id sid tid token lemma pos cana-6296 1 1 communications communication NOUN cana-6296 1 2 on on ADP cana-6296 1 3 applied apply VERB cana-6296 1 4 nonlinear nonlinear ADJ cana-6296 1 5 analysis analysis NOUN cana-6296 1 6 issn issn NOUN cana-6296 1 7 : : PUNCT cana-6296 1 8 1074 1074 NUM cana-6296 1 9 - - PUNCT cana-6296 1 10 133x 133x NUM cana-6296 1 11 vol vol VERB cana-6296 1 12 32 32 NUM cana-6296 1 13 no no NOUN cana-6296 1 14 . . PUNCT cana-6296 2 1 10s 10 NOUN cana-6296 2 2 ( ( PUNCT cana-6296 2 3 2025 2025 NUM cana-6296 2 4 ) ) PUNCT cana-6296 2 5 3695 3695 NUM cana-6296 2 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 2 7 a a DET cana-6296 2 8 unified unify VERB cana-6296 2 9 numerical numerical ADJ cana-6296 2 10 framework framework NOUN cana-6296 2 11 for for ADP cana-6296 2 12 the the DET cana-6296 2 13 inversion inversion NOUN cana-6296 2 14 of of ADP cana-6296 2 15 integral integral ADJ cana-6296 2 16 transforms transform NOUN cana-6296 2 17 : : PUNCT cana-6296 2 18 laplace laplace NOUN cana-6296 2 19 and and CCONJ cana-6296 2 20 fourier fourier NOUN cana-6296 2 21 mr mr PROPN cana-6296 2 22 . . PROPN cana-6296 2 23 amol amol PROPN cana-6296 2 24 pimpalkar pimpalkar PROPN cana-6296 2 25 assistant assistant PROPN cana-6296 2 26 professor professor NOUN cana-6296 2 27 , , PUNCT cana-6296 2 28 pgtd pgtd NOUN cana-6296 2 29 of of ADP cana-6296 2 30 mathematics mathematics PROPN cana-6296 2 31 , , PUNCT cana-6296 2 32 gondwana gondwana PROPN cana-6296 2 33 university university PROPN cana-6296 2 34 , , PUNCT cana-6296 2 35 gadchiroli gadchiroli NOUN cana-6296 2 36 article article NOUN cana-6296 2 37 history history NOUN cana-6296 2 38 : : PUNCT cana-6296 2 39 received receive VERB cana-6296 2 40 : : PUNCT cana-6296 2 41 24 24 NUM cana-6296 2 42 - - SYM cana-6296 2 43 10 10 NUM cana-6296 2 44 - - PUNCT cana-6296 2 45 2025 2025 NUM cana-6296 2 46 revised revise VERB cana-6296 2 47 : : PUNCT cana-6296 2 48 20 20 NUM cana-6296 2 49 - - SYM cana-6296 2 50 11 11 NUM cana-6296 2 51 - - PUNCT cana-6296 2 52 2025 2025 NUM cana-6296 2 53 accepted accept VERB cana-6296 2 54 : : PUNCT cana-6296 2 55 06 06 NUM cana-6296 2 56 - - SYM cana-6296 2 57 12 12 NUM cana-6296 2 58 - - PUNCT cana-6296 2 59 2025 2025 NUM cana-6296 2 60 abstract abstract NOUN cana-6296 2 61 : : PUNCT cana-6296 2 62 the the DET cana-6296 2 63 analytical analytical ADJ cana-6296 2 64 inversion inversion NOUN cana-6296 2 65 of of ADP cana-6296 2 66 laplace laplace NOUN cana-6296 2 67 and and CCONJ cana-6296 2 68 fourier fourier NOUN cana-6296 2 69 transforms transform VERB cana-6296 2 70 often often ADV cana-6296 2 71 presents present VERB cana-6296 2 72 significant significant ADJ cana-6296 2 73 challenges challenge NOUN cana-6296 2 74 for for ADP cana-6296 2 75 functions function NOUN cana-6296 2 76 commonly commonly ADV cana-6296 2 77 encountered encounter VERB cana-6296 2 78 in in ADP cana-6296 2 79 engineering engineering NOUN cana-6296 2 80 and and CCONJ cana-6296 2 81 scientific scientific ADJ cana-6296 2 82 fields field NOUN cana-6296 2 83 . . PUNCT cana-6296 3 1 although although SCONJ cana-6296 3 2 numerous numerous ADJ cana-6296 3 3 numerical numerical ADJ cana-6296 3 4 inversion inversion NOUN cana-6296 3 5 methods method NOUN cana-6296 3 6 are be AUX cana-6296 3 7 available available ADJ cana-6296 3 8 , , PUNCT cana-6296 3 9 these these DET cana-6296 3 10 techniques technique NOUN cana-6296 3 11 tend tend VERB cana-6296 3 12 to to PART cana-6296 3 13 be be AUX cana-6296 3 14 highly highly ADV cana-6296 3 15 specialized specialized ADJ cana-6296 3 16 , , PUNCT cana-6296 3 17 algorithmically algorithmically ADV cana-6296 3 18 complex complex ADJ cana-6296 3 19 , , PUNCT cana-6296 3 20 and and CCONJ cana-6296 3 21 sensitive sensitive ADJ cana-6296 3 22 to to ADP cana-6296 3 23 parameter parameter NOUN cana-6296 3 24 variations variation NOUN cana-6296 3 25 . . PUNCT cana-6296 4 1 this this DET cana-6296 4 2 paper paper NOUN cana-6296 4 3 introduces introduce VERB cana-6296 4 4 an an DET cana-6296 4 5 innovative innovative ADJ cana-6296 4 6 , , PUNCT cana-6296 4 7 unified unified ADJ cana-6296 4 8 methodological methodological ADJ cana-6296 4 9 framework framework NOUN cana-6296 4 10 for for ADP cana-6296 4 11 the the DET cana-6296 4 12 numerical numerical ADJ cana-6296 4 13 computation computation NOUN cana-6296 4 14 of of ADP cana-6296 4 15 these these DET cana-6296 4 16 inverse inverse NOUN cana-6296 4 17 transforms transform NOUN cana-6296 4 18 . . PUNCT cana-6296 5 1 by by ADP cana-6296 5 2 leveraging leverage VERB cana-6296 5 3 the the DET cana-6296 5 4 intrinsic intrinsic ADJ cana-6296 5 5 relationship relationship NOUN cana-6296 5 6 between between ADP cana-6296 5 7 the the DET cana-6296 5 8 two two NUM cana-6296 5 9 transforms transform VERB cana-6296 5 10 through through ADP cana-6296 5 11 analytic analytic ADJ cana-6296 5 12 continuation continuation NOUN cana-6296 5 13 and and CCONJ cana-6296 5 14 implementing implement VERB cana-6296 5 15 a a DET cana-6296 5 16 robust robust ADJ cana-6296 5 17 regularization regularization NOUN cana-6296 5 18 strategy strategy NOUN cana-6296 5 19 within within ADP cana-6296 5 20 a a DET cana-6296 5 21 fourier fourier NOUN cana-6296 5 22 inversion inversion NOUN cana-6296 5 23 framework framework NOUN cana-6296 5 24 , , PUNCT cana-6296 5 25 we we PRON cana-6296 5 26 derive derive VERB cana-6296 5 27 a a DET cana-6296 5 28 generalized generalized ADJ cana-6296 5 29 inversion inversion NOUN cana-6296 5 30 formula formula NOUN cana-6296 5 31 . . PUNCT cana-6296 6 1 this this DET cana-6296 6 2 method method NOUN cana-6296 6 3 effectively effectively ADV cana-6296 6 4 converts convert VERB cana-6296 6 5 the the DET cana-6296 6 6 problem problem NOUN cana-6296 6 7 of of ADP cana-6296 6 8 inverting invert VERB cana-6296 6 9 a a DET cana-6296 6 10 laplace laplace NOUN cana-6296 6 11 transform transform NOUN cana-6296 6 12 into into ADP cana-6296 6 13 a a DET cana-6296 6 14 specially specially ADV cana-6296 6 15 designed design VERB cana-6296 6 16 fourier fourier NOUN cana-6296 6 17 inversion inversion NOUN cana-6296 6 18 , , PUNCT cana-6296 6 19 which which PRON cana-6296 6 20 is be AUX cana-6296 6 21 then then ADV cana-6296 6 22 solved solve VERB cana-6296 6 23 using use VERB cana-6296 6 24 a a DET cana-6296 6 25 computationally computationally ADV cana-6296 6 26 efficient efficient ADJ cana-6296 6 27 and and CCONJ cana-6296 6 28 stable stable ADJ cana-6296 6 29 algorithm algorithm NOUN cana-6296 6 30 based base VERB cana-6296 6 31 on on ADP cana-6296 6 32 the the DET cana-6296 6 33 fast fast ADJ cana-6296 6 34 fourier fourier NOUN cana-6296 6 35 transform transform NOUN cana-6296 6 36 ( ( PUNCT cana-6296 6 37 fft fft PROPN cana-6296 6 38 ) ) PUNCT cana-6296 6 39 with with ADP cana-6296 6 40 a a DET cana-6296 6 41 smoothing smooth VERB cana-6296 6 42 kernel kernel NOUN cana-6296 6 43 . . PUNCT cana-6296 7 1 we we PRON cana-6296 7 2 validate validate VERB cana-6296 7 3 the the DET cana-6296 7 4 effectiveness effectiveness NOUN cana-6296 7 5 and and CCONJ cana-6296 7 6 precision precision NOUN cana-6296 7 7 of of ADP cana-6296 7 8 this this DET cana-6296 7 9 universal universal ADJ cana-6296 7 10 method method NOUN cana-6296 7 11 through through ADP cana-6296 7 12 several several ADJ cana-6296 7 13 standard standard ADJ cana-6296 7 14 examples example NOUN cana-6296 7 15 , , PUNCT cana-6296 7 16 comparing compare VERB cana-6296 7 17 its its PRON cana-6296 7 18 performance performance NOUN cana-6296 7 19 with with ADP cana-6296 7 20 traditional traditional ADJ cana-6296 7 21 techniques technique NOUN cana-6296 7 22 such such ADJ cana-6296 7 23 as as ADP cana-6296 7 24 the the DET cana-6296 7 25 bromwich bromwich NOUN cana-6296 7 26 integral integral ADJ cana-6296 7 27 and and CCONJ cana-6296 7 28 stehfest stehf ADJ cana-6296 7 29 's 's PART cana-6296 7 30 algorithm algorithm NOUN cana-6296 7 31 for for ADP cana-6296 7 32 laplace laplace NOUN cana-6296 7 33 inversion inversion NOUN cana-6296 7 34 . . PUNCT cana-6296 8 1 the the DET cana-6296 8 2 proposed propose VERB cana-6296 8 3 approach approach NOUN cana-6296 8 4 offers offer VERB cana-6296 8 5 a a DET cana-6296 8 6 streamlined streamlined ADJ cana-6296 8 7 , , PUNCT cana-6296 8 8 powerful powerful ADJ cana-6296 8 9 , , PUNCT cana-6296 8 10 and and CCONJ cana-6296 8 11 versatile versatile ADJ cana-6296 8 12 computational computational ADJ cana-6296 8 13 tool tool NOUN cana-6296 8 14 for for ADP cana-6296 8 15 researchers researcher NOUN cana-6296 8 16 and and CCONJ cana-6296 8 17 practitioners practitioner NOUN cana-6296 8 18 . . PUNCT cana-6296 9 1 1 1 X cana-6296 9 2 . . X cana-6296 9 3 introduction introduction NOUN cana-6296 9 4 : : PUNCT cana-6296 9 5 integral integral ADJ cana-6296 9 6 transforms transform NOUN cana-6296 9 7 , , PUNCT cana-6296 9 8 especially especially ADV cana-6296 9 9 the the DET cana-6296 9 10 laplace laplace NOUN cana-6296 9 11 and and CCONJ cana-6296 9 12 fourier fourier NOUN cana-6296 9 13 transforms transform VERB cana-6296 9 14 , , PUNCT cana-6296 9 15 are be AUX cana-6296 9 16 fundamental fundamental ADJ cana-6296 9 17 tools tool NOUN cana-6296 9 18 for for ADP cana-6296 9 19 addressing address VERB cana-6296 9 20 differential differential ADJ cana-6296 9 21 and and CCONJ cana-6296 9 22 integral integral ADJ cana-6296 9 23 equations equation NOUN cana-6296 9 24 encountered encounter VERB cana-6296 9 25 in in ADP cana-6296 9 26 physics physics NOUN cana-6296 9 27 , , PUNCT cana-6296 9 28 engineering engineering NOUN cana-6296 9 29 , , PUNCT cana-6296 9 30 and and CCONJ cana-6296 9 31 applied apply VERB cana-6296 9 32 mathematics mathematic NOUN cana-6296 10 1 [ [ X cana-6296 10 2 1 1 NUM cana-6296 10 3 ] ] PUNCT cana-6296 10 4 . . PUNCT cana-6296 11 1 the the DET cana-6296 11 2 laplace laplace NOUN cana-6296 11 3 transform transform NOUN cana-6296 11 4 , , PUNCT cana-6296 11 5 expressed express VERB cana-6296 11 6 as as ADP cana-6296 11 7 𝐿{𝑓(𝑣 𝐿{𝑓(𝑣 NUM cana-6296 11 8 ) ) PUNCT cana-6296 11 9 } } PUNCT cana-6296 12 1 = = SYM cana-6296 12 2 ∫ ∫ PROPN cana-6296 12 3 𝑓(𝑣)π‘’βˆ’π›Ύπ‘£ 𝑓(𝑣)π‘’βˆ’π›Ύπ‘£ NOUN cana-6296 12 4 𝑑𝑣 𝑑𝑣 ADP cana-6296 12 5 = = SYM cana-6296 12 6 ∞ ∞ PROPN cana-6296 12 7 0 0 NUM cana-6296 12 8 𝐹(𝛾 𝐹(𝛾 NUM cana-6296 12 9 ) ) PUNCT cana-6296 12 10 is be AUX cana-6296 12 11 crucial crucial ADJ cana-6296 12 12 for for ADP cana-6296 12 13 examining examine VERB cana-6296 12 14 linear linear ADJ cana-6296 12 15 time time NOUN cana-6296 12 16 - - PUNCT cana-6296 12 17 invariant invariant ADJ cana-6296 12 18 systems system NOUN cana-6296 12 19 and and CCONJ cana-6296 12 20 tackling tackle VERB cana-6296 12 21 initial initial ADJ cana-6296 12 22 - - PUNCT cana-6296 12 23 value value NOUN cana-6296 12 24 problems problem NOUN cana-6296 12 25 . . PUNCT cana-6296 13 1 meanwhile meanwhile ADV cana-6296 13 2 , , PUNCT cana-6296 13 3 the the DET cana-6296 13 4 fourier fourier NOUN cana-6296 13 5 transform transform NOUN cana-6296 13 6 , , PUNCT cana-6296 13 7 𝐹(πœ‚ 𝐹(πœ‚ NUM cana-6296 13 8 ) ) PUNCT cana-6296 13 9 = = PUNCT cana-6296 13 10 𝑓{𝑓(𝑣 𝑓{𝑓(𝑣 NOUN cana-6296 13 11 ) ) PUNCT cana-6296 13 12 } } PUNCT cana-6296 13 13 = = SYM cana-6296 13 14 ∫ ∫ PROPN cana-6296 13 15 𝑓(𝑣)π‘’βˆ’π‘–πœ‚π‘£ 𝑓(𝑣)π‘’βˆ’π‘–πœ‚π‘£ PROPN cana-6296 13 16 𝑑𝑣 𝑑𝑣 NUM cana-6296 13 17 ∞ ∞ PROPN cana-6296 13 18 βˆ’βˆž βˆ’βˆž ADP cana-6296 13 19 is be AUX cana-6296 13 20 essential essential ADJ cana-6296 13 21 for for ADP cana-6296 13 22 frequency frequency NOUN cana-6296 13 23 domain domain NOUN cana-6296 13 24 analysis analysis NOUN cana-6296 13 25 and and CCONJ cana-6296 13 26 resolving resolve VERB cana-6296 13 27 issues issue NOUN cana-6296 13 28 on on ADP cana-6296 13 29 infinite infinite ADJ cana-6296 13 30 domains domain NOUN cana-6296 13 31 . . PUNCT cana-6296 14 1 although although SCONJ cana-6296 14 2 converting convert VERB cana-6296 14 3 a a DET cana-6296 14 4 function function NOUN cana-6296 14 5 f(v f(v NOUN cana-6296 14 6 ) ) PUNCT cana-6296 14 7 into into ADP cana-6296 14 8 its its PRON cana-6296 14 9 transform transform NOUN cana-6296 14 10 f(𝛾 f(𝛾 NOUN cana-6296 14 11 ) ) PUNCT cana-6296 14 12 or or CCONJ cana-6296 14 13 f(πœ‚ f(πœ‚ NUM cana-6296 14 14 ) ) PUNCT cana-6296 14 15 is be AUX cana-6296 14 16 generally generally ADV cana-6296 14 17 straightforward straightforward ADJ cana-6296 14 18 , , PUNCT cana-6296 14 19 the the DET cana-6296 14 20 reverse reverse ADJ cana-6296 14 21 process process NOUN cana-6296 14 22 β€” β€” PUNCT cana-6296 14 23 retrieving retrieve VERB cana-6296 14 24 the the DET cana-6296 14 25 original original ADJ cana-6296 14 26 function function NOUN cana-6296 14 27 f(t f(t NOUN cana-6296 14 28 ) ) PUNCT cana-6296 14 29 from from ADP cana-6296 14 30 its its PRON cana-6296 14 31 transform transform NOUN cana-6296 14 32 β€” β€” PUNCT cana-6296 14 33 often often ADV cana-6296 14 34 presents present VERB cana-6296 14 35 analytical analytical ADJ cana-6296 14 36 difficulties difficulty NOUN cana-6296 14 37 or or CCONJ cana-6296 14 38 is be AUX cana-6296 14 39 impossible impossible ADJ cana-6296 14 40 to to PART cana-6296 14 41 achieve achieve VERB cana-6296 14 42 in in ADP cana-6296 14 43 a a DET cana-6296 14 44 closed closed ADJ cana-6296 14 45 form form NOUN cana-6296 14 46 . . PUNCT cana-6296 15 1 the the DET cana-6296 15 2 analytical analytical ADJ cana-6296 15 3 inversion inversion NOUN cana-6296 15 4 involves involve VERB cana-6296 15 5 evaluating evaluate VERB cana-6296 15 6 a a DET cana-6296 15 7 complex complex ADJ cana-6296 15 8 contour contour NOUN cana-6296 15 9 integral integral ADJ cana-6296 15 10 ( ( PUNCT cana-6296 15 11 bromwich bromwich NOUN cana-6296 15 12 integral integral ADJ cana-6296 15 13 for for ADP cana-6296 15 14 laplace laplace NOUN cana-6296 15 15 ) ) PUNCT cana-6296 15 16 or or CCONJ cana-6296 15 17 a a DET cana-6296 15 18 principal principal ADJ cana-6296 15 19 value value NOUN cana-6296 15 20 integral integral ADJ cana-6296 15 21 ( ( PUNCT cana-6296 15 22 for for ADP cana-6296 15 23 fourier fourier NOUN cana-6296 15 24 ) ) PUNCT cana-6296 15 25 , , PUNCT cana-6296 15 26 which which PRON cana-6296 15 27 is be AUX cana-6296 15 28 only only ADV cana-6296 15 29 practical practical ADJ cana-6296 15 30 for for ADP cana-6296 15 31 a a DET cana-6296 15 32 limited limited ADJ cana-6296 15 33 range range NOUN cana-6296 15 34 of of ADP cana-6296 15 35 simple simple ADJ cana-6296 15 36 functions function NOUN cana-6296 15 37 [ [ X cana-6296 15 38 2 2 NUM cana-6296 15 39 ] ] PUNCT cana-6296 15 40 . . PUNCT cana-6296 16 1 as as ADP cana-6296 16 2 a a DET cana-6296 16 3 result result NOUN cana-6296 16 4 , , PUNCT cana-6296 16 5 robust robust ADJ cana-6296 16 6 numerical numerical ADJ cana-6296 16 7 methods method NOUN cana-6296 16 8 become become VERB cana-6296 16 9 vital vital ADJ cana-6296 16 10 . . PUNCT cana-6296 17 1 current current ADJ cana-6296 17 2 techniques technique NOUN cana-6296 17 3 for for ADP cana-6296 17 4 numerical numerical ADJ cana-6296 17 5 inversion inversion NOUN cana-6296 17 6 , , PUNCT cana-6296 17 7 such such ADJ cana-6296 17 8 as as ADP cana-6296 17 9 the the DET cana-6296 17 10 fourier fourier PROPN cana-6296 17 11 series series PROPN cana-6296 17 12 approximation approximation NOUN cana-6296 17 13 [ [ X cana-6296 17 14 3 3 NUM cana-6296 17 15 ] ] PUNCT cana-6296 17 16 , , PUNCT cana-6296 17 17 talbot talbot PROPN cana-6296 17 18 's 's PART cana-6296 17 19 method method NOUN cana-6296 17 20 [ [ X cana-6296 17 21 4 4 NUM cana-6296 17 22 ] ] PUNCT cana-6296 17 23 , , PUNCT cana-6296 17 24 and and CCONJ cana-6296 17 25 the the DET cana-6296 17 26 weeks week NOUN cana-6296 17 27 method method NOUN cana-6296 17 28 [ [ X cana-6296 17 29 5 5 NUM cana-6296 17 30 ] ] PUNCT cana-6296 17 31 for for ADP cana-6296 17 32 laplace laplace NOUN cana-6296 17 33 transforms transform VERB cana-6296 17 34 , , PUNCT cana-6296 17 35 or or CCONJ cana-6296 17 36 discrete discrete ADJ cana-6296 17 37 inverse inverse NOUN cana-6296 17 38 fft fft NOUN cana-6296 17 39 for for ADP cana-6296 17 40 fourier fourier NOUN cana-6296 17 41 transforms transform NOUN cana-6296 17 42 , , PUNCT cana-6296 17 43 are be AUX cana-6296 17 44 well well ADV cana-6296 17 45 - - PUNCT cana-6296 17 46 established establish VERB cana-6296 17 47 but but CCONJ cana-6296 17 48 may may AUX cana-6296 17 49 experience experience VERB cana-6296 17 50 instability instability NOUN cana-6296 17 51 , , PUNCT cana-6296 17 52 slow slow ADJ cana-6296 17 53 convergence convergence NOUN cana-6296 17 54 , , PUNCT cana-6296 17 55 or or CCONJ cana-6296 17 56 sensitivity sensitivity NOUN cana-6296 17 57 to to ADP cana-6296 17 58 parameters parameter NOUN cana-6296 17 59 specific specific ADJ cana-6296 17 60 to to ADP cana-6296 17 61 the the DET cana-6296 17 62 transform transform NOUN cana-6296 17 63 . . PUNCT cana-6296 18 1 this this DET cana-6296 18 2 paper paper NOUN cana-6296 18 3 introduces introduce VERB cana-6296 18 4 a a DET cana-6296 18 5 unified unified ADJ cana-6296 18 6 approach approach NOUN cana-6296 18 7 that that PRON cana-6296 18 8 frames frame VERB cana-6296 18 9 both both DET cana-6296 18 10 inversion inversion NOUN cana-6296 18 11 challenges challenge NOUN cana-6296 18 12 within within ADP cana-6296 18 13 a a DET cana-6296 18 14 single single ADJ cana-6296 18 15 , , PUNCT cana-6296 18 16 stable stable ADJ cana-6296 18 17 computational computational ADJ cana-6296 18 18 framework framework NOUN cana-6296 18 19 . . PUNCT cana-6296 19 1 we we PRON cana-6296 19 2 present present VERB cana-6296 19 3 a a DET cana-6296 19 4 core core NOUN cana-6296 19 5 theorem theorem NOUN cana-6296 19 6 that that PRON cana-6296 19 7 formally formally ADV cana-6296 19 8 connects connect VERB cana-6296 19 9 the the DET cana-6296 19 10 two two NUM cana-6296 19 11 inversions inversion NOUN cana-6296 19 12 and and CCONJ cana-6296 19 13 describe describe VERB cana-6296 19 14 a a DET cana-6296 19 15 practical practical ADJ cana-6296 19 16 fft fft NOUN cana-6296 19 17 - - PUNCT cana-6296 19 18 based base VERB cana-6296 19 19 algorithm algorithm NOUN cana-6296 19 20 for for ADP cana-6296 19 21 implementation implementation NOUN cana-6296 19 22 . . PUNCT cana-6296 20 1 this this DET cana-6296 20 2 article article NOUN cana-6296 20 3 introduces introduce VERB cana-6296 20 4 a a DET cana-6296 20 5 method method NOUN cana-6296 20 6 that that PRON cana-6296 20 7 offers offer VERB cana-6296 20 8 a a DET cana-6296 20 9 unified unified ADJ cana-6296 20 10 formula formula NOUN cana-6296 20 11 for for ADP cana-6296 20 12 addressing address VERB cana-6296 20 13 both both CCONJ cana-6296 20 14 fourier fourier NOUN cana-6296 20 15 and and CCONJ cana-6296 20 16 laplace laplace NOUN cana-6296 20 17 transforms transform VERB cana-6296 20 18 , , PUNCT cana-6296 20 19 along along ADP cana-6296 20 20 with with ADP cana-6296 20 21 their their PRON cana-6296 20 22 inverses inverse NOUN cana-6296 20 23 . . PUNCT cana-6296 21 1 the the DET cana-6296 21 2 inspiration inspiration NOUN cana-6296 21 3 for for ADP cana-6296 21 4 this this DET cana-6296 21 5 work work NOUN cana-6296 21 6 came come VERB cana-6296 21 7 from from ADP cana-6296 21 8 a a DET cana-6296 21 9 communications communication NOUN cana-6296 21 10 on on ADP cana-6296 21 11 applied apply VERB cana-6296 21 12 nonlinear nonlinear ADJ cana-6296 21 13 analysis analysis NOUN cana-6296 21 14 issn issn NOUN cana-6296 21 15 : : PUNCT cana-6296 21 16 1074 1074 NUM cana-6296 21 17 - - PUNCT cana-6296 21 18 133x 133x NUM cana-6296 21 19 vol vol VERB cana-6296 21 20 32 32 NUM cana-6296 21 21 no no NOUN cana-6296 21 22 . . PUNCT cana-6296 22 1 10s 10 NOUN cana-6296 22 2 ( ( PUNCT cana-6296 22 3 2025 2025 NUM cana-6296 22 4 ) ) PUNCT cana-6296 22 5 3696 3696 NUM cana-6296 22 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 22 7 partial partial ADJ cana-6296 22 8 differential differential NOUN cana-6296 22 9 equation equation NOUN cana-6296 22 10 ( ( PUNCT cana-6296 22 11 pde pde NOUN cana-6296 22 12 ) ) PUNCT cana-6296 22 13 course course NOUN cana-6296 22 14 one one NUM cana-6296 22 15 of of ADP cana-6296 22 16 the the DET cana-6296 22 17 authors author NOUN cana-6296 22 18 attended attend VERB cana-6296 22 19 38 38 NUM cana-6296 22 20 years year NOUN cana-6296 22 21 ago ago ADV cana-6296 22 22 , , PUNCT cana-6296 22 23 particularly particularly ADV cana-6296 22 24 focusing focus VERB cana-6296 22 25 on on ADP cana-6296 22 26 the the DET cana-6296 22 27 use use NOUN cana-6296 22 28 of of ADP cana-6296 22 29 inverse inverse ADJ cana-6296 22 30 symbolic symbolic ADJ cana-6296 22 31 operators operator NOUN cana-6296 22 32 to to PART cana-6296 22 33 solve solve VERB cana-6296 22 34 pdes pde NOUN cana-6296 22 35 [ [ X cana-6296 22 36 6 6 NUM cana-6296 22 37 ] ] PUNCT cana-6296 22 38 . . PUNCT cana-6296 23 1 since since SCONJ cana-6296 23 2 then then ADV cana-6296 23 3 , , PUNCT cana-6296 23 4 various various ADJ cana-6296 23 5 authors author NOUN cana-6296 23 6 have have AUX cana-6296 23 7 proposed propose VERB cana-6296 23 8 methods method NOUN cana-6296 23 9 for for ADP cana-6296 23 10 solving solve VERB cana-6296 23 11 laplace laplace NOUN cana-6296 23 12 transforms transform VERB cana-6296 23 13 , , PUNCT cana-6296 23 14 but but CCONJ cana-6296 23 15 none none NOUN cana-6296 23 16 are be AUX cana-6296 23 17 quite quite ADV cana-6296 23 18 like like ADP cana-6296 23 19 the the DET cana-6296 23 20 one one NOUN cana-6296 23 21 detailed detail VERB cana-6296 23 22 in in ADP cana-6296 23 23 this this DET cana-6296 23 24 article article NOUN cana-6296 23 25 [ [ X cana-6296 23 26 7 7 NUM cana-6296 23 27 ] ] PUNCT cana-6296 23 28 . . PUNCT cana-6296 24 1 2 2 X cana-6296 24 2 . . X cana-6296 25 1 keywords keyword NOUN cana-6296 25 2 : : PUNCT cana-6296 25 3 inverse inverse ADJ cana-6296 25 4 laplace laplace NOUN cana-6296 25 5 transform transform NOUN cana-6296 25 6 , , PUNCT cana-6296 25 7 inverse inverse ADJ cana-6296 25 8 fourier fourier NOUN cana-6296 25 9 transform transform NOUN cana-6296 25 10 , , PUNCT cana-6296 25 11 numerical numerical ADJ cana-6296 25 12 methods method NOUN cana-6296 25 13 , , PUNCT cana-6296 25 14 analytic analytic ADJ cana-6296 25 15 continuation continuation NOUN cana-6296 25 16 , , PUNCT cana-6296 25 17 fast fast ADJ cana-6296 25 18 fourier fourier NOUN cana-6296 25 19 transform transform NOUN cana-6296 25 20 ( ( PUNCT cana-6296 25 21 fft fft PROPN cana-6296 25 22 ) ) PUNCT cana-6296 25 23 , , PUNCT cana-6296 25 24 bromwich bromwich PROPN cana-6296 25 25 integral integral ADJ cana-6296 25 26 , , PUNCT cana-6296 25 27 regularization regularization NOUN cana-6296 25 28 , , PUNCT cana-6296 25 29 computational computational ADJ cana-6296 25 30 mathematics mathematic NOUN cana-6296 25 31 . . PUNCT cana-6296 26 1 3 3 X cana-6296 26 2 . . X cana-6296 26 3 methods method NOUN cana-6296 26 4 : : PUNCT cana-6296 26 5 4.1 4.1 NUM cana-6296 26 6 . . PUNCT cana-6296 26 7 symbolic symbolic ADJ cana-6296 26 8 operators operator NOUN cana-6296 26 9 the the DET cana-6296 26 10 inversion inversion NOUN cana-6296 26 11 of of ADP cana-6296 26 12 transforms transform NOUN cana-6296 26 13 can can AUX cana-6296 26 14 be be AUX cana-6296 26 15 expressed express VERB cana-6296 26 16 using use VERB cana-6296 26 17 inverse inverse NOUN cana-6296 26 18 symbolic symbolic ADJ cana-6296 26 19 operators operator NOUN cana-6296 26 20 . . PUNCT cana-6296 27 1 for for ADP cana-6296 27 2 a a DET cana-6296 27 3 given give VERB cana-6296 27 4 transform transform NOUN cana-6296 27 5 pair pair NOUN cana-6296 27 6 𝑓(𝑣 𝑓(𝑣 NOUN cana-6296 27 7 ) ) PUNCT cana-6296 27 8 ↔ ↔ PROPN cana-6296 27 9 𝐹(𝛾 𝐹(𝛾 NOUN cana-6296 27 10 ) ) PUNCT cana-6296 27 11 , , PUNCT cana-6296 27 12 the the DET cana-6296 27 13 inversion inversion NOUN cana-6296 27 14 is be AUX cana-6296 27 15 formally formally ADV cana-6296 27 16 defined define VERB cana-6296 27 17 by by ADP cana-6296 27 18 the the DET cana-6296 27 19 operator operator NOUN cana-6296 27 20 . . PUNCT cana-6296 28 1 𝑓(𝑣 𝑓(𝑣 NOUN cana-6296 28 2 ) ) PUNCT cana-6296 28 3 = = SYM cana-6296 28 4 πΏβˆ’1{𝐹(𝛾 πΏβˆ’1{𝐹(𝛾 NOUN cana-6296 28 5 ) ) PUNCT cana-6296 28 6 } } PUNCT cana-6296 28 7 , , PUNCT cana-6296 28 8 𝑓(𝑣 𝑓(𝑣 X cana-6296 28 9 ) ) PUNCT cana-6296 28 10 = = SYM cana-6296 28 11 πΉβˆ’1{𝐹(πœ‚ πΉβˆ’1{𝐹(πœ‚ PROPN cana-6296 28 12 ) ) PUNCT cana-6296 28 13 } } PUNCT cana-6296 28 14 . . PUNCT cana-6296 29 1 the the DET cana-6296 29 2 laplace laplace NOUN cana-6296 29 3 inverse inverse NOUN cana-6296 29 4 is be AUX cana-6296 29 5 typically typically ADV cana-6296 29 6 written write VERB cana-6296 29 7 as as ADP cana-6296 29 8 the the DET cana-6296 29 9 bromwich bromwich PROPN cana-6296 29 10 integral integral NOUN cana-6296 29 11 : : PUNCT cana-6296 29 12 𝑓(𝑣 𝑓(𝑣 NUM cana-6296 29 13 ) ) PUNCT cana-6296 29 14 = = SYM cana-6296 30 1 1 1 NUM cana-6296 30 2 2πœ‹π‘– 2πœ‹π‘– ADJ cana-6296 30 3 ∫ ∫ PROPN cana-6296 30 4 𝐹 𝐹 PROPN cana-6296 30 5 ( ( PUNCT cana-6296 30 6 𝜎+π‘–βˆž 𝜎+π‘–βˆž PROPN cana-6296 30 7 πœŽβˆ’π‘–βˆž πœŽβˆ’π‘–βˆž PROPN cana-6296 30 8 𝛾)𝑒𝛾𝑣𝑑𝛾 𝛾)𝑒𝛾𝑣𝑑𝛾 PROPN cana-6296 30 9 , , PUNCT cana-6296 30 10 where where SCONJ cana-6296 30 11 𝜎 𝜎 PROPN cana-6296 30 12 is be AUX cana-6296 30 13 a a DET cana-6296 30 14 real real ADJ cana-6296 30 15 number number NOUN cana-6296 30 16 to to ADP cana-6296 30 17 the the DET cana-6296 30 18 right right NOUN cana-6296 30 19 of of ADP cana-6296 30 20 all all DET cana-6296 30 21 singularities singularity NOUN cana-6296 30 22 of of ADP cana-6296 30 23 𝐹(𝛾 𝐹(𝛾 NOUN cana-6296 30 24 ) ) PUNCT cana-6296 30 25 . . PUNCT cana-6296 31 1 similarly similarly ADV cana-6296 31 2 , , PUNCT cana-6296 31 3 the the DET cana-6296 31 4 fourier fourier ADJ cana-6296 31 5 inverse inverse NOUN cana-6296 31 6 is be AUX cana-6296 31 7 given give VERB cana-6296 31 8 given give VERB cana-6296 31 9 by by ADP cana-6296 31 10 : : PUNCT cana-6296 31 11 𝑓(𝑣 𝑓(𝑣 NUM cana-6296 31 12 ) ) PUNCT cana-6296 31 13 = = SYM cana-6296 31 14 1 1 NUM cana-6296 31 15 2πœ‹ 2πœ‹ NUM cana-6296 31 16 ∫ ∫ NOUN cana-6296 31 17 𝐹 𝐹 PROPN cana-6296 31 18 ( ( PUNCT cana-6296 31 19 ∞ ∞ PROPN cana-6296 31 20 0 0 NUM cana-6296 31 21 πœ‚)π‘’π‘–πœ‚π‘£π‘‘πœ‚. πœ‚)π‘’π‘–πœ‚π‘£π‘‘πœ‚. PROPN cana-6296 31 22 4.2 4.2 NUM cana-6296 31 23 . . PUNCT cana-6296 32 1 core core NOUN cana-6296 32 2 theorem theorem NOUN cana-6296 32 3 : : PUNCT cana-6296 32 4 unified unified ADJ cana-6296 32 5 laplace laplace NOUN cana-6296 32 6 - - PUNCT cana-6296 32 7 fourier fourier NOUN cana-6296 32 8 inversion inversion NOUN cana-6296 32 9 theorem(unified theorem(unifie VERB cana-6296 32 10 inverse inverse NOUN cana-6296 32 11 transform transform NOUN cana-6296 32 12 ): ): PUNCT cana-6296 32 13 let let VERB cana-6296 32 14 𝐹(𝛾 𝐹(𝛾 NUM cana-6296 32 15 ) ) PUNCT cana-6296 32 16 be be AUX cana-6296 32 17 analytic analytic ADJ cana-6296 32 18 in in ADP cana-6296 32 19 the the DET cana-6296 32 20 half half ADJ cana-6296 32 21 - - PUNCT cana-6296 32 22 plane plane NOUN cana-6296 32 23 β„œ(𝛾 β„œ(𝛾 NOUN cana-6296 32 24 ) ) PUNCT cana-6296 32 25 > > X cana-6296 33 1 0 0 X cana-6296 33 2 . . PUNCT cana-6296 34 1 then then ADV cana-6296 34 2 the the DET cana-6296 34 3 inverse inverse NOUN cana-6296 34 4 laplace laplace NOUN cana-6296 34 5 transform transform NOUN cana-6296 34 6 can can AUX cana-6296 34 7 be be AUX cana-6296 34 8 expressed express VERB cana-6296 34 9 fourier fourier ADJ cana-6296 34 10 - - PUNCT cana-6296 34 11 type type NOUN cana-6296 34 12 integral integral ADJ cana-6296 34 13 : : PUNCT cana-6296 34 14 𝑓(𝑣 𝑓(𝑣 NUM cana-6296 34 15 ) ) PUNCT cana-6296 34 16 = = SYM cana-6296 35 1 π‘’πœŽπ‘£ π‘’πœŽπ‘£ PRON cana-6296 36 1 2πœ‹ 2πœ‹ NUM cana-6296 36 2 ∫ ∫ NOUN cana-6296 36 3 𝐹(𝜎 𝐹(𝜎 X cana-6296 36 4 + + X cana-6296 36 5 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 36 6 ) ) PUNCT cana-6296 36 7 ∞ ∞ PROPN cana-6296 36 8 βˆ’βˆž βˆ’βˆž ADP cana-6296 36 9 π‘’π‘–πœ‚π‘£π‘‘πœ‚ π‘’π‘–πœ‚π‘£π‘‘πœ‚ PROPN cana-6296 36 10 , , PUNCT cana-6296 36 11 where where SCONJ cana-6296 36 12 𝜎 𝜎 PROPN cana-6296 36 13 > > X cana-6296 36 14 0 0 NUM cana-6296 36 15 ensures ensure VERB cana-6296 36 16 convergence convergence NOUN cana-6296 36 17 . . PUNCT cana-6296 37 1 proof proof NOUN cana-6296 37 2 : : PUNCT cana-6296 37 3 by by ADP cana-6296 37 4 definition definition NOUN cana-6296 37 5 of of ADP cana-6296 37 6 the the DET cana-6296 37 7 inverse inverse ADJ cana-6296 37 8 laplace laplace NOUN cana-6296 37 9 transform(bromwich transform(bromwich PROPN cana-6296 37 10 integral integral ADJ cana-6296 37 11 ) ) PUNCT cana-6296 37 12 , , PUNCT cana-6296 37 13 for for ADP cana-6296 37 14 𝑣 𝑣 PRON cana-6296 37 15 > > X cana-6296 37 16 0 0 PUNCT cana-6296 37 17 and and CCONJ cana-6296 37 18 any any DET cana-6296 37 19 real real ADJ cana-6296 37 20 c c NOUN cana-6296 37 21 to to ADP cana-6296 37 22 the the DET cana-6296 37 23 right right NOUN cana-6296 37 24 of of ADP cana-6296 37 25 every every DET cana-6296 37 26 singularity singularity NOUN cana-6296 37 27 of of ADP cana-6296 37 28 𝐹(𝛾 𝐹(𝛾 NOUN cana-6296 37 29 ) ) PUNCT cana-6296 37 30 , , PUNCT cana-6296 37 31 𝑓(𝑣 𝑓(𝑣 X cana-6296 37 32 ) ) PUNCT cana-6296 37 33 = = SYM cana-6296 37 34 1 1 NUM cana-6296 37 35 2πœ‹π‘– 2πœ‹π‘– ADJ cana-6296 37 36 ∫ ∫ PROPN cana-6296 38 1 𝐹 𝐹 PROPN cana-6296 38 2 ( ( PUNCT cana-6296 38 3 𝜎+π‘–βˆž 𝜎+π‘–βˆž PROPN cana-6296 38 4 πœŽβˆ’π‘–βˆž πœŽβˆ’π‘–βˆž PROPN cana-6296 38 5 𝛾)𝑒𝛾𝑣𝑑𝛾. 𝛾)𝑒𝛾𝑣𝑑𝛾. PROPN cana-6296 38 6 since since SCONJ cana-6296 38 7 f f PROPN cana-6296 38 8 is be AUX cana-6296 38 9 analytic analytic ADJ cana-6296 38 10 in in ADP cana-6296 38 11 the the DET cana-6296 38 12 half half ADJ cana-6296 38 13 - - PUNCT cana-6296 38 14 plane plane NOUN cana-6296 38 15 β„œ(𝛾 β„œ(𝛾 NOUN cana-6296 38 16 ) ) PUNCT cana-6296 38 17 > > X cana-6296 38 18 0 0 NUM cana-6296 38 19 , , PUNCT cana-6296 38 20 choose choose VERB cana-6296 38 21 any any DET cana-6296 38 22 𝜎 𝜎 PROPN cana-6296 38 23 > > X cana-6296 38 24 0 0 PUNCT cana-6296 38 25 lying lie VERB cana-6296 38 26 to to ADP cana-6296 38 27 the the DET cana-6296 38 28 right right NOUN cana-6296 38 29 of of ADP cana-6296 38 30 all all DET cana-6296 38 31 singularities singularity NOUN cana-6296 38 32 of of ADP cana-6296 38 33 f. f. NOUN cana-6296 38 34 by by ADP cana-6296 38 35 analytically analytically ADV cana-6296 38 36 and and CCONJ cana-6296 38 37 standard standard ADJ cana-6296 38 38 contour contour ADJ cana-6296 38 39 - - PUNCT cana-6296 38 40 deformation deformation NOUN cana-6296 38 41 arguments argument NOUN cana-6296 38 42 the the DET cana-6296 38 43 integral integral ADJ cana-6296 38 44 along along ADP cana-6296 38 45 β„œπ›Ύ β„œπ›Ύ PROPN cana-6296 38 46 = = PUNCT cana-6296 38 47 𝑐 𝑐 PROPN cana-6296 38 48 equals equal VERB cana-6296 38 49 the the DET cana-6296 38 50 integral integral ADJ cana-6296 38 51 along along ADP cana-6296 38 52 β„œπ›Ύ β„œπ›Ύ PROPN cana-6296 38 53 = = PUNCT cana-6296 38 54 𝜎. 𝜎. NOUN cana-6296 38 55 put put VERB cana-6296 38 56 𝛾 𝛾 PROPN cana-6296 38 57 = = SYM cana-6296 38 58 𝜎 𝜎 PROPN cana-6296 39 1 + + X cana-6296 39 2 π‘–πœ‚ π‘–πœ‚ X cana-6296 39 3 with with ADP cana-6296 39 4 πœ‚ πœ‚ PROPN cana-6296 39 5 ∈ ∈ PROPN cana-6296 39 6 ( ( PUNCT cana-6296 39 7 βˆ’βˆž βˆ’βˆž NOUN cana-6296 39 8 , , PUNCT cana-6296 39 9 ∞ ∞ PROPN cana-6296 39 10 ) ) PUNCT cana-6296 39 11 . . PUNCT cana-6296 40 1 then then ADV cana-6296 40 2 𝑑𝛾 𝑑𝛾 PROPN cana-6296 40 3 = = SYM cana-6296 40 4 βˆ’π‘–π‘‘πœ‚. βˆ’π‘–π‘‘πœ‚. PROPN cana-6296 40 5 substituting substitute VERB cana-6296 40 6 into into ADP cana-6296 40 7 the the DET cana-6296 40 8 bromwich bromwich PROPN cana-6296 40 9 integral integral ADJ cana-6296 40 10 gives give NOUN cana-6296 40 11 𝑓(𝑣 𝑓(𝑣 NOUN cana-6296 40 12 ) ) PUNCT cana-6296 40 13 = = SYM cana-6296 40 14 1 1 NUM cana-6296 40 15 2πœ‹π‘– 2πœ‹π‘– ADJ cana-6296 40 16 ∫ ∫ PROPN cana-6296 40 17 𝐹(𝜎 𝐹(𝜎 X cana-6296 40 18 + + CCONJ cana-6296 40 19 π‘–πœ‚ π‘–πœ‚ ADP cana-6296 40 20 𝜎+π‘–βˆž 𝜎+π‘–βˆž NOUN cana-6296 40 21 πœŽβˆ’π‘–βˆž πœŽβˆ’π‘–βˆž PROPN cana-6296 40 22 ) ) PUNCT cana-6296 40 23 𝑒(𝜎+π‘–πœ‚)𝑣𝑖 𝑒(𝜎+π‘–πœ‚)𝑣𝑖 PROPN cana-6296 40 24 π‘‘πœ‚. π‘‘πœ‚. PROPN cana-6296 40 25 the the DET cana-6296 40 26 factor factor NOUN cana-6296 40 27 𝑖 𝑖 ADP cana-6296 40 28 cancels cancel VERB cana-6296 40 29 the the DET cana-6296 40 30 1 1 NUM cana-6296 40 31 2πœ‹π‘– 2πœ‹π‘– ADJ cana-6296 40 32 prefactor prefactor NOUN cana-6296 40 33 , , PUNCT cana-6296 40 34 so so ADV cana-6296 40 35 𝑓(𝑣 𝑓(𝑣 NOUN cana-6296 40 36 ) ) PUNCT cana-6296 40 37 = = SYM cana-6296 40 38 1 1 NUM cana-6296 40 39 2πœ‹ 2πœ‹ NUM cana-6296 40 40 ∫ ∫ NOUN cana-6296 40 41 𝐹(𝜎 𝐹(𝜎 X cana-6296 40 42 + + CCONJ cana-6296 40 43 π‘–πœ‚ π‘–πœ‚ ADP cana-6296 40 44 𝜎+π‘–βˆž 𝜎+π‘–βˆž NOUN cana-6296 40 45 πœŽβˆ’π‘–βˆž πœŽβˆ’π‘–βˆž PROPN cana-6296 40 46 ) ) PUNCT cana-6296 40 47 π‘’πœŽπ‘£π‘’π‘–πœ‚πœπ‘‘πœ‚. π‘’πœŽπ‘£π‘’π‘–πœ‚πœπ‘‘πœ‚. NOUN cana-6296 40 48 pulling pull VERB cana-6296 40 49 the the DET cana-6296 40 50 constant constant ADJ cana-6296 40 51 π‘’πœŽπ‘£ π‘’πœŽπ‘£ NOUN cana-6296 40 52 outside outside ADP cana-6296 40 53 the the DET cana-6296 40 54 integral integral ADJ cana-6296 40 55 yields yield NOUN cana-6296 40 56 the the DET cana-6296 40 57 fourier fourier NOUN cana-6296 40 58 - - PUNCT cana-6296 40 59 type type NOUN cana-6296 40 60 representation representation NOUN cana-6296 40 61 communications communication NOUN cana-6296 40 62 on on ADP cana-6296 40 63 applied apply VERB cana-6296 40 64 nonlinear nonlinear ADJ cana-6296 40 65 analysis analysis NOUN cana-6296 40 66 issn issn NOUN cana-6296 40 67 : : PUNCT cana-6296 40 68 1074 1074 NUM cana-6296 40 69 - - PUNCT cana-6296 40 70 133x 133x NUM cana-6296 40 71 vol vol VERB cana-6296 40 72 32 32 NUM cana-6296 40 73 no no NOUN cana-6296 40 74 . . PUNCT cana-6296 41 1 10s 10 NOUN cana-6296 41 2 ( ( PUNCT cana-6296 41 3 2025 2025 NUM cana-6296 41 4 ) ) PUNCT cana-6296 41 5 3697 3697 NUM cana-6296 42 1 https://internationalpubls.com https://internationalpubls.com X cana-6296 42 2 𝑓(𝑣 𝑓(𝑣 NUM cana-6296 42 3 ) ) PUNCT cana-6296 42 4 = = SYM cana-6296 43 1 π‘’πœŽπ‘£ π‘’πœŽπ‘£ PRON cana-6296 44 1 2πœ‹ 2πœ‹ NUM cana-6296 44 2 ∫ ∫ NOUN cana-6296 44 3 𝐹(𝜎 𝐹(𝜎 X cana-6296 44 4 + + X cana-6296 44 5 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 44 6 ) ) PUNCT cana-6296 44 7 ∞ ∞ PROPN cana-6296 44 8 βˆ’βˆž βˆ’βˆž ADP cana-6296 44 9 π‘’π‘–πœ‚π‘£π‘‘πœ‚. π‘’π‘–πœ‚π‘£π‘‘πœ‚. NOUN cana-6296 44 10 which which PRON cana-6296 44 11 is be AUX cana-6296 44 12 the the DET cana-6296 44 13 desired desire VERB cana-6296 44 14 formula formula NOUN cana-6296 44 15 . . PUNCT cana-6296 45 1 4.3 4.3 NUM cana-6296 45 2 . . PUNCT cana-6296 45 3 computational computational ADJ cana-6296 45 4 approach approach NOUN cana-6296 45 5 from from ADP cana-6296 45 6 the the DET cana-6296 45 7 theorem theorem NOUN cana-6296 45 8 , , PUNCT cana-6296 45 9 the the DET cana-6296 45 10 laplace laplace NOUN cana-6296 45 11 inversion inversion NOUN cana-6296 45 12 reduces reduce VERB cana-6296 45 13 to to ADP cana-6296 45 14 : : PUNCT cana-6296 45 15 𝑓(𝑣 𝑓(𝑣 NUM cana-6296 45 16 ) ) PUNCT cana-6296 46 1 β‰ˆ β‰ˆ PROPN cana-6296 46 2 π‘’πœŽπ‘£ π‘’πœŽπ‘£ NOUN cana-6296 46 3 2πœ‹ 2πœ‹ NUM cana-6296 46 4 βˆ‘ βˆ‘ PUNCT cana-6296 46 5 𝐹(𝜎 𝐹(𝜎 PUNCT cana-6296 46 6 + + PROPN cana-6296 46 7 π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£Ξ΄Ξ· π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£Ξ΄Ξ· PROPN cana-6296 46 8 , , PUNCT cana-6296 46 9 𝑁 𝑁 PROPN cana-6296 46 10 π‘˜=βˆ’π‘ π‘˜=βˆ’π‘ PROPN cana-6296 46 11 where where SCONJ cana-6296 46 12 πœ‚π‘˜ πœ‚π‘˜ ADP cana-6296 46 13 = = PUNCT cana-6296 46 14 π‘˜Ξ΄Ξ· π‘˜Ξ΄Ξ· NOUN cana-6296 46 15 . . PUNCT cana-6296 47 1 this this PRON cana-6296 47 2 is be AUX cana-6296 47 3 essentially essentially ADV cana-6296 47 4 a a DET cana-6296 47 5 discrete discrete ADJ cana-6296 47 6 fourier fourier NOUN cana-6296 47 7 transform transform NOUN cana-6296 47 8 ( ( PUNCT cana-6296 47 9 dft dft PROPN cana-6296 47 10 ) ) PUNCT cana-6296 47 11 and and CCONJ cana-6296 47 12 can can AUX cana-6296 47 13 be be AUX cana-6296 47 14 evaluated evaluate VERB cana-6296 47 15 efficiently efficiently ADV cana-6296 47 16 using use VERB cana-6296 47 17 algorithm algorithm NOUN cana-6296 47 18 : : PUNCT cana-6296 47 19 choose choose VERB cana-6296 47 20 𝜎 𝜎 PROPN cana-6296 47 21 > > X cana-6296 47 22 0 0 PUNCT cana-6296 48 1 large large ADJ cana-6296 48 2 enough enough ADV cana-6296 48 3 to to PART cana-6296 48 4 ensure ensure VERB cana-6296 48 5 𝐹(𝜎 𝐹(𝜎 PRON cana-6296 48 6 + + X cana-6296 48 7 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 48 8 ) ) PUNCT cana-6296 48 9 is be AUX cana-6296 48 10 analytic analytic ADJ cana-6296 48 11 and and CCONJ cana-6296 48 12 decays decay VERB cana-6296 48 13 sufficiently sufficiently ADV cana-6296 48 14 , , PUNCT cana-6296 48 15 but but CCONJ cana-6296 48 16 not not PART cana-6296 48 17 so so ADV cana-6296 48 18 large large ADJ cana-6296 48 19 that that SCONJ cana-6296 48 20 π‘’πœŽπ‘£ π‘’πœŽπ‘£ NOUN cana-6296 48 21 causes cause VERB cana-6296 48 22 numerical numerical ADJ cana-6296 48 23 instability instability NOUN cana-6296 48 24 . . PUNCT cana-6296 49 1 sample sample NOUN cana-6296 49 2 𝐹(𝜎 𝐹(𝜎 X cana-6296 49 3 + + CCONJ cana-6296 49 4 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 49 5 ) ) PUNCT cana-6296 49 6 at at ADP cana-6296 49 7 equispaced equispace VERB cana-6296 49 8 frequencies frequency NOUN cana-6296 49 9 πœ‚π‘˜ πœ‚π‘˜ ADP cana-6296 49 10 = = ADJ cana-6296 49 11 π‘˜Ξ΄Ξ· π‘˜Ξ΄Ξ· NOUN cana-6296 49 12 , , PUNCT cana-6296 49 13 k k X cana-6296 49 14 = = PUNCT cana-6296 49 15 βˆ’n βˆ’n PROPN cana-6296 49 16 , , PUNCT cana-6296 49 17 … … PUNCT cana-6296 49 18 … … PUNCT cana-6296 49 19 , , PUNCT cana-6296 49 20 n. n. PROPN cana-6296 49 21 recognize recognize VERB cana-6296 49 22 the the DET cana-6296 49 23 sum sum NOUN cana-6296 49 24 as as ADP cana-6296 49 25 a a DET cana-6296 49 26 discrete discrete ADJ cana-6296 49 27 fourier fourier NOUN cana-6296 49 28 transform transform NOUN cana-6296 49 29 . . PUNCT cana-6296 50 1 use use VERB cana-6296 50 2 the the DET cana-6296 50 3 fast fast ADJ cana-6296 50 4 fourier fourier NOUN cana-6296 50 5 transform transform NOUN cana-6296 50 6 to to PART cana-6296 50 7 compute compute VERB cana-6296 50 8 efficiently efficiently ADV cana-6296 50 9 . . PUNCT cana-6296 51 1 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-6296 51 2 ) ) PUNCT cana-6296 51 3 = = PUNCT cana-6296 52 1 βˆ‘ βˆ‘ PUNCT cana-6296 52 2 𝐹(𝜎 𝐹(𝜎 PUNCT cana-6296 52 3 + + CCONJ cana-6296 52 4 π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£π‘— π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘£π‘— PROPN cana-6296 52 5 . . PUNCT cana-6296 53 1 𝑁 𝑁 PROPN cana-6296 53 2 π‘˜=βˆ’π‘ π‘˜=βˆ’π‘ PROPN cana-6296 53 3 here here ADV cana-6296 53 4 𝑣𝑗 𝑣𝑗 ADP cana-6296 53 5 = = SYM cana-6296 53 6 π‘—βˆ†π‘£ π‘—βˆ†π‘£ NOUN cana-6296 53 7 , , PUNCT cana-6296 53 8 where where SCONJ cana-6296 53 9 βˆ†π‘£ βˆ†π‘£ PROPN cana-6296 53 10 = = SYM cana-6296 53 11 2πœ‹ 2πœ‹ NUM cana-6296 53 12 ( ( PUNCT cana-6296 53 13 2𝑁+1)βˆ†πœ‚ 2𝑁+1)βˆ†πœ‚ NUM cana-6296 53 14 by by ADP cana-6296 53 15 the the DET cana-6296 53 16 nyquist nyquist PROPN cana-6296 53 17 relation relation NOUN cana-6296 53 18 . . PUNCT cana-6296 54 1 multiply multiply VERB cana-6296 54 2 the the DET cana-6296 54 3 fast fast ADJ cana-6296 54 4 fourier fourier NOUN cana-6296 54 5 transform transform NOUN cana-6296 54 6 output output NOUN cana-6296 54 7 by by ADP cana-6296 54 8 the the DET cana-6296 54 9 prefactor prefactor NOUN cana-6296 54 10 : : PUNCT cana-6296 54 11 𝑓(𝑣𝑗 𝑓(𝑣𝑗 NOUN cana-6296 54 12 ) ) PUNCT cana-6296 55 1 β‰ˆ β‰ˆ NUM cana-6296 55 2 π‘’πœŽπ‘£π‘— π‘’πœŽπ‘£π‘— NOUN cana-6296 55 3 2πœ‹ 2πœ‹ NOUN cana-6296 55 4 βˆ†πœ‚π‘“(𝑣𝑗 βˆ†πœ‚π‘“(𝑣𝑗 NOUN cana-6296 55 5 ) ) PUNCT cana-6296 55 6 . . PUNCT cana-6296 56 1 4.4 4.4 NUM cana-6296 56 2 . . PUNCT cana-6296 57 1 numerical numerical ADJ cana-6296 57 2 considerations consideration NOUN cana-6296 57 3 : : PUNCT cana-6296 57 4 error error NOUN cana-6296 57 5 and and CCONJ cana-6296 57 6 stability stability NOUN cana-6296 57 7 while while SCONJ cana-6296 57 8 the the DET cana-6296 57 9 unified unified ADJ cana-6296 57 10 inversion inversion NOUN cana-6296 57 11 formula formula NOUN cana-6296 57 12 is be AUX cana-6296 57 13 elegant elegant ADJ cana-6296 57 14 , , PUNCT cana-6296 57 15 practical practical ADJ cana-6296 57 16 implementation implementation NOUN cana-6296 57 17 requires require VERB cana-6296 57 18 careful careful ADJ cana-6296 57 19 treatment treatment NOUN cana-6296 57 20 of of ADP cana-6296 57 21 stability stability NOUN cana-6296 57 22 , , PUNCT cana-6296 57 23 truncation truncation NOUN cana-6296 57 24 and and CCONJ cana-6296 57 25 disecretization disecretization NOUN cana-6296 57 26 errors error NOUN cana-6296 57 27 . . PUNCT cana-6296 58 1 a a DET cana-6296 58 2 ) ) PUNCT cana-6296 58 3 choice choice NOUN cana-6296 58 4 of of ADP cana-6296 58 5 shift shift NOUN cana-6296 58 6 parameter parameter NOUN cana-6296 58 7 𝝈 𝝈 NOUN cana-6296 58 8 i.for i.for ADP cana-6296 58 9 laplace laplace NOUN cana-6296 58 10 inversion inversion NOUN cana-6296 58 11 , , PUNCT cana-6296 58 12 the the DET cana-6296 58 13 factor factor NOUN cana-6296 58 14 π‘’πœŽπ‘£ π‘’πœŽπ‘£ NOUN cana-6296 58 15 grows grow VERB cana-6296 58 16 exponentially exponentially ADV cana-6296 58 17 with with ADP cana-6296 58 18 𝑣. 𝑣. PROPN cana-6296 58 19 ii.a ii.a PROPN cana-6296 58 20 small small PROPN cana-6296 58 21 𝜎 𝜎 PROPN cana-6296 58 22 reduces reduce VERB cana-6296 58 23 amplification amplification NOUN cana-6296 58 24 but but CCONJ cana-6296 58 25 risks risk NOUN cana-6296 58 26 instability instability NOUN cana-6296 58 27 if if SCONJ cana-6296 58 28 poles pole NOUN cana-6296 58 29 of of ADP cana-6296 58 30 𝐹(𝛾)lie 𝐹(𝛾)lie NOUN cana-6296 58 31 too too ADV cana-6296 58 32 close close ADJ cana-6296 58 33 to to ADP cana-6296 58 34 the the DET cana-6296 58 35 contour contour NOUN cana-6296 58 36 . . PUNCT cana-6296 59 1 iii.a iii.a NOUN cana-6296 59 2 large large ADJ cana-6296 59 3 𝜎 𝜎 PROPN cana-6296 59 4 improves improve VERB cana-6296 59 5 convergence convergence NOUN cana-6296 59 6 of of ADP cana-6296 59 7 the the DET cana-6296 59 8 integrand integrand NOUN cana-6296 59 9 but but CCONJ cana-6296 59 10 increases increase VERB cana-6296 59 11 sensitivity sensitivity NOUN cana-6296 59 12 to to PART cana-6296 59 13 round round VERB cana-6296 59 14 - - PUNCT cana-6296 59 15 off off ADP cana-6296 59 16 errors error NOUN cana-6296 59 17 . . PUNCT cana-6296 60 1 b b X cana-6296 60 2 ) ) PUNCT cana-6296 60 3 truncation truncation NOUN cana-6296 60 4 of of ADP cana-6296 60 5 the the DET cana-6296 60 6 fourier fourier NOUN cana-6296 60 7 integral integral ADJ cana-6296 60 8 the the DET cana-6296 60 9 inversion inversion NOUN cana-6296 60 10 formula formula NOUN cana-6296 60 11 requires require VERB cana-6296 60 12 integration integration NOUN cana-6296 60 13 over over ADP cana-6296 60 14 βˆ’βˆž βˆ’βˆž ADP cana-6296 60 15 < < X cana-6296 60 16 πœ‚ πœ‚ X cana-6296 60 17 < < X cana-6296 60 18 ∞. ∞. PROPN cana-6296 60 19 in in ADP cana-6296 60 20 practice practice NOUN cana-6296 60 21 , , PUNCT cana-6296 60 22 we we PRON cana-6296 60 23 truncate truncate VERB cana-6296 60 24 to to PART cana-6296 60 25 |πœ‚| |πœ‚| VERB cana-6296 60 26 ≀ ≀ NOUN cana-6296 60 27 𝐹(𝜎 𝐹(𝜎 PUNCT cana-6296 61 1 + + X cana-6296 61 2 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 61 3 ) ) PUNCT cana-6296 61 4 . . PUNCT cana-6296 62 1 i.truncation i.truncation NOUN cana-6296 62 2 error error NOUN cana-6296 62 3 decreases decrease VERB cana-6296 62 4 as as ADP cana-6296 62 5 πœ‚π‘šπ‘Žπ‘₯ πœ‚π‘šπ‘Žπ‘₯ NOUN cana-6296 62 6 increase increase NOUN cana-6296 62 7 . . PUNCT cana-6296 63 1 ii.foe ii.foe ADJ cana-6296 63 2 smooth smooth NOUN cana-6296 63 3 𝐹(𝜎 𝐹(𝜎 PUNCT cana-6296 63 4 + + X cana-6296 63 5 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 63 6 ) ) PUNCT cana-6296 63 7 , , PUNCT cana-6296 63 8 truncation truncation NOUN cana-6296 63 9 error error NOUN cana-6296 63 10 decays decay VERB cana-6296 63 11 rapidly rapidly ADV cana-6296 63 12 . . PUNCT cana-6296 64 1 iii.for iii.for ADP cana-6296 64 2 oscillatory oscillatory ADJ cana-6296 64 3 or or CCONJ cana-6296 64 4 slowly slowly ADV cana-6296 64 5 decaying decay VERB cana-6296 64 6 transform transform NOUN cana-6296 64 7 , , PUNCT cana-6296 64 8 convergence convergence NOUN cana-6296 64 9 may may AUX cana-6296 64 10 be be AUX cana-6296 64 11 slow slow ADJ cana-6296 64 12 , , PUNCT cana-6296 64 13 requiring require VERB cana-6296 64 14 larger large ADJ cana-6296 64 15 domains domain NOUN cana-6296 64 16 . . PUNCT cana-6296 65 1 c c X cana-6296 65 2 ) ) PUNCT cana-6296 65 3 discretization discretization NOUN cana-6296 65 4 and and CCONJ cana-6296 65 5 aliasing aliase VERB cana-6296 65 6 discretization discretization NOUN cana-6296 65 7 of of ADP cana-6296 65 8 πœ‚ πœ‚ NOUN cana-6296 65 9 with with ADP cana-6296 65 10 step step NOUN cana-6296 65 11 size size NOUN cana-6296 65 12 Ξ΄πœ‚ Ξ΄πœ‚ NOUN cana-6296 65 13 introduces introduce NOUN cana-6296 65 14 aliasing aliase VERB cana-6296 65 15 errors error NOUN cana-6296 65 16 . . PUNCT cana-6296 66 1 since since SCONJ cana-6296 66 2 the the DET cana-6296 66 3 fft fft PROPN cana-6296 66 4 approximates approximate VERB cana-6296 66 5 a a DET cana-6296 66 6 periodic periodic ADJ cana-6296 66 7 fourier fourier NOUN cana-6296 66 8 series series NOUN cana-6296 66 9 , , PUNCT cana-6296 66 10 the the DET cana-6296 66 11 computed compute VERB cana-6296 66 12 𝑓(𝑣 𝑓(𝑣 NOUN cana-6296 66 13 ) ) PUNCT cana-6296 66 14 may may AUX cana-6296 66 15 be be AUX cana-6296 66 16 slow slow ADJ cana-6296 66 17 - - PUNCT cana-6296 66 18 around around ADP cana-6296 66 19 artifacts artifact NOUN cana-6296 66 20 . . PUNCT cana-6296 67 1 i.remedy i.remedy NOUN cana-6296 67 2 : : PUNCT cana-6296 67 3 zero zero NUM cana-6296 67 4 - - PUNCT cana-6296 67 5 padding padding NOUN cana-6296 67 6 and and CCONJ cana-6296 67 7 oversampling oversample VERB cana-6296 67 8 in in ADP cana-6296 67 9 frequency frequency NOUN cana-6296 67 10 spaces space NOUN cana-6296 67 11 . . PUNCT cana-6296 68 1 ii.nyquist ii.nyquist NOUN cana-6296 68 2 condition condition NOUN cana-6296 68 3 : : PUNCT cana-6296 68 4 Ξ΄πœ‚ Ξ΄πœ‚ X cana-6296 68 5 = = SYM cana-6296 68 6 πœ‹ πœ‹ ADP cana-6296 68 7 π‘£π‘šπ‘Žπ‘₯ π‘£π‘šπ‘Žπ‘₯ NOUN cana-6296 68 8 to to PART cana-6296 68 9 resolve resolve VERB cana-6296 68 10 oscillations oscillation NOUN cana-6296 68 11 correctly correctly ADV cana-6296 68 12 . . PUNCT cana-6296 69 1 d d X cana-6296 69 2 ) ) PUNCT cana-6296 69 3 round round ADJ cana-6296 69 4 - - PUNCT cana-6296 69 5 off off ADP cana-6296 69 6 errors error NOUN cana-6296 69 7 and and CCONJ cana-6296 69 8 regularization regularization NOUN cana-6296 69 9 communications communication NOUN cana-6296 69 10 on on ADP cana-6296 69 11 applied apply VERB cana-6296 69 12 nonlinear nonlinear ADJ cana-6296 69 13 analysis analysis NOUN cana-6296 69 14 issn issn NOUN cana-6296 69 15 : : PUNCT cana-6296 69 16 1074 1074 NUM cana-6296 69 17 - - PUNCT cana-6296 69 18 133x 133x NUM cana-6296 69 19 vol vol VERB cana-6296 69 20 32 32 NUM cana-6296 69 21 no no NOUN cana-6296 69 22 . . PUNCT cana-6296 70 1 10s 10 NOUN cana-6296 70 2 ( ( PUNCT cana-6296 70 3 2025 2025 NUM cana-6296 70 4 ) ) PUNCT cana-6296 70 5 3698 3698 NUM cana-6296 70 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 71 1 i.the i.the DET cana-6296 71 2 exponential exponential ADJ cana-6296 71 3 factor factor NOUN cana-6296 71 4 π‘’πœŽπ‘£ π‘’πœŽπ‘£ NOUN cana-6296 71 5 can can AUX cana-6296 71 6 mgnify mgnify VERB cana-6296 71 7 floating float VERB cana-6296 71 8 - - PUNCT cana-6296 71 9 point point NOUN cana-6296 71 10 errors error NOUN cana-6296 71 11 for for ADP cana-6296 71 12 large large ADJ cana-6296 71 13 𝑣. 𝑣. NOUN cana-6296 71 14 ii.scaling ii.scaling NOUN cana-6296 71 15 techniques technique NOUN cana-6296 71 16 mitigate mitigate VERB cana-6296 71 17 overflow overflow NOUN cana-6296 71 18 / / SYM cana-6296 71 19 underflow underflow NOUN cana-6296 71 20 issues issue NOUN cana-6296 71 21 . . PUNCT cana-6296 72 1 e e X cana-6296 72 2 ) ) PUNCT cana-6296 72 3 stability stability NOUN cana-6296 72 4 summary summary NOUN cana-6296 72 5 i.laplace i.laplace NOUN cana-6296 72 6 inversion inversion NOUN cana-6296 72 7 : : PUNCT cana-6296 72 8 main main ADJ cana-6296 72 9 risk risk NOUN cana-6296 72 10 = = SYM cana-6296 72 11 exponential exponential ADJ cana-6296 72 12 growth growth NOUN cana-6296 72 13 from from ADP cana-6296 72 14 π‘’πœŽπ‘£. π‘’πœŽπ‘£. ADJ cana-6296 72 15 stability stability NOUN cana-6296 72 16 requires require VERB cana-6296 72 17 careful careful ADJ cana-6296 72 18 balancing balancing NOUN cana-6296 72 19 of of ADP cana-6296 72 20 𝜎. 𝜎. NOUN cana-6296 72 21 ii.fourier ii.fouri ADJ cana-6296 72 22 inversion inversion NOUN cana-6296 72 23 : : PUNCT cana-6296 72 24 main main ADJ cana-6296 72 25 risk risk NOUN cana-6296 72 26 = = SYM cana-6296 72 27 aliasing aliasing NOUN cana-6296 72 28 and and CCONJ cana-6296 72 29 truncation truncation NOUN cana-6296 72 30 errors error NOUN cana-6296 72 31 . . PUNCT cana-6296 73 1 stability stability NOUN cana-6296 73 2 requires require VERB cana-6296 73 3 sufficient sufficient ADJ cana-6296 73 4 sampling sampling NOUN cana-6296 73 5 . . PUNCT cana-6296 74 1 4 4 X cana-6296 74 2 . . X cana-6296 74 3 examples example NOUN cana-6296 74 4 for for ADP cana-6296 74 5 inverse inverse NOUN cana-6296 74 6 of of ADP cana-6296 74 7 laplace laplace NOUN cana-6296 74 8 and and CCONJ cana-6296 74 9 fourier fourier NOUN cana-6296 74 10 transform transform VERB cana-6296 74 11 5.1 5.1 NUM cana-6296 74 12 . . PUNCT cana-6296 75 1 laplace laplace NOUN cana-6296 75 2 transform transform VERB cana-6296 75 3 inversion inversion NOUN cana-6296 75 4 test test NOUN cana-6296 75 5 case case NOUN cana-6296 75 6 : : PUNCT cana-6296 75 7 consider consider VERB cana-6296 75 8 the the DET cana-6296 75 9 laplace laplace NOUN cana-6296 75 10 transform transform VERB cana-6296 75 11 𝐹(𝑠 𝐹(𝑠 NUM cana-6296 75 12 ) ) PUNCT cana-6296 75 13 = = SYM cana-6296 75 14 1 1 NUM cana-6296 75 15 𝑠2 𝑠2 NOUN cana-6296 75 16 + + CCONJ cana-6296 75 17 1 1 NUM cana-6296 75 18 the the DET cana-6296 75 19 analytical analytical ADJ cana-6296 75 20 inverse inverse NOUN cana-6296 75 21 is be AUX cana-6296 75 22 known know VERB cana-6296 75 23 : : PUNCT cana-6296 75 24 𝑓(𝑑 𝑓(𝑑 PROPN cana-6296 75 25 ) ) PUNCT cana-6296 75 26 = = SYM cana-6296 75 27 sin(𝑑 sin(𝑑 PROPN cana-6296 75 28 ) ) PUNCT cana-6296 75 29 , , PUNCT cana-6296 75 30 𝑑 𝑑 PROPN cana-6296 75 31 β‰₯ β‰₯ NOUN cana-6296 75 32 0 0 NUM cana-6296 75 33 . . PUNCT cana-6296 76 1 numerical numerical ADJ cana-6296 76 2 inversion inversion NOUN cana-6296 76 3 using use VERB cana-6296 76 4 the the DET cana-6296 76 5 unified unify VERB cana-6296 76 6 fft fft NOUN cana-6296 76 7 framework framework NOUN cana-6296 76 8 : : PUNCT cana-6296 76 9 β€’ β€’ NUM cana-6296 76 10 shift shift NOUN cana-6296 76 11 parameter parameter NOUN cana-6296 76 12 : : PUNCT cana-6296 76 13 choose choose VERB cana-6296 76 14 𝜎 𝜎 PROPN cana-6296 76 15 > > X cana-6296 76 16 0 0 PROPN cana-6296 76 17 , , PUNCT cana-6296 76 18 𝑒. 𝑒. PROPN cana-6296 76 19 𝑔. 𝑔. PROPN cana-6296 76 20 , , PUNCT cana-6296 76 21 𝜎 𝜎 PROPN cana-6296 76 22 = = SYM cana-6296 76 23 0.5 0.5 NUM cana-6296 76 24 . . NOUN cana-6296 76 25 β€’ β€’ NUM cana-6296 76 26 frequency frequency NOUN cana-6296 76 27 sampling sampling NOUN cana-6296 76 28 : : PUNCT cana-6296 76 29 sample sample NOUN cana-6296 76 30 𝐹(𝜎 𝐹(𝜎 X cana-6296 76 31 + + NUM cana-6296 76 32 π‘–πœ‚π‘˜ π‘–πœ‚π‘˜ NOUN cana-6296 76 33 ) ) PUNCT cana-6296 76 34 at at ADP cana-6296 76 35 equispaced equispace VERB cana-6296 76 36 frequencies frequency NOUN cana-6296 76 37 πœ‚π‘˜ πœ‚π‘˜ ADP cana-6296 76 38 = = ADJ cana-6296 76 39 π‘˜Ξ΄Ξ· π‘˜Ξ΄Ξ· NOUN cana-6296 76 40 , , PUNCT cana-6296 76 41 k k X cana-6296 76 42 = = PUNCT cana-6296 76 43 βˆ’n βˆ’n PROPN cana-6296 76 44 , , PUNCT cana-6296 76 45 … … PUNCT cana-6296 76 46 … … PUNCT cana-6296 76 47 , , PUNCT cana-6296 77 1 n. n. NOUN cana-6296 77 2 β€’ β€’ NOUN cana-6296 77 3 fft fft PROPN cana-6296 77 4 computation computation NOUN cana-6296 77 5 : : PUNCT cana-6296 77 6 recognize recognize VERB cana-6296 77 7 the the DET cana-6296 77 8 inversion inversion NOUN cana-6296 77 9 as as ADP cana-6296 77 10 a a DET cana-6296 77 11 discrete discrete ADJ cana-6296 77 12 fourier fourier NOUN cana-6296 77 13 sum sum NOUN cana-6296 77 14 𝑓(𝑑𝑗 𝑓(𝑑𝑗 PROPN cana-6296 77 15 ) ) PUNCT cana-6296 77 16 = = PUNCT cana-6296 77 17 βˆ‘ βˆ‘ PUNCT cana-6296 77 18 𝐹(𝜎 𝐹(𝜎 X cana-6296 77 19 + + CCONJ cana-6296 77 20 π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— PROPN cana-6296 77 21 . . PUNCT cana-6296 78 1 𝑁 𝑁 PROPN cana-6296 78 2 π‘˜=βˆ’π‘ π‘˜=βˆ’π‘ PROPN cana-6296 78 3 β€’ β€’ NOUN cana-6296 78 4 scaling scaling NOUN cana-6296 78 5 : : PUNCT cana-6296 78 6 multiply multiply ADV cana-6296 78 7 by by ADP cana-6296 78 8 the the DET cana-6296 78 9 prefactor prefactor NOUN cana-6296 78 10 to to PART cana-6296 78 11 recover recover VERB cana-6296 78 12 the the DET cana-6296 78 13 approximation approximation NOUN cana-6296 78 14 𝑓(𝑑𝑗 𝑓(𝑑𝑗 PROPN cana-6296 78 15 ) ) PUNCT cana-6296 79 1 β‰ˆ β‰ˆ PROPN cana-6296 79 2 δη δη PROPN cana-6296 79 3 2πœ‹ 2πœ‹ PROPN cana-6296 79 4 π‘’πœŽπ‘‘π‘—π‘“(𝑑𝑗 π‘’πœŽπ‘‘π‘—π‘“(𝑑𝑗 PROPN cana-6296 79 5 ) ) PUNCT cana-6296 79 6 . . PUNCT cana-6296 80 1 results result NOUN cana-6296 80 2 : : PUNCT cana-6296 80 3 t t PROPN cana-6296 80 4 analytical analytical ADJ cana-6296 80 5 sin(t sin(t PROPN cana-6296 80 6 ) ) PUNCT cana-6296 80 7 numerical numerical PROPN cana-6296 80 8 fft fft PROPN cana-6296 80 9 approximation approximation NOUN cana-6296 80 10 error error NOUN cana-6296 80 11 0 0 NUM cana-6296 80 12 0.0000 0.0000 NUM cana-6296 80 13 0.0000 0.0000 NUM cana-6296 80 14 0 0 NUM cana-6296 80 15 1 1 NUM cana-6296 80 16 0.8415 0.8415 NUM cana-6296 81 1 0.8421 0.8421 NUM cana-6296 81 2 6 6 NUM cana-6296 81 3 Γ— Γ— NOUN cana-6296 81 4 10 10 NUM cana-6296 81 5 - - SYM cana-6296 81 6 4 4 NUM cana-6296 81 7 2 2 NUM cana-6296 81 8 0.9093 0.9093 NUM cana-6296 81 9 0.9087 0.9087 NUM cana-6296 81 10 6 6 NUM cana-6296 81 11 Γ— Γ— NOUN cana-6296 81 12 10 10 NUM cana-6296 81 13 - - SYM cana-6296 81 14 4 4 NUM cana-6296 81 15 3 3 NUM cana-6296 81 16 0.1411 0.1411 NUM cana-6296 81 17 0.1409 0.1409 NUM cana-6296 81 18 2 2 NUM cana-6296 81 19 Γ— Γ— NOUN cana-6296 81 20 10 10 NUM cana-6296 81 21 - - SYM cana-6296 81 22 4 4 NUM cana-6296 81 23 observation observation NOUN cana-6296 81 24 : : PUNCT cana-6296 81 25 the the DET cana-6296 81 26 fft fft PROPN cana-6296 81 27 - - PUNCT cana-6296 81 28 based base VERB cana-6296 81 29 universal universal ADJ cana-6296 81 30 inversion inversion NOUN cana-6296 81 31 method method NOUN cana-6296 81 32 yields yield NOUN cana-6296 81 33 results result NOUN cana-6296 81 34 that that PRON cana-6296 81 35 are be AUX cana-6296 81 36 in in ADP cana-6296 81 37 excellent excellent ADJ cana-6296 81 38 agreement agreement NOUN cana-6296 81 39 with with ADP cana-6296 81 40 the the DET cana-6296 81 41 analytical analytical ADJ cana-6296 81 42 solution solution NOUN cana-6296 81 43 . . PUNCT cana-6296 82 1 the the DET cana-6296 82 2 errors error NOUN cana-6296 82 3 are be AUX cana-6296 82 4 very very ADV cana-6296 82 5 small small ADJ cana-6296 82 6 , , PUNCT cana-6296 82 7 confirming confirm VERB cana-6296 82 8 both both CCONJ cana-6296 82 9 the the DET cana-6296 82 10 accuracy accuracy NOUN cana-6296 82 11 and and CCONJ cana-6296 82 12 computational computational ADJ cana-6296 82 13 efficiency efficiency NOUN cana-6296 82 14 of of ADP cana-6296 82 15 the the DET cana-6296 82 16 framework framework NOUN cana-6296 82 17 for for ADP cana-6296 82 18 laplace laplace NOUN cana-6296 82 19 inversion inversion NOUN cana-6296 82 20 problems problem NOUN cana-6296 82 21 . . PUNCT cana-6296 83 1 5.2 5.2 NUM cana-6296 83 2 . . PUNCT cana-6296 84 1 a a DET cana-6296 84 2 traditional traditional ADJ cana-6296 84 3 method method NOUN cana-6296 84 4 example example NOUN cana-6296 84 5 to to PART cana-6296 84 6 provide provide VERB cana-6296 84 7 a a DET cana-6296 84 8 comparison comparison NOUN cana-6296 84 9 with with ADP cana-6296 84 10 the the DET cana-6296 84 11 proposed propose VERB cana-6296 84 12 fft fft PROPN cana-6296 84 13 - - PUNCT cana-6296 84 14 based base VERB cana-6296 84 15 universal universal ADJ cana-6296 84 16 framework framework NOUN cana-6296 84 17 , , PUNCT cana-6296 84 18 we we PRON cana-6296 84 19 apply apply VERB cana-6296 84 20 talbot talbot PROPN cana-6296 84 21 ’s ’s PART cana-6296 84 22 method method NOUN cana-6296 84 23 for for ADP cana-6296 84 24 the the DET cana-6296 84 25 same same ADJ cana-6296 84 26 laplace laplace NOUN cana-6296 84 27 transform transform NOUN cana-6296 84 28 : : PUNCT cana-6296 84 29 𝐹(𝑠 𝐹(𝑠 NUM cana-6296 84 30 ) ) PUNCT cana-6296 84 31 = = SYM cana-6296 84 32 1 1 NUM cana-6296 84 33 𝑠2 𝑠2 NOUN cana-6296 84 34 + + SYM cana-6296 84 35 1 1 NUM cana-6296 84 36 . . PUNCT cana-6296 85 1 communications communication NOUN cana-6296 85 2 on on ADP cana-6296 85 3 applied apply VERB cana-6296 85 4 nonlinear nonlinear ADJ cana-6296 85 5 analysis analysis NOUN cana-6296 85 6 issn issn NOUN cana-6296 85 7 : : PUNCT cana-6296 85 8 1074 1074 NUM cana-6296 85 9 - - PUNCT cana-6296 85 10 133x 133x NUM cana-6296 85 11 vol vol VERB cana-6296 85 12 32 32 NUM cana-6296 85 13 no no NOUN cana-6296 85 14 . . PUNCT cana-6296 86 1 10s 10 NOUN cana-6296 86 2 ( ( PUNCT cana-6296 86 3 2025 2025 NUM cana-6296 86 4 ) ) PUNCT cana-6296 86 5 3699 3699 NUM cana-6296 86 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 86 7 talbot talbot PROPN cana-6296 86 8 ’s ’s PART cana-6296 86 9 method method PROPN cana-6296 86 10 overview overview NOUN cana-6296 86 11 : : PUNCT cana-6296 86 12 talbot talbot PROPN cana-6296 86 13 ’s ’s PART cana-6296 86 14 method method NOUN cana-6296 86 15 deforms deform VERB cana-6296 86 16 the the DET cana-6296 86 17 bromwich bromwich NOUN cana-6296 86 18 integral integral ADJ cana-6296 86 19 contour contour NOUN cana-6296 86 20 into into ADP cana-6296 86 21 a a DET cana-6296 86 22 special special ADJ cana-6296 86 23 curve curve NOUN cana-6296 86 24 in in ADP cana-6296 86 25 the the DET cana-6296 86 26 complex complex ADJ cana-6296 86 27 plane plane NOUN cana-6296 86 28 that that PRON cana-6296 86 29 avoids avoid VERB cana-6296 86 30 oscillations oscillation NOUN cana-6296 86 31 and and CCONJ cana-6296 86 32 accelerates accelerate VERB cana-6296 86 33 convergence convergence NOUN cana-6296 86 34 . . PUNCT cana-6296 87 1 the the DET cana-6296 87 2 inversion inversion NOUN cana-6296 87 3 formula formula NOUN cana-6296 87 4 can can AUX cana-6296 87 5 be be AUX cana-6296 87 6 approximated approximate VERB cana-6296 87 7 as as ADP cana-6296 87 8 𝑓(𝑑 𝑓(𝑑 NOUN cana-6296 87 9 ) ) PUNCT cana-6296 88 1 β‰ˆ β‰ˆ PROPN cana-6296 88 2 1 1 NUM cana-6296 88 3 𝑀 𝑀 PROPN cana-6296 88 4 βˆ‘ βˆ‘ PUNCT cana-6296 88 5 β„œ β„œ PROPN cana-6296 89 1 [ [ PUNCT cana-6296 89 2 π‘€π‘šπΉ π‘€π‘šπΉ ADV cana-6296 89 3 ( ( PUNCT cana-6296 89 4 πœƒπ‘š πœƒπ‘š ADV cana-6296 89 5 𝑑 𝑑 PROPN cana-6296 89 6 ) ) PUNCT cana-6296 89 7 ] ] PUNCT cana-6296 89 8 𝑀 𝑀 PROPN cana-6296 89 9 π‘š=1 π‘š=1 PUNCT cana-6296 89 10 where where SCONJ cana-6296 89 11 πœƒπ‘š πœƒπ‘š ADV cana-6296 89 12 are be AUX cana-6296 89 13 contour contour NOUN cana-6296 89 14 nodes node NOUN cana-6296 89 15 and and CCONJ cana-6296 89 16 π‘€π‘šare π‘€π‘šare ADJ cana-6296 89 17 complex complex ADJ cana-6296 89 18 weights weight NOUN cana-6296 89 19 designed design VERB cana-6296 89 20 to to PART cana-6296 89 21 capture capture VERB cana-6296 89 22 the the DET cana-6296 89 23 contribution contribution NOUN cana-6296 89 24 of of ADP cana-6296 89 25 the the DET cana-6296 89 26 laplace laplace NOUN cana-6296 89 27 integrand integrand PROPN cana-6296 89 28 along along ADP cana-6296 89 29 talbot talbot PROPN cana-6296 89 30 ’s ’s PART cana-6296 89 31 contour contour NOUN cana-6296 89 32 . . PUNCT cana-6296 90 1 this this DET cana-6296 90 2 method method NOUN cana-6296 90 3 is be AUX cana-6296 90 4 highly highly ADV cana-6296 90 5 accurate accurate ADJ cana-6296 90 6 for for ADP cana-6296 90 7 many many ADJ cana-6296 90 8 smooth smooth ADJ cana-6296 90 9 transform transform NOUN cana-6296 90 10 but but CCONJ cana-6296 90 11 requires require VERB cana-6296 90 12 careful careful ADJ cana-6296 90 13 contour contour NOUN cana-6296 90 14 construction construction NOUN cana-6296 90 15 and and CCONJ cana-6296 90 16 parameter parameter NOUN cana-6296 90 17 tuning tuning NOUN cana-6296 90 18 . . PUNCT cana-6296 91 1 results result NOUN cana-6296 91 2 : : PUNCT cana-6296 91 3 t t PROPN cana-6296 91 4 analytical analytical ADJ cana-6296 91 5 sin(t sin(t PROPN cana-6296 91 6 ) ) PUNCT cana-6296 91 7 numerical numerical PROPN cana-6296 91 8 fft fft PROPN cana-6296 91 9 approximation approximation NOUN cana-6296 91 10 error error NOUN cana-6296 91 11 0 0 NUM cana-6296 91 12 0.0000 0.0000 NUM cana-6296 91 13 0.0000 0.0000 NUM cana-6296 91 14 0 0 NUM cana-6296 91 15 1 1 NUM cana-6296 91 16 0.8415 0.8415 NUM cana-6296 91 17 0.8416 0.8416 NUM cana-6296 91 18 1 1 NUM cana-6296 91 19 Γ— Γ— NOUN cana-6296 91 20 10 10 NUM cana-6296 91 21 - - SYM cana-6296 91 22 4 4 NUM cana-6296 91 23 2 2 NUM cana-6296 91 24 0.9093 0.9093 NUM cana-6296 91 25 0.9094 0.9094 NUM cana-6296 91 26 1 1 NUM cana-6296 91 27 Γ— Γ— NOUN cana-6296 91 28 10 10 NUM cana-6296 91 29 - - SYM cana-6296 91 30 4 4 NUM cana-6296 91 31 3 3 NUM cana-6296 91 32 0.1411 0.1411 NUM cana-6296 91 33 0.1412 0.1412 NUM cana-6296 91 34 1 1 NUM cana-6296 91 35 Γ— Γ— NOUN cana-6296 91 36 10 10 NUM cana-6296 91 37 - - SYM cana-6296 91 38 4 4 NUM cana-6296 91 39 observation observation NOUN cana-6296 91 40 : : PUNCT cana-6296 91 41 while while SCONJ cana-6296 91 42 traditional traditional ADJ cana-6296 91 43 numerical numerical ADJ cana-6296 91 44 methods method NOUN cana-6296 91 45 ( ( PUNCT cana-6296 91 46 e.g. e.g. ADV cana-6296 91 47 stehfest stehf ADJ cana-6296 91 48 or or CCONJ cana-6296 91 49 talbot talbot PROPN cana-6296 91 50 ) ) PUNCT cana-6296 91 51 can can AUX cana-6296 91 52 reproduce reproduce VERB cana-6296 91 53 the the DET cana-6296 91 54 correct correct ADJ cana-6296 91 55 inversion inversion NOUN cana-6296 91 56 , , PUNCT cana-6296 91 57 they they PRON cana-6296 91 58 often often ADV cana-6296 91 59 require require VERB cana-6296 91 60 careful careful ADJ cana-6296 91 61 parameter parameter NOUN cana-6296 91 62 tuning tuning NOUN cana-6296 91 63 and and CCONJ cana-6296 91 64 may may AUX cana-6296 91 65 becomes become VERB cana-6296 91 66 unstable unstable ADJ cana-6296 91 67 for for ADP cana-6296 91 68 larger large ADJ cana-6296 91 69 time time NOUN cana-6296 91 70 values value NOUN cana-6296 91 71 . . PUNCT cana-6296 92 1 this this PRON cana-6296 92 2 highlights highlight VERB cana-6296 92 3 the the DET cana-6296 92 4 motivation motivation NOUN cana-6296 92 5 for for ADP cana-6296 92 6 a a DET cana-6296 92 7 more more ADV cana-6296 92 8 stable stable ADJ cana-6296 92 9 and and CCONJ cana-6296 92 10 universal universal ADJ cana-6296 92 11 approach approach NOUN cana-6296 92 12 . . PUNCT cana-6296 93 1 5.3 5.3 NUM cana-6296 93 2 . . PUNCT cana-6296 94 1 the the DET cana-6296 94 2 universal universal ADJ cana-6296 94 3 equation equation NOUN cana-6296 94 4 the the DET cana-6296 94 5 examples example NOUN cana-6296 94 6 above above ADP cana-6296 94 7 illustrate illustrate VERB cana-6296 94 8 how how SCONJ cana-6296 94 9 both both DET cana-6296 94 10 fft fft NOUN cana-6296 94 11 - - PUNCT cana-6296 94 12 based base VERB cana-6296 94 13 framework framework NOUN cana-6296 94 14 ( ( PUNCT cana-6296 94 15 section section NOUN cana-6296 94 16 5.1 5.1 NUM cana-6296 94 17 ) ) PUNCT cana-6296 94 18 and and CCONJ cana-6296 94 19 talbot talbot PROPN cana-6296 94 20 ’s ’s PART cana-6296 94 21 traditional traditional ADJ cana-6296 94 22 contour contour NOUN cana-6296 94 23 method method NOUN cana-6296 94 24 ( ( PUNCT cana-6296 94 25 section section NOUN cana-6296 94 26 5.2 5.2 NUM cana-6296 94 27 ) ) PUNCT cana-6296 94 28 can can AUX cana-6296 94 29 successfully successfully ADV cana-6296 94 30 invert invert VERB cana-6296 94 31 laplace laplace NOUN cana-6296 94 32 transform transform NOUN cana-6296 94 33 . . PUNCT cana-6296 95 1 however however ADV cana-6296 95 2 , , PUNCT cana-6296 95 3 the the DET cana-6296 95 4 true true ADJ cana-6296 95 5 strength strength NOUN cana-6296 95 6 of of ADP cana-6296 95 7 the the DET cana-6296 95 8 proposed propose VERB cana-6296 95 9 approach approach NOUN cana-6296 95 10 lies lie VERB cana-6296 95 11 in in ADP cana-6296 95 12 it it PRON cana-6296 95 13 ’s ’ VERB cana-6296 95 14 universality universality NOUN cana-6296 95 15 : : PUNCT cana-6296 95 16 the the DET cana-6296 95 17 same same ADJ cana-6296 95 18 numerical numerical ADJ cana-6296 95 19 formula formula NOUN cana-6296 95 20 governs govern VERB cana-6296 95 21 the the DET cana-6296 95 22 inversion inversion NOUN cana-6296 95 23 both both DET cana-6296 95 24 laplace laplace NOUN cana-6296 95 25 and and CCONJ cana-6296 95 26 fourier fourier NOUN cana-6296 95 27 transforms transform VERB cana-6296 95 28 . . PUNCT cana-6296 96 1 unified unified ADJ cana-6296 96 2 inversion inversion NOUN cana-6296 96 3 formula formula NOUN cana-6296 96 4 : : PUNCT cana-6296 96 5 from from ADP cana-6296 96 6 the the DET cana-6296 96 7 theorem theorem NOUN cana-6296 96 8 in in ADP cana-6296 96 9 section section NOUN cana-6296 96 10 4.2 4.2 NUM cana-6296 96 11 , , PUNCT cana-6296 96 12 the the DET cana-6296 96 13 laplace laplace NOUN cana-6296 96 14 inverse inverse NOUN cana-6296 96 15 can can AUX cana-6296 96 16 be be AUX cana-6296 96 17 recast recast VERB cana-6296 96 18 as as ADP cana-6296 96 19 a a DET cana-6296 96 20 fourier fourier NOUN cana-6296 96 21 - - PUNCT cana-6296 96 22 type type NOUN cana-6296 96 23 integral integral ADJ cana-6296 96 24 : : PUNCT cana-6296 96 25 𝑓(𝑑 𝑓(𝑑 PROPN cana-6296 96 26 ) ) PUNCT cana-6296 97 1 = = SYM cana-6296 97 2 π‘’πœŽπ‘£ π‘’πœŽπ‘£ PRON cana-6296 97 3 2πœ‹ 2πœ‹ NUM cana-6296 97 4 ∫ ∫ NOUN cana-6296 97 5 𝐹(𝜎 𝐹(𝜎 X cana-6296 97 6 + + X cana-6296 97 7 π‘–πœ‚ π‘–πœ‚ NOUN cana-6296 97 8 ) ) PUNCT cana-6296 97 9 ∞ ∞ PROPN cana-6296 97 10 βˆ’βˆž βˆ’βˆž ADP cana-6296 97 11 π‘’π‘–πœ‚π‘‘π‘‘πœ‚ π‘’π‘–πœ‚π‘‘π‘‘πœ‚ PROPN cana-6296 97 12 , , PUNCT cana-6296 97 13 𝜎 𝜎 PROPN cana-6296 97 14 > > X cana-6296 97 15 0 0 X cana-6296 97 16 . . PUNCT cana-6296 98 1 by by ADP cana-6296 98 2 discretizing discretize VERB cana-6296 98 3 the the DET cana-6296 98 4 integral integral ADJ cana-6296 98 5 , , PUNCT cana-6296 98 6 we we PRON cana-6296 98 7 obtain obtain VERB cana-6296 98 8 the the DET cana-6296 98 9 numerical numerical ADJ cana-6296 98 10 approximation approximation NOUN cana-6296 98 11 𝑓(𝑑𝑗 𝑓(𝑑𝑗 PROPN cana-6296 98 12 ) ) PUNCT cana-6296 99 1 β‰ˆ β‰ˆ PROPN cana-6296 99 2 δη δη PROPN cana-6296 99 3 2πœ‹ 2πœ‹ NOUN cana-6296 99 4 π‘’πœŽπ‘‘π‘— π‘’πœŽπ‘‘π‘— ADJ cana-6296 99 5 βˆ‘ βˆ‘ PROPN cana-6296 99 6 𝐹(𝜎 𝐹(𝜎 PUNCT cana-6296 99 7 + + CCONJ cana-6296 99 8 π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— π‘–πœ‚π‘˜)π‘’π‘–πœ‚π‘˜π‘‘π‘— PROPN cana-6296 99 9 . . PUNCT cana-6296 100 1 𝑁 𝑁 PROPN cana-6296 100 2 π‘˜=βˆ’π‘ π‘˜=βˆ’π‘ PROPN cana-6296 100 3 this this PRON cana-6296 100 4 is be AUX cana-6296 100 5 structurally structurally ADV cana-6296 100 6 identical identical ADJ cana-6296 100 7 to to ADP cana-6296 100 8 the the DET cana-6296 100 9 inverse inverse ADJ cana-6296 100 10 fourier fourier NOUN cana-6296 100 11 transform transform NOUN cana-6296 100 12 , , PUNCT cana-6296 100 13 differing differ VERB cana-6296 100 14 only only ADV cana-6296 100 15 by by ADP cana-6296 100 16 the the DET cana-6296 100 17 exponential exponential ADJ cana-6296 100 18 shift shift NOUN cana-6296 100 19 factor factor NOUN cana-6296 100 20 π‘’πœŽπ‘‘. π‘’πœŽπ‘‘. X cana-6296 101 1 interpretation interpretation NOUN cana-6296 101 2 : : PUNCT cana-6296 101 3 i.the i.the DET cana-6296 101 4 shift shift NOUN cana-6296 101 5 parameter parameter NOUN cana-6296 101 6 𝜎 𝜎 PROPN cana-6296 101 7 ensures ensure VERB cana-6296 101 8 convergence convergence NOUN cana-6296 101 9 of of ADP cana-6296 101 10 the the DET cana-6296 101 11 laplace laplace NOUN cana-6296 101 12 inversion inversion NOUN cana-6296 101 13 by by ADP cana-6296 101 14 moving move VERB cana-6296 101 15 the the DET cana-6296 101 16 contour contour NOUN cana-6296 101 17 to to ADP cana-6296 101 18 the the DET cana-6296 101 19 right right NOUN cana-6296 101 20 of of ADP cana-6296 101 21 all all DET cana-6296 101 22 singularities singularity NOUN cana-6296 101 23 . . PUNCT cana-6296 102 1 ii.when ii.when ADV cana-6296 102 2 𝜎 𝜎 X cana-6296 102 3 = = SYM cana-6296 102 4 0 0 PROPN cana-6296 102 5 , , PUNCT cana-6296 102 6 the the DET cana-6296 102 7 same same ADJ cana-6296 102 8 formula formula NOUN cana-6296 102 9 reduces reduce VERB cana-6296 102 10 directly directly ADV cana-6296 102 11 to to ADP cana-6296 102 12 the the DET cana-6296 102 13 standard standard ADJ cana-6296 102 14 inverse inverse ADJ cana-6296 102 15 fourier fourier NOUN cana-6296 102 16 transform transform NOUN cana-6296 102 17 . . PUNCT cana-6296 103 1 communications communication NOUN cana-6296 103 2 on on ADP cana-6296 103 3 applied apply VERB cana-6296 103 4 nonlinear nonlinear ADJ cana-6296 103 5 analysis analysis NOUN cana-6296 103 6 issn issn NOUN cana-6296 103 7 : : PUNCT cana-6296 103 8 1074 1074 NUM cana-6296 103 9 - - PUNCT cana-6296 103 10 133x 133x NUM cana-6296 103 11 vol vol VERB cana-6296 103 12 32 32 NUM cana-6296 103 13 no no NOUN cana-6296 103 14 . . PUNCT cana-6296 104 1 10s 10 NOUN cana-6296 104 2 ( ( PUNCT cana-6296 104 3 2025 2025 NUM cana-6296 104 4 ) ) PUNCT cana-6296 104 5 3700 3700 NUM cana-6296 104 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 104 7 iii.thus iii.thu NOUN cana-6296 104 8 , , PUNCT cana-6296 104 9 laplace laplace NOUN cana-6296 104 10 and and CCONJ cana-6296 104 11 fourier fourier ADJ cana-6296 104 12 inversion inversion NOUN cana-6296 104 13 are be AUX cana-6296 104 14 unified unify VERB cana-6296 104 15 under under ADP cana-6296 104 16 one one NUM cana-6296 104 17 discrete discrete ADJ cana-6296 104 18 fourier fourier NOUN cana-6296 104 19 - - PUNCT cana-6296 104 20 type type NOUN cana-6296 104 21 framework framework NOUN cana-6296 104 22 . . PUNCT cana-6296 105 1 advantages advantage NOUN cana-6296 105 2 of of ADP cana-6296 105 3 the the DET cana-6296 105 4 universal universal ADJ cana-6296 105 5 equation equation NOUN cana-6296 105 6 : : PUNCT cana-6296 105 7 i.single i.single NOUN cana-6296 105 8 algorithm algorithm NOUN cana-6296 105 9 : : PUNCT cana-6296 105 10 both both CCONJ cana-6296 105 11 laplace laplace NOUN cana-6296 105 12 and and CCONJ cana-6296 105 13 fourier fouri ADJ cana-6296 105 14 inversions inversion NOUN cana-6296 105 15 are be AUX cana-6296 105 16 computed compute VERB cana-6296 105 17 using use VERB cana-6296 105 18 the the DET cana-6296 105 19 same same ADJ cana-6296 105 20 fftbased fftbase VERB cana-6296 105 21 routine routine NOUN cana-6296 105 22 . . PUNCT cana-6296 106 1 ii.simplicity ii.simplicity NOUN cana-6296 106 2 : : PUNCT cana-6296 106 3 only only ADV cana-6296 106 4 equispaced equispace VERB cana-6296 106 5 frequency frequency NOUN cana-6296 106 6 sampling sampling NOUN cana-6296 106 7 is be AUX cana-6296 106 8 required require VERB cana-6296 106 9 ; ; PUNCT cana-6296 106 10 no no DET cana-6296 106 11 special special ADJ cana-6296 106 12 contours contours NOUN cana-6296 106 13 or or CCONJ cana-6296 106 14 weight weight NOUN cana-6296 106 15 functions function NOUN cana-6296 106 16 are be AUX cana-6296 106 17 needed need VERB cana-6296 106 18 . . PUNCT cana-6296 107 1 iii.stability iii.stability NOUN cana-6296 107 2 : : PUNCT cana-6296 107 3 the the DET cana-6296 107 4 exponential exponential ADJ cana-6296 107 5 shift shift NOUN cana-6296 107 6 and and CCONJ cana-6296 107 7 optional optional ADJ cana-6296 107 8 smoothing smoothing NOUN cana-6296 107 9 kernels kernel NOUN cana-6296 107 10 provide provide VERB cana-6296 107 11 numerical numerical ADJ cana-6296 107 12 robustness robustness NOUN cana-6296 107 13 . . PUNCT cana-6296 108 1 iv.versatility iv.versatility NOUN cana-6296 108 2 : : PUNCT cana-6296 108 3 the the DET cana-6296 108 4 method method NOUN cana-6296 108 5 generalizes generalize VERB cana-6296 108 6 across across ADP cana-6296 108 7 application application NOUN cana-6296 108 8 domains domain NOUN cana-6296 108 9 where where SCONJ cana-6296 108 10 either either CCONJ cana-6296 108 11 laplace laplace NOUN cana-6296 108 12 and and CCONJ cana-6296 108 13 fourier fourier NOUN cana-6296 108 14 transforms transform VERB cana-6296 108 15 appear appear VERB cana-6296 108 16 . . PUNCT cana-6296 109 1 observation observation NOUN cana-6296 109 2 : : PUNCT cana-6296 109 3 the the DET cana-6296 109 4 proposed propose VERB cana-6296 109 5 universal universal ADJ cana-6296 109 6 inversion inversion NOUN cana-6296 109 7 formula formula NOUN cana-6296 109 8 provides provide VERB cana-6296 109 9 a a DET cana-6296 109 10 single single ADJ cana-6296 109 11 computational computational ADJ cana-6296 109 12 framework framework NOUN cana-6296 109 13 for for ADP cana-6296 109 14 both both CCONJ cana-6296 109 15 laplace laplace NOUN cana-6296 109 16 and and CCONJ cana-6296 109 17 fourier fourier NOUN cana-6296 109 18 transforms transform VERB cana-6296 109 19 . . PUNCT cana-6296 110 1 its its PRON cana-6296 110 2 fft fft NOUN cana-6296 110 3 - - PUNCT cana-6296 110 4 based base VERB cana-6296 110 5 structure structure NOUN cana-6296 110 6 ensures ensure VERB cana-6296 110 7 stability stability NOUN cana-6296 110 8 and and CCONJ cana-6296 110 9 efficiency efficiency NOUN cana-6296 110 10 while while SCONJ cana-6296 110 11 avoiding avoid VERB cana-6296 110 12 the the DET cana-6296 110 13 complexities complexity NOUN cana-6296 110 14 of of ADP cana-6296 110 15 method method NOUN cana-6296 110 16 - - PUNCT cana-6296 110 17 specific specific ADJ cana-6296 110 18 parameter parameter NOUN cana-6296 110 19 selection selection NOUN cana-6296 110 20 . . PUNCT cana-6296 111 1 5.4 5.4 NUM cana-6296 111 2 . . PUNCT cana-6296 112 1 fourier fourier PROPN cana-6296 112 2 transform transform VERB cana-6296 112 3 inversion inversion NOUN cana-6296 112 4 test test NOUN cana-6296 112 5 case case NOUN cana-6296 112 6 : : PUNCT cana-6296 112 7 consider consider VERB cana-6296 112 8 the the DET cana-6296 112 9 fourier fourier ADJ cana-6296 112 10 transform transform NOUN cana-6296 112 11 of of ADP cana-6296 112 12 a a DET cana-6296 112 13 gaussian gaussian NOUN cana-6296 112 14 : : PUNCT cana-6296 112 15 𝐹(𝑀 𝐹(𝑀 NUM cana-6296 112 16 ) ) PUNCT cana-6296 112 17 = = PRON cana-6296 112 18 π‘’βˆ’π‘€2 π‘’βˆ’π‘€2 VERB cana-6296 112 19 4⁄ 4⁄ PROPN cana-6296 112 20 the the DET cana-6296 112 21 analytical analytical ADJ cana-6296 112 22 inverse inverse NOUN cana-6296 112 23 is be AUX cana-6296 112 24 also also ADV cana-6296 112 25 gaussian gaussian ADJ cana-6296 112 26 : : PUNCT cana-6296 112 27 𝑓(𝑑 𝑓(𝑑 PROPN cana-6296 112 28 ) ) PUNCT cana-6296 112 29 = = SYM cana-6296 112 30 1 1 NUM cana-6296 112 31 βˆšπœ‹ βˆšπœ‹ PUNCT cana-6296 112 32 π‘’βˆ’π‘‘2 π‘’βˆ’π‘‘2 ADV cana-6296 112 33 , , PUNCT cana-6296 112 34 βˆ’βˆž βˆ’βˆž X cana-6296 112 35 < < X cana-6296 112 36 𝑑 𝑑 X cana-6296 112 37 < < X cana-6296 112 38 ∞ ∞ PROPN cana-6296 112 39 numerical numerical ADJ cana-6296 112 40 inversion inversion NOUN cana-6296 112 41 using use VERB cana-6296 112 42 the the DET cana-6296 112 43 unified unify VERB cana-6296 112 44 fft fft NOUN cana-6296 112 45 framework framework NOUN cana-6296 112 46 : : PUNCT cana-6296 112 47 since since SCONJ cana-6296 112 48 the the DET cana-6296 112 49 fourier fourier NOUN cana-6296 112 50 transform transform NOUN cana-6296 112 51 inversion inversion NOUN cana-6296 112 52 is be AUX cana-6296 112 53 a a DET cana-6296 112 54 soecial soecial ADJ cana-6296 112 55 case case NOUN cana-6296 112 56 of of ADP cana-6296 112 57 the the DET cana-6296 112 58 universal universal ADJ cana-6296 112 59 equation equation NOUN cana-6296 112 60 ( ( PUNCT cana-6296 112 61 with with ADP cana-6296 112 62 𝜎 𝜎 PROPN cana-6296 112 63 = = SYM cana-6296 112 64 0 0 NUM cana-6296 112 65 ) ) PUNCT cana-6296 112 66 , , PUNCT cana-6296 112 67 we we PRON cana-6296 112 68 directly directly ADV cana-6296 112 69 apply apply VERB cana-6296 112 70 𝑓(𝑑𝑗 𝑓(𝑑𝑗 PROPN cana-6296 112 71 ) ) PUNCT cana-6296 113 1 β‰ˆ β‰ˆ PROPN cana-6296 113 2 Ξ΄w Ξ΄w ADP cana-6296 113 3 2πœ‹ 2πœ‹ PROPN cana-6296 113 4 βˆ‘ βˆ‘ PUNCT cana-6296 113 5 𝐹(π‘€π‘˜)π‘’π‘–π‘€π‘˜π‘‘π‘— 𝐹(π‘€π‘˜)π‘’π‘–π‘€π‘˜π‘‘π‘— X cana-6296 113 6 . . PUNCT cana-6296 114 1 𝑁 𝑁 PROPN cana-6296 114 2 π‘˜=βˆ’π‘ π‘˜=βˆ’π‘ PROPN cana-6296 114 3 where where SCONJ cana-6296 114 4 π‘€π‘˜ π‘€π‘˜ ADV cana-6296 114 5 = = SYM cana-6296 115 1 π‘˜βˆ†π‘€. π‘˜βˆ†π‘€. NOUN cana-6296 115 2 results result VERB cana-6296 115 3 : : PUNCT cana-6296 115 4 t t PROPN cana-6296 115 5 analytical analytical ADJ cana-6296 115 6 sin(t sin(t PROPN cana-6296 115 7 ) ) PUNCT cana-6296 115 8 numerical numerical PROPN cana-6296 115 9 fft fft PROPN cana-6296 115 10 approximation approximation NOUN cana-6296 115 11 error error NOUN cana-6296 115 12 0 0 NUM cana-6296 115 13 0.5642 0.5642 NUM cana-6296 115 14 0.5640 0.5640 NUM cana-6296 115 15 2 2 NUM cana-6296 115 16 Γ— Γ— NOUN cana-6296 115 17 10 10 NUM cana-6296 115 18 - - SYM cana-6296 115 19 4 4 NUM cana-6296 115 20 1 1 NUM cana-6296 115 21 0.2076 0.2076 NUM cana-6296 115 22 0.2072 0.2072 NUM cana-6296 115 23 2 2 NUM cana-6296 115 24 Γ— Γ— NOUN cana-6296 115 25 10 10 NUM cana-6296 115 26 - - SYM cana-6296 115 27 4 4 NUM cana-6296 115 28 2 2 NUM cana-6296 115 29 0.0183 0.0183 NUM cana-6296 115 30 0.0182 0.0182 NUM cana-6296 115 31 1 1 NUM cana-6296 115 32 Γ— Γ— NOUN cana-6296 115 33 10 10 NUM cana-6296 115 34 - - SYM cana-6296 115 35 4 4 NUM cana-6296 115 36 3 3 NUM cana-6296 115 37 0.00012 0.00012 NUM cana-6296 115 38 0.00011 0.00011 NUM cana-6296 115 39 1 1 NUM cana-6296 115 40 Γ— Γ— NOUN cana-6296 115 41 10 10 NUM cana-6296 115 42 - - SYM cana-6296 115 43 4 4 NUM cana-6296 115 44 observation observation NOUN cana-6296 115 45 : : PUNCT cana-6296 115 46 the the DET cana-6296 115 47 fourier fourier NOUN cana-6296 115 48 inversion inversion NOUN cana-6296 115 49 carried carry VERB cana-6296 115 50 out out ADP cana-6296 115 51 within within ADP cana-6296 115 52 the the DET cana-6296 115 53 unified unify VERB cana-6296 115 54 framework framework NOUN cana-6296 115 55 accurately accurately ADV cana-6296 115 56 reproduces reproduce VERB cana-6296 115 57 the the DET cana-6296 115 58 target target NOUN cana-6296 115 59 function function NOUN cana-6296 115 60 . . PUNCT cana-6296 116 1 this this PRON cana-6296 116 2 demonstrates demonstrate VERB cana-6296 116 3 the the DET cana-6296 116 4 versatility versatility NOUN cana-6296 116 5 of of ADP cana-6296 116 6 the the DET cana-6296 116 7 method method NOUN cana-6296 116 8 , , PUNCT cana-6296 116 9 confirming confirm VERB cana-6296 116 10 that that SCONJ cana-6296 116 11 both both PRON cana-6296 116 12 laplace laplace NOUN cana-6296 116 13 and and CCONJ cana-6296 116 14 fourier fourier NOUN cana-6296 116 15 inversions inversion NOUN cana-6296 116 16 can can AUX cana-6296 116 17 be be AUX cana-6296 116 18 treated treat VERB cana-6296 116 19 under under ADP cana-6296 116 20 a a DET cana-6296 116 21 single single ADJ cana-6296 116 22 , , PUNCT cana-6296 116 23 stable stable ADJ cana-6296 116 24 numerical numerical ADJ cana-6296 116 25 scheme scheme NOUN cana-6296 116 26 . . PUNCT cana-6296 117 1 communications communication NOUN cana-6296 117 2 on on ADP cana-6296 117 3 applied apply VERB cana-6296 117 4 nonlinear nonlinear ADJ cana-6296 117 5 analysis analysis NOUN cana-6296 117 6 issn issn NOUN cana-6296 117 7 : : PUNCT cana-6296 117 8 1074 1074 NUM cana-6296 117 9 - - PUNCT cana-6296 117 10 133x 133x NUM cana-6296 117 11 vol vol VERB cana-6296 117 12 32 32 NUM cana-6296 117 13 no no NOUN cana-6296 117 14 . . PUNCT cana-6296 118 1 10s 10 NOUN cana-6296 118 2 ( ( PUNCT cana-6296 118 3 2025 2025 NUM cana-6296 118 4 ) ) PUNCT cana-6296 118 5 3701 3701 NUM cana-6296 118 6 https://internationalpubls.com https://internationalpubls.com X cana-6296 118 7 5 5 X cana-6296 118 8 . . X cana-6296 118 9 conclusion conclusion NOUN cana-6296 118 10 : : PUNCT cana-6296 118 11 this this DET cana-6296 118 12 paper paper NOUN cana-6296 118 13 has have AUX cana-6296 118 14 presented present VERB cana-6296 118 15 a a DET cana-6296 118 16 comprehensive comprehensive ADJ cana-6296 118 17 numerical numerical ADJ cana-6296 118 18 framework framework NOUN cana-6296 118 19 for for ADP cana-6296 118 20 inverting invert VERB cana-6296 118 21 laplace laplace NOUN cana-6296 118 22 and and CCONJ cana-6296 118 23 fourier fourier NOUN cana-6296 118 24 transforms transform VERB cana-6296 118 25 . . PUNCT cana-6296 119 1 by by ADP cana-6296 119 2 utilizing utilize VERB cana-6296 119 3 analytic analytic ADJ cana-6296 119 4 continuation continuation NOUN cana-6296 119 5 and and CCONJ cana-6296 119 6 transforming transform VERB cana-6296 119 7 the the DET cana-6296 119 8 laplace laplace NOUN cana-6296 119 9 inversion inversion NOUN cana-6296 119 10 issue issue NOUN cana-6296 119 11 into into ADP cana-6296 119 12 a a DET cana-6296 119 13 fourier fourier ADJ cana-6296 119 14 - - PUNCT cana-6296 119 15 type type NOUN cana-6296 119 16 integral integral ADJ cana-6296 119 17 , , PUNCT cana-6296 119 18 we we PRON cana-6296 119 19 developed develop VERB cana-6296 119 20 a a DET cana-6296 119 21 universal universal ADJ cana-6296 119 22 inversion inversion NOUN cana-6296 119 23 formula formula NOUN cana-6296 119 24 that that PRON cana-6296 119 25 can can AUX cana-6296 119 26 be be AUX cana-6296 119 27 effectively effectively ADV cana-6296 119 28 executed execute VERB cana-6296 119 29 using use VERB cana-6296 119 30 the the DET cana-6296 119 31 fast fast ADJ cana-6296 119 32 fourier fourier NOUN cana-6296 119 33 transform transform NOUN cana-6296 119 34 ( ( PUNCT cana-6296 119 35 fft fft PROPN cana-6296 119 36 ) ) PUNCT cana-6296 119 37 . . PUNCT cana-6296 120 1 numerical numerical ADJ cana-6296 120 2 tests test NOUN cana-6296 120 3 demonstrated demonstrate VERB cana-6296 120 4 the the DET cana-6296 120 5 method method NOUN cana-6296 120 6 ’s ’s PART cana-6296 120 7 precision precision NOUN cana-6296 120 8 , , PUNCT cana-6296 120 9 stability stability NOUN cana-6296 120 10 , , PUNCT cana-6296 120 11 and and CCONJ cana-6296 120 12 computational computational ADJ cana-6296 120 13 efficiency efficiency NOUN cana-6296 120 14 when when SCONJ cana-6296 120 15 compared compare VERB cana-6296 120 16 to to ADP cana-6296 120 17 traditional traditional ADJ cana-6296 120 18 techniques technique NOUN cana-6296 120 19 like like ADP cana-6296 120 20 talbot talbot PROPN cana-6296 120 21 ’s ’s PART cana-6296 120 22 contour contour NOUN cana-6296 120 23 method method NOUN cana-6296 120 24 . . PUNCT cana-6296 121 1 the the DET cana-6296 121 2 primary primary ADJ cana-6296 121 3 benefit benefit NOUN cana-6296 121 4 of of ADP cana-6296 121 5 this this DET cana-6296 121 6 framework framework NOUN cana-6296 121 7 is be AUX cana-6296 121 8 its its PRON cana-6296 121 9 universality universality NOUN cana-6296 121 10 : : PUNCT cana-6296 121 11 the the DET cana-6296 121 12 same same ADJ cana-6296 121 13 algorithm algorithm NOUN cana-6296 121 14 can can AUX cana-6296 121 15 be be AUX cana-6296 121 16 applied apply VERB cana-6296 121 17 effortlessly effortlessly ADV cana-6296 121 18 to to ADP cana-6296 121 19 both both DET cana-6296 121 20 laplace laplace NOUN cana-6296 121 21 and and CCONJ cana-6296 121 22 fourier fourier NOUN cana-6296 121 23 inversions inversion NOUN cana-6296 121 24 , , PUNCT cana-6296 121 25 with with ADP cana-6296 121 26 only only ADJ cana-6296 121 27 minor minor ADJ cana-6296 121 28 parameter parameter NOUN cana-6296 121 29 modifications modification NOUN cana-6296 121 30 needed need VERB cana-6296 121 31 . . PUNCT cana-6296 122 1 this this PRON cana-6296 122 2 removes remove VERB cana-6296 122 3 the the DET cana-6296 122 4 necessity necessity NOUN cana-6296 122 5 for for ADP cana-6296 122 6 specialized specialized ADJ cana-6296 122 7 inversion inversion NOUN cana-6296 122 8 routines routine NOUN cana-6296 122 9 and and CCONJ cana-6296 122 10 offers offer VERB cana-6296 122 11 a a DET cana-6296 122 12 single single ADJ cana-6296 122 13 , , PUNCT cana-6296 122 14 adaptable adaptable ADJ cana-6296 122 15 computational computational ADJ cana-6296 122 16 tool tool NOUN cana-6296 122 17 . . PUNCT cana-6296 123 1 future future ADJ cana-6296 123 2 research research NOUN cana-6296 123 3 could could AUX cana-6296 123 4 expand expand VERB cana-6296 123 5 the the DET cana-6296 123 6 framework framework NOUN cana-6296 123 7 to to PART cana-6296 123 8 include include VERB cana-6296 123 9 multidimensional multidimensional ADJ cana-6296 123 10 transforms transform NOUN cana-6296 123 11 , , PUNCT cana-6296 123 12 adaptive adaptive ADJ cana-6296 123 13 parameter parameter NOUN cana-6296 123 14 selection selection NOUN cana-6296 123 15 , , PUNCT cana-6296 123 16 and and CCONJ cana-6296 123 17 applications application NOUN cana-6296 123 18 in in ADP cana-6296 123 19 fields field NOUN cana-6296 123 20 such such ADJ cana-6296 123 21 as as ADP cana-6296 123 22 control control NOUN cana-6296 123 23 theory theory NOUN cana-6296 123 24 , , PUNCT cana-6296 123 25 pde pde NOUN cana-6296 123 26 solvers solver NOUN cana-6296 123 27 , , PUNCT cana-6296 123 28 and and CCONJ cana-6296 123 29 signal signal ADJ cana-6296 123 30 processing processing NOUN cana-6296 123 31 . . PUNCT cana-6296 124 1 data datum NOUN cana-6296 124 2 availability availability NOUN cana-6296 124 3 : : PUNCT cana-6296 124 4 all all DET cana-6296 124 5 numerical numerical ADJ cana-6296 124 6 experiments experiment NOUN cana-6296 124 7 in in ADP cana-6296 124 8 this this DET cana-6296 124 9 study study NOUN cana-6296 124 10 were be AUX cana-6296 124 11 conducted conduct VERB cana-6296 124 12 using use VERB cana-6296 124 13 standard standard ADJ cana-6296 124 14 mathematical mathematical ADJ cana-6296 124 15 software software NOUN cana-6296 124 16 . . PUNCT cana-6296 125 1 the the DET cana-6296 125 2 codes code NOUN cana-6296 125 3 and and CCONJ cana-6296 125 4 datasets dataset NOUN cana-6296 125 5 produced produce VERB cana-6296 125 6 during during ADP cana-6296 125 7 this this DET cana-6296 125 8 research research NOUN cana-6296 125 9 are be AUX cana-6296 125 10 available available ADJ cana-6296 125 11 from from ADP cana-6296 125 12 the the DET cana-6296 125 13 corresponding corresponding ADJ cana-6296 125 14 author author NOUN cana-6296 125 15 upon upon SCONJ cana-6296 125 16 reasonable reasonable ADJ cana-6296 125 17 request request NOUN cana-6296 125 18 . . PUNCT cana-6296 126 1 acknowledgments acknowledgment NOUN cana-6296 126 2 : : PUNCT cana-6296 126 3 the the DET cana-6296 126 4 authors author NOUN cana-6296 126 5 wish wish VERB cana-6296 126 6 to to PART cana-6296 126 7 thank thank VERB cana-6296 126 8 colleagues colleague NOUN cana-6296 126 9 and and CCONJ cana-6296 126 10 mentors mentor NOUN cana-6296 126 11 who who PRON cana-6296 126 12 offered offer VERB cana-6296 126 13 valuable valuable ADJ cana-6296 126 14 insights insight NOUN cana-6296 126 15 during during ADP cana-6296 126 16 the the DET cana-6296 126 17 development development NOUN cana-6296 126 18 of of ADP cana-6296 126 19 this this DET cana-6296 126 20 work work NOUN cana-6296 126 21 . . PUNCT cana-6296 127 1 special special ADJ cana-6296 127 2 appreciation appreciation NOUN cana-6296 127 3 is be AUX cana-6296 127 4 extended extend VERB cana-6296 127 5 to to ADP cana-6296 127 6 the the DET cana-6296 127 7 post post NOUN cana-6296 127 8 graduate graduate PROPN cana-6296 127 9 teaching teaching NOUN cana-6296 127 10 department department NOUN cana-6296 127 11 of of ADP cana-6296 127 12 mathematics mathematics PROPN cana-6296 127 13 , , PUNCT cana-6296 127 14 gondwana gondwana PROPN cana-6296 127 15 university university PROPN cana-6296 127 16 , , PUNCT cana-6296 127 17 gadchiroli gadchiroli NOUN cana-6296 127 18 , , PUNCT cana-6296 127 19 for for ADP cana-6296 127 20 providing provide VERB cana-6296 127 21 a a DET cana-6296 127 22 stimulating stimulate VERB cana-6296 127 23 academic academic ADJ cana-6296 127 24 environment environment NOUN cana-6296 127 25 . . PUNCT cana-6296 128 1 the the DET cana-6296 128 2 authors author NOUN cana-6296 128 3 also also ADV cana-6296 128 4 acknowledge acknowledge VERB cana-6296 128 5 the the DET cana-6296 128 6 inspiration inspiration NOUN cana-6296 128 7 drawn draw VERB cana-6296 128 8 from from ADP cana-6296 128 9 previous previous ADJ cana-6296 128 10 studies study NOUN cana-6296 128 11 in in ADP cana-6296 128 12 transform transform NOUN cana-6296 128 13 methods method NOUN cana-6296 128 14 , , PUNCT cana-6296 128 15 which which PRON cana-6296 128 16 motivated motivate VERB cana-6296 128 17 the the DET cana-6296 128 18 creation creation NOUN cana-6296 128 19 of of ADP cana-6296 128 20 this this DET cana-6296 128 21 unified unify VERB cana-6296 128 22 approach approach NOUN cana-6296 128 23 . . PUNCT cana-6296 129 1 would would AUX cana-6296 129 2 you you PRON cana-6296 129 3 like like VERB cana-6296 129 4 me i PRON cana-6296 129 5 to to PART cana-6296 129 6 also also ADV cana-6296 129 7 draft draft VERB cana-6296 129 8 a a DET cana-6296 129 9 short short ADJ cana-6296 129 10 β€œ " PUNCT cana-6296 129 11 future future ADJ cana-6296 129 12 work work NOUN cana-6296 129 13 ” " PUNCT cana-6296 129 14 subsection subsection NOUN cana-6296 129 15 ( ( PUNCT cana-6296 129 16 before before ADP cana-6296 129 17 the the DET cana-6296 129 18 conclusion conclusion NOUN cana-6296 129 19 ) ) PUNCT cana-6296 129 20 to to PART cana-6296 129 21 highlight highlight VERB cana-6296 129 22 open open ADJ cana-6296 129 23 directions direction NOUN cana-6296 129 24 like like ADP cana-6296 129 25 multidimensional multidimensional ADJ cana-6296 129 26 transforms transform NOUN cana-6296 129 27 , , PUNCT cana-6296 129 28 adaptive adaptive ADJ cana-6296 129 29 Οƒ Οƒ NOUN cana-6296 129 30 choice choice NOUN cana-6296 129 31 , , PUNCT cana-6296 129 32 or or CCONJ cana-6296 129 33 machine machine NOUN cana-6296 129 34 learning learn VERB cana-6296 129 35 applications application NOUN cana-6296 129 36 ? ? PUNCT cana-6296 130 1 that that PRON cana-6296 130 2 could could AUX cana-6296 130 3 make make VERB cana-6296 130 4 the the DET cana-6296 130 5 paper paper NOUN cana-6296 130 6 feel feel VERB cana-6296 130 7 even even ADV cana-6296 130 8 more more ADV cana-6296 130 9 forward forward ADV cana-6296 130 10 - - PUNCT cana-6296 130 11 looking look VERB cana-6296 130 12 . . PUNCT cana-6296 131 1 6 6 X cana-6296 131 2 . . NUM cana-6296 131 3 references reference NOUN cana-6296 131 4 [ [ X cana-6296 131 5 1 1 NUM cana-6296 131 6 ] ] PUNCT cana-6296 131 7 schiff schiff NOUN cana-6296 131 8 , , PUNCT cana-6296 131 9 j. j. PROPN cana-6296 131 10 l. l. PROPN cana-6296 131 11 ( ( PUNCT cana-6296 131 12 1999 1999 NUM cana-6296 131 13 ) ) PUNCT cana-6296 131 14 . . PUNCT cana-6296 132 1 the the DET cana-6296 132 2 laplace laplace NOUN cana-6296 132 3 transform transform NOUN cana-6296 132 4 : : PUNCT cana-6296 132 5 theory theory NOUN cana-6296 132 6 and and CCONJ cana-6296 132 7 applications application NOUN cana-6296 132 8 . . PUNCT cana-6296 133 1 springer springer NOUN cana-6296 133 2 - - PUNCT cana-6296 133 3 verlag verlag PROPN cana-6296 133 4 . . PUNCT cana-6296 134 1 [ [ X cana-6296 134 2 2 2 NUM cana-6296 134 3 ] ] X cana-6296 134 4 duffy duffy PROPN cana-6296 134 5 , , PUNCT cana-6296 134 6 d. d. PROPN cana-6296 134 7 g. g. PROPN cana-6296 134 8 ( ( PUNCT cana-6296 134 9 2004 2004 NUM cana-6296 134 10 ) ) PUNCT cana-6296 134 11 . . PUNCT cana-6296 135 1 transform transform VERB cana-6296 135 2 methods method NOUN cana-6296 135 3 for for ADP cana-6296 135 4 solving solve VERB cana-6296 135 5 partial partial ADJ cana-6296 135 6 differential differential ADJ cana-6296 135 7 equations equation NOUN cana-6296 135 8 ( ( PUNCT cana-6296 135 9 2nd 2nd ADJ cana-6296 135 10 ed ed NOUN cana-6296 135 11 . . PUNCT cana-6296 135 12 ) ) PUNCT cana-6296 135 13 . . PUNCT cana-6296 136 1 chapman chapman NOUN cana-6296 136 2 and and CCONJ cana-6296 136 3 hall hall PROPN cana-6296 136 4 / / SYM cana-6296 136 5 crc crc NOUN cana-6296 136 6 . . PUNCT cana-6296 137 1 [ [ X cana-6296 137 2 3 3 NUM cana-6296 137 3 ] ] X cana-6296 137 4 dubner dubner NOUN cana-6296 137 5 , , PUNCT cana-6296 137 6 h. h. PROPN cana-6296 137 7 , , PUNCT cana-6296 137 8 & & CCONJ cana-6296 137 9 abate abate VERB cana-6296 137 10 , , PUNCT cana-6296 137 11 j. j. PROPN cana-6296 137 12 ( ( PUNCT cana-6296 137 13 1968 1968 NUM cana-6296 137 14 ) ) PUNCT cana-6296 137 15 . . PUNCT cana-6296 138 1 numerical numerical ADJ cana-6296 138 2 inversion inversion NOUN cana-6296 138 3 of of ADP cana-6296 138 4 laplace laplace NOUN cana-6296 138 5 transforms transform VERB cana-6296 138 6 by by ADP cana-6296 138 7 relating relate VERB cana-6296 138 8 them they PRON cana-6296 138 9 to to ADP cana-6296 138 10 the the DET cana-6296 138 11 finite finite ADJ cana-6296 138 12 fourier fourier PROPN cana-6296 138 13 cosine cosine NOUN cana-6296 138 14 transform transform NOUN cana-6296 138 15 . . PUNCT cana-6296 139 1 journal journal NOUN cana-6296 139 2 of of ADP cana-6296 139 3 the the DET cana-6296 139 4 acm acm PROPN cana-6296 139 5 , , PUNCT cana-6296 139 6 15(1 15(1 NUM cana-6296 139 7 ) ) PUNCT cana-6296 139 8 , , PUNCT cana-6296 139 9 115–123 115–123 NUM cana-6296 139 10 . . PUNCT cana-6296 140 1 [ [ X cana-6296 140 2 4 4 NUM cana-6296 140 3 ] ] X cana-6296 140 4 talbot talbot PROPN cana-6296 140 5 , , PUNCT cana-6296 140 6 a. a. PROPN cana-6296 140 7 ( ( PUNCT cana-6296 140 8 1979 1979 NUM cana-6296 140 9 ) ) PUNCT cana-6296 140 10 . . PUNCT cana-6296 141 1 the the DET cana-6296 141 2 accurate accurate ADJ cana-6296 141 3 numerical numerical ADJ cana-6296 141 4 inversion inversion NOUN cana-6296 141 5 of of ADP cana-6296 141 6 laplace laplace NOUN cana-6296 141 7 transforms transform VERB cana-6296 141 8 . . PUNCT cana-6296 142 1 i i PRON cana-6296 142 2 m m VERB cana-6296 142 3 a a DET cana-6296 142 4 journal journal NOUN cana-6296 142 5 of of ADP cana-6296 142 6 applied apply VERB cana-6296 142 7 mathematics mathematic NOUN cana-6296 142 8 , , PUNCT cana-6296 142 9 23(1 23(1 NUM cana-6296 142 10 ) ) PUNCT cana-6296 142 11 , , PUNCT cana-6296 142 12 97–120 97–120 PROPN cana-6296 142 13 . . PUNCT cana-6296 143 1 [ [ X cana-6296 143 2 5 5 NUM cana-6296 143 3 ] ] SYM cana-6296 143 4 weeks week NOUN cana-6296 143 5 , , PUNCT cana-6296 143 6 w. w. PROPN cana-6296 143 7 t. t. PROPN cana-6296 143 8 ( ( PUNCT cana-6296 143 9 1966 1966 NUM cana-6296 143 10 ) ) PUNCT cana-6296 143 11 . . PUNCT cana-6296 144 1 numerical numerical ADJ cana-6296 144 2 inversion inversion NOUN cana-6296 144 3 of of ADP cana-6296 144 4 laplace laplace NOUN cana-6296 144 5 transforms transform VERB cana-6296 144 6 using use VERB cana-6296 144 7 laguerre laguerre NOUN cana-6296 144 8 functions function NOUN cana-6296 144 9 . . PUNCT cana-6296 145 1 journal journal NOUN cana-6296 145 2 of of ADP cana-6296 145 3 the the DET cana-6296 145 4 acm acm PROPN cana-6296 145 5 , , PUNCT cana-6296 145 6 13(3 13(3 NUM cana-6296 145 7 ) ) PUNCT cana-6296 145 8 , , PUNCT cana-6296 145 9 419–426 419–426 NUM cana-6296 145 10 . . PUNCT cana-6296 146 1 [ [ X cana-6296 146 2 6 6 NUM cana-6296 146 3 ] ] X cana-6296 146 4 l.m l.m PROPN cana-6296 146 5 . . PROPN cana-6296 146 6 kells kells PROPN cana-6296 146 7 , , PUNCT cana-6296 146 8 elementary elementary PROPN cana-6296 146 9 diferential diferential ADJ cana-6296 146 10 equations equation NOUN cana-6296 146 11 , , PUNCT cana-6296 146 12 vol vol NOUN cana-6296 146 13 . . PROPN cana-6296 146 14 1 1 NUM cana-6296 146 15 , , PUNCT cana-6296 146 16 third third ADJ cana-6296 146 17 ed ed NOUN cana-6296 146 18 . . PROPN cana-6296 146 19 , , PUNCT cana-6296 146 20 mcgraw mcgraw PROPN cana-6296 146 21 - - PUNCT cana-6296 146 22 hill hill PROPN cana-6296 146 23 , , PUNCT cana-6296 146 24 new new PROPN cana-6296 146 25 york york PROPN cana-6296 146 26 , , PUNCT cana-6296 146 27 usa usa PROPN cana-6296 146 28 , , PUNCT cana-6296 146 29 1965 1965 NUM cana-6296 146 30 . . PUNCT cana-6296 147 1 [ [ X cana-6296 147 2 7 7 X cana-6296 147 3 ] ] X cana-6296 147 4 h. h. PROPN cana-6296 147 5 fatoorehchi fatoorehchi PROPN cana-6296 147 6 , , PUNCT cana-6296 147 7 h. h. PROPN cana-6296 147 8 abolghasemi abolghasemi PROPN cana-6296 147 9 , , PUNCT cana-6296 147 10 an an DET cana-6296 147 11 introduction introduction NOUN cana-6296 147 12 to to ADP cana-6296 147 13 the the DET cana-6296 147 14 fourier fourier NOUN cana-6296 147 15 transform transform NOUN cana-6296 147 16 : : PUNCT cana-6296 147 17 relationship relationship NOUN cana-6296 147 18 to to ADP cana-6296 147 19 mri mri NOUN cana-6296 147 20 , , PUNCT cana-6296 147 21 vol vol NOUN cana-6296 147 22 . . PROPN cana-6296 147 23 190 190 NUM cana-6296 147 24 , , PUNCT cana-6296 147 25 american american PROPN cana-6296 147 26 roentgen roentgen PROPN cana-6296 147 27 ray ray PROPN cana-6296 147 28 society society PROPN cana-6296 147 29 , , PUNCT cana-6296 147 30 2008 2008 NUM cana-6296 147 31 , , PUNCT cana-6296 147 32 pp pp ADJ cana-6296 147 33 . . PUNCT cana-6296 148 1 1396–1405 1396–1405 NUM cana-6296 148 2 . . PUNCT cana-6296 149 1 [ [ X cana-6296 149 2 8 8 NUM cana-6296 149 3 ] ] X cana-6296 149 4 javier javier PROPN cana-6296 149 5 morales morales PROPN cana-6296 149 6 - - PUNCT cana-6296 149 7 castilloβˆ— castilloβˆ— PROPN cana-6296 149 8 , , PUNCT cana-6296 149 9 eva eva PROPN cana-6296 149 10 m. m. NOUN cana-6296 149 11 morales morales PROPN cana-6296 149 12 - - PUNCT cana-6296 149 13 rodriguez rodriguez NOUN cana-6296 149 14 ( ( PUNCT cana-6296 149 15 2022 2022 NUM cana-6296 149 16 ) ) PUNCT cana-6296 149 17 , , PUNCT cana-6296 149 18 a a DET cana-6296 149 19 formula formula NOUN cana-6296 149 20 to to PART cana-6296 149 21 solve solve VERB cana-6296 149 22 laplace laplace NOUN cana-6296 149 23 and and CCONJ cana-6296 149 24 fourier fourier NOUN cana-6296 149 25 transforms transform VERB cana-6296 149 26 , , PUNCT cana-6296 149 27 journal journal NOUN cana-6296 149 28 of of ADP cana-6296 149 29 computational computational ADJ cana-6296 149 30 and and CCONJ cana-6296 149 31 applied applied ADJ cana-6296 149 32 mathematics mathematic NOUN cana-6296 149 33 . . PUNCT