id sid tid token lemma pos cana-936 1 1 communications communication NOUN cana-936 1 2 on on ADP cana-936 1 3 applied apply VERB cana-936 1 4 nonlinear nonlinear ADJ cana-936 1 5 analysis analysis NOUN cana-936 1 6 issn issn NOUN cana-936 1 7 : : PUNCT cana-936 1 8 1074 1074 NUM cana-936 1 9 - - PUNCT cana-936 1 10 133x 133x NUM cana-936 1 11 vol vol NOUN cana-936 1 12 31 31 NUM cana-936 1 13 no no NOUN cana-936 1 14 . . PUNCT cana-936 2 1 4s 4s NUM cana-936 2 2 ( ( PUNCT cana-936 2 3 2024 2024 NUM cana-936 2 4 ) ) PUNCT cana-936 2 5 425 425 NUM cana-936 2 6 https://internationalpubls.com https://internationalpubls.com X cana-936 2 7 numerical numerical ADJ cana-936 2 8 and and CCONJ cana-936 2 9 analytical analytical ADJ cana-936 2 10 approaches approach NOUN cana-936 2 11 to to ADP cana-936 2 12 fractional fractional ADJ cana-936 2 13 quantum quantum ADJ cana-936 2 14 mechanics mechanic NOUN cana-936 2 15 vikas vikas PROPN cana-936 2 16 kumar1 kumar1 PROPN cana-936 2 17 , , PUNCT cana-936 2 18 dr dr PROPN cana-936 2 19 nand nand PROPN cana-936 2 20 kumar2 kumar2 PROPN cana-936 2 21 1 1 NUM cana-936 2 22 assistant assistant NOUN cana-936 2 23 professor professor NOUN cana-936 2 24 school school NOUN cana-936 2 25 of of ADP cana-936 2 26 applied applied ADJ cana-936 2 27 and and CCONJ cana-936 2 28 life life NOUN cana-936 2 29 sciences science NOUN cana-936 2 30 , , PUNCT cana-936 2 31 uttaranchal uttaranchal PROPN cana-936 2 32 university university NOUN cana-936 2 33 , , PUNCT cana-936 2 34 dehradun dehradun PROPN cana-936 2 35 , , PUNCT cana-936 2 36 248007 248007 NUM cana-936 2 37 , , PUNCT cana-936 2 38 india india PROPN cana-936 2 39 . . PUNCT cana-936 3 1 orcid orcid PROPN cana-936 3 2 00000003 00000003 NUM cana-936 3 3 - - SYM cana-936 3 4 3170 3170 NUM cana-936 3 5 - - SYM cana-936 3 6 7933 7933 NUM cana-936 3 7 . . PUNCT cana-936 4 1 2 2 NUM cana-936 4 2 assistant assistant NOUN cana-936 4 3 professor professor NOUN cana-936 4 4 , , PUNCT cana-936 4 5 department department NOUN cana-936 4 6 of of ADP cana-936 4 7 physics physics PROPN cana-936 4 8 , , PUNCT cana-936 4 9 jawaharlal jawaharlal PROPN cana-936 4 10 nehru nehru PROPN cana-936 4 11 memorial memorial PROPN cana-936 4 12 pg pg PROPN cana-936 4 13 college college PROPN cana-936 4 14 , , PUNCT cana-936 4 15 barabanki barabanki PROPN cana-936 4 16 , , PUNCT cana-936 4 17 up up ADP cana-936 4 18 , , PUNCT cana-936 4 19 india india PROPN cana-936 4 20 . . PUNCT cana-936 5 1 email:nandkumar350@gmail.com email:nandkumar350@gmail.com X cana-936 5 2 . . PUNCT cana-936 6 1 orcid orcid NOUN cana-936 6 2 : : PUNCT cana-936 6 3 0009 0009 NUM cana-936 6 4 - - SYM cana-936 6 5 0004 0004 NUM cana-936 6 6 - - PUNCT cana-936 6 7 3139 3139 NUM cana-936 6 8 - - PUNCT cana-936 6 9 3385 3385 NUM cana-936 6 10 . . PUNCT cana-936 7 1 article article NOUN cana-936 7 2 history history NOUN cana-936 7 3 : : PUNCT cana-936 7 4 received receive VERB cana-936 7 5 : : PUNCT cana-936 7 6 10 10 NUM cana-936 7 7 - - PUNCT cana-936 7 8 04 04 NUM cana-936 7 9 - - PUNCT cana-936 7 10 2024 2024 NUM cana-936 7 11 revised revise VERB cana-936 7 12 : : PUNCT cana-936 7 13 20 20 NUM cana-936 7 14 - - SYM cana-936 7 15 06 06 NUM cana-936 7 16 - - PUNCT cana-936 7 17 2024 2024 NUM cana-936 7 18 accepted accept VERB cana-936 7 19 : : PUNCT cana-936 7 20 21 21 NUM cana-936 7 21 - - SYM cana-936 7 22 06 06 NUM cana-936 7 23 - - PUNCT cana-936 7 24 2024 2024 NUM cana-936 7 25 abstract abstract NOUN cana-936 7 26 : : PUNCT cana-936 7 27 our our PRON cana-936 7 28 research research NOUN cana-936 7 29 meticulously meticulously ADV cana-936 7 30 navigates navigate VERB cana-936 7 31 the the DET cana-936 7 32 realms realm NOUN cana-936 7 33 of of ADP cana-936 7 34 fractional fractional ADJ cana-936 7 35 quantum quantum ADJ cana-936 7 36 mechanics mechanic NOUN cana-936 7 37 ( ( PUNCT cana-936 7 38 fqm fqm NOUN cana-936 7 39 ) ) PUNCT cana-936 7 40 , , PUNCT cana-936 7 41 focusing focus VERB cana-936 7 42 on on ADP cana-936 7 43 a a DET cana-936 7 44 critical critical ADJ cana-936 7 45 examination examination NOUN cana-936 7 46 of of ADP cana-936 7 47 both both CCONJ cana-936 7 48 numerical numerical ADJ cana-936 7 49 and and CCONJ cana-936 7 50 analytical analytical ADJ cana-936 7 51 methods method NOUN cana-936 7 52 that that PRON cana-936 7 53 harness harness VERB cana-936 7 54 the the DET cana-936 7 55 potential potential NOUN cana-936 7 56 of of ADP cana-936 7 57 fractional fractional ADJ cana-936 7 58 calculus calculus NOUN cana-936 7 59 to to PART cana-936 7 60 illuminate illuminate VERB cana-936 7 61 the the DET cana-936 7 62 quantum quantum ADJ cana-936 7 63 world world NOUN cana-936 7 64 's 's PART cana-936 7 65 complexities complexity NOUN cana-936 7 66 . . PUNCT cana-936 8 1 by by ADP cana-936 8 2 embarking embark VERB cana-936 8 3 on on ADP cana-936 8 4 this this DET cana-936 8 5 scholarly scholarly ADJ cana-936 8 6 journey journey NOUN cana-936 8 7 , , PUNCT cana-936 8 8 we we PRON cana-936 8 9 aim aim VERB cana-936 8 10 to to PART cana-936 8 11 decode decode VERB cana-936 8 12 the the DET cana-936 8 13 intricate intricate ADJ cana-936 8 14 dynamics dynamic NOUN cana-936 8 15 that that PRON cana-936 8 16 fractional fractional ADJ cana-936 8 17 equations equation NOUN cana-936 8 18 reveal reveal VERB cana-936 8 19 about about ADP cana-936 8 20 quantum quantum NOUN cana-936 8 21 systems system NOUN cana-936 8 22 , , PUNCT cana-936 8 23 pushing push VERB cana-936 8 24 the the DET cana-936 8 25 boundaries boundary NOUN cana-936 8 26 of of ADP cana-936 8 27 conventional conventional ADJ cana-936 8 28 quantum quantum ADJ cana-936 8 29 mechanics mechanic NOUN cana-936 8 30 . . PUNCT cana-936 9 1 this this DET cana-936 9 2 comparative comparative ADJ cana-936 9 3 study study NOUN cana-936 9 4 meticulously meticulously ADV cana-936 9 5 evaluates evaluate VERB cana-936 9 6 the the DET cana-936 9 7 efficacy efficacy NOUN cana-936 9 8 and and CCONJ cana-936 9 9 insights insight NOUN cana-936 9 10 provided provide VERB cana-936 9 11 by by ADP cana-936 9 12 these these DET cana-936 9 13 two two NUM cana-936 9 14 distinct distinct ADJ cana-936 9 15 approaches approach NOUN cana-936 9 16 , , PUNCT cana-936 9 17 highlighting highlight VERB cana-936 9 18 their their PRON cana-936 9 19 contributions contribution NOUN cana-936 9 20 to to ADP cana-936 9 21 a a DET cana-936 9 22 deeper deep ADJ cana-936 9 23 understanding understanding NOUN cana-936 9 24 of of ADP cana-936 9 25 quantum quantum ADJ cana-936 9 26 phenomena phenomenon NOUN cana-936 9 27 . . PUNCT cana-936 10 1 as as SCONJ cana-936 10 2 we we PRON cana-936 10 3 traverse traverse VERB cana-936 10 4 through through ADP cana-936 10 5 the the DET cana-936 10 6 layers layer NOUN cana-936 10 7 of of ADP cana-936 10 8 quantum quantum NOUN cana-936 10 9 dynamics dynamic NOUN cana-936 10 10 , , PUNCT cana-936 10 11 our our PRON cana-936 10 12 work work NOUN cana-936 10 13 seeks seek VERB cana-936 10 14 to to PART cana-936 10 15 contribute contribute VERB cana-936 10 16 significantly significantly ADV cana-936 10 17 to to ADP cana-936 10 18 the the DET cana-936 10 19 theoretical theoretical ADJ cana-936 10 20 framework framework NOUN cana-936 10 21 of of ADP cana-936 10 22 fqm fqm NOUN cana-936 10 23 , , PUNCT cana-936 10 24 offering offer VERB cana-936 10 25 innovative innovative ADJ cana-936 10 26 perspectives perspective NOUN cana-936 10 27 and and CCONJ cana-936 10 28 methodologies methodology NOUN cana-936 10 29 . . PUNCT cana-936 11 1 the the DET cana-936 11 2 essence essence NOUN cana-936 11 3 of of ADP cana-936 11 4 this this DET cana-936 11 5 research research NOUN cana-936 11 6 lies lie VERB cana-936 11 7 in in ADP cana-936 11 8 its its PRON cana-936 11 9 potential potential NOUN cana-936 11 10 to to PART cana-936 11 11 forge forge VERB cana-936 11 12 new new ADJ cana-936 11 13 theoretical theoretical ADJ cana-936 11 14 pathways pathway NOUN cana-936 11 15 and and CCONJ cana-936 11 16 inspire inspire VERB cana-936 11 17 further further ADJ cana-936 11 18 exploration exploration NOUN cana-936 11 19 in in ADP cana-936 11 20 the the DET cana-936 11 21 quantum quantum ADJ cana-936 11 22 domain domain NOUN cana-936 11 23 , , PUNCT cana-936 11 24 leveraging leverage VERB cana-936 11 25 the the DET cana-936 11 26 unique unique ADJ cana-936 11 27 capabilities capability NOUN cana-936 11 28 of of ADP cana-936 11 29 fractional fractional ADJ cana-936 11 30 calculus calculus NOUN cana-936 11 31 . . PUNCT cana-936 12 1 by by ADP cana-936 12 2 enriching enrich VERB cana-936 12 3 the the DET cana-936 12 4 scientific scientific ADJ cana-936 12 5 community community NOUN cana-936 12 6 with with ADP cana-936 12 7 our our PRON cana-936 12 8 findings finding NOUN cana-936 12 9 , , PUNCT cana-936 12 10 we we PRON cana-936 12 11 aspire aspire VERB cana-936 12 12 to to PART cana-936 12 13 open open VERB cana-936 12 14 new new ADJ cana-936 12 15 horizons horizon NOUN cana-936 12 16 in in ADP cana-936 12 17 the the DET cana-936 12 18 study study NOUN cana-936 12 19 of of ADP cana-936 12 20 quantum quantum ADJ cana-936 12 21 mechanics mechanic NOUN cana-936 12 22 , , PUNCT cana-936 12 23 marking mark VERB cana-936 12 24 a a DET cana-936 12 25 step step NOUN cana-936 12 26 forward forward ADV cana-936 12 27 in in ADP cana-936 12 28 the the DET cana-936 12 29 ongoing ongoing ADJ cana-936 12 30 quest quest NOUN cana-936 12 31 to to PART cana-936 12 32 unravel unravel VERB cana-936 12 33 the the DET cana-936 12 34 mysteries mystery NOUN cana-936 12 35 of of ADP cana-936 12 36 the the DET cana-936 12 37 quantum quantum ADJ cana-936 12 38 universe universe NOUN cana-936 12 39 . . PUNCT cana-936 13 1 keywords keyword NOUN cana-936 13 2 : : PUNCT cana-936 13 3 fractional fractional ADJ cana-936 13 4 quantum quantum ADJ cana-936 13 5 mechanics mechanic NOUN cana-936 13 6 , , PUNCT cana-936 13 7 fractional fractional ADJ cana-936 13 8 calculus calculus NOUN cana-936 13 9 , , PUNCT cana-936 13 10 numerical numerical ADJ cana-936 13 11 methods method NOUN cana-936 13 12 , , PUNCT cana-936 13 13 analytical analytical ADJ cana-936 13 14 methods method NOUN cana-936 13 15 , , PUNCT cana-936 13 16 riesz riesz VERB cana-936 13 17 fractional fractional ADJ cana-936 13 18 derivative derivative ADJ cana-936 13 19 , , PUNCT cana-936 13 20 quantum quantum ADJ cana-936 13 21 tunneling tunneling NOUN cana-936 13 22 , , PUNCT cana-936 13 23 fractional fractional ADJ cana-936 13 24 schrΓΆdinger schrΓΆdinger ADJ cana-936 13 25 equation equation NOUN cana-936 13 26 , , PUNCT cana-936 13 27 numerical numerical ADJ cana-936 13 28 simulations simulation NOUN cana-936 13 29 , , PUNCT cana-936 13 30 quantum quantum ADJ cana-936 13 31 harmonic harmonic ADJ cana-936 13 32 oscillator oscillator NOUN cana-936 13 33 , , PUNCT cana-936 13 34 mathematical mathematical ADJ cana-936 13 35 physics physics NOUN cana-936 13 36 . . PUNCT cana-936 14 1 1 1 X cana-936 14 2 . . X cana-936 14 3 introduction introduction NOUN cana-936 14 4 the the DET cana-936 14 5 advent advent NOUN cana-936 14 6 of of ADP cana-936 14 7 fractional fractional ADJ cana-936 14 8 quantum quantum ADJ cana-936 14 9 mechanics mechanic NOUN cana-936 14 10 ( ( PUNCT cana-936 14 11 fqm fqm NOUN cana-936 14 12 ) ) PUNCT cana-936 14 13 has have AUX cana-936 14 14 heralded herald VERB cana-936 14 15 a a DET cana-936 14 16 new new ADJ cana-936 14 17 era era NOUN cana-936 14 18 in in ADP cana-936 14 19 the the DET cana-936 14 20 exploration exploration NOUN cana-936 14 21 of of ADP cana-936 14 22 quantum quantum ADJ cana-936 14 23 phenomena phenomenon NOUN cana-936 14 24 , , PUNCT cana-936 14 25 introducing introduce VERB cana-936 14 26 a a DET cana-936 14 27 framework framework NOUN cana-936 14 28 that that PRON cana-936 14 29 extends extend VERB cana-936 14 30 beyond beyond ADP cana-936 14 31 the the DET cana-936 14 32 confines confine NOUN cana-936 14 33 of of ADP cana-936 14 34 classical classical ADJ cana-936 14 35 calculus calculus NOUN cana-936 14 36 to to PART cana-936 14 37 embrace embrace VERB cana-936 14 38 the the DET cana-936 14 39 complexities complexity NOUN cana-936 14 40 and and CCONJ cana-936 14 41 intricacies intricacy NOUN cana-936 14 42 inherent inherent ADJ cana-936 14 43 in in ADP cana-936 14 44 the the DET cana-936 14 45 quantum quantum ADJ cana-936 14 46 world world NOUN cana-936 14 47 . . PUNCT cana-936 15 1 this this DET cana-936 15 2 groundbreaking groundbreake VERB cana-936 15 3 approach approach NOUN cana-936 15 4 leverages leverage VERB cana-936 15 5 the the DET cana-936 15 6 mathematical mathematical ADJ cana-936 15 7 sophistication sophistication NOUN cana-936 15 8 of of ADP cana-936 15 9 fractional fractional ADJ cana-936 15 10 calculus calculus NOUN cana-936 15 11 , , PUNCT cana-936 15 12 employing employ VERB cana-936 15 13 derivatives derivative NOUN cana-936 15 14 and and CCONJ cana-936 15 15 integrals integral NOUN cana-936 15 16 of of ADP cana-936 15 17 non non ADJ cana-936 15 18 - - ADJ cana-936 15 19 integer integer ADJ cana-936 15 20 orders order NOUN cana-936 15 21 to to PART cana-936 15 22 provide provide VERB cana-936 15 23 a a DET cana-936 15 24 more more ADV cana-936 15 25 nuanced nuanced ADJ cana-936 15 26 and and CCONJ cana-936 15 27 comprehensive comprehensive ADJ cana-936 15 28 understanding understanding NOUN cana-936 15 29 of of ADP cana-936 15 30 quantum quantum ADJ cana-936 15 31 dynamics dynamic NOUN cana-936 15 32 . . PUNCT cana-936 16 1 the the DET cana-936 16 2 introduction introduction NOUN cana-936 16 3 of of ADP cana-936 16 4 fqm fqm NOUN cana-936 16 5 has have AUX cana-936 16 6 not not PART cana-936 16 7 only only ADV cana-936 16 8 expanded expand VERB cana-936 16 9 the the DET cana-936 16 10 theoretical theoretical ADJ cana-936 16 11 landscape landscape NOUN cana-936 16 12 of of ADP cana-936 16 13 quantum quantum ADJ cana-936 16 14 mechanics mechanic NOUN cana-936 16 15 but but CCONJ cana-936 16 16 also also ADV cana-936 16 17 offered offer VERB cana-936 16 18 new new ADJ cana-936 16 19 perspectives perspective NOUN cana-936 16 20 and and CCONJ cana-936 16 21 methodologies methodology NOUN cana-936 16 22 for for ADP cana-936 16 23 investigating investigate VERB cana-936 16 24 quantum quantum ADJ cana-936 16 25 systems system NOUN cana-936 16 26 . . PUNCT cana-936 17 1 fractional fractional ADJ cana-936 17 2 calculus calculus NOUN cana-936 17 3 : : PUNCT cana-936 17 4 bridging bridge VERB cana-936 17 5 the the DET cana-936 17 6 gap gap NOUN cana-936 17 7 in in ADP cana-936 17 8 quantum quantum ADJ cana-936 17 9 mechanics mechanic NOUN cana-936 17 10 at at ADP cana-936 17 11 the the DET cana-936 17 12 heart heart NOUN cana-936 17 13 of of ADP cana-936 17 14 fqm fqm NOUN cana-936 17 15 lies lie VERB cana-936 17 16 fractional fractional ADJ cana-936 17 17 calculus calculus NOUN cana-936 17 18 , , PUNCT cana-936 17 19 a a DET cana-936 17 20 branch branch NOUN cana-936 17 21 of of ADP cana-936 17 22 mathematical mathematical ADJ cana-936 17 23 analysis analysis NOUN cana-936 17 24 that that PRON cana-936 17 25 generalizes generalize VERB cana-936 17 26 the the DET cana-936 17 27 concept concept NOUN cana-936 17 28 of of ADP cana-936 17 29 taking take VERB cana-936 17 30 derivatives derivative NOUN cana-936 17 31 and and CCONJ cana-936 17 32 integrals integral NOUN cana-936 17 33 to to ADP cana-936 17 34 fractional fractional ADJ cana-936 17 35 orders order NOUN cana-936 17 36 . . PUNCT cana-936 18 1 this this DET cana-936 18 2 mathematical mathematical ADJ cana-936 18 3 toolset toolset NOUN cana-936 18 4 allows allow VERB cana-936 18 5 for for ADP cana-936 18 6 the the DET cana-936 18 7 description description NOUN cana-936 18 8 of of ADP cana-936 18 9 processes process NOUN cana-936 18 10 and and CCONJ cana-936 18 11 phenomena phenomenon NOUN cana-936 18 12 that that PRON cana-936 18 13 are be AUX cana-936 18 14 not not PART cana-936 18 15 adequately adequately ADV cana-936 18 16 captured capture VERB cana-936 18 17 by by ADP cana-936 18 18 traditional traditional ADJ cana-936 18 19 integerorder integerorder NOUN cana-936 18 20 calculus calculus NOUN cana-936 18 21 , , PUNCT cana-936 18 22 offering offer VERB cana-936 18 23 a a DET cana-936 18 24 more more ADV cana-936 18 25 flexible flexible ADJ cana-936 18 26 and and CCONJ cana-936 18 27 general general ADJ cana-936 18 28 framework framework NOUN cana-936 18 29 for for ADP cana-936 18 30 modeling model VERB cana-936 18 31 physical physical ADJ cana-936 18 32 systems system NOUN cana-936 18 33 . . PUNCT cana-936 19 1 the the DET cana-936 19 2 communications communication NOUN cana-936 19 3 on on ADP cana-936 19 4 applied apply VERB cana-936 19 5 nonlinear nonlinear ADJ cana-936 19 6 analysis analysis NOUN cana-936 19 7 issn issn NOUN cana-936 19 8 : : PUNCT cana-936 19 9 1074 1074 NUM cana-936 19 10 - - PUNCT cana-936 19 11 133x 133x NUM cana-936 19 12 vol vol NOUN cana-936 19 13 31 31 NUM cana-936 19 14 no no NOUN cana-936 19 15 . . PUNCT cana-936 20 1 4s 4s NUM cana-936 20 2 ( ( PUNCT cana-936 20 3 2024 2024 NUM cana-936 20 4 ) ) PUNCT cana-936 20 5 426 426 NUM cana-936 20 6 https://internationalpubls.com https://internationalpubls.com X cana-936 20 7 application application NOUN cana-936 20 8 of of ADP cana-936 20 9 fractional fractional ADJ cana-936 20 10 calculus calculus NOUN cana-936 20 11 in in ADP cana-936 20 12 quantum quantum ADJ cana-936 20 13 mechanics mechanic NOUN cana-936 20 14 represents represent VERB cana-936 20 15 a a DET cana-936 20 16 significant significant ADJ cana-936 20 17 shift shift NOUN cana-936 20 18 in in ADP cana-936 20 19 how how SCONJ cana-936 20 20 researchers researcher NOUN cana-936 20 21 approach approach VERB cana-936 20 22 the the DET cana-936 20 23 study study NOUN cana-936 20 24 of of ADP cana-936 20 25 quantum quantum NOUN cana-936 20 26 systems system NOUN cana-936 20 27 , , PUNCT cana-936 20 28 providing provide VERB cana-936 20 29 a a DET cana-936 20 30 versatile versatile ADJ cana-936 20 31 tool tool NOUN cana-936 20 32 for for ADP cana-936 20 33 exploring explore VERB cana-936 20 34 the the DET cana-936 20 35 subtleties subtlety NOUN cana-936 20 36 of of ADP cana-936 20 37 quantum quantum NOUN cana-936 20 38 dynamics dynamic NOUN cana-936 20 39 . . PUNCT cana-936 21 1 the the DET cana-936 21 2 foundation foundation NOUN cana-936 21 3 of of ADP cana-936 21 4 fractional fractional ADJ cana-936 21 5 quantum quantum ADJ cana-936 21 6 mechanics mechanic NOUN cana-936 21 7 a a DET cana-936 21 8 pivotal pivotal ADJ cana-936 21 9 element element NOUN cana-936 21 10 in in ADP cana-936 21 11 fqm fqm NOUN cana-936 21 12 is be AUX cana-936 21 13 the the DET cana-936 21 14 formulation formulation NOUN cana-936 21 15 of of ADP cana-936 21 16 quantum quantum ADJ cana-936 21 17 dynamics dynamic NOUN cana-936 21 18 through through ADP cana-936 21 19 fractional fractional ADJ cana-936 21 20 differential differential ADJ cana-936 21 21 equations equation NOUN cana-936 21 22 . . PUNCT cana-936 22 1 these these DET cana-936 22 2 equations equation NOUN cana-936 22 3 , , PUNCT cana-936 22 4 characterized characterize VERB cana-936 22 5 by by ADP cana-936 22 6 their their PRON cana-936 22 7 fractional fractional ADJ cana-936 22 8 derivatives derivative NOUN cana-936 22 9 , , PUNCT cana-936 22 10 encapsulate encapsulate VERB cana-936 22 11 the the DET cana-936 22 12 essence essence NOUN cana-936 22 13 of of ADP cana-936 22 14 quantum quantum ADJ cana-936 22 15 behavior behavior NOUN cana-936 22 16 in in ADP cana-936 22 17 a a DET cana-936 22 18 way way NOUN cana-936 22 19 that that PRON cana-936 22 20 traditional traditional ADJ cana-936 22 21 differential differential ADJ cana-936 22 22 equations equation NOUN cana-936 22 23 can can AUX cana-936 22 24 not not PART cana-936 22 25 . . PUNCT cana-936 23 1 for for ADP cana-936 23 2 instance instance NOUN cana-936 23 3 , , PUNCT cana-936 23 4 the the DET cana-936 23 5 caputo caputo PROPN cana-936 23 6 fractional fractional PROPN cana-936 23 7 derivative derivative NOUN cana-936 23 8 , , PUNCT cana-936 23 9 represented represent VERB cana-936 23 10 by by ADP cana-936 23 11 : : PUNCT cana-936 23 12 π·π‘Ž π·π‘Ž NOUN cana-936 23 13 π‘žπ‘“(𝑑 π‘žπ‘“(𝑑 NUM cana-936 23 14 ) ) PUNCT cana-936 23 15 = = SYM cana-936 23 16 1 1 NUM cana-936 23 17 𝛀(𝑛 𝛀(𝑛 NUM cana-936 23 18 βˆ’ βˆ’ PROPN cana-936 23 19 π‘ž π‘ž NOUN cana-936 23 20 ) ) PUNCT cana-936 23 21 ∫ ∫ PROPN cana-936 23 22 𝑓(𝑛)(𝜏 𝑓(𝑛)(𝜏 NUM cana-936 23 23 ) ) PUNCT cana-936 23 24 ( ( PUNCT cana-936 23 25 𝑑 𝑑 PROPN cana-936 23 26 βˆ’ βˆ’ PROPN cana-936 23 27 𝜏)π‘žβˆ’π‘›+1 𝜏)π‘žβˆ’π‘›+1 NOUN cana-936 23 28 𝑑 𝑑 PROPN cana-936 23 29 π‘Ž π‘Ž PROPN cana-936 23 30 π‘‘πœ π‘‘πœ NOUN cana-936 23 31 , , PUNCT cana-936 23 32 serves serve VERB cana-936 23 33 as as ADP cana-936 23 34 a a DET cana-936 23 35 fundamental fundamental ADJ cana-936 23 36 mathematical mathematical ADJ cana-936 23 37 tool tool NOUN cana-936 23 38 in in ADP cana-936 23 39 fqm fqm NOUN cana-936 23 40 . . PUNCT cana-936 24 1 here here ADV cana-936 24 2 , , PUNCT cana-936 24 3 q q PROPN cana-936 24 4 denotes denote VERB cana-936 24 5 the the DET cana-936 24 6 order order NOUN cana-936 24 7 of of ADP cana-936 24 8 the the DET cana-936 24 9 derivative derivative NOUN cana-936 24 10 , , PUNCT cana-936 24 11 providing provide VERB cana-936 24 12 a a DET cana-936 24 13 measure measure NOUN cana-936 24 14 of of ADP cana-936 24 15 the the DET cana-936 24 16 system system NOUN cana-936 24 17 's 's PART cana-936 24 18 deviation deviation NOUN cana-936 24 19 from from ADP cana-936 24 20 classical classical ADJ cana-936 24 21 behavior behavior NOUN cana-936 24 22 . . PUNCT cana-936 25 1 this this DET cana-936 25 2 equation equation NOUN cana-936 25 3 exemplifies exemplify VERB cana-936 25 4 the the DET cana-936 25 5 core core ADJ cana-936 25 6 principles principle NOUN cana-936 25 7 of of ADP cana-936 25 8 fractional fractional ADJ cana-936 25 9 calculus calculus NOUN cana-936 25 10 and and CCONJ cana-936 25 11 its its PRON cana-936 25 12 application application NOUN cana-936 25 13 in in ADP cana-936 25 14 quantum quantum ADJ cana-936 25 15 mechanics mechanic NOUN cana-936 25 16 research research NOUN cana-936 25 17 aims aim VERB cana-936 25 18 and and CCONJ cana-936 25 19 objectives objective VERB cana-936 25 20 the the DET cana-936 25 21 primary primary ADJ cana-936 25 22 aim aim NOUN cana-936 25 23 of of ADP cana-936 25 24 this this DET cana-936 25 25 research research NOUN cana-936 25 26 is be AUX cana-936 25 27 to to PART cana-936 25 28 undertake undertake VERB cana-936 25 29 a a DET cana-936 25 30 comparative comparative ADJ cana-936 25 31 study study NOUN cana-936 25 32 of of ADP cana-936 25 33 numerical numerical ADJ cana-936 25 34 and and CCONJ cana-936 25 35 analytical analytical ADJ cana-936 25 36 approaches approach NOUN cana-936 25 37 in in ADP cana-936 25 38 fqm fqm NOUN cana-936 25 39 , , PUNCT cana-936 25 40 with with ADP cana-936 25 41 the the DET cana-936 25 42 objective objective NOUN cana-936 25 43 of of ADP cana-936 25 44 elucidating elucidate VERB cana-936 25 45 the the DET cana-936 25 46 quantum quantum ADJ cana-936 25 47 dynamics dynamic NOUN cana-936 25 48 of of ADP cana-936 25 49 systems system NOUN cana-936 25 50 described describe VERB cana-936 25 51 by by ADP cana-936 25 52 fractional fractional ADJ cana-936 25 53 differential differential ADJ cana-936 25 54 equations equation NOUN cana-936 25 55 . . PUNCT cana-936 26 1 by by ADP cana-936 26 2 systematically systematically ADV cana-936 26 3 exploring explore VERB cana-936 26 4 both both DET cana-936 26 5 methodologies methodology NOUN cana-936 26 6 , , PUNCT cana-936 26 7 this this DET cana-936 26 8 study study NOUN cana-936 26 9 seeks seek VERB cana-936 26 10 to to PART cana-936 26 11 uncover uncover VERB cana-936 26 12 the the DET cana-936 26 13 advantages advantage NOUN cana-936 26 14 and and CCONJ cana-936 26 15 limitations limitation NOUN cana-936 26 16 inherent inherent ADJ cana-936 26 17 in in ADP cana-936 26 18 each each DET cana-936 26 19 approach approach NOUN cana-936 26 20 , , PUNCT cana-936 26 21 thereby thereby ADV cana-936 26 22 contributing contribute VERB cana-936 26 23 to to ADP cana-936 26 24 a a DET cana-936 26 25 deeper deep ADJ cana-936 26 26 and and CCONJ cana-936 26 27 more more ADV cana-936 26 28 comprehensive comprehensive ADJ cana-936 26 29 understanding understanding NOUN cana-936 26 30 of of ADP cana-936 26 31 fqm fqm NOUN cana-936 26 32 . . PUNCT cana-936 27 1 understanding understanding NOUN cana-936 27 2 fqm fqm VERB cana-936 27 3 through through ADP cana-936 27 4 fractional fractional ADJ cana-936 27 5 calculus calculus NOUN cana-936 27 6 the the DET cana-936 27 7 application application NOUN cana-936 27 8 of of ADP cana-936 27 9 fractional fractional ADJ cana-936 27 10 derivatives derivative NOUN cana-936 27 11 in in ADP cana-936 27 12 quantum quantum ADJ cana-936 27 13 mechanics mechanic NOUN cana-936 27 14 enables enable VERB cana-936 27 15 the the DET cana-936 27 16 description description NOUN cana-936 27 17 of of ADP cana-936 27 18 quantum quantum ADJ cana-936 27 19 paths path NOUN cana-936 27 20 and and CCONJ cana-936 27 21 states state NOUN cana-936 27 22 in in ADP cana-936 27 23 a a DET cana-936 27 24 way way NOUN cana-936 27 25 that that PRON cana-936 27 26 conventional conventional ADJ cana-936 27 27 calculus calculus NOUN cana-936 27 28 can can AUX cana-936 27 29 not not PART cana-936 27 30 . . PUNCT cana-936 28 1 by by ADP cana-936 28 2 allowing allow VERB cana-936 28 3 the the DET cana-936 28 4 order order NOUN cana-936 28 5 of of ADP cana-936 28 6 the the DET cana-936 28 7 derivative derivative NOUN cana-936 28 8 to to PART cana-936 28 9 vary vary VERB cana-936 28 10 , , PUNCT cana-936 28 11 fractional fractional ADJ cana-936 28 12 calculus calculus NOUN cana-936 28 13 provides provide VERB cana-936 28 14 a a DET cana-936 28 15 more more ADV cana-936 28 16 flexible flexible ADJ cana-936 28 17 framework framework NOUN cana-936 28 18 for for ADP cana-936 28 19 modeling model VERB cana-936 28 20 quantum quantum ADJ cana-936 28 21 behavior behavior NOUN cana-936 28 22 , , PUNCT cana-936 28 23 particularly particularly ADV cana-936 28 24 in in ADP cana-936 28 25 systems system NOUN cana-936 28 26 where where SCONJ cana-936 28 27 classical classical ADJ cana-936 28 28 mechanics mechanic NOUN cana-936 28 29 fails fail VERB cana-936 28 30 to to PART cana-936 28 31 offer offer VERB cana-936 28 32 accurate accurate ADJ cana-936 28 33 predictions prediction NOUN cana-936 28 34 . . PUNCT cana-936 29 1 this this DET cana-936 29 2 exploration exploration NOUN cana-936 29 3 into into ADP cana-936 29 4 the the DET cana-936 29 5 foundational foundational ADJ cana-936 29 6 aspects aspect NOUN cana-936 29 7 of of ADP cana-936 29 8 fqm fqm NOUN cana-936 29 9 sets set VERB cana-936 29 10 the the DET cana-936 29 11 stage stage NOUN cana-936 29 12 for for ADP cana-936 29 13 deeper deep ADJ cana-936 29 14 discussions discussion NOUN cana-936 29 15 on on ADP cana-936 29 16 how how SCONJ cana-936 29 17 fractional fractional ADJ cana-936 29 18 calculus calculus NOUN cana-936 29 19 is be AUX cana-936 29 20 applied apply VERB cana-936 29 21 in in ADP cana-936 29 22 various various ADJ cana-936 29 23 quantum quantum ADJ cana-936 29 24 mechanical mechanical ADJ cana-936 29 25 problems problem NOUN cana-936 29 26 , , PUNCT cana-936 29 27 paving pave VERB cana-936 29 28 the the DET cana-936 29 29 way way NOUN cana-936 29 30 for for ADP cana-936 29 31 groundbreaking groundbreake VERB cana-936 29 32 research research NOUN cana-936 29 33 and and CCONJ cana-936 29 34 insights insight NOUN cana-936 29 35 into into ADP cana-936 29 36 the the DET cana-936 29 37 quantum quantum ADJ cana-936 29 38 world world NOUN cana-936 29 39 . . PUNCT cana-936 30 1 table table NOUN cana-936 30 2 1 1 NUM cana-936 30 3 : : PUNCT cana-936 30 4 comparison comparison NOUN cana-936 30 5 of of ADP cana-936 30 6 fractional fractional ADJ cana-936 30 7 and and CCONJ cana-936 30 8 integer integer NOUN cana-936 30 9 order order NOUN cana-936 30 10 derivatives derivative NOUN cana-936 30 11 feature feature VERB cana-936 30 12 integer integer NOUN cana-936 30 13 order order NOUN cana-936 30 14 derivative derivative ADJ cana-936 30 15 fractional fractional ADJ cana-936 30 16 order order NOUN cana-936 30 17 derivative derivative ADJ cana-936 30 18 definition definition NOUN cana-936 30 19 based base VERB cana-936 30 20 on on ADP cana-936 30 21 integer integer NOUN cana-936 30 22 orders order NOUN cana-936 30 23 extends extend VERB cana-936 30 24 to to ADP cana-936 30 25 non non ADJ cana-936 30 26 - - ADJ cana-936 30 27 integer integer ADJ cana-936 30 28 orders order NOUN cana-936 30 29 mathematical mathematical ADJ cana-936 30 30 model model NOUN cana-936 30 31 less less ADV cana-936 30 32 flexible flexible ADJ cana-936 30 33 , , PUNCT cana-936 30 34 specific specific ADJ cana-936 30 35 cases case NOUN cana-936 30 36 more more ADV cana-936 30 37 flexible flexible ADJ cana-936 30 38 , , PUNCT cana-936 30 39 general general ADJ cana-936 30 40 cases case NOUN cana-936 30 41 application application NOUN cana-936 30 42 in in ADP cana-936 30 43 physics physics NOUN cana-936 30 44 classical classical ADJ cana-936 30 45 mechanics mechanic NOUN cana-936 30 46 quantum quantum NOUN cana-936 30 47 mechanics mechanic NOUN cana-936 30 48 , , PUNCT cana-936 30 49 particularly particularly ADV cana-936 30 50 fqm fqm VERB cana-936 30 51 this this DET cana-936 30 52 table table NOUN cana-936 30 53 succinctly succinctly ADV cana-936 30 54 contrasts contrast VERB cana-936 30 55 the the DET cana-936 30 56 characteristics characteristic NOUN cana-936 30 57 and and CCONJ cana-936 30 58 applications application NOUN cana-936 30 59 of of ADP cana-936 30 60 integer integer NOUN cana-936 30 61 and and CCONJ cana-936 30 62 fractional fractional ADJ cana-936 30 63 order order NOUN cana-936 30 64 derivatives derivative NOUN cana-936 30 65 , , PUNCT cana-936 30 66 underscoring underscore VERB cana-936 30 67 the the DET cana-936 30 68 versatility versatility NOUN cana-936 30 69 and and CCONJ cana-936 30 70 breadth breadth NOUN cana-936 30 71 that that DET cana-936 30 72 fractional fractional ADJ cana-936 30 73 calculus calculus NOUN cana-936 30 74 introduces introduce NOUN cana-936 30 75 to to ADP cana-936 30 76 quantum quantum ADJ cana-936 30 77 mechanics mechanic NOUN cana-936 30 78 . . PUNCT cana-936 31 1 communications communication NOUN cana-936 31 2 on on ADP cana-936 31 3 applied apply VERB cana-936 31 4 nonlinear nonlinear ADJ cana-936 31 5 analysis analysis NOUN cana-936 31 6 issn issn NOUN cana-936 31 7 : : PUNCT cana-936 31 8 1074 1074 NUM cana-936 31 9 - - PUNCT cana-936 31 10 133x 133x NUM cana-936 31 11 vol vol NOUN cana-936 31 12 31 31 NUM cana-936 31 13 no no NOUN cana-936 31 14 . . PUNCT cana-936 32 1 4s 4s NUM cana-936 32 2 ( ( PUNCT cana-936 32 3 2024 2024 NUM cana-936 32 4 ) ) PUNCT cana-936 32 5 427 427 NUM cana-936 33 1 https://internationalpubls.com https://internationalpubls.com SYM cana-936 33 2 2 2 NUM cana-936 33 3 . . NOUN cana-936 33 4 methodologies methodology NOUN cana-936 33 5 analytical analytical ADJ cana-936 33 6 approaches approach VERB cana-936 33 7 analytical analytical ADJ cana-936 33 8 methods method NOUN cana-936 33 9 in in ADP cana-936 33 10 fractional fractional ADJ cana-936 33 11 quantum quantum ADJ cana-936 33 12 mechanics mechanic NOUN cana-936 33 13 ( ( PUNCT cana-936 33 14 fqm fqm NOUN cana-936 33 15 ) ) PUNCT cana-936 33 16 hinge hinge NOUN cana-936 33 17 on on ADP cana-936 33 18 leveraging leverage VERB cana-936 33 19 the the DET cana-936 33 20 mathematical mathematical ADJ cana-936 33 21 elegance elegance NOUN cana-936 33 22 of of ADP cana-936 33 23 fractional fractional ADJ cana-936 33 24 calculus calculus NOUN cana-936 33 25 to to PART cana-936 33 26 solve solve VERB cana-936 33 27 quantum quantum ADJ cana-936 33 28 mechanical mechanical ADJ cana-936 33 29 problems problem NOUN cana-936 33 30 with with ADP cana-936 33 31 exact exact ADJ cana-936 33 32 solutions solution NOUN cana-936 33 33 . . PUNCT cana-936 34 1 these these DET cana-936 34 2 approaches approach NOUN cana-936 34 3 are be AUX cana-936 34 4 invaluable invaluable ADJ cana-936 34 5 for for ADP cana-936 34 6 understanding understand VERB cana-936 34 7 the the DET cana-936 34 8 theoretical theoretical ADJ cana-936 34 9 underpinnings underpinning NOUN cana-936 34 10 of of ADP cana-936 34 11 fqm fqm NOUN cana-936 34 12 , , PUNCT cana-936 34 13 providing provide VERB cana-936 34 14 insights insight NOUN cana-936 34 15 into into ADP cana-936 34 16 the the DET cana-936 34 17 behavior behavior NOUN cana-936 34 18 of of ADP cana-936 34 19 quantum quantum NOUN cana-936 34 20 systems system NOUN cana-936 34 21 . . PUNCT cana-936 35 1 1 1 X cana-936 35 2 . . X cana-936 35 3 fractional fractional ADJ cana-936 35 4 schrΓΆdinger schrΓΆdinger ADJ cana-936 35 5 equation equation NOUN cana-936 35 6 the the DET cana-936 35 7 fractional fractional ADJ cana-936 35 8 schrΓΆdinger schrΓΆdinger ADJ cana-936 35 9 equation equation NOUN cana-936 35 10 is be AUX cana-936 35 11 a a DET cana-936 35 12 cornerstone cornerstone NOUN cana-936 35 13 of of ADP cana-936 35 14 fqm fqm NOUN cana-936 35 15 , , PUNCT cana-936 35 16 extending extend VERB cana-936 35 17 the the DET cana-936 35 18 classical classical ADJ cana-936 35 19 schrΓΆdinger schrΓΆdinger ADJ cana-936 35 20 equation equation NOUN cana-936 35 21 to to PART cana-936 35 22 incorporate incorporate VERB cana-936 35 23 fractional fractional ADJ cana-936 35 24 derivatives derivative NOUN cana-936 35 25 , , PUNCT cana-936 35 26 thus thus ADV cana-936 35 27 capturing capture VERB cana-936 35 28 quantum quantum ADJ cana-936 35 29 phenomena phenomenon NOUN cana-936 35 30 more more ADV cana-936 35 31 accurately accurately ADV cana-936 35 32 in in ADP cana-936 35 33 certain certain ADJ cana-936 35 34 contexts context NOUN cana-936 35 35 . . PUNCT cana-936 36 1 equation equation NOUN cana-936 36 2 2 2 NUM cana-936 36 3 : : PUNCT cana-936 36 4 fractional fractional ADJ cana-936 36 5 schrΓΆdinger schrΓΆdinger ADJ cana-936 36 6 equation equation NOUN cana-936 36 7 𝑖ℏ 𝑖ℏ ADV cana-936 36 8 πœ•πœ“(π‘₯ πœ•πœ“(π‘₯ PROPN cana-936 36 9 , , PUNCT cana-936 36 10 𝑑 𝑑 PROPN cana-936 36 11 ) ) PUNCT cana-936 36 12 πœ•π‘‘ πœ•π‘‘ NOUN cana-936 37 1 = = PUNCT cana-936 38 1 [ [ X cana-936 38 2 𝐷π‘₯ 𝐷π‘₯ PROPN cana-936 38 3 π‘Ž π‘Ž X cana-936 38 4 + + X cana-936 38 5 𝑉(π‘₯)]πœ“(π‘₯ 𝑉(π‘₯)]πœ“(π‘₯ NUM cana-936 38 6 , , PUNCT cana-936 38 7 𝑑 𝑑 NOUN cana-936 38 8 ) ) PUNCT cana-936 38 9 , , PUNCT cana-936 38 10 where where SCONJ cana-936 38 11 i i PRON cana-936 38 12 is be AUX cana-936 38 13 the the DET cana-936 38 14 imaginary imaginary ADJ cana-936 38 15 unit unit NOUN cana-936 38 16 , , PUNCT cana-936 38 17 ℏℏ ℏℏ ADP cana-936 38 18 is be AUX cana-936 38 19 the the DET cana-936 38 20 reduced reduce VERB cana-936 38 21 planck planck NOUN cana-936 38 22 's 's PART cana-936 38 23 constant constant ADJ cana-936 38 24 , , PUNCT cana-936 38 25 ψ(x ψ(x PROPN cana-936 38 26 , , PUNCT cana-936 38 27 t t PROPN cana-936 38 28 ) ) PUNCT cana-936 38 29 is be AUX cana-936 38 30 the the DET cana-936 38 31 wave wave NOUN cana-936 38 32 function function NOUN cana-936 38 33 of of ADP cana-936 38 34 the the DET cana-936 38 35 particle particle NOUN cana-936 38 36 , , PUNCT cana-936 38 37 𝐷π‘₯ 𝐷π‘₯ PROPN cana-936 38 38 π‘Ž π‘Ž NOUN cana-936 38 39 represents represent VERB cana-936 38 40 the the DET cana-936 38 41 fractional fractional ADJ cana-936 38 42 derivative derivative NOUN cana-936 38 43 with with ADP cana-936 38 44 respect respect NOUN cana-936 38 45 to to ADP cana-936 38 46 x x PUNCT cana-936 38 47 of of ADP cana-936 38 48 order order NOUN cana-936 38 49 Ξ± Ξ± NOUN cana-936 38 50 , , PUNCT cana-936 38 51 and and CCONJ cana-936 38 52 v(x v(x PROPN cana-936 38 53 ) ) PUNCT cana-936 38 54 is be AUX cana-936 38 55 the the DET cana-936 38 56 potential potential ADJ cana-936 38 57 energy energy NOUN cana-936 38 58 . . PUNCT cana-936 39 1 2 2 X cana-936 39 2 . . X cana-936 39 3 fractional fractional ADJ cana-936 39 4 path path NOUN cana-936 39 5 integral integral ADJ cana-936 39 6 formulation formulation NOUN cana-936 39 7 the the DET cana-936 39 8 path path NOUN cana-936 39 9 integral integral ADJ cana-936 39 10 formulation formulation NOUN cana-936 39 11 of of ADP cana-936 39 12 quantum quantum ADJ cana-936 39 13 mechanics mechanic NOUN cana-936 39 14 can can AUX cana-936 39 15 also also ADV cana-936 39 16 be be AUX cana-936 39 17 extended extend VERB cana-936 39 18 to to ADP cana-936 39 19 fractional fractional ADJ cana-936 39 20 orders order NOUN cana-936 39 21 , , PUNCT cana-936 39 22 providing provide VERB cana-936 39 23 a a DET cana-936 39 24 framework framework NOUN cana-936 39 25 for for ADP cana-936 39 26 calculating calculate VERB cana-936 39 27 amplitudes amplitude NOUN cana-936 39 28 by by ADP cana-936 39 29 summing sum VERB cana-936 39 30 over over ADP cana-936 39 31 all all DET cana-936 39 32 possible possible ADJ cana-936 39 33 paths path NOUN cana-936 39 34 , , PUNCT cana-936 39 35 with with ADP cana-936 39 36 paths path NOUN cana-936 39 37 weighted weight VERB cana-936 39 38 by by ADP cana-936 39 39 a a DET cana-936 39 40 phase phase NOUN cana-936 39 41 factor factor NOUN cana-936 39 42 . . PUNCT cana-936 40 1 equation equation NOUN cana-936 40 2 3 3 NUM cana-936 40 3 : : PUNCT cana-936 40 4 fractional fractional ADJ cana-936 40 5 path path NOUN cana-936 40 6 integral integral ADJ cana-936 40 7 𝐾(π‘₯ 𝐾(π‘₯ SYM cana-936 40 8 , , PUNCT cana-936 40 9 𝑑 𝑑 PROPN cana-936 40 10 ; ; PUNCT cana-936 40 11 π‘₯0 π‘₯0 NOUN cana-936 40 12 , , PUNCT cana-936 40 13 𝑑0 𝑑0 PROPN cana-936 40 14 ) ) PUNCT cana-936 40 15 = = SYM cana-936 41 1 ∫ ∫ PROPN cana-936 41 2 𝐷[π‘₯(𝑑)]𝑒 𝐷[π‘₯(𝑑)]𝑒 NOUN cana-936 41 3 𝑖 𝑖 PRON cana-936 41 4 ℏ𝑆[π‘₯(𝑑 ℏ𝑆[π‘₯(𝑑 ADP cana-936 41 5 ) ) PUNCT cana-936 41 6 ] ] PUNCT cana-936 41 7 , , PUNCT cana-936 41 8 π‘₯(𝑑)=π‘₯ π‘₯(𝑑)=π‘₯ PROPN cana-936 41 9 π‘₯(𝑑0)=π‘₯0 π‘₯(𝑑0)=π‘₯0 PROPN cana-936 41 10 where where SCONJ cana-936 41 11 k(x k(x PROPN cana-936 41 12 , , PUNCT cana-936 41 13 t;x0,t0 t;x0,t0 NUM cana-936 41 14 ) ) PUNCT cana-936 41 15 is be AUX cana-936 41 16 the the DET cana-936 41 17 kernel kernel NOUN cana-936 41 18 or or CCONJ cana-936 41 19 propagator propagator NOUN cana-936 41 20 from from ADP cana-936 41 21 ( ( PUNCT cana-936 41 22 x0,t0 x0,t0 PROPN cana-936 41 23 ) ) PUNCT cana-936 41 24 to to ADP cana-936 41 25 ( ( PUNCT cana-936 41 26 x x X cana-936 41 27 , , PUNCT cana-936 41 28 t t PROPN cana-936 41 29 ) ) PUNCT cana-936 41 30 , , PUNCT cana-936 41 31 d[x(t d[x(t PROPN cana-936 41 32 ) ) PUNCT cana-936 41 33 ] ] PUNCT cana-936 41 34 denotes denote VERB cana-936 41 35 the the DET cana-936 41 36 measure measure NOUN cana-936 41 37 over over ADP cana-936 41 38 all all DET cana-936 41 39 paths path NOUN cana-936 41 40 , , PUNCT cana-936 41 41 and and CCONJ cana-936 41 42 s[x(t s[x(t PROPN cana-936 41 43 ) ) PUNCT cana-936 41 44 ] ] PUNCT cana-936 41 45 is be AUX cana-936 41 46 the the DET cana-936 41 47 action action NOUN cana-936 41 48 , , PUNCT cana-936 41 49 extended extend VERB cana-936 41 50 to to PART cana-936 41 51 incorporate incorporate VERB cana-936 41 52 fractional fractional ADJ cana-936 41 53 dynamics dynamic NOUN cana-936 41 54 . . PUNCT cana-936 42 1 limitations limitation NOUN cana-936 42 2 of of ADP cana-936 42 3 analytical analytical ADJ cana-936 42 4 approaches approach NOUN cana-936 42 5 while while SCONJ cana-936 42 6 analytical analytical ADJ cana-936 42 7 methods method NOUN cana-936 42 8 offer offer VERB cana-936 42 9 precise precise ADJ cana-936 42 10 solutions solution NOUN cana-936 42 11 , , PUNCT cana-936 42 12 their their PRON cana-936 42 13 applicability applicability NOUN cana-936 42 14 is be AUX cana-936 42 15 often often ADV cana-936 42 16 limited limit VERB cana-936 42 17 to to ADP cana-936 42 18 simpler simple ADJ cana-936 42 19 systems system NOUN cana-936 42 20 or or CCONJ cana-936 42 21 those those PRON cana-936 42 22 for for ADP cana-936 42 23 which which PRON cana-936 42 24 exact exact ADJ cana-936 42 25 solutions solution NOUN cana-936 42 26 are be AUX cana-936 42 27 feasible feasible ADJ cana-936 42 28 . . PUNCT cana-936 43 1 complex complex ADJ cana-936 43 2 or or CCONJ cana-936 43 3 non non ADJ cana-936 43 4 - - ADJ cana-936 43 5 linear linear ADJ cana-936 43 6 systems system NOUN cana-936 43 7 frequently frequently ADV cana-936 43 8 defy defy VERB cana-936 43 9 analytical analytical ADJ cana-936 43 10 solution solution NOUN cana-936 43 11 , , PUNCT cana-936 43 12 necessitating necessitate VERB cana-936 43 13 numerical numerical ADJ cana-936 43 14 or or CCONJ cana-936 43 15 approximate approximate ADJ cana-936 43 16 methods method NOUN cana-936 43 17 . . PUNCT cana-936 44 1 numerical numerical PROPN cana-936 44 2 approaches approaches PROPN cana-936 44 3 numerical numerical ADJ cana-936 44 4 simulations simulation NOUN cana-936 44 5 play play VERB cana-936 44 6 a a DET cana-936 44 7 pivotal pivotal ADJ cana-936 44 8 role role NOUN cana-936 44 9 in in ADP cana-936 44 10 fqm fqm NOUN cana-936 44 11 , , PUNCT cana-936 44 12 especially especially ADV cana-936 44 13 when when SCONJ cana-936 44 14 dealing deal VERB cana-936 44 15 with with ADP cana-936 44 16 complex complex ADJ cana-936 44 17 systems system NOUN cana-936 44 18 where where SCONJ cana-936 44 19 analytical analytical ADJ cana-936 44 20 solutions solution NOUN cana-936 44 21 are be AUX cana-936 44 22 unattainable unattainable ADJ cana-936 44 23 . . PUNCT cana-936 45 1 these these DET cana-936 45 2 methods method NOUN cana-936 45 3 rely rely VERB cana-936 45 4 on on ADP cana-936 45 5 discretizing discretize VERB cana-936 45 6 the the DET cana-936 45 7 problem problem NOUN cana-936 45 8 and and CCONJ cana-936 45 9 employing employ VERB cana-936 45 10 computational computational ADJ cana-936 45 11 algorithms algorithm NOUN cana-936 45 12 to to PART cana-936 45 13 approximate approximate ADJ cana-936 45 14 solutions solution NOUN cana-936 45 15 . . PUNCT cana-936 46 1 1 1 X cana-936 46 2 . . X cana-936 46 3 finite finite ADJ cana-936 46 4 difference difference NOUN cana-936 46 5 method method NOUN cana-936 46 6 ( ( PUNCT cana-936 46 7 fdm fdm NOUN cana-936 46 8 ) ) PUNCT cana-936 46 9 the the DET cana-936 46 10 fdm fdm NOUN cana-936 46 11 approximates approximate VERB cana-936 46 12 fractional fractional ADJ cana-936 46 13 derivatives derivative NOUN cana-936 46 14 by by ADP cana-936 46 15 discretizing discretize VERB cana-936 46 16 the the DET cana-936 46 17 differential differential ADJ cana-936 46 18 equation equation NOUN cana-936 46 19 , , PUNCT cana-936 46 20 allowing allow VERB cana-936 46 21 for for ADP cana-936 46 22 numerical numerical ADJ cana-936 46 23 solution solution NOUN cana-936 46 24 of of ADP cana-936 46 25 the the DET cana-936 46 26 fractional fractional ADJ cana-936 46 27 schrΓΆdinger schrΓΆdinger ADJ cana-936 46 28 equation equation NOUN cana-936 46 29 and and CCONJ cana-936 46 30 other other ADJ cana-936 46 31 fqm fqm NOUN cana-936 46 32 formulations formulation NOUN cana-936 46 33 . . PUNCT cana-936 47 1 communications communication NOUN cana-936 47 2 on on ADP cana-936 47 3 applied apply VERB cana-936 47 4 nonlinear nonlinear ADJ cana-936 47 5 analysis analysis NOUN cana-936 47 6 issn issn NOUN cana-936 47 7 : : PUNCT cana-936 47 8 1074 1074 NUM cana-936 47 9 - - PUNCT cana-936 47 10 133x 133x NUM cana-936 47 11 vol vol NOUN cana-936 47 12 31 31 NUM cana-936 47 13 no no NOUN cana-936 47 14 . . PUNCT cana-936 48 1 4s 4s NUM cana-936 48 2 ( ( PUNCT cana-936 48 3 2024 2024 NUM cana-936 48 4 ) ) PUNCT cana-936 48 5 428 428 NUM cana-936 48 6 https://internationalpubls.com https://internationalpubls.com X cana-936 48 7 2 2 X cana-936 48 8 . . PUNCT cana-936 48 9 spectral spectral ADJ cana-936 48 10 methods method NOUN cana-936 48 11 spectral spectral ADJ cana-936 48 12 methods method NOUN cana-936 48 13 involve involve VERB cana-936 48 14 representing represent VERB cana-936 48 15 the the DET cana-936 48 16 solution solution NOUN cana-936 48 17 as as ADP cana-936 48 18 a a DET cana-936 48 19 sum sum NOUN cana-936 48 20 of of ADP cana-936 48 21 basis basis NOUN cana-936 48 22 functions function NOUN cana-936 48 23 ( ( PUNCT cana-936 48 24 e.g. e.g. ADV cana-936 48 25 , , PUNCT cana-936 48 26 fourier fourier ADJ cana-936 48 27 series series NOUN cana-936 48 28 ) ) PUNCT cana-936 48 29 , , PUNCT cana-936 48 30 enabling enable VERB cana-936 48 31 the the DET cana-936 48 32 efficient efficient ADJ cana-936 48 33 computation computation NOUN cana-936 48 34 of of ADP cana-936 48 35 fractional fractional ADJ cana-936 48 36 derivatives derivative NOUN cana-936 48 37 and and CCONJ cana-936 48 38 the the DET cana-936 48 39 solution solution NOUN cana-936 48 40 of of ADP cana-936 48 41 fqm fqm NOUN cana-936 48 42 equations equation NOUN cana-936 48 43 in in ADP cana-936 48 44 the the DET cana-936 48 45 frequency frequency NOUN cana-936 48 46 domain domain NOUN cana-936 48 47 . . PUNCT cana-936 49 1 table table NOUN cana-936 49 2 1 1 NUM cana-936 49 3 : : PUNCT cana-936 49 4 efficiency efficiency NOUN cana-936 49 5 and and CCONJ cana-936 49 6 applicability applicability NOUN cana-936 49 7 of of ADP cana-936 49 8 numerical numerical ADJ cana-936 49 9 methods method NOUN cana-936 49 10 method method VERB cana-936 49 11 efficiency efficiency NOUN cana-936 49 12 applicability applicability NOUN cana-936 49 13 limitations limitation NOUN cana-936 49 14 fdm fdm VERB cana-936 49 15 high high ADV cana-936 49 16 for for ADP cana-936 49 17 lowdimensional lowdimensional ADJ cana-936 49 18 systems system NOUN cana-936 49 19 suitable suitable ADJ cana-936 49 20 for for ADP cana-936 49 21 a a DET cana-936 49 22 wide wide ADJ cana-936 49 23 range range NOUN cana-936 49 24 of of ADP cana-936 49 25 problems problem NOUN cana-936 49 26 becomes become VERB cana-936 49 27 computationally computationally ADV cana-936 49 28 expensive expensive ADJ cana-936 49 29 in in ADP cana-936 49 30 higher high ADJ cana-936 49 31 dimensions dimension NOUN cana-936 49 32 spectral spectral ADJ cana-936 49 33 methods method NOUN cana-936 49 34 very very ADV cana-936 49 35 high high ADJ cana-936 49 36 in in ADP cana-936 49 37 frequency frequency NOUN cana-936 49 38 domain domain NOUN cana-936 49 39 analyses analysis NOUN cana-936 49 40 best good ADJ cana-936 49 41 for for ADP cana-936 49 42 problems problem NOUN cana-936 49 43 with with ADP cana-936 49 44 periodic periodic ADJ cana-936 49 45 boundary boundary ADJ cana-936 49 46 conditions condition NOUN cana-936 49 47 requires require VERB cana-936 49 48 transformation transformation NOUN cana-936 49 49 of of ADP cana-936 49 50 the the DET cana-936 49 51 problem problem NOUN cana-936 49 52 into into ADP cana-936 49 53 the the DET cana-936 49 54 frequency frequency NOUN cana-936 49 55 domain domain NOUN cana-936 49 56 comparative comparative NOUN cana-936 49 57 strengths strength NOUN cana-936 49 58 numerical numerical ADJ cana-936 49 59 methods method NOUN cana-936 49 60 offer offer VERB cana-936 49 61 the the DET cana-936 49 62 flexibility flexibility NOUN cana-936 49 63 to to PART cana-936 49 64 tackle tackle VERB cana-936 49 65 a a DET cana-936 49 66 wide wide ADJ cana-936 49 67 array array NOUN cana-936 49 68 of of ADP cana-936 49 69 problems problem NOUN cana-936 49 70 in in ADP cana-936 49 71 fqm fqm NOUN cana-936 49 72 , , PUNCT cana-936 49 73 from from ADP cana-936 49 74 simple simple ADJ cana-936 49 75 to to ADP cana-936 49 76 highly highly ADV cana-936 49 77 complex complex ADJ cana-936 49 78 systems system NOUN cana-936 49 79 . . PUNCT cana-936 50 1 their their PRON cana-936 50 2 comparative comparative ADJ cana-936 50 3 strengths strength NOUN cana-936 50 4 lie lie VERB cana-936 50 5 in in ADP cana-936 50 6 their their PRON cana-936 50 7 adaptability adaptability NOUN cana-936 50 8 and and CCONJ cana-936 50 9 scalability scalability NOUN cana-936 50 10 , , PUNCT cana-936 50 11 allowing allow VERB cana-936 50 12 researchers researcher NOUN cana-936 50 13 to to PART cana-936 50 14 explore explore VERB cana-936 50 15 systems system NOUN cana-936 50 16 beyond beyond ADP cana-936 50 17 the the DET cana-936 50 18 reach reach NOUN cana-936 50 19 of of ADP cana-936 50 20 analytical analytical ADJ cana-936 50 21 solutions solution NOUN cana-936 50 22 . . PUNCT cana-936 51 1 however however ADV cana-936 51 2 , , PUNCT cana-936 51 3 the the DET cana-936 51 4 choice choice NOUN cana-936 51 5 of of ADP cana-936 51 6 method method NOUN cana-936 51 7 depends depend VERB cana-936 51 8 on on ADP cana-936 51 9 the the DET cana-936 51 10 specific specific ADJ cana-936 51 11 characteristics characteristic NOUN cana-936 51 12 of of ADP cana-936 51 13 the the DET cana-936 51 14 problem problem NOUN cana-936 51 15 , , PUNCT cana-936 51 16 including include VERB cana-936 51 17 boundary boundary ADJ cana-936 51 18 conditions condition NOUN cana-936 51 19 , , PUNCT cana-936 51 20 dimensions dimension NOUN cana-936 51 21 , , PUNCT cana-936 51 22 and and CCONJ cana-936 51 23 the the DET cana-936 51 24 desired desire VERB cana-936 51 25 accuracy accuracy NOUN cana-936 51 26 . . PUNCT cana-936 52 1 3 3 X cana-936 52 2 . . NUM cana-936 52 3 analytical analytical ADJ cana-936 52 4 case case NOUN cana-936 52 5 study study NOUN cana-936 52 6 : : PUNCT cana-936 52 7 riesz riesz VERB cana-936 52 8 fractional fractional ADJ cana-936 52 9 derivative derivative NOUN cana-936 52 10 in in ADP cana-936 52 11 quantum quantum NOUN cana-936 52 12 tunneling tunnel VERB cana-936 52 13 the the DET cana-936 52 14 riesz riesz NOUN cana-936 52 15 fractional fractional ADJ cana-936 52 16 derivative derivative ADJ cana-936 52 17 and and CCONJ cana-936 52 18 quantum quantum NOUN cana-936 52 19 tunneling tunneling NOUN cana-936 52 20 one one NUM cana-936 52 21 of of ADP cana-936 52 22 the the DET cana-936 52 23 quintessential quintessential ADJ cana-936 52 24 problems problem NOUN cana-936 52 25 in in ADP cana-936 52 26 quantum quantum ADJ cana-936 52 27 mechanics mechanic NOUN cana-936 52 28 is be AUX cana-936 52 29 quantum quantum ADJ cana-936 52 30 tunneling tunneling NOUN cana-936 52 31 , , PUNCT cana-936 52 32 where where SCONJ cana-936 52 33 a a DET cana-936 52 34 particle particle NOUN cana-936 52 35 traverses traverse VERB cana-936 52 36 a a DET cana-936 52 37 potential potential ADJ cana-936 52 38 barrier barrier NOUN cana-936 52 39 higher high ADJ cana-936 52 40 than than ADP cana-936 52 41 its its PRON cana-936 52 42 kinetic kinetic ADJ cana-936 52 43 energy energy NOUN cana-936 52 44 , , PUNCT cana-936 52 45 a a DET cana-936 52 46 phenomenon phenomenon NOUN cana-936 52 47 inexplicable inexplicable ADJ cana-936 52 48 by by ADP cana-936 52 49 classical classical ADJ cana-936 52 50 physics physic NOUN cana-936 52 51 . . PUNCT cana-936 53 1 the the DET cana-936 53 2 application application NOUN cana-936 53 3 of of ADP cana-936 53 4 the the DET cana-936 53 5 riesz riesz PROPN cana-936 53 6 fractional fractional ADJ cana-936 53 7 derivative derivative NOUN cana-936 53 8 , , PUNCT cana-936 53 9 a a DET cana-936 53 10 cornerstone cornerstone NOUN cana-936 53 11 of of ADP cana-936 53 12 fqm fqm NOUN cana-936 53 13 , , PUNCT cana-936 53 14 offers offer VERB cana-936 53 15 a a DET cana-936 53 16 fresh fresh ADJ cana-936 53 17 perspective perspective NOUN cana-936 53 18 on on ADP cana-936 53 19 this this DET cana-936 53 20 well well ADV cana-936 53 21 - - PUNCT cana-936 53 22 studied study VERB cana-936 53 23 phenomenon phenomenon NOUN cana-936 53 24 . . PUNCT cana-936 54 1 equation equation NOUN cana-936 54 2 4 4 NUM cana-936 54 3 : : PUNCT cana-936 54 4 riesz riesz VERB cana-936 54 5 fractional fractional ADJ cana-936 54 6 derivative derivative ADJ cana-936 55 1 𝐷𝑅𝑖𝑒𝑠𝑧 𝐷𝑅𝑖𝑒𝑠𝑧 PROPN cana-936 55 2 π‘ž π‘ž X cana-936 55 3 πœ“(π‘₯ πœ“(π‘₯ ADV cana-936 55 4 ) ) PUNCT cana-936 55 5 = = SYM cana-936 55 6 βˆ’ βˆ’ PROPN cana-936 55 7 1 1 NUM cana-936 55 8 2πœ‹ 2πœ‹ NUM cana-936 55 9 ∫ ∫ PROPN cana-936 56 1 ⬚ ⬚ PROPN cana-936 56 2 + + PROPN cana-936 56 3 ∞ ∞ PROPN cana-936 56 4 βˆ’βˆž βˆ’βˆž ADP cana-936 56 5 ∣ ∣ NOUN cana-936 56 6 π‘˜ π‘˜ DET cana-936 56 7 βˆ£π‘ž βˆ£π‘ž PROPN cana-936 56 8 π‘’π‘–π‘˜π‘₯ π‘’π‘–π‘˜π‘₯ NOUN cana-936 56 9 ^ ^ PUNCT cana-936 56 10 πœ“ πœ“ PROPN cana-936 56 11 ( ( PUNCT cana-936 56 12 π‘˜)π‘‘π‘˜ π‘˜)π‘‘π‘˜ PROPN cana-936 56 13 , , PUNCT cana-936 56 14 where where SCONJ cana-936 56 15 𝐷𝑅𝑖𝑒𝑠𝑧 𝐷𝑅𝑖𝑒𝑠𝑧 PROPN cana-936 56 16 π‘ž π‘ž X cana-936 56 17 denotes denote VERB cana-936 56 18 the the DET cana-936 56 19 riesz riesz PROPN cana-936 56 20 fractional fractional ADJ cana-936 56 21 derivative derivative NOUN cana-936 56 22 of of ADP cana-936 56 23 order order NOUN cana-936 56 24 ψ(x ψ(x NOUN cana-936 56 25 ) ) PUNCT cana-936 56 26 is be AUX cana-936 56 27 the the DET cana-936 56 28 wave wave NOUN cana-936 56 29 function function NOUN cana-936 56 30 , , PUNCT cana-936 56 31 k k PROPN cana-936 56 32 is be AUX cana-936 56 33 the the DET cana-936 56 34 wave wave NOUN cana-936 56 35 number number NOUN cana-936 56 36 , , PUNCT cana-936 56 37 and and CCONJ cana-936 56 38 ^ ^ PUNCT cana-936 56 39 πœ“ πœ“ PROPN cana-936 56 40 ( ( PUNCT cana-936 56 41 π‘˜ π‘˜ NOUN cana-936 56 42 ) ) PUNCT cana-936 56 43 is be AUX cana-936 56 44 the the DET cana-936 56 45 fourier fourier NOUN cana-936 56 46 transform transform NOUN cana-936 56 47 of of ADP cana-936 56 48 ψ(x ψ(x NOUN cana-936 56 49 ) ) PUNCT cana-936 56 50 . . PUNCT cana-936 57 1 by by ADP cana-936 57 2 applying apply VERB cana-936 57 3 the the DET cana-936 57 4 riesz riesz NOUN cana-936 57 5 fractional fractional ADJ cana-936 57 6 derivative derivative NOUN cana-936 57 7 to to ADP cana-936 57 8 the the DET cana-936 57 9 quantum quantum ADJ cana-936 57 10 tunneling tunneling NOUN cana-936 57 11 problem problem NOUN cana-936 57 12 , , PUNCT cana-936 57 13 we we PRON cana-936 57 14 obtain obtain VERB cana-936 57 15 a a DET cana-936 57 16 modified modify VERB cana-936 57 17 wave wave NOUN cana-936 57 18 function function NOUN cana-936 57 19 that that PRON cana-936 57 20 reflects reflect VERB cana-936 57 21 the the DET cana-936 57 22 non non ADJ cana-936 57 23 - - ADJ cana-936 57 24 local local ADJ cana-936 57 25 properties property NOUN cana-936 57 26 of of ADP cana-936 57 27 fractional fractional ADJ cana-936 57 28 derivatives derivative NOUN cana-936 57 29 , , PUNCT cana-936 57 30 offering offer VERB cana-936 57 31 a a DET cana-936 57 32 nuanced nuanced ADJ cana-936 57 33 understanding understanding NOUN cana-936 57 34 of of ADP cana-936 57 35 the the DET cana-936 57 36 tunneling tunneling NOUN cana-936 57 37 process process NOUN cana-936 57 38 and and CCONJ cana-936 57 39 potentially potentially ADV cana-936 57 40 revising revise VERB cana-936 57 41 the the DET cana-936 57 42 predicted predict VERB cana-936 57 43 tunneling tunneling NOUN cana-936 57 44 rates rate NOUN cana-936 57 45 . . PUNCT cana-936 58 1 4 4 X cana-936 58 2 . . X cana-936 58 3 numerical numerical PROPN cana-936 58 4 case case NOUN cana-936 58 5 study study NOUN cana-936 58 6 : : PUNCT cana-936 58 7 fractional fractional ADJ cana-936 58 8 quantum quantum ADJ cana-936 58 9 harmonic harmonic NOUN cana-936 58 10 oscillator oscillator NOUN cana-936 58 11 introduction introduction NOUN cana-936 58 12 to to ADP cana-936 58 13 the the DET cana-936 58 14 fractional fractional ADJ cana-936 58 15 quantum quantum ADJ cana-936 58 16 harmonic harmonic NOUN cana-936 58 17 oscillator oscillator NOUN cana-936 58 18 the the DET cana-936 58 19 harmonic harmonic ADJ cana-936 58 20 oscillator oscillator NOUN cana-936 58 21 model model NOUN cana-936 58 22 is be AUX cana-936 58 23 pivotal pivotal ADJ cana-936 58 24 in in ADP cana-936 58 25 quantum quantum ADJ cana-936 58 26 mechanics mechanic NOUN cana-936 58 27 , , PUNCT cana-936 58 28 describing describe VERB cana-936 58 29 a a DET cana-936 58 30 wide wide ADJ cana-936 58 31 array array NOUN cana-936 58 32 of of ADP cana-936 58 33 physical physical ADJ cana-936 58 34 systems system NOUN cana-936 58 35 . . PUNCT cana-936 59 1 extending extend VERB cana-936 59 2 this this DET cana-936 59 3 model model NOUN cana-936 59 4 to to PART cana-936 59 5 fqm fqm NOUN cana-936 59 6 involves involve VERB cana-936 59 7 incorporating incorporate VERB cana-936 59 8 fractional fractional ADJ cana-936 59 9 derivatives derivative NOUN cana-936 59 10 , , PUNCT cana-936 59 11 which which PRON cana-936 59 12 complicates complicate VERB cana-936 59 13 its its PRON cana-936 59 14 analytical analytical ADJ cana-936 59 15 solution solution NOUN cana-936 59 16 but but CCONJ cana-936 59 17 opens open VERB cana-936 59 18 the the DET cana-936 59 19 door door NOUN cana-936 59 20 for for ADP cana-936 59 21 numerical numerical ADJ cana-936 59 22 exploration exploration NOUN cana-936 59 23 . . PUNCT cana-936 60 1 communications communication NOUN cana-936 60 2 on on ADP cana-936 60 3 applied apply VERB cana-936 60 4 nonlinear nonlinear ADJ cana-936 60 5 analysis analysis NOUN cana-936 60 6 issn issn NOUN cana-936 60 7 : : PUNCT cana-936 60 8 1074 1074 NUM cana-936 60 9 - - PUNCT cana-936 60 10 133x 133x NUM cana-936 60 11 vol vol NOUN cana-936 60 12 31 31 NUM cana-936 60 13 no no NOUN cana-936 60 14 . . PUNCT cana-936 61 1 4s 4s NUM cana-936 61 2 ( ( PUNCT cana-936 61 3 2024 2024 NUM cana-936 61 4 ) ) PUNCT cana-936 61 5 429 429 NUM cana-936 61 6 https://internationalpubls.com https://internationalpubls.com X cana-936 61 7 numerical numerical ADJ cana-936 61 8 approaches approach VERB cana-936 61 9 various various ADJ cana-936 61 10 numerical numerical ADJ cana-936 61 11 methods method NOUN cana-936 61 12 , , PUNCT cana-936 61 13 such such ADJ cana-936 61 14 as as ADP cana-936 61 15 the the DET cana-936 61 16 finite finite ADJ cana-936 61 17 difference difference NOUN cana-936 61 18 method method NOUN cana-936 61 19 ( ( PUNCT cana-936 61 20 fdm fdm NOUN cana-936 61 21 ) ) PUNCT cana-936 61 22 and and CCONJ cana-936 61 23 the the DET cana-936 61 24 spectral spectral ADJ cana-936 61 25 method method NOUN cana-936 61 26 , , PUNCT cana-936 61 27 are be AUX cana-936 61 28 employed employ VERB cana-936 61 29 to to PART cana-936 61 30 approximate approximate VERB cana-936 61 31 the the DET cana-936 61 32 solution solution NOUN cana-936 61 33 to to ADP cana-936 61 34 the the DET cana-936 61 35 fractional fractional ADJ cana-936 61 36 quantum quantum ADJ cana-936 61 37 harmonic harmonic NOUN cana-936 61 38 oscillator oscillator NOUN cana-936 61 39 problem problem NOUN cana-936 61 40 . . PUNCT cana-936 62 1 these these DET cana-936 62 2 methods method NOUN cana-936 62 3 enable enable VERB cana-936 62 4 the the DET cana-936 62 5 exploration exploration NOUN cana-936 62 6 of of ADP cana-936 62 7 the the DET cana-936 62 8 oscillator oscillator NOUN cana-936 62 9 's 's PART cana-936 62 10 properties property NOUN cana-936 62 11 under under ADP cana-936 62 12 fractional fractional ADJ cana-936 62 13 quantum quantum ADJ cana-936 62 14 mechanics mechanic NOUN cana-936 62 15 , , PUNCT cana-936 62 16 including include VERB cana-936 62 17 alterations alteration NOUN cana-936 62 18 in in ADP cana-936 62 19 its its PRON cana-936 62 20 energy energy NOUN cana-936 62 21 levels level NOUN cana-936 62 22 and and CCONJ cana-936 62 23 wave wave NOUN cana-936 62 24 functions function NOUN cana-936 62 25 . . PUNCT cana-936 63 1 table table NOUN cana-936 63 2 2 2 NUM cana-936 63 3 : : PUNCT cana-936 63 4 comparative comparative ADJ cana-936 63 5 outcomes outcome NOUN cana-936 63 6 of of ADP cana-936 63 7 numerical numerical ADJ cana-936 63 8 methods method NOUN cana-936 63 9 for for ADP cana-936 63 10 the the DET cana-936 63 11 fractional fractional ADJ cana-936 63 12 quantum quantum ADJ cana-936 63 13 harmonic harmonic NOUN cana-936 63 14 oscillator oscillator NOUN cana-936 63 15 method method NOUN cana-936 63 16 computed compute VERB cana-936 63 17 energy energy NOUN cana-936 63 18 levels level NOUN cana-936 63 19 wave wave VERB cana-936 63 20 function function NOUN cana-936 63 21 characteristics characteristics PROPN cana-936 63 22 computational computational ADJ cana-936 63 23 efficiency efficiency NOUN cana-936 63 24 fdm fdm NOUN cana-936 63 25 close close ADJ cana-936 63 26 approximation approximation NOUN cana-936 63 27 to to ADP cana-936 63 28 analytical analytical ADJ cana-936 63 29 predictions prediction NOUN cana-936 63 30 accurately accurately ADV cana-936 63 31 captures capture VERB cana-936 63 32 the the DET cana-936 63 33 asymmetry asymmetry NOUN cana-936 63 34 introduced introduce VERB cana-936 63 35 by by ADP cana-936 63 36 fractional fractional ADJ cana-936 63 37 derivatives derivative NOUN cana-936 63 38 high high ADJ cana-936 63 39 for for ADP cana-936 63 40 low low ADJ cana-936 63 41 - - PUNCT cana-936 63 42 dimensional dimensional ADJ cana-936 63 43 systems system NOUN cana-936 63 44 spectral spectral ADJ cana-936 63 45 method method NOUN cana-936 63 46 highly highly ADV cana-936 63 47 accurate accurate ADJ cana-936 63 48 energy energy NOUN cana-936 63 49 levels level NOUN cana-936 63 50 provides provide VERB cana-936 63 51 detailed detailed ADJ cana-936 63 52 insight insight NOUN cana-936 63 53 into into ADP cana-936 63 54 the the DET cana-936 63 55 wave wave NOUN cana-936 63 56 function function NOUN cana-936 63 57 's 's PART cana-936 63 58 behavior behavior NOUN cana-936 63 59 very very ADV cana-936 63 60 efficient efficient ADJ cana-936 63 61 for for ADP cana-936 63 62 problems problem NOUN cana-936 63 63 with with ADP cana-936 63 64 smooth smooth ADJ cana-936 63 65 solutions solution NOUN cana-936 63 66 discussion discussion NOUN cana-936 63 67 the the DET cana-936 63 68 analytical analytical ADJ cana-936 63 69 case case NOUN cana-936 63 70 study study NOUN cana-936 63 71 demonstrates demonstrate VERB cana-936 63 72 the the DET cana-936 63 73 power power NOUN cana-936 63 74 of of ADP cana-936 63 75 the the DET cana-936 63 76 riesz riesz PROPN cana-936 63 77 fractional fractional ADJ cana-936 63 78 derivative derivative NOUN cana-936 63 79 in in ADP cana-936 63 80 offering offer VERB cana-936 63 81 new new ADJ cana-936 63 82 insights insight NOUN cana-936 63 83 into into ADP cana-936 63 84 traditional traditional ADJ cana-936 63 85 quantum quantum ADJ cana-936 63 86 mechanics mechanic NOUN cana-936 63 87 problems problem NOUN cana-936 63 88 , , PUNCT cana-936 63 89 such such ADJ cana-936 63 90 as as ADP cana-936 63 91 quantum quantum ADJ cana-936 63 92 tunneling tunneling NOUN cana-936 63 93 . . PUNCT cana-936 64 1 this this DET cana-936 64 2 approach approach NOUN cana-936 64 3 highlights highlight VERB cana-936 64 4 the the DET cana-936 64 5 importance importance NOUN cana-936 64 6 of of ADP cana-936 64 7 non non ADJ cana-936 64 8 - - ADJ cana-936 64 9 local local ADJ cana-936 64 10 derivatives derivative NOUN cana-936 64 11 in in ADP cana-936 64 12 capturing capture VERB cana-936 64 13 the the DET cana-936 64 14 essence essence NOUN cana-936 64 15 of of ADP cana-936 64 16 quantum quantum NOUN cana-936 64 17 phenomena phenomenon NOUN cana-936 64 18 more more ADV cana-936 64 19 accurately accurately ADV cana-936 64 20 . . PUNCT cana-936 65 1 on on ADP cana-936 65 2 the the DET cana-936 65 3 other other ADJ cana-936 65 4 hand hand NOUN cana-936 65 5 , , PUNCT cana-936 65 6 the the DET cana-936 65 7 numerical numerical PROPN cana-936 65 8 case case NOUN cana-936 65 9 study study NOUN cana-936 65 10 on on ADP cana-936 65 11 the the DET cana-936 65 12 fractional fractional ADJ cana-936 65 13 quantum quantum ADJ cana-936 65 14 harmonic harmonic NOUN cana-936 65 15 oscillator oscillator NOUN cana-936 65 16 underscores underscore VERB cana-936 65 17 the the DET cana-936 65 18 practical practical ADJ cana-936 65 19 implications implication NOUN cana-936 65 20 of of ADP cana-936 65 21 numerical numerical ADJ cana-936 65 22 methods method NOUN cana-936 65 23 in in ADP cana-936 65 24 fqm fqm NOUN cana-936 65 25 . . PUNCT cana-936 66 1 by by ADP cana-936 66 2 comparing compare VERB cana-936 66 3 the the DET cana-936 66 4 outcomes outcome NOUN cana-936 66 5 from from ADP cana-936 66 6 various various ADJ cana-936 66 7 numerical numerical ADJ cana-936 66 8 methods method NOUN cana-936 66 9 , , PUNCT cana-936 66 10 this this DET cana-936 66 11 case case NOUN cana-936 66 12 study study NOUN cana-936 66 13 reveals reveal VERB cana-936 66 14 the the DET cana-936 66 15 strengths strength NOUN cana-936 66 16 and and CCONJ cana-936 66 17 limitations limitation NOUN cana-936 66 18 of of ADP cana-936 66 19 each each DET cana-936 66 20 approach approach NOUN cana-936 66 21 , , PUNCT cana-936 66 22 showcasing showcase VERB cana-936 66 23 their their PRON cana-936 66 24 utility utility NOUN cana-936 66 25 in in ADP cana-936 66 26 investigating investigate VERB cana-936 66 27 complex complex ADJ cana-936 66 28 quantum quantum NOUN cana-936 66 29 systems system NOUN cana-936 66 30 where where SCONJ cana-936 66 31 analytical analytical ADJ cana-936 66 32 solutions solution NOUN cana-936 66 33 are be AUX cana-936 66 34 not not PART cana-936 66 35 feasible feasible ADJ cana-936 66 36 . . PUNCT cana-936 67 1 together together ADV cana-936 67 2 , , PUNCT cana-936 67 3 these these DET cana-936 67 4 case case NOUN cana-936 67 5 studies study NOUN cana-936 67 6 not not PART cana-936 67 7 only only ADV cana-936 67 8 illustrate illustrate VERB cana-936 67 9 the the DET cana-936 67 10 application application NOUN cana-936 67 11 of of ADP cana-936 67 12 fractional fractional ADJ cana-936 67 13 calculus calculus NOUN cana-936 67 14 in in ADP cana-936 67 15 quantum quantum ADJ cana-936 67 16 mechanics mechanic NOUN cana-936 67 17 but but CCONJ cana-936 67 18 also also ADV cana-936 67 19 emphasize emphasize VERB cana-936 67 20 the the DET cana-936 67 21 complementary complementary ADJ cana-936 67 22 nature nature NOUN cana-936 67 23 of of ADP cana-936 67 24 analytical analytical ADJ cana-936 67 25 and and CCONJ cana-936 67 26 numerical numerical ADJ cana-936 67 27 methods method NOUN cana-936 67 28 in in ADP cana-936 67 29 advancing advance VERB cana-936 67 30 our our PRON cana-936 67 31 understanding understanding NOUN cana-936 67 32 of of ADP cana-936 67 33 the the DET cana-936 67 34 quantum quantum ADJ cana-936 67 35 world world NOUN cana-936 67 36 . . PUNCT cana-936 68 1 through through ADP cana-936 68 2 these these DET cana-936 68 3 investigations investigation NOUN cana-936 68 4 , , PUNCT cana-936 68 5 fqm fqm NOUN cana-936 68 6 emerges emerge VERB cana-936 68 7 as as ADP cana-936 68 8 a a DET cana-936 68 9 robust robust ADJ cana-936 68 10 framework framework NOUN cana-936 68 11 for for ADP cana-936 68 12 exploring explore VERB cana-936 68 13 quantum quantum ADJ cana-936 68 14 phenomena phenomenon NOUN cana-936 68 15 , , PUNCT cana-936 68 16 with with ADP cana-936 68 17 the the DET cana-936 68 18 potential potential NOUN cana-936 68 19 to to PART cana-936 68 20 uncover uncover VERB cana-936 68 21 new new ADJ cana-936 68 22 physical physical ADJ cana-936 68 23 insights insight NOUN cana-936 68 24 and and CCONJ cana-936 68 25 contribute contribute VERB cana-936 68 26 significantly significantly ADV cana-936 68 27 to to ADP cana-936 68 28 the the DET cana-936 68 29 field field NOUN cana-936 68 30 of of ADP cana-936 68 31 quantum quantum ADJ cana-936 68 32 mechanics mechanic NOUN cana-936 68 33 . . PUNCT cana-936 69 1 5 5 X cana-936 69 2 . . X cana-936 69 3 summary summary NOUN cana-936 69 4 of of ADP cana-936 69 5 findings finding NOUN cana-936 69 6 analytical analytical ADJ cana-936 69 7 insights insight NOUN cana-936 69 8 from from ADP cana-936 69 9 the the DET cana-936 69 10 riesz riesz PROPN cana-936 69 11 fractional fractional PROPN cana-936 69 12 derivative derivative NOUN cana-936 69 13 the the DET cana-936 69 14 application application NOUN cana-936 69 15 of of ADP cana-936 69 16 the the DET cana-936 69 17 riesz riesz PROPN cana-936 69 18 fractional fractional ADJ cana-936 69 19 derivative derivative NOUN cana-936 69 20 to to ADP cana-936 69 21 quantum quantum ADJ cana-936 69 22 tunneling tunneling NOUN cana-936 69 23 has have AUX cana-936 69 24 elucidated elucidate VERB cana-936 69 25 non non ADJ cana-936 69 26 - - ADJ cana-936 69 27 local local ADJ cana-936 69 28 effects effect NOUN cana-936 69 29 inherent inherent ADJ cana-936 69 30 in in ADP cana-936 69 31 quantum quantum ADJ cana-936 69 32 processes process NOUN cana-936 69 33 , , PUNCT cana-936 69 34 providing provide VERB cana-936 69 35 a a DET cana-936 69 36 more more ADV cana-936 69 37 nuanced nuanced ADJ cana-936 69 38 understanding understanding NOUN cana-936 69 39 of of ADP cana-936 69 40 phenomena phenomenon NOUN cana-936 69 41 like like ADP cana-936 69 42 tunneling tunneling NOUN cana-936 69 43 rates rate NOUN cana-936 69 44 and and CCONJ cana-936 69 45 barrier barrier NOUN cana-936 69 46 penetration penetration NOUN cana-936 69 47 . . PUNCT cana-936 70 1 this this DET cana-936 70 2 analytical analytical ADJ cana-936 70 3 approach approach NOUN cana-936 70 4 underlines underline VERB cana-936 70 5 the the DET cana-936 70 6 potential potential NOUN cana-936 70 7 of of ADP cana-936 70 8 fractional fractional ADJ cana-936 70 9 calculus calculus NOUN cana-936 70 10 to to PART cana-936 70 11 refine refine VERB cana-936 70 12 our our PRON cana-936 70 13 comprehension comprehension NOUN cana-936 70 14 of of ADP cana-936 70 15 quantum quantum ADJ cana-936 70 16 mechanics mechanic NOUN cana-936 70 17 , , PUNCT cana-936 70 18 particularly particularly ADV cana-936 70 19 in in ADP cana-936 70 20 scenarios scenario NOUN cana-936 70 21 where where SCONJ cana-936 70 22 traditional traditional ADJ cana-936 70 23 models model NOUN cana-936 70 24 offer offer VERB cana-936 70 25 limited limited ADJ cana-936 70 26 insight insight NOUN cana-936 70 27 . . PUNCT cana-936 71 1 numerical numerical ADJ cana-936 71 2 discoveries discovery NOUN cana-936 71 3 in in ADP cana-936 71 4 the the DET cana-936 71 5 fractional fractional ADJ cana-936 71 6 quantum quantum NOUN cana-936 71 7 harmonic harmonic NOUN cana-936 71 8 oscillator oscillator NOUN cana-936 71 9 the the DET cana-936 71 10 exploration exploration NOUN cana-936 71 11 of of ADP cana-936 71 12 the the DET cana-936 71 13 fractional fractional ADJ cana-936 71 14 quantum quantum ADJ cana-936 71 15 harmonic harmonic NOUN cana-936 71 16 oscillator oscillator NOUN cana-936 71 17 through through ADP cana-936 71 18 numerical numerical ADJ cana-936 71 19 simulations simulation NOUN cana-936 71 20 has have AUX cana-936 71 21 revealed reveal VERB cana-936 71 22 alterations alteration NOUN cana-936 71 23 in in ADP cana-936 71 24 energy energy NOUN cana-936 71 25 levels level NOUN cana-936 71 26 and and CCONJ cana-936 71 27 wave wave NOUN cana-936 71 28 functions function NOUN cana-936 71 29 , , PUNCT cana-936 71 30 attributing attribute VERB cana-936 71 31 these these DET cana-936 71 32 changes change NOUN cana-936 71 33 to to ADP cana-936 71 34 the the DET cana-936 71 35 inclusion inclusion NOUN cana-936 71 36 of of ADP cana-936 71 37 fractional fractional ADJ cana-936 71 38 derivatives derivative NOUN cana-936 71 39 . . PUNCT cana-936 72 1 the the DET cana-936 72 2 comparison comparison NOUN cana-936 72 3 of of ADP cana-936 72 4 different different ADJ cana-936 72 5 numerical numerical ADJ cana-936 72 6 methods method NOUN cana-936 72 7 highlighted highlight VERB cana-936 72 8 the the DET cana-936 72 9 spectral spectral ADJ cana-936 72 10 communications communication NOUN cana-936 72 11 on on ADP cana-936 72 12 applied apply VERB cana-936 72 13 nonlinear nonlinear ADJ cana-936 72 14 analysis analysis NOUN cana-936 72 15 issn issn NOUN cana-936 72 16 : : PUNCT cana-936 72 17 1074 1074 NUM cana-936 72 18 - - PUNCT cana-936 72 19 133x 133x NUM cana-936 72 20 vol vol NOUN cana-936 72 21 31 31 NUM cana-936 72 22 no no NOUN cana-936 72 23 . . PUNCT cana-936 73 1 4s 4s NUM cana-936 73 2 ( ( PUNCT cana-936 73 3 2024 2024 NUM cana-936 73 4 ) ) PUNCT cana-936 73 5 430 430 NUM cana-936 73 6 https://internationalpubls.com https://internationalpubls.com X cana-936 73 7 method method NOUN cana-936 73 8 's 's PART cana-936 73 9 efficiency efficiency NOUN cana-936 73 10 in in ADP cana-936 73 11 capturing capture VERB cana-936 73 12 detailed detailed ADJ cana-936 73 13 wave wave NOUN cana-936 73 14 function function NOUN cana-936 73 15 behavior behavior NOUN cana-936 73 16 , , PUNCT cana-936 73 17 while while SCONJ cana-936 73 18 the the DET cana-936 73 19 finite finite ADJ cana-936 73 20 difference difference NOUN cana-936 73 21 method method NOUN cana-936 73 22 ( ( PUNCT cana-936 73 23 fdm fdm NOUN cana-936 73 24 ) ) PUNCT cana-936 73 25 offered offer VERB cana-936 73 26 a a DET cana-936 73 27 balance balance NOUN cana-936 73 28 between between ADP cana-936 73 29 accuracy accuracy NOUN cana-936 73 30 and and CCONJ cana-936 73 31 computational computational ADJ cana-936 73 32 feasibility feasibility NOUN cana-936 73 33 . . PUNCT cana-936 74 1 comparative comparative ADJ cana-936 74 2 analysis analysis NOUN cana-936 74 3 the the DET cana-936 74 4 divergent divergent ADJ cana-936 74 5 methodologies methodology NOUN cana-936 74 6 applied apply VERB cana-936 74 7 in in ADP cana-936 74 8 the the DET cana-936 74 9 case case NOUN cana-936 74 10 studies study NOUN cana-936 74 11 β€” β€” PUNCT cana-936 74 12 analytical analytical ADJ cana-936 74 13 vs. vs. X cana-936 74 14 numerical numerical ADJ cana-936 74 15 β€” β€” PUNCT cana-936 74 16 unveil unveil VERB cana-936 74 17 the the DET cana-936 74 18 dual dual ADJ cana-936 74 19 facets facet NOUN cana-936 74 20 of of ADP cana-936 74 21 fqm fqm NOUN cana-936 74 22 research research NOUN cana-936 74 23 . . PUNCT cana-936 75 1 analytical analytical ADJ cana-936 75 2 methods method NOUN cana-936 75 3 , , PUNCT cana-936 75 4 with with ADP cana-936 75 5 their their PRON cana-936 75 6 precise precise ADJ cana-936 75 7 solutions solution NOUN cana-936 75 8 , , PUNCT cana-936 75 9 underscore underscore VERB cana-936 75 10 the the DET cana-936 75 11 theoretical theoretical ADJ cana-936 75 12 potential potential NOUN cana-936 75 13 of of ADP cana-936 75 14 fqm fqm NOUN cana-936 75 15 , , PUNCT cana-936 75 16 while while SCONJ cana-936 75 17 numerical numerical ADJ cana-936 75 18 methods method NOUN cana-936 75 19 showcase showcase VERB cana-936 75 20 the the DET cana-936 75 21 practical practical ADJ cana-936 75 22 applicability applicability NOUN cana-936 75 23 in in ADP cana-936 75 24 complex complex ADJ cana-936 75 25 systems system NOUN cana-936 75 26 , , PUNCT cana-936 75 27 underscoring underscore VERB cana-936 75 28 the the DET cana-936 75 29 symbiotic symbiotic ADJ cana-936 75 30 relationship relationship NOUN cana-936 75 31 between between ADP cana-936 75 32 theory theory NOUN cana-936 75 33 and and CCONJ cana-936 75 34 computation computation NOUN cana-936 75 35 in in ADP cana-936 75 36 quantum quantum ADJ cana-936 75 37 mechanics mechanic NOUN cana-936 75 38 . . PUNCT cana-936 76 1 implications implication NOUN cana-936 76 2 for for ADP cana-936 76 3 fqm fqm NOUN cana-936 76 4 research research NOUN cana-936 76 5 the the DET cana-936 76 6 insights insight NOUN cana-936 76 7 gained gain VERB cana-936 76 8 from from ADP cana-936 76 9 both both DET cana-936 76 10 case case NOUN cana-936 76 11 studies study NOUN cana-936 76 12 accentuate accentuate VERB cana-936 76 13 the the DET cana-936 76 14 importance importance NOUN cana-936 76 15 of of ADP cana-936 76 16 fractional fractional ADJ cana-936 76 17 calculus calculus NOUN cana-936 76 18 in in ADP cana-936 76 19 extending extend VERB cana-936 76 20 the the DET cana-936 76 21 boundaries boundary NOUN cana-936 76 22 of of ADP cana-936 76 23 quantum quantum ADJ cana-936 76 24 mechanics mechanic NOUN cana-936 76 25 . . PUNCT cana-936 77 1 they they PRON cana-936 77 2 suggest suggest VERB cana-936 77 3 a a DET cana-936 77 4 fertile fertile ADJ cana-936 77 5 ground ground NOUN cana-936 77 6 for for ADP cana-936 77 7 future future ADJ cana-936 77 8 research research NOUN cana-936 77 9 , , PUNCT cana-936 77 10 especially especially ADV cana-936 77 11 in in ADP cana-936 77 12 unraveling unravel VERB cana-936 77 13 quantum quantum ADJ cana-936 77 14 systems system NOUN cana-936 77 15 ' ' PART cana-936 77 16 complexities complexity NOUN cana-936 77 17 through through ADP cana-936 77 18 the the DET cana-936 77 19 lens lens NOUN cana-936 77 20 of of ADP cana-936 77 21 fqm fqm NOUN cana-936 77 22 . . PUNCT cana-936 78 1 these these DET cana-936 78 2 findings finding NOUN cana-936 78 3 advocate advocate VERB cana-936 78 4 for for ADP cana-936 78 5 a a DET cana-936 78 6 balanced balanced ADJ cana-936 78 7 approach approach NOUN cana-936 78 8 , , PUNCT cana-936 78 9 leveraging leverage VERB cana-936 78 10 both both DET cana-936 78 11 analytical analytical ADJ cana-936 78 12 clarity clarity NOUN cana-936 78 13 and and CCONJ cana-936 78 14 numerical numerical ADJ cana-936 78 15 versatility versatility NOUN cana-936 78 16 to to PART cana-936 78 17 navigate navigate VERB cana-936 78 18 the the DET cana-936 78 19 challenges challenge NOUN cana-936 78 20 inherent inherent ADJ cana-936 78 21 in in ADP cana-936 78 22 the the DET cana-936 78 23 quantum quantum ADJ cana-936 78 24 domain domain NOUN cana-936 78 25 . . PUNCT cana-936 79 1 numerical numerical ADJ cana-936 79 2 discoveries discovery NOUN cana-936 79 3 : : PUNCT cana-936 79 4 empirical empirical ADJ cana-936 79 5 insights insight NOUN cana-936 79 6 through through ADP cana-936 79 7 computational computational ADJ cana-936 79 8 methods method NOUN cana-936 79 9 contrastingly contrastingly ADV cana-936 79 10 , , PUNCT cana-936 79 11 the the DET cana-936 79 12 numerical numerical ADJ cana-936 79 13 investigation investigation NOUN cana-936 79 14 into into ADP cana-936 79 15 the the DET cana-936 79 16 fractional fractional ADJ cana-936 79 17 quantum quantum ADJ cana-936 79 18 harmonic harmonic NOUN cana-936 79 19 oscillator oscillator NOUN cana-936 79 20 showcases showcase VERB cana-936 79 21 the the DET cana-936 79 22 practicality practicality NOUN cana-936 79 23 and and CCONJ cana-936 79 24 adaptability adaptability NOUN cana-936 79 25 of of ADP cana-936 79 26 computational computational ADJ cana-936 79 27 methods method NOUN cana-936 79 28 in in ADP cana-936 79 29 fqm fqm NOUN cana-936 79 30 . . PUNCT cana-936 80 1 table table NOUN cana-936 80 2 2 2 NUM cana-936 80 3 : : PUNCT cana-936 80 4 comparative comparative ADJ cana-936 80 5 efficacy efficacy NOUN cana-936 80 6 of of ADP cana-936 80 7 numerical numerical ADJ cana-936 80 8 methods method NOUN cana-936 80 9 method method VERB cana-936 80 10 energy energy NOUN cana-936 80 11 levels level NOUN cana-936 80 12 accuracy accuracy NOUN cana-936 80 13 wave wave NOUN cana-936 80 14 function function NOUN cana-936 80 15 insights insight NOUN cana-936 80 16 computational computational ADJ cana-936 80 17 demand demand NOUN cana-936 80 18 finite finite ADJ cana-936 80 19 difference difference NOUN cana-936 80 20 method method NOUN cana-936 80 21 ( ( PUNCT cana-936 80 22 fdm fdm NOUN cana-936 80 23 ) ) PUNCT cana-936 80 24 high high ADJ cana-936 80 25 for for ADP cana-936 80 26 simple simple ADJ cana-936 80 27 systems system NOUN cana-936 80 28 accurate accurate ADJ cana-936 80 29 for for ADP cana-936 80 30 basic basic ADJ cana-936 80 31 models model NOUN cana-936 80 32 moderate moderate ADJ cana-936 80 33 , , PUNCT cana-936 80 34 increases increase VERB cana-936 80 35 with with ADP cana-936 80 36 system system NOUN cana-936 80 37 complexity complexity NOUN cana-936 80 38 spectral spectral ADJ cana-936 80 39 method method NOUN cana-936 80 40 very very ADV cana-936 80 41 high high ADV cana-936 80 42 detailed detailed ADJ cana-936 80 43 , , PUNCT cana-936 80 44 especially especially ADV cana-936 80 45 for for ADP cana-936 80 46 smooth smooth ADJ cana-936 80 47 potentials potential NOUN cana-936 80 48 high high ADJ cana-936 80 49 , , PUNCT cana-936 80 50 efficient efficient ADJ cana-936 80 51 for for ADP cana-936 80 52 specific specific ADJ cana-936 80 53 boundary boundary ADJ cana-936 80 54 conditions condition NOUN cana-936 80 55 this this DET cana-936 80 56 table table NOUN cana-936 80 57 provides provide VERB cana-936 80 58 a a DET cana-936 80 59 succinct succinct ADJ cana-936 80 60 comparison comparison NOUN cana-936 80 61 of of ADP cana-936 80 62 how how SCONJ cana-936 80 63 different different ADJ cana-936 80 64 numerical numerical ADJ cana-936 80 65 methods method NOUN cana-936 80 66 perform perform VERB cana-936 80 67 in in ADP cana-936 80 68 addressing address VERB cana-936 80 69 the the DET cana-936 80 70 fractional fractional ADJ cana-936 80 71 quantum quantum ADJ cana-936 80 72 harmonic harmonic NOUN cana-936 80 73 oscillator oscillator NOUN cana-936 80 74 problem problem NOUN cana-936 80 75 , , PUNCT cana-936 80 76 highlighting highlight VERB cana-936 80 77 the the DET cana-936 80 78 trade trade NOUN cana-936 80 79 - - PUNCT cana-936 80 80 offs off NOUN cana-936 80 81 between between ADP cana-936 80 82 accuracy accuracy NOUN cana-936 80 83 , , PUNCT cana-936 80 84 insight insight NOUN cana-936 80 85 , , PUNCT cana-936 80 86 and and CCONJ cana-936 80 87 computational computational ADJ cana-936 80 88 resources resource NOUN cana-936 80 89 . . PUNCT cana-936 81 1 bridging bridge VERB cana-936 81 2 analytical analytical ADJ cana-936 81 3 and and CCONJ cana-936 81 4 numerical numerical PROPN cana-936 81 5 findings finding NOUN cana-936 81 6 the the DET cana-936 81 7 interplay interplay NOUN cana-936 81 8 between between ADP cana-936 81 9 the the DET cana-936 81 10 refined refined ADJ cana-936 81 11 theoretical theoretical ADJ cana-936 81 12 insights insight NOUN cana-936 81 13 provided provide VERB cana-936 81 14 by by ADP cana-936 81 15 analytical analytical ADJ cana-936 81 16 methods method NOUN cana-936 81 17 and and CCONJ cana-936 81 18 the the DET cana-936 81 19 empirical empirical ADJ cana-936 81 20 discoveries discovery NOUN cana-936 81 21 facilitated facilitate VERB cana-936 81 22 by by ADP cana-936 81 23 numerical numerical ADJ cana-936 81 24 methods method NOUN cana-936 81 25 enriches enrich VERB cana-936 81 26 the the DET cana-936 81 27 field field NOUN cana-936 81 28 of of ADP cana-936 81 29 fqm fqm NOUN cana-936 81 30 . . PUNCT cana-936 82 1 this this DET cana-936 82 2 dual dual ADJ cana-936 82 3 approach approach NOUN cana-936 82 4 not not PART cana-936 82 5 only only ADV cana-936 82 6 broadens broaden VERB cana-936 82 7 the the DET cana-936 82 8 scope scope NOUN cana-936 82 9 of of ADP cana-936 82 10 fqm fqm NOUN cana-936 82 11 research research NOUN cana-936 82 12 but but CCONJ cana-936 82 13 also also ADV cana-936 82 14 emphasizes emphasize VERB cana-936 82 15 the the DET cana-936 82 16 complementary complementary ADJ cana-936 82 17 nature nature NOUN cana-936 82 18 of of ADP cana-936 82 19 theoretical theoretical ADJ cana-936 82 20 precision precision NOUN cana-936 82 21 and and CCONJ cana-936 82 22 computational computational ADJ cana-936 82 23 flexibility flexibility NOUN cana-936 82 24 . . PUNCT cana-936 83 1 expanding expand VERB cana-936 83 2 the the DET cana-936 83 3 frontier frontier NOUN cana-936 83 4 of of ADP cana-936 83 5 fqm fqm NOUN cana-936 83 6 research research NOUN cana-936 83 7 the the DET cana-936 83 8 contributions contribution NOUN cana-936 83 9 from from ADP cana-936 83 10 both both DET cana-936 83 11 case case NOUN cana-936 83 12 studies study NOUN cana-936 83 13 signal signal VERB cana-936 83 14 a a DET cana-936 83 15 significant significant ADJ cana-936 83 16 leap leap NOUN cana-936 83 17 forward forward ADV cana-936 83 18 in in ADP cana-936 83 19 the the DET cana-936 83 20 understanding understanding NOUN cana-936 83 21 and and CCONJ cana-936 83 22 application application NOUN cana-936 83 23 of of ADP cana-936 83 24 fqm fqm NOUN cana-936 83 25 . . PUNCT cana-936 84 1 they they PRON cana-936 84 2 demonstrate demonstrate VERB cana-936 84 3 the the DET cana-936 84 4 invaluable invaluable ADJ cana-936 84 5 role role NOUN cana-936 84 6 of of ADP cana-936 84 7 fractional fractional ADJ cana-936 84 8 calculus calculus NOUN cana-936 84 9 in in ADP cana-936 84 10 modeling model VERB cana-936 84 11 quantum quantum ADJ cana-936 84 12 phenomena phenomenon NOUN cana-936 84 13 with with ADP cana-936 84 14 an an DET cana-936 84 15 unprecedented unprecedented ADJ cana-936 84 16 level level NOUN cana-936 84 17 of of ADP cana-936 84 18 detail detail NOUN cana-936 84 19 and and CCONJ cana-936 84 20 accuracy accuracy NOUN cana-936 84 21 . . PUNCT cana-936 85 1 navigating navigate VERB cana-936 85 2 challenges challenge NOUN cana-936 85 3 : : PUNCT cana-936 85 4 while while SCONJ cana-936 85 5 the the DET cana-936 85 6 journey journey NOUN cana-936 85 7 through through ADP cana-936 85 8 fqm fqm NOUN cana-936 85 9 's 's PART cana-936 85 10 analytical analytical ADJ cana-936 85 11 and and CCONJ cana-936 85 12 numerical numerical ADJ cana-936 85 13 landscapes landscape NOUN cana-936 85 14 has have AUX cana-936 85 15 been be AUX cana-936 85 16 enlightening enlightening ADJ cana-936 85 17 , , PUNCT cana-936 85 18 it it PRON cana-936 85 19 has have AUX cana-936 85 20 not not PART cana-936 85 21 been be AUX cana-936 85 22 without without ADP cana-936 85 23 its its PRON cana-936 85 24 hurdles hurdle NOUN cana-936 85 25 : : PUNCT cana-936 85 26 communications communication NOUN cana-936 85 27 on on ADP cana-936 85 28 applied apply VERB cana-936 85 29 nonlinear nonlinear ADJ cana-936 85 30 analysis analysis NOUN cana-936 85 31 issn issn NOUN cana-936 85 32 : : PUNCT cana-936 85 33 1074 1074 NUM cana-936 85 34 - - PUNCT cana-936 85 35 133x 133x NUM cana-936 85 36 vol vol NOUN cana-936 85 37 31 31 NUM cana-936 85 38 no no NOUN cana-936 85 39 . . PUNCT cana-936 86 1 4s 4s NUM cana-936 86 2 ( ( PUNCT cana-936 86 3 2024 2024 NUM cana-936 86 4 ) ) PUNCT cana-936 86 5 431 431 NUM cana-936 87 1 https://internationalpubls.com https://internationalpubls.com X cana-936 87 2 1 1 X cana-936 87 3 . . PUNCT cana-936 87 4 complexity complexity NOUN cana-936 87 5 of of ADP cana-936 87 6 fractional fractional ADJ cana-936 87 7 calculus calculus NOUN cana-936 87 8 : : PUNCT cana-936 87 9 the the DET cana-936 87 10 sophisticated sophisticated ADJ cana-936 87 11 nature nature NOUN cana-936 87 12 of of ADP cana-936 87 13 fractional fractional ADJ cana-936 87 14 calculus calculus NOUN cana-936 87 15 introduces introduce VERB cana-936 87 16 a a DET cana-936 87 17 steep steep ADJ cana-936 87 18 learning learn VERB cana-936 87 19 curve curve NOUN cana-936 87 20 and and CCONJ cana-936 87 21 mathematical mathematical ADJ cana-936 87 22 challenges challenge NOUN cana-936 87 23 , , PUNCT cana-936 87 24 especially especially ADV cana-936 87 25 when when SCONJ cana-936 87 26 extending extend VERB cana-936 87 27 to to ADP cana-936 87 28 non non ADJ cana-936 87 29 - - ADJ cana-936 87 30 linear linear ADJ cana-936 87 31 or or CCONJ cana-936 87 32 multidimensional multidimensional ADJ cana-936 87 33 systems system NOUN cana-936 87 34 . . PUNCT cana-936 88 1 2 2 X cana-936 88 2 . . X cana-936 88 3 computational computational ADJ cana-936 88 4 intensity intensity NOUN cana-936 88 5 : : PUNCT cana-936 88 6 the the DET cana-936 88 7 detailed detailed ADJ cana-936 88 8 and and CCONJ cana-936 88 9 often often ADV cana-936 88 10 intensive intensive ADJ cana-936 88 11 computational computational ADJ cana-936 88 12 requirements requirement NOUN cana-936 88 13 of of ADP cana-936 88 14 numerical numerical ADJ cana-936 88 15 methods method NOUN cana-936 88 16 pose pose VERB cana-936 88 17 significant significant ADJ cana-936 88 18 challenges challenge NOUN cana-936 88 19 , , PUNCT cana-936 88 20 particularly particularly ADV cana-936 88 21 for for ADP cana-936 88 22 larger large ADJ cana-936 88 23 or or CCONJ cana-936 88 24 more more ADV cana-936 88 25 complex complex ADJ cana-936 88 26 systems system NOUN cana-936 88 27 . . PUNCT cana-936 89 1 3 3 X cana-936 89 2 . . X cana-936 89 3 modeling modeling NOUN cana-936 89 4 precision precision NOUN cana-936 89 5 : : PUNCT cana-936 89 6 the the DET cana-936 89 7 accuracy accuracy NOUN cana-936 89 8 of of ADP cana-936 89 9 fractional fractional ADJ cana-936 89 10 models model NOUN cana-936 89 11 and and CCONJ cana-936 89 12 the the DET cana-936 89 13 physical physical ADJ cana-936 89 14 interpretability interpretability NOUN cana-936 89 15 of of ADP cana-936 89 16 their their PRON cana-936 89 17 results result NOUN cana-936 89 18 continue continue VERB cana-936 89 19 to to PART cana-936 89 20 be be AUX cana-936 89 21 areas area NOUN cana-936 89 22 needing need VERB cana-936 89 23 refinement refinement NOUN cana-936 89 24 , , PUNCT cana-936 89 25 highlighting highlight VERB cana-936 89 26 the the DET cana-936 89 27 need need NOUN cana-936 89 28 for for ADP cana-936 89 29 ongoing ongoing ADJ cana-936 89 30 theoretical theoretical ADJ cana-936 89 31 and and CCONJ cana-936 89 32 computational computational ADJ cana-936 89 33 innovation innovation NOUN cana-936 89 34 . . PUNCT cana-936 90 1 conclusion conclusion NOUN cana-936 90 2 and and CCONJ cana-936 90 3 future future ADJ cana-936 90 4 directions direction NOUN cana-936 90 5 this this DET cana-936 90 6 study study NOUN cana-936 90 7 embarked embark VERB cana-936 90 8 on on ADP cana-936 90 9 a a DET cana-936 90 10 comprehensive comprehensive ADJ cana-936 90 11 exploration exploration NOUN cana-936 90 12 of of ADP cana-936 90 13 fractional fractional ADJ cana-936 90 14 quantum quantum ADJ cana-936 90 15 mechanics mechanic NOUN cana-936 90 16 ( ( PUNCT cana-936 90 17 fqm fqm NOUN cana-936 90 18 ) ) PUNCT cana-936 90 19 , , PUNCT cana-936 90 20 utilizing utilize VERB cana-936 90 21 both both CCONJ cana-936 90 22 analytical analytical ADJ cana-936 90 23 and and CCONJ cana-936 90 24 numerical numerical ADJ cana-936 90 25 approaches approach NOUN cana-936 90 26 to to PART cana-936 90 27 delve delve VERB cana-936 90 28 into into ADP cana-936 90 29 the the DET cana-936 90 30 intricacies intricacy NOUN cana-936 90 31 of of ADP cana-936 90 32 quantum quantum NOUN cana-936 90 33 systems system NOUN cana-936 90 34 through through ADP cana-936 90 35 the the DET cana-936 90 36 lens lens NOUN cana-936 90 37 of of ADP cana-936 90 38 fractional fractional ADJ cana-936 90 39 calculus calculus NOUN cana-936 90 40 . . PUNCT cana-936 91 1 by by ADP cana-936 91 2 investigating investigate VERB cana-936 91 3 the the DET cana-936 91 4 riesz riesz PROPN cana-936 91 5 fractional fractional ADJ cana-936 91 6 derivative derivative NOUN cana-936 91 7 's 's PART cana-936 91 8 application application NOUN cana-936 91 9 in in ADP cana-936 91 10 quantum quantum NOUN cana-936 91 11 tunneling tunneling NOUN cana-936 91 12 and and CCONJ cana-936 91 13 employing employ VERB cana-936 91 14 numerical numerical ADJ cana-936 91 15 methods method NOUN cana-936 91 16 to to PART cana-936 91 17 analyze analyze VERB cana-936 91 18 the the DET cana-936 91 19 fractional fractional ADJ cana-936 91 20 quantum quantum ADJ cana-936 91 21 harmonic harmonic ADJ cana-936 91 22 oscillator oscillator NOUN cana-936 91 23 , , PUNCT cana-936 91 24 this this DET cana-936 91 25 research research NOUN cana-936 91 26 has have AUX cana-936 91 27 illuminated illuminate VERB cana-936 91 28 the the DET cana-936 91 29 potential potential NOUN cana-936 91 30 and and CCONJ cana-936 91 31 challenges challenge NOUN cana-936 91 32 of of ADP cana-936 91 33 fqm fqm NOUN cana-936 91 34 , , PUNCT cana-936 91 35 marking mark VERB cana-936 91 36 significant significant ADJ cana-936 91 37 strides stride NOUN cana-936 91 38 in in ADP cana-936 91 39 the the DET cana-936 91 40 understanding understanding NOUN cana-936 91 41 of of ADP cana-936 91 42 quantum quantum ADJ cana-936 91 43 phenomena phenomenon NOUN cana-936 91 44 . . PUNCT cana-936 92 1 6 6 X cana-936 92 2 . . X cana-936 92 3 key key ADJ cana-936 92 4 points point NOUN cana-936 92 5 recap recap NOUN cana-936 92 6 1 1 NUM cana-936 92 7 . . PUNCT cana-936 93 1 analytical analytical ADJ cana-936 93 2 insights insight NOUN cana-936 93 3 : : PUNCT cana-936 93 4 the the DET cana-936 93 5 use use NOUN cana-936 93 6 of of ADP cana-936 93 7 the the DET cana-936 93 8 riesz riesz PROPN cana-936 93 9 fractional fractional ADJ cana-936 93 10 derivative derivative NOUN cana-936 93 11 has have AUX cana-936 93 12 provided provide VERB cana-936 93 13 a a DET cana-936 93 14 deeper deep ADJ cana-936 93 15 understanding understanding NOUN cana-936 93 16 of of ADP cana-936 93 17 non non ADJ cana-936 93 18 - - ADJ cana-936 93 19 local local ADJ cana-936 93 20 quantum quantum NOUN cana-936 93 21 dynamics dynamic NOUN cana-936 93 22 , , PUNCT cana-936 93 23 particularly particularly ADV cana-936 93 24 in in ADP cana-936 93 25 phenomena phenomenon NOUN cana-936 93 26 such such ADJ cana-936 93 27 as as ADP cana-936 93 28 quantum quantum ADJ cana-936 93 29 tunneling tunneling NOUN cana-936 93 30 , , PUNCT cana-936 93 31 highlighting highlight VERB cana-936 93 32 the the DET cana-936 93 33 utility utility NOUN cana-936 93 34 of of ADP cana-936 93 35 fractional fractional ADJ cana-936 93 36 calculus calculus NOUN cana-936 93 37 in in ADP cana-936 93 38 revealing reveal VERB cana-936 93 39 nuanced nuanced ADJ cana-936 93 40 aspects aspect NOUN cana-936 93 41 of of ADP cana-936 93 42 quantum quantum ADJ cana-936 93 43 mechanics mechanic NOUN cana-936 93 44 that that PRON cana-936 93 45 traditional traditional ADJ cana-936 93 46 approaches approach NOUN cana-936 93 47 may may AUX cana-936 93 48 overlook overlook VERB cana-936 93 49 . . PUNCT cana-936 94 1 2 2 X cana-936 94 2 . . NOUN cana-936 94 3 numerical numerical ADJ cana-936 94 4 discoveries discovery NOUN cana-936 94 5 : : PUNCT cana-936 94 6 through through ADP cana-936 94 7 the the DET cana-936 94 8 examination examination NOUN cana-936 94 9 of of ADP cana-936 94 10 the the DET cana-936 94 11 fractional fractional ADJ cana-936 94 12 quantum quantum ADJ cana-936 94 13 harmonic harmonic ADJ cana-936 94 14 oscillator oscillator NOUN cana-936 94 15 , , PUNCT cana-936 94 16 this this DET cana-936 94 17 study study NOUN cana-936 94 18 demonstrated demonstrate VERB cana-936 94 19 the the DET cana-936 94 20 effectiveness effectiveness NOUN cana-936 94 21 of of ADP cana-936 94 22 numerical numerical ADJ cana-936 94 23 methods method NOUN cana-936 94 24 in in ADP cana-936 94 25 exploring explore VERB cana-936 94 26 complex complex ADJ cana-936 94 27 quantum quantum ADJ cana-936 94 28 systems system NOUN cana-936 94 29 where where SCONJ cana-936 94 30 analytical analytical ADJ cana-936 94 31 solutions solution NOUN cana-936 94 32 are be AUX cana-936 94 33 not not PART cana-936 94 34 feasible feasible ADJ cana-936 94 35 , , PUNCT cana-936 94 36 showcasing showcase VERB cana-936 94 37 the the DET cana-936 94 38 adaptability adaptability NOUN cana-936 94 39 and and CCONJ cana-936 94 40 potential potential NOUN cana-936 94 41 of of ADP cana-936 94 42 computational computational ADJ cana-936 94 43 approaches approach NOUN cana-936 94 44 in in ADP cana-936 94 45 fqm fqm NOUN cana-936 94 46 research research NOUN cana-936 94 47 . . PUNCT cana-936 95 1 7 7 X cana-936 95 2 . . X cana-936 95 3 contributions contribution NOUN cana-936 95 4 to to PART cana-936 95 5 fqm fqm VERB cana-936 95 6 this this DET cana-936 95 7 research research NOUN cana-936 95 8 contributes contribute VERB cana-936 95 9 to to ADP cana-936 95 10 the the DET cana-936 95 11 burgeoning burgeon VERB cana-936 95 12 field field NOUN cana-936 95 13 of of ADP cana-936 95 14 fqm fqm NOUN cana-936 95 15 by by ADP cana-936 95 16 : : PUNCT cana-936 95 17 β€’ β€’ NOUN cana-936 95 18 demonstrating demonstrate VERB cana-936 95 19 the the DET cana-936 95 20 practical practical ADJ cana-936 95 21 applications application NOUN cana-936 95 22 of of ADP cana-936 95 23 fractional fractional ADJ cana-936 95 24 calculus calculus NOUN cana-936 95 25 in in ADP cana-936 95 26 addressing address VERB cana-936 95 27 quantum quantum ADJ cana-936 95 28 mechanical mechanical ADJ cana-936 95 29 problems problem NOUN cana-936 95 30 , , PUNCT cana-936 95 31 thereby thereby ADV cana-936 95 32 enriching enrich VERB cana-936 95 33 the the DET cana-936 95 34 theoretical theoretical ADJ cana-936 95 35 framework framework NOUN cana-936 95 36 of of ADP cana-936 95 37 quantum quantum ADJ cana-936 95 38 mechanics mechanic NOUN cana-936 95 39 . . PUNCT cana-936 96 1 β€’ β€’ NOUN cana-936 97 1 highlighting highlight VERB cana-936 97 2 the the DET cana-936 97 3 complementary complementary ADJ cana-936 97 4 nature nature NOUN cana-936 97 5 of of ADP cana-936 97 6 analytical analytical ADJ cana-936 97 7 and and CCONJ cana-936 97 8 numerical numerical ADJ cana-936 97 9 methods method NOUN cana-936 97 10 in in ADP cana-936 97 11 exploring explore VERB cana-936 97 12 quantum quantum ADJ cana-936 97 13 systems system NOUN cana-936 97 14 , , PUNCT cana-936 97 15 thus thus ADV cana-936 97 16 advocating advocate VERB cana-936 97 17 for for ADP cana-936 97 18 a a DET cana-936 97 19 multifaceted multifaceted ADJ cana-936 97 20 approach approach NOUN cana-936 97 21 to to ADP cana-936 97 22 research research NOUN cana-936 97 23 in in ADP cana-936 97 24 quantum quantum ADJ cana-936 97 25 mechanics mechanic NOUN cana-936 97 26 . . PUNCT cana-936 98 1 β€’ β€’ NUM cana-936 98 2 identifying identify VERB cana-936 98 3 the the DET cana-936 98 4 challenges challenge NOUN cana-936 98 5 and and CCONJ cana-936 98 6 limitations limitation NOUN cana-936 98 7 inherent inherent ADJ cana-936 98 8 in in ADP cana-936 98 9 applying apply VERB cana-936 98 10 fractional fractional ADJ cana-936 98 11 calculus calculus NOUN cana-936 98 12 to to ADP cana-936 98 13 quantum quantum ADJ cana-936 98 14 mechanics mechanic NOUN cana-936 98 15 , , PUNCT cana-936 98 16 thereby thereby ADV cana-936 98 17 setting set VERB cana-936 98 18 the the DET cana-936 98 19 stage stage NOUN cana-936 98 20 for for ADP cana-936 98 21 future future ADJ cana-936 98 22 advancements advancement NOUN cana-936 98 23 in in ADP cana-936 98 24 the the DET cana-936 98 25 field field NOUN cana-936 98 26 . . PUNCT cana-936 99 1 8 8 X cana-936 99 2 . . X cana-936 99 3 suggestions suggestion NOUN cana-936 99 4 for for ADP cana-936 99 5 future future ADJ cana-936 99 6 research research NOUN cana-936 99 7 the the DET cana-936 99 8 findings finding NOUN cana-936 99 9 from from ADP cana-936 99 10 this this DET cana-936 99 11 study study NOUN cana-936 99 12 pave pave VERB cana-936 99 13 the the DET cana-936 99 14 way way NOUN cana-936 99 15 for for ADP cana-936 99 16 several several ADJ cana-936 99 17 promising promising ADJ cana-936 99 18 directions direction NOUN cana-936 99 19 for for ADP cana-936 99 20 future future ADJ cana-936 99 21 research research NOUN cana-936 99 22 in in ADP cana-936 99 23 fqm fqm NOUN cana-936 99 24 : : PUNCT cana-936 99 25 1 1 X cana-936 99 26 . . X cana-936 99 27 exploring explore VERB cana-936 99 28 more more ADJ cana-936 99 29 quantum quantum ADJ cana-936 99 30 systems system NOUN cana-936 99 31 : : PUNCT cana-936 99 32 future future ADJ cana-936 99 33 studies study NOUN cana-936 99 34 could could AUX cana-936 99 35 extend extend VERB cana-936 99 36 the the DET cana-936 99 37 application application NOUN cana-936 99 38 of of ADP cana-936 99 39 fractional fractional ADJ cana-936 99 40 calculus calculus NOUN cana-936 99 41 to to ADP cana-936 99 42 other other ADJ cana-936 99 43 quantum quantum NOUN cana-936 99 44 systems system NOUN cana-936 99 45 , , PUNCT cana-936 99 46 such such ADJ cana-936 99 47 as as ADP cana-936 99 48 quantum quantum ADJ cana-936 99 49 field field NOUN cana-936 99 50 theory theory NOUN cana-936 99 51 and and CCONJ cana-936 99 52 quantum quantum NOUN cana-936 99 53 gravity gravity NOUN cana-936 99 54 , , PUNCT cana-936 99 55 to to PART cana-936 99 56 uncover uncover VERB cana-936 99 57 potential potential ADJ cana-936 99 58 insights insight NOUN cana-936 99 59 into into ADP cana-936 99 60 the the DET cana-936 99 61 fundamental fundamental ADJ cana-936 99 62 aspects aspect NOUN cana-936 99 63 of of ADP cana-936 99 64 the the DET cana-936 99 65 universe universe NOUN cana-936 99 66 . . PUNCT cana-936 100 1 communications communication NOUN cana-936 100 2 on on ADP cana-936 100 3 applied apply VERB cana-936 100 4 nonlinear nonlinear ADJ cana-936 100 5 analysis analysis NOUN cana-936 100 6 issn issn NOUN cana-936 100 7 : : PUNCT cana-936 100 8 1074 1074 NUM cana-936 100 9 - - PUNCT cana-936 100 10 133x 133x NUM cana-936 100 11 vol vol NOUN cana-936 100 12 31 31 NUM cana-936 100 13 no no NOUN cana-936 100 14 . . PUNCT cana-936 101 1 4s 4s NUM cana-936 101 2 ( ( PUNCT cana-936 101 3 2024 2024 NUM cana-936 101 4 ) ) PUNCT cana-936 101 5 432 432 NUM cana-936 102 1 https://internationalpubls.com https://internationalpubls.com X cana-936 102 2 2 2 X cana-936 102 3 . . PUNCT cana-936 102 4 advancing advance VERB cana-936 102 5 numerical numerical ADJ cana-936 102 6 methods method NOUN cana-936 102 7 : : PUNCT cana-936 102 8 there there PRON cana-936 102 9 is be VERB cana-936 102 10 a a DET cana-936 102 11 need need NOUN cana-936 102 12 for for ADP cana-936 102 13 the the DET cana-936 102 14 development development NOUN cana-936 102 15 of of ADP cana-936 102 16 more more ADV cana-936 102 17 sophisticated sophisticated ADJ cana-936 102 18 numerical numerical ADJ cana-936 102 19 algorithms algorithm NOUN cana-936 102 20 that that PRON cana-936 102 21 can can AUX cana-936 102 22 efficiently efficiently ADV cana-936 102 23 tackle tackle VERB cana-936 102 24 the the DET cana-936 102 25 complexity complexity NOUN cana-936 102 26 of of ADP cana-936 102 27 fractional fractional ADJ cana-936 102 28 differential differential ADJ cana-936 102 29 equations equation NOUN cana-936 102 30 , , PUNCT cana-936 102 31 potentially potentially ADV cana-936 102 32 enhancing enhance VERB cana-936 102 33 the the DET cana-936 102 34 scalability scalability NOUN cana-936 102 35 and and CCONJ cana-936 102 36 accuracy accuracy NOUN cana-936 102 37 of of ADP cana-936 102 38 simulations simulation NOUN cana-936 102 39 in in ADP cana-936 102 40 fqm fqm NOUN cana-936 102 41 . . PUNCT cana-936 103 1 3 3 X cana-936 103 2 . . X cana-936 103 3 interdisciplinary interdisciplinary ADJ cana-936 103 4 applications application NOUN cana-936 103 5 : : PUNCT cana-936 103 6 exploring explore VERB cana-936 103 7 the the DET cana-936 103 8 interdisciplinary interdisciplinary ADJ cana-936 103 9 applications application NOUN cana-936 103 10 of of ADP cana-936 103 11 fqm fqm NOUN cana-936 103 12 in in ADP cana-936 103 13 areas area NOUN cana-936 103 14 such such ADJ cana-936 103 15 as as ADP cana-936 103 16 quantum quantum NOUN cana-936 103 17 biology biology NOUN cana-936 103 18 , , PUNCT cana-936 103 19 quantum quantum NOUN cana-936 103 20 chemistry chemistry NOUN cana-936 103 21 , , PUNCT cana-936 103 22 and and CCONJ cana-936 103 23 nanotechnology nanotechnology NOUN cana-936 103 24 could could AUX cana-936 103 25 open open VERB cana-936 103 26 new new ADJ cana-936 103 27 avenues avenue NOUN cana-936 103 28 for for ADP cana-936 103 29 research research NOUN cana-936 103 30 , , PUNCT cana-936 103 31 leveraging leverage VERB cana-936 103 32 the the DET cana-936 103 33 unique unique ADJ cana-936 103 34 insights insight NOUN cana-936 103 35 provided provide VERB cana-936 103 36 by by ADP cana-936 103 37 fractional fractional ADJ cana-936 103 38 calculus calculus NOUN cana-936 103 39 to to PART cana-936 103 40 solve solve VERB cana-936 103 41 complex complex ADJ cana-936 103 42 problems problem NOUN cana-936 103 43 across across ADP cana-936 103 44 various various ADJ cana-936 103 45 fields field NOUN cana-936 103 46 . . PUNCT cana-936 104 1 4 4 X cana-936 104 2 . . X cana-936 104 3 theoretical theoretical ADJ cana-936 104 4 developments development NOUN cana-936 104 5 : : PUNCT cana-936 104 6 further further ADJ cana-936 104 7 theoretical theoretical ADJ cana-936 104 8 advancements advancement NOUN cana-936 104 9 in in ADP cana-936 104 10 fractional fractional ADJ cana-936 104 11 calculus calculus NOUN cana-936 104 12 itself itself PRON cana-936 104 13 , , PUNCT cana-936 104 14 including include VERB cana-936 104 15 the the DET cana-936 104 16 refinement refinement NOUN cana-936 104 17 of of ADP cana-936 104 18 fractional fractional ADJ cana-936 104 19 derivatives derivative NOUN cana-936 104 20 and and CCONJ cana-936 104 21 integrals integral NOUN cana-936 104 22 , , PUNCT cana-936 104 23 could could AUX cana-936 104 24 provide provide VERB cana-936 104 25 a a DET cana-936 104 26 more more ADV cana-936 104 27 solid solid ADJ cana-936 104 28 foundation foundation NOUN cana-936 104 29 for for ADP cana-936 104 30 fqm fqm NOUN cana-936 104 31 , , PUNCT cana-936 104 32 enabling enable VERB cana-936 104 33 more more ADV cana-936 104 34 precise precise ADJ cana-936 104 35 and and CCONJ cana-936 104 36 comprehensive comprehensive ADJ cana-936 104 37 modeling modeling NOUN cana-936 104 38 of of ADP cana-936 104 39 quantum quantum NOUN cana-936 104 40 systems system NOUN cana-936 104 41 . . PUNCT cana-936 105 1 9 9 X cana-936 105 2 . . X cana-936 105 3 concluding conclude VERB cana-936 105 4 remarks remark NOUN cana-936 105 5 in in ADP cana-936 105 6 conclusion conclusion NOUN cana-936 105 7 , , PUNCT cana-936 105 8 this this DET cana-936 105 9 study study NOUN cana-936 105 10 underscores underscore VERB cana-936 105 11 the the DET cana-936 105 12 significant significant ADJ cana-936 105 13 potential potential NOUN cana-936 105 14 of of ADP cana-936 105 15 fractional fractional ADJ cana-936 105 16 quantum quantum ADJ cana-936 105 17 mechanics mechanic NOUN cana-936 105 18 as as ADP cana-936 105 19 a a DET cana-936 105 20 frontier frontier NOUN cana-936 105 21 in in ADP cana-936 105 22 the the DET cana-936 105 23 exploration exploration NOUN cana-936 105 24 of of ADP cana-936 105 25 quantum quantum ADJ cana-936 105 26 phenomena phenomenon NOUN cana-936 105 27 , , PUNCT cana-936 105 28 offering offer VERB cana-936 105 29 a a DET cana-936 105 30 fresh fresh ADJ cana-936 105 31 perspective perspective NOUN cana-936 105 32 on on ADP cana-936 105 33 the the DET cana-936 105 34 complexities complexity NOUN cana-936 105 35 of of ADP cana-936 105 36 the the DET cana-936 105 37 quantum quantum ADJ cana-936 105 38 world world NOUN cana-936 105 39 . . PUNCT cana-936 106 1 by by ADP cana-936 106 2 harnessing harness VERB cana-936 106 3 the the DET cana-936 106 4 power power NOUN cana-936 106 5 of of ADP cana-936 106 6 fractional fractional ADJ cana-936 106 7 calculus calculus NOUN cana-936 106 8 , , PUNCT cana-936 106 9 researchers researcher NOUN cana-936 106 10 can can AUX cana-936 106 11 push push VERB cana-936 106 12 the the DET cana-936 106 13 boundaries boundary NOUN cana-936 106 14 of of ADP cana-936 106 15 conventional conventional ADJ cana-936 106 16 quantum quantum ADJ cana-936 106 17 mechanics mechanic NOUN cana-936 106 18 , , PUNCT cana-936 106 19 exploring explore VERB cana-936 106 20 new new ADJ cana-936 106 21 dimensions dimension NOUN cana-936 106 22 of of ADP cana-936 106 23 quantum quantum ADJ cana-936 106 24 behavior behavior NOUN cana-936 106 25 and and CCONJ cana-936 106 26 potentially potentially ADV cana-936 106 27 unlocking unlock VERB cana-936 106 28 secrets secret NOUN cana-936 106 29 of of ADP cana-936 106 30 the the DET cana-936 106 31 quantum quantum ADJ cana-936 106 32 universe universe NOUN cana-936 106 33 . . PUNCT cana-936 107 1 as as SCONJ cana-936 107 2 the the DET cana-936 107 3 field field NOUN cana-936 107 4 of of ADP cana-936 107 5 fqm fqm NOUN cana-936 107 6 continues continue VERB cana-936 107 7 to to PART cana-936 107 8 evolve evolve VERB cana-936 107 9 , , PUNCT cana-936 107 10 it it PRON cana-936 107 11 promises promise VERB cana-936 107 12 to to PART cana-936 107 13 catalyze catalyze NOUN cana-936 107 14 breakthroughs breakthrough NOUN cana-936 107 15 that that PRON cana-936 107 16 could could AUX cana-936 107 17 redefine redefine VERB cana-936 107 18 our our PRON cana-936 107 19 understanding understanding NOUN cana-936 107 20 of of ADP cana-936 107 21 quantum quantum ADJ cana-936 107 22 mechanics mechanic NOUN cana-936 107 23 and and CCONJ cana-936 107 24 its its PRON cana-936 107 25 applications application NOUN cana-936 107 26 in in ADP cana-936 107 27 the the DET cana-936 107 28 broader broad ADJ cana-936 107 29 realm realm NOUN cana-936 107 30 of of ADP cana-936 107 31 science science NOUN cana-936 107 32 . . PUNCT cana-936 108 1 references reference NOUN cana-936 108 2 [ [ X cana-936 108 3 1 1 NUM cana-936 108 4 ] ] X cana-936 108 5 herrmann herrmann PROPN cana-936 108 6 , , PUNCT cana-936 108 7 r. r. PROPN cana-936 108 8 ( ( PUNCT cana-936 108 9 2011 2011 NUM cana-936 108 10 ) ) PUNCT cana-936 108 11 . . PUNCT cana-936 109 1 fractional fractional ADJ cana-936 109 2 calculus calculus NOUN cana-936 109 3 : : PUNCT cana-936 109 4 an an DET cana-936 109 5 introduction introduction NOUN cana-936 109 6 for for ADP cana-936 109 7 physicists physicist NOUN cana-936 109 8 . . PUNCT cana-936 110 1 world world NOUN cana-936 110 2 scientific scientific ADJ cana-936 110 3 publishing publishing NOUN cana-936 110 4 . . PUNCT cana-936 111 1 [ [ X cana-936 111 2 2 2 NUM cana-936 111 3 ] ] PUNCT cana-936 111 4 laskin laskin NOUN cana-936 111 5 , , PUNCT cana-936 111 6 n. n. NOUN cana-936 111 7 ( ( PUNCT cana-936 111 8 2000 2000 NUM cana-936 111 9 ) ) PUNCT cana-936 111 10 . . PUNCT cana-936 112 1 " " PUNCT cana-936 112 2 fractional fractional ADJ cana-936 112 3 quantum quantum ADJ cana-936 112 4 mechanics mechanic NOUN cana-936 112 5 and and CCONJ cana-936 112 6 lΓ©vy lΓ©vy ADJ cana-936 112 7 path path NOUN cana-936 112 8 integrals integral NOUN cana-936 112 9 . . PUNCT cana-936 112 10 " " PUNCT cana-936 113 1 physics physics NOUN cana-936 113 2 letters letter NOUN cana-936 113 3 a a PRON cana-936 113 4 , , PUNCT cana-936 113 5 268(4 268(4 NUM cana-936 113 6 - - SYM cana-936 113 7 6 6 NUM cana-936 113 8 ) ) PUNCT cana-936 113 9 , , PUNCT cana-936 113 10 298 298 NUM cana-936 113 11 - - SYM cana-936 113 12 305 305 NUM cana-936 113 13 . . PUNCT cana-936 114 1 [ [ X cana-936 114 2 3 3 NUM cana-936 114 3 ] ] X cana-936 114 4 kilbas kilbas PROPN cana-936 114 5 , , PUNCT cana-936 114 6 a. a. NOUN cana-936 114 7 a. a. PROPN cana-936 114 8 , , PUNCT cana-936 114 9 srivastava srivastava PROPN cana-936 114 10 , , PUNCT cana-936 114 11 h. h. PROPN cana-936 114 12 m. m. PROPN cana-936 114 13 , , PUNCT cana-936 114 14 & & CCONJ cana-936 114 15 trujillo trujillo PROPN cana-936 114 16 , , PUNCT cana-936 114 17 j. j. PROPN cana-936 114 18 j. j. PROPN cana-936 114 19 ( ( PUNCT cana-936 114 20 2006 2006 NUM cana-936 114 21 ) ) PUNCT cana-936 114 22 . . PUNCT cana-936 115 1 theory theory NOUN cana-936 115 2 and and CCONJ cana-936 115 3 applications application NOUN cana-936 115 4 of of ADP cana-936 115 5 fractional fractional ADJ cana-936 115 6 differential differential ADJ cana-936 115 7 equations equation NOUN cana-936 115 8 . . PUNCT cana-936 116 1 elsevier elsevier NOUN cana-936 116 2 . . PUNCT cana-936 117 1 [ [ X cana-936 117 2 4 4 NUM cana-936 117 3 ] ] X cana-936 117 4 mainardi mainardi PROPN cana-936 117 5 , , PUNCT cana-936 117 6 f. f. PROPN cana-936 117 7 ( ( PUNCT cana-936 117 8 2010 2010 NUM cana-936 117 9 ) ) PUNCT cana-936 117 10 . . PUNCT cana-936 118 1 fractional fractional ADJ cana-936 118 2 calculus calculus NOUN cana-936 118 3 and and CCONJ cana-936 118 4 waves wave NOUN cana-936 118 5 in in ADP cana-936 118 6 linear linear PROPN cana-936 118 7 viscoelasticity viscoelasticity NOUN cana-936 118 8 : : PUNCT cana-936 118 9 an an DET cana-936 118 10 introduction introduction NOUN cana-936 118 11 to to ADP cana-936 118 12 mathematical mathematical ADJ cana-936 118 13 models model NOUN cana-936 118 14 . . PUNCT cana-936 119 1 imperial imperial ADJ cana-936 119 2 college college NOUN cana-936 119 3 press press NOUN cana-936 119 4 . . PUNCT cana-936 120 1 [ [ X cana-936 120 2 5 5 NUM cana-936 120 3 ] ] SYM cana-936 120 4 podlubny podlubny NOUN cana-936 120 5 , , PUNCT cana-936 120 6 i. i. NOUN cana-936 120 7 ( ( PUNCT cana-936 120 8 1998 1998 NUM cana-936 120 9 ) ) PUNCT cana-936 120 10 . . PUNCT cana-936 121 1 fractional fractional ADJ cana-936 121 2 differential differential ADJ cana-936 121 3 equations equation NOUN cana-936 121 4 . . PUNCT cana-936 122 1 academic academic ADJ cana-936 122 2 press press NOUN cana-936 122 3 . . PUNCT cana-936 123 1 [ [ X cana-936 123 2 6 6 NUM cana-936 123 3 ] ] PUNCT cana-936 123 4 tarasov tarasov NOUN cana-936 123 5 , , PUNCT cana-936 123 6 v. v. PROPN cana-936 123 7 e. e. PROPN cana-936 123 8 ( ( PUNCT cana-936 123 9 2011 2011 NUM cana-936 123 10 ) ) PUNCT cana-936 123 11 . . PUNCT cana-936 124 1 quantum quantum ADJ cana-936 124 2 mechanics mechanic NOUN cana-936 124 3 of of ADP cana-936 124 4 non non ADJ cana-936 124 5 - - ADJ cana-936 124 6 hamiltonian hamiltonian ADJ cana-936 124 7 and and CCONJ cana-936 124 8 dissipative dissipative ADJ cana-936 124 9 systems system NOUN cana-936 124 10 . . PUNCT cana-936 125 1 elsevier elsevier NOUN cana-936 125 2 . . PUNCT cana-936 126 1 [ [ X cana-936 126 2 7 7 X cana-936 126 3 ] ] X cana-936 126 4 west west PROPN cana-936 126 5 , , PUNCT cana-936 126 6 b. b. PROPN cana-936 126 7 j. j. PROPN cana-936 126 8 , , PUNCT cana-936 126 9 bologna bologna NOUN cana-936 126 10 , , PUNCT cana-936 126 11 m. m. NOUN cana-936 126 12 , , PUNCT cana-936 126 13 & & CCONJ cana-936 126 14 grigolini grigolini PROPN cana-936 126 15 , , PUNCT cana-936 126 16 p. p. NOUN cana-936 126 17 ( ( PUNCT cana-936 126 18 2003 2003 NUM cana-936 126 19 ) ) PUNCT cana-936 126 20 . . PUNCT cana-936 127 1 physics physics PROPN cana-936 127 2 of of ADP cana-936 127 3 fractal fractal PROPN cana-936 127 4 operators operator NOUN cana-936 127 5 . . PUNCT cana-936 128 1 springer springer NOUN cana-936 128 2 . . PUNCT cana-936 129 1 [ [ X cana-936 129 2 8 8 NUM cana-936 129 3 ] ] X cana-936 129 4 zaslavsky zaslavsky NOUN cana-936 129 5 , , PUNCT cana-936 129 6 g. g. PROPN cana-936 129 7 m. m. PROPN cana-936 129 8 ( ( PUNCT cana-936 129 9 2005 2005 NUM cana-936 129 10 ) ) PUNCT cana-936 129 11 . . PUNCT cana-936 130 1 " " PUNCT cana-936 130 2 hamiltonian hamiltonian ADJ cana-936 130 3 chaos chaos NOUN cana-936 130 4 and and CCONJ cana-936 130 5 fractional fractional ADJ cana-936 130 6 dynamics dynamic NOUN cana-936 130 7 . . PUNCT cana-936 130 8 " " PUNCT cana-936 131 1 oxford oxford PROPN cana-936 131 2 university university NOUN cana-936 131 3 press press NOUN cana-936 131 4 . . PUNCT cana-936 132 1 [ [ X cana-936 132 2 9 9 NUM cana-936 132 3 ] ] X cana-936 132 4 zhang zhang PROPN cana-936 132 5 , , PUNCT cana-936 132 6 y. y. PROPN cana-936 132 7 , , PUNCT cana-936 132 8 meerschaert meerschaert NOUN cana-936 132 9 , , PUNCT cana-936 132 10 m. m. NOUN cana-936 132 11 m. m. NOUN cana-936 132 12 , , PUNCT cana-936 132 13 & & CCONJ cana-936 132 14 packman packman NOUN cana-936 132 15 , , PUNCT cana-936 132 16 a. a. NOUN cana-936 132 17 i. i. PROPN cana-936 132 18 ( ( PUNCT cana-936 132 19 2016 2016 NUM cana-936 132 20 ) ) PUNCT cana-936 132 21 . . PUNCT cana-936 133 1 " " PUNCT cana-936 133 2 linking link VERB cana-936 133 3 particle particle NOUN cana-936 133 4 and and CCONJ cana-936 133 5 pore pore ADJ cana-936 133 6 - - PUNCT cana-936 133 7 scale scale NOUN cana-936 133 8 models model NOUN cana-936 133 9 of of ADP cana-936 133 10 colloidal colloidal NOUN cana-936 133 11 dispersion dispersion NOUN cana-936 133 12 and and CCONJ cana-936 133 13 deposition deposition NOUN cana-936 133 14 in in ADP cana-936 133 15 porous porous ADJ cana-936 133 16 media medium NOUN cana-936 133 17 . . PUNCT cana-936 133 18 " " PUNCT cana-936 134 1 water water NOUN cana-936 134 2 resources resource NOUN cana-936 134 3 research research NOUN cana-936 134 4 , , PUNCT cana-936 134 5 52(3 52(3 NOUN cana-936 134 6 ) ) PUNCT cana-936 134 7 , , PUNCT cana-936 134 8 1903 1903 NUM cana-936 134 9 - - SYM cana-936 134 10 1922 1922 NUM cana-936 134 11 . . PUNCT cana-936 135 1 [ [ X cana-936 135 2 10 10 NUM cana-936 135 3 ] ] SYM cana-936 135 4 li li PROPN cana-936 135 5 , , PUNCT cana-936 135 6 c. c. PROPN cana-936 135 7 , , PUNCT cana-936 135 8 & & CCONJ cana-936 135 9 zeng zeng PROPN cana-936 135 10 , , PUNCT cana-936 135 11 f. f. PROPN cana-936 135 12 ( ( PUNCT cana-936 135 13 2015 2015 NUM cana-936 135 14 ) ) PUNCT cana-936 135 15 . . PUNCT cana-936 136 1 numerical numerical ADJ cana-936 136 2 methods method NOUN cana-936 136 3 for for ADP cana-936 136 4 fractional fractional ADJ cana-936 136 5 calculus calculus NOUN cana-936 136 6 . . PUNCT cana-936 137 1 crc crc PROPN cana-936 137 2 press press PROPN cana-936 137 3 . . PUNCT