART cana-4595 194 7 assess assess VERB cana-4595 194 8 the the DET cana-4595 194 9 accuracy accuracy NOUN cana-4595 194 10 of of ADP cana-4595 194 11 the the DET cana-4595 194 12 proposed propose VERB cana-4595 194 13 method method NOUN cana-4595 194 14 , , PUNCT cana-4595 194 15 we we PRON cana-4595 194 16 compute compute VERB cana-4595 194 17 the the DET cana-4595 194 18 approximation approximation NOUN cana-4595 194 19 error error NOUN cana-4595 194 20 : : PUNCT cana-4595 194 21 ๐ธ(๐‘ฅ ๐ธ(๐‘ฅ NOUN cana-4595 194 22 ) ) PUNCT cana-4595 194 23 = = PUNCT cana-4595 194 24 |๐‘ฆexact |๐‘ฆexact ADJ cana-4595 194 25 ( ( PUNCT cana-4595 194 26 ๐‘ฅ ๐‘ฅ NOUN cana-4595 194 27 ) ) PUNCT cana-4595 194 28 โˆ’ โˆ’ PROPN cana-4595 195 1 ๐‘ฆapprox ๐‘ฆapprox NOUN cana-4595 195 2 ( ( PUNCT cana-4595 195 3 ๐‘ฅ)| ๐‘ฅ)| ADP cana-4595 195 4 the the DET cana-4595 195 5 results result NOUN cana-4595 195 6 indicate indicate VERB cana-4595 195 7 exponential exponential ADJ cana-4595 195 8 convergence convergence NOUN cana-4595 195 9 as as ADP cana-4595 195 10 the the DET cana-4595 195 11 polynomial polynomial ADJ cana-4595 195 12 order order NOUN cana-4595 195 13 increases increase NOUN cana-4595 195 14 , , PUNCT cana-4595 195 15 confirming confirm VERB cana-4595 195 16 the the DET cana-4595 195 17 efficiency efficiency NOUN cana-4595 195 18 of of ADP cana-4595 195 19 the the DET cana-4595 195 20 gnocchi gnocchi NOUN cana-4595 195 21 polynomial polynomial ADJ cana-4595 195 22 - - PUNCT cana-4595 195 23 based base VERB cana-4595 195 24 approach approach NOUN cana-4595 195 25 . . PUNCT cana-4595 196 1 the the DET cana-4595 196 2 error error NOUN cana-4595 196 3 plot plot NOUN cana-4595 196 4 in in ADP cana-4595 196 5 figure figure NOUN cana-4595 196 6 1 1 NUM cana-4595 196 7 visualizes visualize VERB cana-4595 196 8 the the DET cana-4595 196 9 trend trend NOUN cana-4595 196 10 . . PUNCT cana-4595 197 1 figure figure NOUN cana-4595 197 2 1 1 NUM cana-4595 197 3 : : PUNCT cana-4595 197 4 error error NOUN cana-4595 197 5 analysis analysis NOUN cana-4595 197 6 of of ADP cana-4595 197 7 the the DET cana-4595 197 8 gnocchi gnocchi NOUN cana-4595 197 9 polynomial polynomial ADJ cana-4595 197 10 approximation approximation NOUN cana-4595 197 11 6.1.2 6.1.2 NUM cana-4595 197 12 convergence convergence NOUN cana-4595 197 13 rate rate NOUN cana-4595 197 14 vs. vs. ADP cana-4595 197 15 polynomial polynomial ADJ cana-4595 197 16 order order NOUN cana-4595 197 17 to to PART cana-4595 197 18 further far ADV cana-4595 197 19 validate validate VERB cana-4595 197 20 spectral spectral ADJ cana-4595 197 21 accuracy accuracy NOUN cana-4595 197 22 of of ADP cana-4595 197 23 the the DET cana-4595 197 24 gnocchi gnocchi NOUN cana-4595 197 25 method method NOUN cana-4595 197 26 , , PUNCT cana-4595 197 27 we we PRON cana-4595 197 28 analyze analyze VERB cana-4595 197 29 how how SCONJ cana-4595 197 30 the the DET cana-4595 197 31 maximum maximum ADJ cana-4595 197 32 approximation approximation NOUN cana-4595 197 33 error error NOUN cana-4595 197 34 decays decay NOUN cana-4595 197 35 as as ADP cana-4595 197 36 polynomial polynomial ADJ cana-4595 197 37 order order NOUN cana-4595 197 38 ๐‘ ๐‘ NOUN cana-4595 197 39 increases increase NOUN cana-4595 197 40 . . PUNCT cana-4595 198 1 the the DET cana-4595 198 2 results result NOUN cana-4595 198 3 in in ADP cana-4595 198 4 figure figure NOUN cana-4595 198 5 2 2 NUM cana-4595 198 6 show show VERB cana-4595 198 7 an an DET cana-4595 198 8 exponential exponential ADJ cana-4595 198 9 decrease decrease NOUN cana-4595 198 10 in in ADP cana-4595 198 11 error error NOUN cana-4595 198 12 , , PUNCT cana-4595 198 13 confirming confirm VERB cana-4595 198 14 that that SCONJ cana-4595 198 15 gnocchi gnocchi NOUN cana-4595 198 16 polynomial polynomial ADJ cana-4595 198 17 - - PUNCT cana-4595 198 18 based base VERB cana-4595 198 19 expansion expansion NOUN cana-4595 198 20 provides provide VERB cana-4595 198 21 highly highly ADV cana-4595 198 22 accurate accurate ADJ cana-4595 198 23 results result NOUN cana-4595 198 24 for for ADP cana-4595 198 25 fractional fractional ADJ cana-4595 198 26 operators operator NOUN cana-4595 198 27 . . PUNCT cana-4595 199 1 communications communication NOUN cana-4595 199 2 on on ADP cana-4595 199 3 applied apply VERB cana-4595 199 4 nonlinear nonlinear ADJ cana-4595 199 5 analysis analysis NOUN cana-4595 199 6 issn issn NOUN cana-4595 199 7 : : PUNCT cana-4595 199 8 1074 1074 NUM cana-4595 199 9 - - PUNCT cana-4595 199 10 133x 133x NUM cana-4595 199 11 vol vol NOUN cana-4595 199 12 32 32 NUM cana-4595 199 13 no no NOUN cana-4595 199 14 . . PUNCT cana-4595 200 1 9s 9s NUM cana-4595 200 2 ( ( PUNCT cana-4595 200 3 2025 2025 NUM cana-4595 200 4 ) ) PUNCT cana-4595 200 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 200 6 2979 2979 NUM cana-4595 200 7 figure figure NOUN cana-4595 200 8 2 2 NUM cana-4595 200 9 : : PUNCT cana-4595 200 10 convergence convergence NOUN cana-4595 200 11 rate rate NOUN cana-4595 200 12 of of ADP cana-4595 200 13 gnocchi gnocchi NOUN cana-4595 200 14 polynomial polynomial ADJ cana-4595 200 15 approximation approximation NOUN cana-4595 200 16 6.1.3 6.1.3 NUM cana-4595 200 17 computational computational ADJ cana-4595 200 18 efficiency efficiency NOUN cana-4595 200 19 analysis analysis NOUN cana-4595 200 20 the the DET cana-4595 200 21 computational computational ADJ cana-4595 200 22 complexity complexity NOUN cana-4595 200 23 of of ADP cana-4595 200 24 traditional traditional ADJ cana-4595 200 25 polynomial polynomial ADJ cana-4595 200 26 - - PUNCT cana-4595 200 27 based base VERB cana-4595 200 28 methods method NOUN cana-4595 200 29 often often ADV cana-4595 200 30 scales scale VERB cana-4595 200 31 as as ADP cana-4595 200 32 ๐‘‚(๐‘2 ๐‘‚(๐‘2 NOUN cana-4595 200 33 ) ) PUNCT cana-4595 200 34 , , PUNCT cana-4595 200 35 making make VERB cana-4595 200 36 them they PRON cana-4595 200 37 inefficient inefficient ADJ cana-4595 200 38 for for ADP cana-4595 200 39 large large ADJ cana-4595 200 40 - - PUNCT cana-4595 200 41 scale scale NOUN cana-4595 200 42 problems problem NOUN cana-4595 200 43 . . PUNCT cana-4595 201 1 however however ADV cana-4595 201 2 , , PUNCT cana-4595 201 3 the the DET cana-4595 201 4 gnocchi gnocchi NOUN cana-4595 201 5 - - PUNCT cana-4595 201 6 based base VERB cana-4595 201 7 operational operational ADJ cana-4595 201 8 matrix matrix NOUN cana-4595 201 9 exhibits exhibit VERB cana-4595 201 10 a a DET cana-4595 201 11 sparse sparse ADJ cana-4595 201 12 structure structure NOUN cana-4595 201 13 , , PUNCT cana-4595 201 14 reducing reduce VERB cana-4595 201 15 computational computational ADJ cana-4595 201 16 complexity complexity NOUN cana-4595 201 17 to to PART cana-4595 201 18 ๐‘‚(๐‘ ๐‘‚(๐‘ VERB cana-4595 201 19 ) ) PUNCT cana-4595 201 20 . . PUNCT cana-4595 202 1 figure figure NOUN cana-4595 202 2 3 3 NUM cana-4595 202 3 illustrates illustrate VERB cana-4595 202 4 the the DET cana-4595 202 5 relationship relationship NOUN cana-4595 202 6 between between ADP cana-4595 202 7 polynomial polynomial ADJ cana-4595 202 8 order order NOUN cana-4595 202 9 and and CCONJ cana-4595 202 10 computation computation NOUN cana-4595 202 11 time time NOUN cana-4595 202 12 , , PUNCT cana-4595 202 13 demonstrating demonstrate VERB cana-4595 202 14 that that SCONJ cana-4595 202 15 the the DET cana-4595 202 16 gnocchi gnocchi NOUN cana-4595 202 17 - - PUNCT cana-4595 202 18 based base VERB cana-4595 202 19 approach approach NOUN cana-4595 202 20 significantly significantly ADV cana-4595 202 21 improves improve VERB cana-4595 202 22 efficiency efficiency NOUN cana-4595 202 23 . . PUNCT cana-4595 203 1 figure figure VERB cana-4595 203 2 3 3 NUM cana-4595 203 3 : : PUNCT cana-4595 203 4 computational computational ADJ cana-4595 203 5 efficiency efficiency NOUN cana-4595 203 6 of of ADP cana-4595 203 7 gnocchi gnocchi NOUN cana-4595 203 8 polynomial polynomial ADJ cana-4595 203 9 approximation approximation NOUN cana-4595 203 10 6.2 6.2 NUM cana-4595 203 11 conclusion conclusion NOUN cana-4595 203 12 in in ADP cana-4595 203 13 this this DET cana-4595 203 14 study study NOUN cana-4595 203 15 , , PUNCT cana-4595 203 16 a a DET cana-4595 203 17 new new ADJ cana-4595 203 18 operational operational ADJ cana-4595 203 19 matrix matrix NOUN cana-4595 203 20 based base VERB cana-4595 203 21 on on ADP cana-4595 203 22 gnocchi gnocchi NOUN cana-4595 203 23 polynomial polynomial PROPN cana-4595 203 24 was be AUX cana-4595 203 25 proposed propose VERB cana-4595 203 26 for for ADP cana-4595 203 27 solving solve VERB cana-4595 203 28 non non ADJ cana-4595 203 29 linear linear ADJ cana-4595 203 30 fractional fractional ADJ cana-4595 203 31 differential differential ADJ cana-4595 203 32 equations equation NOUN cana-4595 203 33 ( ( PUNCT cana-4595 203 34 nfdes nfde NOUN cana-4595 203 35 ) ) PUNCT cana-4595 203 36 . . PUNCT cana-4595 204 1 in in ADP cana-4595 204 2 this this DET cana-4595 204 3 method method NOUN cana-4595 204 4 , , PUNCT cana-4595 204 5 the the DET cana-4595 204 6 orthogonality orthogonality NOUN cana-4595 204 7 and and CCONJ cana-4595 204 8 recurrence recurrence NOUN cana-4595 204 9 structure structure NOUN cana-4595 204 10 of of ADP cana-4595 204 11 gnocchi gnocchi NOUN cana-4595 204 12 polynomials polynomial NOUN cana-4595 204 13 are be AUX cana-4595 204 14 embedded embed VERB cana-4595 204 15 in in ADP cana-4595 204 16 the the DET cana-4595 204 17 operational operational ADJ cana-4595 204 18 framework framework NOUN cana-4595 204 19 , , PUNCT cana-4595 204 20 which which PRON cana-4595 204 21 transforms transform VERB cana-4595 204 22 complex complex ADJ cana-4595 204 23 nfdes nfde NOUN cana-4595 204 24 to to ADP cana-4595 204 25 a a DET cana-4595 204 26 tractable tractable ADJ cana-4595 204 27 algebraic algebraic ADJ cana-4595 204 28 system system NOUN cana-4595 204 29 , , PUNCT cana-4595 204 30 based base VERB cana-4595 204 31 on on ADP cana-4595 204 32 which which PRON cana-4595 204 33 complex complex ADJ cana-4595 204 34 problems problem NOUN cana-4595 204 35 can can AUX cana-4595 204 36 be be AUX cana-4595 204 37 handled handle VERB cana-4595 204 38 with with ADP cana-4595 204 39 improved improved ADJ cana-4595 204 40 computational computational ADJ cana-4595 204 41 performance performance NOUN cana-4595 204 42 . . PUNCT cana-4595 205 1 the the DET cana-4595 205 2 method method NOUN cana-4595 205 3 's 's PART cana-4595 205 4 mathematical mathematical ADJ cana-4595 205 5 foundation foundation NOUN cana-4595 205 6 rests rest VERB cana-4595 205 7 on on ADP cana-4595 205 8 the the DET cana-4595 205 9 key key ADJ cana-4595 205 10 approximation approximation NOUN cana-4595 205 11 : : PUNCT cana-4595 205 12 communications communication NOUN cana-4595 205 13 on on ADP cana-4595 205 14 applied apply VERB cana-4595 205 15 nonlinear nonlinear ADJ cana-4595 205 16 analysis analysis NOUN cana-4595 205 17 issn issn NOUN cana-4595 205 18 : : PUNCT cana-4595 205 19 1074 1074 NUM cana-4595 205 20 - - PUNCT cana-4595 205 21 133x 133x NUM cana-4595 205 22 vol vol NOUN cana-4595 205 23 32 32 NUM cana-4595 206 1 no no NOUN cana-4595 206 2 . . PUNCT cana-4595 207 1 9s 9s NUM cana-4595 207 2 ( ( PUNCT cana-4595 207 3 2025 2025 NUM cana-4595 207 4 ) ) PUNCT cana-4595 207 5 https://internationalpubls.com https://internationalpubls.com X cana-4595 207 6 2980 2980 NUM cana-4595 207 7 ๐ท๐›ผ๐‘ฆ(๐‘ฅ ๐ท๐›ผ๐‘ฆ(๐‘ฅ NOUN cana-4595 207 8 ) ) PUNCT cana-4595 208 1 โ‰ˆ โ‰ˆ PROPN cana-4595 208 2 ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ ๐‘ƒ(๐›ผ)๐ถ๐‘‡๐บ(๐‘ฅ PROPN cana-4595 208 3 ) ) PUNCT cana-4595 208 4 where where SCONJ cana-4595 208 5 ๐‘ƒ(๐›ผ ๐‘ƒ(๐›ผ NUM cana-4595 208 6 ) ) PUNCT cana-4595 208 7 is be AUX cana-4595 208 8 the the DET cana-4595 208 9 operational operational ADJ cana-4595 208 10 matrix matrix NOUN cana-4595 208 11 , , PUNCT cana-4595 208 12 and and CCONJ cana-4595 208 13 ๐บ(๐‘ฅ ๐บ(๐‘ฅ NOUN cana-4595 208 14 ) ) PUNCT cana-4595 208 15 is be AUX cana-4595 208 16 the the DET cana-4595 208 17 gnocchi gnocchi NOUN cana-4595 208 18 polynomial polynomial ADJ cana-4595 208 19 basis basis NOUN cana-4595 208 20 . . PUNCT cana-4595 209 1 this this DET cana-4595 209 2 formulation formulation NOUN cana-4595 209 3 is be AUX cana-4595 209 4 not not PART cana-4595 209 5 only only ADV cana-4595 209 6 computationally computationally ADV cana-4595 209 7 elegant elegant ADJ cana-4595 209 8 but but CCONJ cana-4595 209 9 also also ADV cana-4595 209 10 well well ADV cana-4595 209 11 - - PUNCT cana-4595 209 12 suited suit VERB cana-4595 209 13 for for ADP cana-4595 209 14 high high ADJ cana-4595 209 15 - - PUNCT cana-4595 209 16 precision precision NOUN cana-4595 209 17 fractional fractional ADJ cana-4595 209 18 modeling modeling NOUN cana-4595 209 19 . . PUNCT cana-4595 210 1 its its PRON cana-4595 210 2 numerical numerical ADJ cana-4595 210 3 effectiveness effectiveness NOUN cana-4595 210 4 has have AUX cana-4595 210 5 been be AUX cana-4595 210 6 confirmed confirm VERB cana-4595 210 7 through through ADP cana-4595 210 8 multiple multiple ADJ cana-4595 210 9 validation validation NOUN cana-4595 210 10 steps step NOUN cana-4595 210 11 . . PUNCT cana-4595 211 1 figure figure VERB cana-4595 211 2 4 4 NUM cana-4595 211 3 illustrates illustrate VERB cana-4595 211 4 the the DET cana-4595 211 5 shape shape NOUN cana-4595 211 6 and and CCONJ cana-4595 211 7 behavior behavior NOUN cana-4595 211 8 of of ADP cana-4595 211 9 the the DET cana-4595 211 10 first first ADJ cana-4595 211 11 few few ADJ cana-4595 211 12 gnocchi gnocchi NOUN cana-4595 211 13 basis basis NOUN cana-4595 211 14 functions function NOUN cana-4595 211 15 , , PUNCT cana-4595 211 16 reinforcing reinforce VERB cana-4595 211 17 their their PRON cana-4595 211 18 role role NOUN cana-4595 211 19 in in ADP cana-4595 211 20 smooth smooth ADJ cana-4595 211 21 function function NOUN cana-4595 211 22 representation representation NOUN cana-4595 211 23 . . PUNCT cana-4595 212 1 figure figure VERB cana-4595 212 2 4 4 NUM cana-4595 212 3 : : PUNCT cana-4595 212 4 sample sample NOUN cana-4595 212 5 gnocchi gnocchi NOUN cana-4595 212 6 polynomial polynomial ADJ cana-4595 212 7 basis basis NOUN cana-4595 212 8 functions function NOUN cana-4595 212 9 this this DET cana-4595 212 10 plot plot NOUN cana-4595 212 11 illustrates illustrate VERB cana-4595 212 12 the the DET cana-4595 212 13 behavior behavior NOUN cana-4595 212 14 of of ADP cana-4595 212 15 the the DET cana-4595 212 16 first first ADJ cana-4595 212 17 few few ADJ cana-4595 212 18 gnocchi gnocchi NOUN cana-4595 212 19 polynomials polynomial NOUN cana-4595 212 20 ๐บ0(๐‘ฅ ๐บ0(๐‘ฅ NUM cana-4595 212 21 ) ) PUNCT cana-4595 212 22 , , PUNCT cana-4595 212 23 ๐บ1(๐‘ฅ ๐บ1(๐‘ฅ NUM cana-4595 212 24 ) ) PUNCT cana-4595 212 25 , , PUNCT cana-4595 212 26 ๐บ2(๐‘ฅ ๐บ2(๐‘ฅ NUM cana-4595 212 27 ) ) PUNCT cana-4595 212 28 , , PUNCT cana-4595 212 29 ๐บ3(๐‘ฅ ๐บ3(๐‘ฅ NUM cana-4595 212 30 ) ) PUNCT cana-4595 212 31 over over ADP cana-4595 212 32 the the DET cana-4595 212 33 interval interval NOUN cana-4595 213 1 [ [ X cana-4595 213 2 โˆ’1,1 โˆ’1,1 X cana-4595 213 3 ] ] X cana-4595 213 4 , , PUNCT cana-4595 213 5 demonstrating demonstrate VERB cana-4595 213 6 their their PRON cana-4595 213 7 structure structure NOUN cana-4595 213 8 and and CCONJ cana-4595 213 9 oscillatory oscillatory ADJ cana-4595 213 10 behavior behavior NOUN cana-4595 213 11 . . PUNCT cana-4595 214 1 it it PRON cana-4595 214 2 supports support VERB cana-4595 214 3 the the DET cana-4595 214 4 basis basis NOUN cana-4595 214 5 expansion expansion NOUN cana-4595 214 6 concept concept NOUN cana-4595 214 7 used use VERB cana-4595 214 8 in in ADP cana-4595 214 9 the the DET cana-4595 214 10 proposed propose VERB cana-4595 214 11 operational operational ADJ cana-4595 214 12 matrix matrix NOUN cana-4595 214 13 formulation formulation NOUN cana-4595 214 14 . . PUNCT cana-4595 215 1 figure figure NOUN cana-4595 215 2 5 5 NUM cana-4595 215 3 also also ADV cana-4595 215 4 validates validate VERB cana-4595 215 5 the the DET cana-4595 215 6 spectral spectral ADJ cana-4595 215 7 characteristics characteristic NOUN cana-4595 215 8 of of ADP cana-4595 215 9 the the DET cana-4595 215 10 proposed propose VERB cana-4595 215 11 method method NOUN cana-4595 215 12 , , PUNCT cana-4595 215 13 illustrating illustrate VERB cana-4595 215 14 that that SCONJ cana-4595 215 15 the the DET cana-4595 215 16 spectral spectral ADJ cana-4595 215 17 radius radius NOUN cana-4595 215 18 of of ADP cana-4595 215 19 the the DET cana-4595 215 20 operational operational ADJ cana-4595 215 21 matrix matrix NOUN cana-4595 215 22 grows grow VERB cana-4595 215 23 slowly slowly ADV cana-4595 215 24 with with ADP cana-4595 215 25 polynomial polynomial ADJ cana-4595 215 26 order order NOUN cana-4595 215 27 , , PUNCT cana-4595 215 28 which which PRON cana-4595 215 29 indicates indicate VERB cana-4595 215 30 strong strong ADJ cana-4595 215 31 numerical numerical ADJ cana-4595 215 32 stability stability NOUN cana-4595 215 33 on on ADP cana-4595 215 34 larger large ADJ cana-4595 215 35 systems system NOUN cana-4595 215 36 . . PUNCT cana-4595 216 1 figure figure VERB cana-4595 216 2 5 5 NUM cana-4595 216 3 : : PUNCT cana-4595 216 4 spectral spectral ADJ cana-4595 216 5 radius radius NOUN cana-4595 216 6 of of ADP cana-4595 216 7 operational operational ADJ cana-4595 216 8 matrix matrix NOUN cana-4595 216 9 vs. vs. ADP cana-4595 216 10 polynomial polynomial ADJ cana-4595 216 11 order order NOUN cana-4595 216 12 communications communication NOUN cana-4595 216 13 on on ADP cana-4595 216 14 applied apply VERB cana-4595 216 15 nonlinear nonlinear ADJ cana-4595 216 16 analysis analysis NOUN cana-4595 216 17 issn issn NOUN cana-4595 216 18 : : PUNCT cana-4595 216 19 1074 1074 NUM cana-4595 216 20 - - PUNCT cana-4595 216 21 133x 133x NUM cana-4595 216 22 vol vol NOUN cana-4595 216 23 32 32 NUM cana-4595 217 1 no no NOUN cana-4595 217 2 . . PUNCT cana-4595 218 1 9s 9s NUM cana-4595 218 2 ( ( PUNCT cana-4595 218 3 2025 2025 NUM cana-4595 218 4 ) ) PUNCT cana-4595 219 1 https://internationalpubls.com https://internationalpubls.com X cana-4595 219 2 2981 2981 NUM cana-4595 219 3 this this DET cana-4595 219 4 figure figure NOUN cana-4595 219 5 illustrates illustrate VERB cana-4595 219 6 the the DET cana-4595 219 7 evolution evolution NOUN cana-4595 219 8 of of ADP cana-4595 219 9 the the DET cana-4595 219 10 spectral spectral ADJ cana-4595 219 11 radius radius NOUN cana-4595 219 12 of of ADP cana-4595 219 13 the the DET cana-4595 219 14 operational operational ADJ cana-4595 219 15 matrix matrix NOUN cana-4595 219 16 with with ADP cana-4595 219 17 increasing increase VERB cana-4595 219 18 polynomial polynomial ADJ cana-4595 219 19 order order NOUN cana-4595 219 20 . . PUNCT cana-4595 220 1 it it PRON cana-4595 220 2 indicates indicate VERB cana-4595 220 3 visually visually ADV cana-4595 220 4 how how SCONJ cana-4595 220 5 stable stable ADJ cana-4595 220 6 the the DET cana-4595 220 7 method method NOUN cana-4595 220 8 is be AUX cana-4595 220 9 numerically numerically ADV cana-4595 220 10 , , PUNCT cana-4595 220 11 and and CCONJ cana-4595 220 12 it it PRON cana-4595 220 13 shows show VERB cana-4595 220 14 that that SCONJ cana-4595 220 15 the the DET cana-4595 220 16 gnocchi gnocchi NOUN cana-4595 220 17 based base VERB cana-4595 220 18 matrix matrix NOUN cana-4595 220 19 stays stay VERB cana-4595 220 20 well well ADV cana-4595 220 21 conditioned condition VERB cana-4595 220 22 for for ADP cana-4595 220 23 larger large ADJ cana-4595 220 24 systems system NOUN cana-4595 220 25 . . PUNCT cana-4595 221 1 additionally additionally ADV cana-4595 221 2 , , PUNCT cana-4595 221 3 the the DET cana-4595 221 4 earlier early ADJ cana-4595 221 5 figures figure NOUN cana-4595 221 6 ( ( PUNCT cana-4595 221 7 figures figure NOUN cana-4595 221 8 1 1 NUM cana-4595 221 9 to to PART cana-4595 221 10 3 3 NUM cana-4595 221 11 ) ) PUNCT cana-4595 221 12 also also ADV cana-4595 221 13 confirmed confirm VERB cana-4595 221 14 the the DET cana-4595 221 15 exponential exponential ADJ cana-4595 221 16 convergence convergence NOUN cana-4595 221 17 , , PUNCT cana-4595 221 18 error error NOUN cana-4595 221 19 decay decay NOUN cana-4595 221 20 and and CCONJ cana-4595 221 21 efficiency efficiency NOUN cana-4595 221 22 of of ADP cana-4595 221 23 the the DET cana-4595 221 24 runtime runtime NOUN cana-4595 221 25 of of ADP cana-4595 221 26 the the DET cana-4595 221 27 method method NOUN cana-4595 221 28 , , PUNCT cana-4595 221 29 showing show VERB cana-4595 221 30 that that SCONJ cana-4595 221 31 the the DET cana-4595 221 32 method method NOUN cana-4595 221 33 outperforms outperform VERB cana-4595 221 34 classical classical ADJ cana-4595 221 35 approaches approach NOUN cana-4595 221 36 such such ADJ cana-4595 221 37 as as ADP cana-4595 221 38 legendre legendre PROPN cana-4595 221 39 and and CCONJ cana-4595 221 40 chebyshev chebyshev PROPN cana-4595 221 41 based base VERB cana-4595 221 42 matrices matrix NOUN cana-4595 221 43 . . PUNCT cana-4595 222 1 summing sum VERB cana-4595 222 2 up up ADP cana-4595 222 3 all all PRON cana-4595 222 4 , , PUNCT cana-4595 222 5 the the DET cana-4595 222 6 gnocchi gnocchi NOUN cana-4595 222 7 polynomial polynomial ADJ cana-4595 222 8 - - PUNCT cana-4595 222 9 based base VERB cana-4595 222 10 operational operational ADJ cana-4595 222 11 matrix matrix NOUN cana-4595 222 12 offers offer NOUN cana-4595 222 13 : : PUNCT cana-4595 222 14 โ€ข โ€ข NUM cana-4595 222 15 spectral spectral ADJ cana-4595 222 16 - - PUNCT cana-4595 222 17 level level NOUN cana-4595 222 18 convergence convergence NOUN cana-4595 222 19 for for ADP cana-4595 222 20 smooth smooth ADJ cana-4595 222 21 solutions solution NOUN cana-4595 222 22 โ€ข โ€ข ADV cana-4595 222 23 sparse sparse ADJ cana-4595 222 24 and and CCONJ cana-4595 222 25 efficient efficient ADJ cana-4595 222 26 matrix matrix NOUN cana-4595 222 27 structure structure NOUN cana-4595 222 28 โ€ข โ€ข ADP cana-4595 222 29 improved improve VERB cana-4595 222 30 stability stability NOUN cana-4595 222 31 for for ADP cana-4595 222 32 non non ADJ cana-4595 222 33 - - ADJ cana-4595 222 34 linear linear ADJ cana-4595 222 35 fractional fractional ADJ cana-4595 222 36 operators operator NOUN cana-4595 222 37 the the DET cana-4595 222 38 method method NOUN cana-4595 222 39 is be AUX cana-4595 222 40 shown show VERB cana-4595 222 41 to to PART cana-4595 222 42 have have VERB cana-4595 222 43 great great ADJ cana-4595 222 44 promise promise NOUN cana-4595 222 45 for for ADP cana-4595 222 46 solving solve VERB cana-4595 222 47 large large ADJ cana-4595 222 48 scale scale NOUN cana-4595 222 49 nfdes nfde VERB cana-4595 222 50 efficiently efficiently ADV cana-4595 222 51 . . PUNCT cana-4595 223 1 this this DET cana-4595 223 2 approach approach NOUN cana-4595 223 3 can can AUX cana-4595 223 4 be be AUX cana-4595 223 5 further far ADV cana-4595 223 6 extended extend VERB cana-4595 223 7 to to PART cana-4595 223 8 multi multi VERB cana-4595 223 9 dimensional dimensional ADJ cana-4595 223 10 fractional fractional ADJ cana-4595 223 11 pdes pde NOUN cana-4595 223 12 and and CCONJ cana-4595 223 13 also also ADV cana-4595 223 14 combined combine VERB cana-4595 223 15 with with ADP cana-4595 223 16 adaptive adaptive ADJ cana-4595 223 17 spectral spectral ADJ cana-4595 223 18 methods method NOUN cana-4595 223 19 and and CCONJ cana-4595 223 20 potentially potentially ADV cana-4595 223 21 be be AUX cana-4595 223 22 integrated integrate VERB cana-4595 223 23 with with ADP cana-4595 223 24 machine machine NOUN cana-4595 223 25 learning learning NOUN cana-4595 223 26 based base VERB cana-4595 223 27 basis basis NOUN cana-4595 223 28 selection selection NOUN cana-4595 223 29 techniques technique NOUN cana-4595 223 30 for for ADP cana-4595 223 31 handling handle VERB cana-4595 223 32 singularities singularity NOUN cana-4595 223 33 or or CCONJ cana-4595 223 34 real real ADJ cana-4595 223 35 time time NOUN cana-4595 223 36 applications application NOUN cana-4595 223 37 . . PUNCT cana-4595 224 1 references reference NOUN cana-4595 224 2 1 1 NUM cana-4595 224 3 . . PUNCT cana-4595 225 1 rigi rigi PROPN cana-4595 225 2 , , PUNCT cana-4595 225 3 f. f. PROPN cana-4595 225 4 , , PUNCT cana-4595 225 5 & & CCONJ cana-4595 225 6 tajadodi tajadodi PROPN cana-4595 225 7 , , PUNCT cana-4595 225 8 h. h. PROPN cana-4595 225 9 ( ( PUNCT cana-4595 225 10 2019 2019 NUM cana-4595 225 11 ) ) PUNCT cana-4595 225 12 . . PUNCT cana-4595 226 1 numerical numerical ADJ cana-4595 226 2 approach approach NOUN cana-4595 226 3 of of ADP cana-4595 226 4 fractional fractional PROPN cana-4595 226 5 abel abel PROPN cana-4595 226 6 differential differential NOUN cana-4595 226 7 equation equation NOUN cana-4595 226 8 by by ADP cana-4595 226 9 genocchi genocchi PROPN cana-4595 226 10 polynomials polynomial NOUN cana-4595 226 11 . . PUNCT cana-4595 227 1 international international ADJ cana-4595 227 2 journal journal PROPN cana-4595 227 3 of of ADP cana-4595 227 4 applied applied ADJ cana-4595 227 5 and and CCONJ cana-4595 227 6 computational computational ADJ cana-4595 227 7 mathematics mathematic NOUN cana-4595 227 8 , , PUNCT cana-4595 227 9 5(5 5(5 NUM cana-4595 227 10 ) ) PUNCT cana-4595 227 11 , , PUNCT cana-4595 227 12 134 134 NUM cana-4595 227 13 . . NOUN cana-4595 227 14 2 2 NUM cana-4595 227 15 . . X cana-4595 228 1 medina medina PROPN cana-4595 228 2 - - PUNCT cana-4595 228 3 ramos ramos PROPN cana-4595 228 4 , , PUNCT cana-4595 228 5 c. c. PROPN cana-4595 228 6 , , PUNCT cana-4595 228 7 carbonel carbonel NOUN cana-4595 228 8 - - PUNCT cana-4595 228 9 olazabal olazabal NOUN cana-4595 228 10 , , PUNCT cana-4595 228 11 d. d. PROPN cana-4595 228 12 , , PUNCT cana-4595 228 13 betetta betetta PROPN cana-4595 228 14 - - PUNCT cana-4595 228 15 gomez gomez PROPN cana-4595 228 16 , , PUNCT cana-4595 228 17 j. j. PROPN cana-4595 228 18 , , PUNCT cana-4595 228 19 & & CCONJ cana-4595 228 20 gonzales gonzales PROPN cana-4595 228 21 - - PUNCT cana-4595 228 22 torres torre NOUN cana-4595 228 23 , , PUNCT cana-4595 228 24 c. c. PROPN cana-4595 228 25 ( ( PUNCT cana-4595 228 26 2024 2024 NUM cana-4595 228 27 , , PUNCT cana-4595 228 28 september september PROPN cana-4595 228 29 ) ) PUNCT cana-4595 228 30 . . PUNCT cana-4595 229 1 genocchi genocchi PROPN cana-4595 229 2 polynomials polynomial VERB cana-4595 229 3 in in ADP cana-4595 229 4 volterra volterra PROPN cana-4595 229 5 model model NOUN cana-4595 229 6 to to PART cana-4595 229 7 identify identify VERB cana-4595 229 8 non non ADJ cana-4595 229 9 - - ADJ cana-4595 229 10 linear linear ADJ cana-4595 229 11 systems system NOUN cana-4595 229 12 , , PUNCT cana-4595 229 13 case case NOUN cana-4595 229 14 of of ADP cana-4595 229 15 world world NOUN cana-4595 229 16 ocean ocean PROPN cana-4595 229 17 fishery fishery PROPN cana-4595 229 18 . . PUNCT cana-4595 230 1 in in ADP cana-4595 230 2 2024 2024 NUM cana-4595 230 3 ieee ieee NOUN cana-4595 230 4 biennial biennial ADJ cana-4595 230 5 congress congress PROPN cana-4595 230 6 of of ADP cana-4595 230 7 argentina argentina PROPN cana-4595 230 8 ( ( PUNCT cana-4595 230 9 argencon argencon PROPN cana-4595 230 10 ) ) PUNCT cana-4595 230 11 ( ( PUNCT cana-4595 230 12 pp pp X cana-4595 230 13 . . PUNCT cana-4595 231 1 1 1 NUM cana-4595 231 2 - - SYM cana-4595 231 3 8) 8) NUM cana-4595 231 4 . . PUNCT cana-4595 231 5 ieee ieee NOUN cana-4595 231 6 . . PUNCT cana-4595 232 1 3 3 NUM cana-4595 232 2 . . X cana-4595 232 3 graglia graglia PROPN cana-4595 232 4 , , PUNCT cana-4595 232 5 r. r. PROPN cana-4595 232 6 , , PUNCT cana-4595 232 7 pelosi pelosi PROPN cana-4595 232 8 , , PUNCT cana-4595 232 9 g. g. PROPN cana-4595 232 10 , , PUNCT cana-4595 232 11 & & CCONJ cana-4595 232 12 selleri selleri PROPN cana-4595 232 13 , , PUNCT cana-4595 232 14 s. s. PROPN cana-4595 232 15 ( ( PUNCT cana-4595 232 16 2016 2016 NUM cana-4595 232 17 ) ) PUNCT cana-4595 232 18 . . PUNCT cana-4595 233 1 international international ADJ cana-4595 233 2 workshop workshop NOUN cana-4595 233 3 on on ADP cana-4595 233 4 finite finite ADJ cana-4595 233 5 elements element NOUN cana-4595 233 6 for for ADP cana-4595 233 7 microwave microwave NOUN cana-4595 233 8 engineering engineering NOUN cana-4595 233 9 : : PUNCT cana-4595 233 10 from from ADP cana-4595 233 11 1992 1992 NUM cana-4595 233 12 to to PART cana-4595 233 13 present present VERB cana-4595 233 14 & & CCONJ cana-4595 233 15 proceedings proceeding NOUN cana-4595 233 16 of of ADP cana-4595 233 17 the the DET cana-4595 233 18 13th 13th NOUN cana-4595 233 19 workshop workshop NOUN cana-4595 233 20 ( ( PUNCT cana-4595 233 21 p. p. NOUN cana-4595 233 22 212 212 NUM cana-4595 233 23 ) ) PUNCT cana-4595 233 24 . . PUNCT cana-4595 234 1 firenze firenze PROPN cana-4595 234 2 university university PROPN cana-4595 234 3 press press NOUN cana-4595 234 4 . . PUNCT cana-4595 235 1 4 4 X cana-4595 235 2 . . X cana-4595 235 3 castiglioni castiglioni PROPN cana-4595 235 4 , , PUNCT cana-4595 235 5 p. p. PROPN cana-4595 235 6 , , PUNCT cana-4595 235 7 & & CCONJ cana-4595 235 8 faini faini PROPN cana-4595 235 9 , , PUNCT cana-4595 235 10 a. a. NOUN cana-4595 235 11 ( ( PUNCT cana-4595 235 12 2019 2019 NUM cana-4595 235 13 ) ) PUNCT cana-4595 235 14 . . PUNCT cana-4595 236 1 a a DET cana-4595 236 2 fast fast ADJ cana-4595 236 3 dfa dfa NOUN cana-4595 236 4 algorithm algorithm NOUN cana-4595 236 5 for for ADP cana-4595 236 6 multifractal multifractal ADJ cana-4595 236 7 multiscale multiscale ADJ cana-4595 236 8 analysis analysis NOUN cana-4595 236 9 of of ADP cana-4595 236 10 physiological physiological ADJ cana-4595 236 11 time time NOUN cana-4595 236 12 series series NOUN cana-4595 236 13 . . PUNCT cana-4595 237 1 frontiers frontier NOUN cana-4595 237 2 in in ADP cana-4595 237 3 physiology physiology NOUN cana-4595 237 4 , , PUNCT cana-4595 237 5 10 10 NUM cana-4595 237 6 , , PUNCT cana-4595 237 7 115 115 NUM cana-4595 237 8 . . PUNCT cana-4595 238 1 5 5 NUM cana-4595 238 2 . . X cana-4595 238 3 fantoni fantoni PROPN cana-4595 238 4 , , PUNCT cana-4595 238 5 a. a. NOUN cana-4595 238 6 ( ( PUNCT cana-4595 238 7 2023 2023 NUM cana-4595 238 8 ) ) PUNCT cana-4595 238 9 . . PUNCT cana-4595 239 1 assessment assessment NOUN cana-4595 239 2 of of ADP cana-4595 239 3 vocal vocal ADJ cana-4595 239 4 fatigue fatigue NOUN cana-4595 239 5 of of ADP cana-4595 239 6 multiple multiple ADJ cana-4595 239 7 sclerosis sclerosis NOUN cana-4595 239 8 patients patient NOUN cana-4595 239 9 . . PUNCT cana-4595 240 1 validation validation NOUN cana-4595 240 2 of of ADP cana-4595 240 3 a a DET cana-4595 240 4 contact contact NOUN cana-4595 240 5 microphone microphone NOUN cana-4595 240 6 - - PUNCT cana-4595 240 7 based base VERB cana-4595 240 8 device device NOUN cana-4595 240 9 for for ADP cana-4595 240 10 long long ADJ cana-4595 240 11 - - PUNCT cana-4595 240 12 term term NOUN cana-4595 240 13 monitoring monitoring NOUN cana-4595 240 14 . . PUNCT cana-4595 241 1 doctoral doctoral ADJ cana-4595 241 2 dissertation dissertation NOUN cana-4595 241 3 , , PUNCT cana-4595 241 4 politecnico politecnico PROPN cana-4595 241 5 di di PROPN cana-4595 241 6 torino torino PROPN cana-4595 241 7 . . PUNCT cana-4595 242 1 6 6 NUM cana-4595 242 2 . . X cana-4595 242 3 giliberti giliberti PROPN cana-4595 242 4 , , PUNCT cana-4595 242 5 e. e. PROPN cana-4595 242 6 ( ( PUNCT cana-4595 242 7 2020 2020 NUM cana-4595 242 8 ) ) PUNCT cana-4595 242 9 . . PUNCT cana-4595 243 1 on on ADP cana-4595 243 2 neutron neutron NOUN cana-4595 243 3 stars star NOUN cana-4595 243 4 ' ' PART cana-4595 243 5 crust crust NOUN cana-4595 243 6 breaking breaking NOUN cana-4595 243 7 and and CCONJ cana-4595 243 8 gravitational gravitational ADJ cana-4595 243 9 waves wave NOUN cana-4595 243 10 emission emission NOUN cana-4595 243 11 . . PUNCT cana-4595 244 1 7 7 X cana-4595 244 2 . . X cana-4595 244 3 honkapohja honkapohja NOUN cana-4595 244 4 , , PUNCT cana-4595 244 5 s. s. PROPN cana-4595 244 6 , , PUNCT cana-4595 244 7 & & CCONJ cana-4595 244 8 mcclung mcclung PROPN cana-4595 244 9 , , PUNCT cana-4595 244 10 n. n. PROPN cana-4595 244 11 ( ( PUNCT cana-4595 244 12 2024 2024 NUM cana-4595 244 13 ) ) PUNCT cana-4595 244 14 . . PUNCT cana-4595 245 1 on on ADP cana-4595 245 2 robustness robustness NOUN cana-4595 245 3 of of ADP cana-4595 245 4 average average ADJ cana-4595 245 5 inflation inflation NOUN cana-4595 245 6 targeting target VERB cana-4595 245 7 . . PUNCT cana-4595 245 8 available available ADJ cana-4595 245 9 at at ADP cana-4595 245 10 ssrn ssrn PROPN cana-4595 245 11 4021712 4021712 NUM cana-4595 245 12 . . PUNCT cana-4595 246 1 8 8 NUM cana-4595 246 2 . . X cana-4595 246 3 ramos ramos PROPN cana-4595 246 4 , , PUNCT cana-4595 246 5 h. h. PROPN cana-4595 246 6 s. s. PROPN cana-4595 246 7 d. d. PROPN cana-4595 246 8 c. c. PROPN cana-4595 246 9 f. f. PROPN cana-4595 246 10 ( ( PUNCT cana-4595 246 11 2018 2018 NUM cana-4595 246 12 ) ) PUNCT cana-4595 246 13 . . PUNCT cana-4595 247 1 three three NUM cana-4595 247 2 dimensional dimensional ADJ cana-4595 247 3 equation equation NOUN cana-4595 247 4 of of ADP cana-4595 247 5 state state NOUN cana-4595 247 6 for for ADP cana-4595 247 7 core core NOUN cana-4595 247 8 - - PUNCT cana-4595 247 9 collapse collapse NOUN cana-4595 247 10 supernova supernova NOUN cana-4595 247 11 matter matter NOUN cana-4595 247 12 . . PUNCT cana-4595 248 1 universidade universidade PROPN cana-4595 248 2 de de PROPN cana-4595 248 3 coimbra coimbra PROPN cana-4595 248 4 ( ( PUNCT cana-4595 248 5 portugal portugal PROPN cana-4595 248 6 ) ) PUNCT cana-4595 248 7 . . PUNCT cana-4595 249 1 9 9 X cana-4595 249 2 . . X cana-4595 249 3 capineri capineri PROPN cana-4595 249 4 , , PUNCT cana-4595 249 5 l. l. PROPN cana-4595 249 6 ( ( PUNCT cana-4595 249 7 2021 2021 NUM cana-4595 249 8 ) ) PUNCT cana-4595 249 9 . . PUNCT cana-4595 250 1 advanced advanced ADJ cana-4595 250 2 technologies technology NOUN cana-4595 250 3 for for ADP cana-4595 250 4 piezoelectric piezoelectric ADJ cana-4595 250 5 sensors sensor NOUN cana-4595 250 6 in in ADP cana-4595 250 7 shm shm NOUN cana-4595 250 8 systems system NOUN cana-4595 250 9 : : PUNCT cana-4595 250 10 a a DET cana-4595 250 11 review review NOUN cana-4595 250 12 . . PUNCT cana-4595 251 1 in in ADP cana-4595 251 2 sms sms PROPN cana-4595 251 3 - - PUNCT cana-4595 251 4 sensors sensor NOUN cana-4595 251 5 - - PUNCT cana-4595 251 6 egf egf NOUN cana-4595 251 7 - - PUNCT cana-4595 251 8 nanomed-2021 nanomed-2021 NOUN cana-4595 251 9 - - PUNCT cana-4595 251 10 joint joint NOUN cana-4595 251 11 - - PUNCT cana-4595 251 12 conferences conference NOUN cana-4595 251 13 - - PUNCT cana-4595 251 14 book book NOUN cana-4595 251 15 - - PUNCT cana-4595 251 16 of of ADP cana-4595 251 17 - - PUNCT cana-4595 251 18 abstracts abstract NOUN cana-4595 251 19 ( ( PUNCT cana-4595 251 20 pp pp ADJ cana-4595 251 21 . . PUNCT cana-4595 252 1 28 28 NUM cana-4595 252 2 - - SYM cana-4595 252 3 28 28 NUM cana-4595 252 4 ) ) PUNCT cana-4595 252 5 . . PUNCT cana-4595 253 1 mdpi mdpi PROPN cana-4595 253 2 . . PUNCT cana-4595 254 1 10 10 NUM cana-4595 254 2 . . X cana-4595 254 3 danilatou danilatou PROPN cana-4595 254 4 , , PUNCT cana-4595 254 5 v. v. ADP cana-4595 254 6 a. a. PROPN cana-4595 254 7 s. s. PROPN cana-4595 254 8 i. i. PROPN cana-4595 254 9 l. l. PROPN cana-4595 254 10 i. i. PROPN cana-4595 254 11 k. k. PROPN cana-4595 254 12 i. i. PROPN cana-4595 254 13 ( ( PUNCT cana-4595 254 14 2021 2021 NUM cana-4595 254 15 ) ) PUNCT cana-4595 254 16 . . PUNCT cana-4595 255 1 risk risk NOUN cana-4595 255 2 assessment assessment NOUN cana-4595 255 3 and and CCONJ cana-4595 255 4 mortality mortality NOUN cana-4595 255 5 prediction prediction NOUN cana-4595 255 6 in in ADP cana-4595 255 7 patients patient NOUN cana-4595 255 8 with with ADP cana-4595 255 9 venous venous ADJ cana-4595 255 10 thromboembolism thromboembolism NOUN cana-4595 255 11 using use VERB cana-4595 255 12 big big ADJ cana-4595 255 13 data datum NOUN cana-4595 255 14 and and CCONJ cana-4595 255 15 machine machine NOUN cana-4595 255 16 learning learning NOUN cana-4595 255 17 . . PUNCT cana-4595 256 1 doctoral doctoral ADJ cana-4595 256 2 dissertation dissertation NOUN cana-4595 256 3 , , PUNCT cana-4595 256 4 bournemouth bournemouth PROPN cana-4595 256 5 university university NOUN cana-4595 256 6 . . PUNCT cana-4595 257 1 11 11 NUM cana-4595 257 2 . . X cana-4595 257 3 mccomb mccomb PROPN cana-4595 257 4 , , PUNCT cana-4595 257 5 m. m. NOUN cana-4595 257 6 c. c. PROPN cana-4595 257 7 ( ( PUNCT cana-4595 257 8 2020 2020 NUM cana-4595 257 9 ) ) PUNCT cana-4595 257 10 . . PUNCT cana-4595 258 1 machine machine NOUN cana-4595 258 2 learning learning NOUN cana-4595 258 3 - - PUNCT cana-4595 258 4 guided guide VERB cana-4595 258 5 , , PUNCT cana-4595 258 6 biomarker biomarker NOUN cana-4595 258 7 - - PUNCT cana-4595 258 8 enabled enable VERB cana-4595 258 9 disease disease NOUN cana-4595 258 10 progression progression NOUN cana-4595 258 11 modeling modeling NOUN cana-4595 258 12 . . PUNCT cana-4595 259 1 doctoral doctoral ADJ cana-4595 259 2 dissertation dissertation NOUN cana-4595 259 3 , , PUNCT cana-4595 259 4 state state NOUN cana-4595 259 5 university university PROPN cana-4595 259 6 of of ADP cana-4595 259 7 new new PROPN cana-4595 259 8 york york PROPN cana-4595 259 9 at at ADP cana-4595 259 10 buffalo buffalo PROPN cana-4595 259 11 . . PUNCT