id sid tid token lemma pos american_scientific_journal-4152 1 1 11 11 NUM american_scientific_journal-4152 1 2 american american ADJ american_scientific_journal-4152 1 3 scientific scientific ADJ american_scientific_journal-4152 1 4 research research NOUN american_scientific_journal-4152 1 5 journal journal NOUN american_scientific_journal-4152 1 6 for for ADP american_scientific_journal-4152 1 7 engineering engineering NOUN american_scientific_journal-4152 1 8 , , PUNCT american_scientific_journal-4152 1 9 technology technology NOUN american_scientific_journal-4152 1 10 , , PUNCT american_scientific_journal-4152 1 11 and and CCONJ american_scientific_journal-4152 1 12 sciences science NOUN american_scientific_journal-4152 1 13 ( ( PUNCT american_scientific_journal-4152 1 14 asrjets asrjet NOUN american_scientific_journal-4152 1 15 ) ) PUNCT american_scientific_journal-4152 1 16 issn issn PROPN american_scientific_journal-4152 1 17 ( ( PUNCT american_scientific_journal-4152 1 18 print print NOUN american_scientific_journal-4152 1 19 ) ) PUNCT american_scientific_journal-4152 1 20 2313 2313 NUM american_scientific_journal-4152 1 21 - - SYM american_scientific_journal-4152 1 22 4410 4410 NUM american_scientific_journal-4152 1 23 , , PUNCT american_scientific_journal-4152 1 24 issn issn PROPN american_scientific_journal-4152 1 25 ( ( PUNCT american_scientific_journal-4152 1 26 online online ADJ american_scientific_journal-4152 1 27 ) ) PUNCT american_scientific_journal-4152 1 28 2313 2313 NUM american_scientific_journal-4152 1 29 - - SYM american_scientific_journal-4152 1 30 4402 4402 NUM american_scientific_journal-4152 1 31 Β© Β© PROPN american_scientific_journal-4152 1 32 global global ADJ american_scientific_journal-4152 1 33 society society NOUN american_scientific_journal-4152 1 34 of of ADP american_scientific_journal-4152 1 35 scientific scientific ADJ american_scientific_journal-4152 1 36 research research NOUN american_scientific_journal-4152 1 37 and and CCONJ american_scientific_journal-4152 1 38 researchers researcher NOUN american_scientific_journal-4152 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-4152 1 40 the the DET american_scientific_journal-4152 1 41 relationship relationship NOUN american_scientific_journal-4152 1 42 of of ADP american_scientific_journal-4152 1 43 fractional fractional ADJ american_scientific_journal-4152 1 44 laplace laplace NOUN american_scientific_journal-4152 1 45 transform transform NOUN american_scientific_journal-4152 1 46 with with ADP american_scientific_journal-4152 1 47 fractional fractional ADJ american_scientific_journal-4152 1 48 fourier fourier NOUN american_scientific_journal-4152 1 49 , , PUNCT american_scientific_journal-4152 1 50 mellin mellin PROPN american_scientific_journal-4152 1 51 and and CCONJ american_scientific_journal-4152 1 52 sumudu sumudu NOUN american_scientific_journal-4152 1 53 transforms transform VERB american_scientific_journal-4152 1 54 maria maria PROPN american_scientific_journal-4152 1 55 balocha balocha PROPN american_scientific_journal-4152 1 56 , , PUNCT american_scientific_journal-4152 1 57 saleem saleem PROPN american_scientific_journal-4152 1 58 iqbalb iqbalb PROPN american_scientific_journal-4152 1 59 * * SYM american_scientific_journal-4152 1 60 , , PUNCT american_scientific_journal-4152 1 61 farhana farhana PROPN american_scientific_journal-4152 1 62 sarwarc sarwarc PROPN american_scientific_journal-4152 1 63 , , PUNCT american_scientific_journal-4152 1 64 abdul abdul PROPN american_scientific_journal-4152 1 65 rehmand rehmand PROPN american_scientific_journal-4152 1 66 a a DET american_scientific_journal-4152 1 67 , , PUNCT american_scientific_journal-4152 1 68 b b NOUN american_scientific_journal-4152 1 69 , , PUNCT american_scientific_journal-4152 1 70 ddepartment ddepartment NOUN american_scientific_journal-4152 1 71 of of ADP american_scientific_journal-4152 1 72 mathematics mathematic NOUN american_scientific_journal-4152 1 73 , , PUNCT american_scientific_journal-4152 1 74 university university NOUN american_scientific_journal-4152 1 75 of of ADP american_scientific_journal-4152 1 76 balochistan balochistan PROPN american_scientific_journal-4152 1 77 , , PUNCT american_scientific_journal-4152 1 78 quetta quetta PROPN american_scientific_journal-4152 1 79 87300 87300 NUM american_scientific_journal-4152 1 80 , , PUNCT american_scientific_journal-4152 1 81 pakistan pakistan PROPN american_scientific_journal-4152 1 82 cdepartment cdepartment NOUN american_scientific_journal-4152 1 83 of of ADP american_scientific_journal-4152 1 84 mathematics mathematics PROPN american_scientific_journal-4152 1 85 f.g.girls f.g.girl NOUN american_scientific_journal-4152 1 86 degree degree NOUN american_scientific_journal-4152 1 87 college college NOUN american_scientific_journal-4152 1 88 , , PUNCT american_scientific_journal-4152 1 89 madrissa madrissa NOUN american_scientific_journal-4152 1 90 road road NOUN american_scientific_journal-4152 1 91 , , PUNCT american_scientific_journal-4152 1 92 quetta quetta PROPN american_scientific_journal-4152 1 93 , , PUNCT american_scientific_journal-4152 1 94 cantt cantt PROPN american_scientific_journal-4152 1 95 , , PUNCT american_scientific_journal-4152 1 96 87300 87300 NUM american_scientific_journal-4152 1 97 , , PUNCT american_scientific_journal-4152 1 98 pakistan pakistan PROPN american_scientific_journal-4152 1 99 aemail aemail NOUN american_scientific_journal-4152 1 100 : : PUNCT american_scientific_journal-4152 1 101 maria.umbreen@yahoo.com maria.umbreen@yahoo.com X american_scientific_journal-4152 1 102 bemail bemail NOUN american_scientific_journal-4152 1 103 : : PUNCT american_scientific_journal-4152 1 104 saleemiqbal81@yahoo.com saleemiqbal81@yahoo.com X american_scientific_journal-4152 1 105 cemail cemail NOUN american_scientific_journal-4152 1 106 : : PUNCT american_scientific_journal-4152 1 107 f_saleem10@yahoo.com f_saleem10@yahoo.com X american_scientific_journal-4152 1 108 eemail eemail NOUN american_scientific_journal-4152 1 109 : : PUNCT american_scientific_journal-4152 2 1 abdul_maths@yahoo.com abdul_maths@yahoo.com X american_scientific_journal-4152 3 1 abstract abstract ADJ american_scientific_journal-4152 3 2 we we PRON american_scientific_journal-4152 3 3 have have AUX american_scientific_journal-4152 3 4 developed develop VERB american_scientific_journal-4152 3 5 in in ADP american_scientific_journal-4152 3 6 this this DET american_scientific_journal-4152 3 7 research research NOUN american_scientific_journal-4152 3 8 paper paper NOUN american_scientific_journal-4152 3 9 , , PUNCT american_scientific_journal-4152 3 10 some some PRON american_scientific_journal-4152 3 11 of of ADP american_scientific_journal-4152 3 12 the the DET american_scientific_journal-4152 3 13 fundamental fundamental ADJ american_scientific_journal-4152 3 14 relationship relationship NOUN american_scientific_journal-4152 3 15 of of ADP american_scientific_journal-4152 3 16 fractional fractional ADJ american_scientific_journal-4152 3 17 laplace laplace NOUN american_scientific_journal-4152 3 18 transform transform NOUN american_scientific_journal-4152 3 19 with with ADP american_scientific_journal-4152 3 20 fractional fractional ADJ american_scientific_journal-4152 3 21 fourier fourier NOUN american_scientific_journal-4152 3 22 , , PUNCT american_scientific_journal-4152 3 23 fractional fractional ADJ american_scientific_journal-4152 3 24 mellin mellin PROPN american_scientific_journal-4152 3 25 and and CCONJ american_scientific_journal-4152 3 26 fractional fractional ADJ american_scientific_journal-4152 3 27 sumudu sumudu NOUN american_scientific_journal-4152 3 28 transforms transform VERB american_scientific_journal-4152 3 29 . . PUNCT american_scientific_journal-4152 4 1 these these DET american_scientific_journal-4152 4 2 results result NOUN american_scientific_journal-4152 4 3 are be AUX american_scientific_journal-4152 4 4 expressed express VERB american_scientific_journal-4152 4 5 mathematically mathematically ADV american_scientific_journal-4152 4 6 , , PUNCT american_scientific_journal-4152 4 7 and and CCONJ american_scientific_journal-4152 4 8 such such ADJ american_scientific_journal-4152 4 9 relationships relationship NOUN american_scientific_journal-4152 4 10 should should AUX american_scientific_journal-4152 4 11 be be AUX american_scientific_journal-4152 4 12 very very ADV american_scientific_journal-4152 4 13 useful useful ADJ american_scientific_journal-4152 4 14 in in ADP american_scientific_journal-4152 4 15 applications application NOUN american_scientific_journal-4152 4 16 to to PART american_scientific_journal-4152 4 17 signal signal VERB american_scientific_journal-4152 4 18 processing processing NOUN american_scientific_journal-4152 4 19 and and CCONJ american_scientific_journal-4152 4 20 optics optic NOUN american_scientific_journal-4152 4 21 . . PUNCT american_scientific_journal-4152 5 1 keywords keyword NOUN american_scientific_journal-4152 5 2 : : PUNCT american_scientific_journal-4152 5 3 fractional fractional ADJ american_scientific_journal-4152 5 4 laplace laplace NOUN american_scientific_journal-4152 5 5 transform transform NOUN american_scientific_journal-4152 5 6 ; ; PUNCT american_scientific_journal-4152 5 7 fractional fractional ADJ american_scientific_journal-4152 5 8 mellin mellin PROPN american_scientific_journal-4152 5 9 transform transform NOUN american_scientific_journal-4152 5 10 ; ; PUNCT american_scientific_journal-4152 5 11 fractional fractional ADJ american_scientific_journal-4152 5 12 fourier fourier NOUN american_scientific_journal-4152 5 13 transform transform NOUN american_scientific_journal-4152 5 14 ; ; PUNCT american_scientific_journal-4152 5 15 fractional fractional ADJ american_scientific_journal-4152 5 16 sumudu sumudu NOUN american_scientific_journal-4152 5 17 transform transform NOUN american_scientific_journal-4152 5 18 . . PUNCT american_scientific_journal-4152 6 1 1 1 X american_scientific_journal-4152 6 2 . . X american_scientific_journal-4152 6 3 introduction introduction NOUN american_scientific_journal-4152 6 4 fractional fractional ADJ american_scientific_journal-4152 6 5 integral integral ADJ american_scientific_journal-4152 6 6 transforms transform NOUN american_scientific_journal-4152 6 7 provide provide VERB american_scientific_journal-4152 6 8 a a DET american_scientific_journal-4152 6 9 well well ADV american_scientific_journal-4152 6 10 - - PUNCT american_scientific_journal-4152 6 11 established establish VERB american_scientific_journal-4152 6 12 and and CCONJ american_scientific_journal-4152 6 13 valuable valuable ADJ american_scientific_journal-4152 6 14 method method NOUN american_scientific_journal-4152 6 15 for for ADP american_scientific_journal-4152 6 16 solving solve VERB american_scientific_journal-4152 6 17 problems problem NOUN american_scientific_journal-4152 6 18 in in ADP american_scientific_journal-4152 6 19 many many ADJ american_scientific_journal-4152 6 20 areas area NOUN american_scientific_journal-4152 6 21 of of ADP american_scientific_journal-4152 6 22 applied applied ADJ american_scientific_journal-4152 6 23 mathematics mathematic NOUN american_scientific_journal-4152 6 24 , , PUNCT american_scientific_journal-4152 6 25 physics physics NOUN american_scientific_journal-4152 6 26 like like ADP american_scientific_journal-4152 6 27 optics optic NOUN american_scientific_journal-4152 6 28 , , PUNCT american_scientific_journal-4152 6 29 signal signal ADJ american_scientific_journal-4152 6 30 processing processing NOUN american_scientific_journal-4152 6 31 , , PUNCT american_scientific_journal-4152 6 32 quantum quantum NOUN american_scientific_journal-4152 6 33 mechanics mechanic NOUN american_scientific_journal-4152 6 34 . . PUNCT american_scientific_journal-4152 7 1 since since SCONJ american_scientific_journal-4152 7 2 the the DET american_scientific_journal-4152 7 3 introduction introduction NOUN american_scientific_journal-4152 7 4 of of ADP american_scientific_journal-4152 7 5 fractional fractional ADJ american_scientific_journal-4152 7 6 fourier fourier NOUN american_scientific_journal-4152 7 7 transform transform NOUN american_scientific_journal-4152 7 8 by by ADP american_scientific_journal-4152 7 9 namias namia NOUN american_scientific_journal-4152 7 10 in in ADP american_scientific_journal-4152 7 11 1980 1980 NUM american_scientific_journal-4152 7 12 [ [ X american_scientific_journal-4152 7 13 1 1 NUM american_scientific_journal-4152 7 14 ] ] PUNCT american_scientific_journal-4152 7 15 , , PUNCT american_scientific_journal-4152 7 16 the the DET american_scientific_journal-4152 7 17 applied apply VERB american_scientific_journal-4152 7 18 mathematicians mathematician NOUN american_scientific_journal-4152 7 19 , , PUNCT american_scientific_journal-4152 7 20 physicists physicist NOUN american_scientific_journal-4152 7 21 are be AUX american_scientific_journal-4152 7 22 paying pay VERB american_scientific_journal-4152 7 23 their their PRON american_scientific_journal-4152 7 24 attention attention NOUN american_scientific_journal-4152 7 25 not not PART american_scientific_journal-4152 7 26 only only ADV american_scientific_journal-4152 7 27 on on ADP american_scientific_journal-4152 7 28 fractional fractional ADJ american_scientific_journal-4152 7 29 fourier fourier NOUN american_scientific_journal-4152 7 30 transforms transform VERB american_scientific_journal-4152 7 31 but but CCONJ american_scientific_journal-4152 7 32 also also ADV american_scientific_journal-4152 7 33 working work VERB american_scientific_journal-4152 7 34 on on ADP american_scientific_journal-4152 7 35 many many ADJ american_scientific_journal-4152 7 36 other other ADJ american_scientific_journal-4152 7 37 transforms transform NOUN american_scientific_journal-4152 7 38 like like ADP american_scientific_journal-4152 7 39 fractional fractional ADJ american_scientific_journal-4152 7 40 hilbert hilbert NOUN american_scientific_journal-4152 7 41 transform transform VERB american_scientific_journal-4152 7 42 fractional fractional ADJ american_scientific_journal-4152 7 43 mellin mellin PROPN american_scientific_journal-4152 7 44 transform transform VERB american_scientific_journal-4152 7 45 fractional fractional ADJ american_scientific_journal-4152 7 46 laplace laplace NOUN american_scientific_journal-4152 7 47 transform transform NOUN american_scientific_journal-4152 7 48 and and CCONJ american_scientific_journal-4152 7 49 fractional fractional ADJ american_scientific_journal-4152 7 50 sumudu sumudu NOUN american_scientific_journal-4152 7 51 transform transform NOUN american_scientific_journal-4152 7 52 . . PUNCT american_scientific_journal-4152 8 1 in in ADP american_scientific_journal-4152 8 2 recent recent ADJ american_scientific_journal-4152 8 3 times time NOUN american_scientific_journal-4152 8 4 the the DET american_scientific_journal-4152 8 5 fractional fractional ADJ american_scientific_journal-4152 8 6 integral integral ADJ american_scientific_journal-4152 8 7 transforms transform NOUN american_scientific_journal-4152 8 8 have have AUX american_scientific_journal-4152 8 9 become become VERB american_scientific_journal-4152 8 10 a a DET american_scientific_journal-4152 8 11 very very ADV american_scientific_journal-4152 8 12 important important ADJ american_scientific_journal-4152 8 13 tool tool NOUN american_scientific_journal-4152 8 14 and and CCONJ american_scientific_journal-4152 8 15 are be AUX american_scientific_journal-4152 8 16 playing play VERB american_scientific_journal-4152 8 17 a a DET american_scientific_journal-4152 8 18 key key ADJ american_scientific_journal-4152 8 19 role role NOUN american_scientific_journal-4152 8 20 in in ADP american_scientific_journal-4152 8 21 various various ADJ american_scientific_journal-4152 8 22 branches branch NOUN american_scientific_journal-4152 8 23 of of ADP american_scientific_journal-4152 8 24 applied apply VERB american_scientific_journal-4152 8 25 mathematics mathematic NOUN american_scientific_journal-4152 8 26 and and CCONJ american_scientific_journal-4152 8 27 physics physics NOUN american_scientific_journal-4152 8 28 . . PUNCT american_scientific_journal-4152 9 1 fractional fractional ADJ american_scientific_journal-4152 9 2 laplace laplace NOUN american_scientific_journal-4152 9 3 transform transform NOUN american_scientific_journal-4152 9 4 was be AUX american_scientific_journal-4152 9 5 introduced introduce VERB american_scientific_journal-4152 9 6 by by ADP american_scientific_journal-4152 9 7 many many ADJ american_scientific_journal-4152 9 8 researchers researcher NOUN american_scientific_journal-4152 9 9 in in ADP american_scientific_journal-4152 9 10 their their PRON american_scientific_journal-4152 9 11 research research NOUN american_scientific_journal-4152 9 12 articles article NOUN american_scientific_journal-4152 9 13 in in ADP american_scientific_journal-4152 9 14 different different ADJ american_scientific_journal-4152 9 15 ways way NOUN american_scientific_journal-4152 9 16 , , PUNCT american_scientific_journal-4152 9 17 in in ADP american_scientific_journal-4152 9 18 2003 2003 NUM american_scientific_journal-4152 9 19 it it PRON american_scientific_journal-4152 9 20 was be AUX american_scientific_journal-4152 9 21 first first ADV american_scientific_journal-4152 9 22 defined define VERB american_scientific_journal-4152 9 23 by by ADP american_scientific_journal-4152 9 24 a. a. NOUN american_scientific_journal-4152 9 25 torre torre PROPN american_scientific_journal-4152 9 26 as as ADP american_scientific_journal-4152 9 27 a a DET american_scientific_journal-4152 9 28 special special ADJ american_scientific_journal-4152 9 29 case case NOUN american_scientific_journal-4152 9 30 of of ADP american_scientific_journal-4152 9 31 canonical canonical ADJ american_scientific_journal-4152 9 32 transform transform NOUN american_scientific_journal-4152 9 33 with with ADP american_scientific_journal-4152 9 34 characteristic characteristic ADJ american_scientific_journal-4152 9 35 matrix matrix NOUN american_scientific_journal-4152 9 36 and and CCONJ american_scientific_journal-4152 9 37 its its PRON american_scientific_journal-4152 9 38 relation relation NOUN american_scientific_journal-4152 9 39 to to ADP american_scientific_journal-4152 9 40 canonical canonical ADJ american_scientific_journal-4152 9 41 transform transform NOUN american_scientific_journal-4152 9 42 , , PUNCT american_scientific_journal-4152 9 43 and and CCONJ american_scientific_journal-4152 9 44 parabolic parabolic ADJ american_scientific_journal-4152 9 45 differential differential NOUN american_scientific_journal-4152 9 46 equations equation NOUN american_scientific_journal-4152 9 47 are be AUX american_scientific_journal-4152 9 48 discussed discuss VERB american_scientific_journal-4152 9 49 in in ADP american_scientific_journal-4152 9 50 [ [ X american_scientific_journal-4152 9 51 2 2 NUM american_scientific_journal-4152 9 52 ] ] PUNCT american_scientific_journal-4152 9 53 , , PUNCT american_scientific_journal-4152 9 54 the the DET american_scientific_journal-4152 9 55 properties property NOUN american_scientific_journal-4152 9 56 of of ADP american_scientific_journal-4152 9 57 fractional fractional ADJ american_scientific_journal-4152 9 58 laplace laplace NOUN american_scientific_journal-4152 9 59 transform transform NOUN american_scientific_journal-4152 9 60 are be AUX american_scientific_journal-4152 9 61 also also ADV american_scientific_journal-4152 9 62 developed develop VERB american_scientific_journal-4152 9 63 [ [ X american_scientific_journal-4152 9 64 3 3 NUM american_scientific_journal-4152 9 65 ] ] PUNCT american_scientific_journal-4152 9 66 . . PUNCT american_scientific_journal-4152 10 1 ---------------------------------------------------------------------- ---------------------------------------------------------------------- PUNCT american_scientific_journal-4152 11 1 * * PUNCT american_scientific_journal-4152 11 2 corresponding correspond VERB american_scientific_journal-4152 11 3 author author NOUN american_scientific_journal-4152 11 4 . . PUNCT american_scientific_journal-4152 12 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-4152 12 2 american american PROPN american_scientific_journal-4152 12 3 scientific scientific PROPN american_scientific_journal-4152 12 4 research research NOUN american_scientific_journal-4152 12 5 journal journal NOUN american_scientific_journal-4152 12 6 for for ADP american_scientific_journal-4152 12 7 engineering engineering NOUN american_scientific_journal-4152 12 8 , , PUNCT american_scientific_journal-4152 12 9 technology technology NOUN american_scientific_journal-4152 12 10 , , PUNCT american_scientific_journal-4152 12 11 and and CCONJ american_scientific_journal-4152 12 12 sciences science NOUN american_scientific_journal-4152 12 13 ( ( PUNCT american_scientific_journal-4152 12 14 asrjets asrjet NOUN american_scientific_journal-4152 12 15 ) ) PUNCT american_scientific_journal-4152 12 16 ( ( PUNCT american_scientific_journal-4152 12 17 2019 2019 NUM american_scientific_journal-4152 12 18 ) ) PUNCT american_scientific_journal-4152 12 19 volume volume NOUN american_scientific_journal-4152 12 20 55 55 NUM american_scientific_journal-4152 12 21 , , PUNCT american_scientific_journal-4152 12 22 no no DET american_scientific_journal-4152 12 23 1 1 NUM american_scientific_journal-4152 12 24 , , PUNCT american_scientific_journal-4152 12 25 pp pp ADV american_scientific_journal-4152 12 26 11 11 NUM american_scientific_journal-4152 12 27 - - SYM american_scientific_journal-4152 12 28 16 16 NUM american_scientific_journal-4152 12 29 12 12 NUM american_scientific_journal-4152 12 30 in in ADP american_scientific_journal-4152 12 31 2009 2009 NUM american_scientific_journal-4152 12 32 guy guy NOUN american_scientific_journal-4152 12 33 jumarie jumarie PROPN american_scientific_journal-4152 12 34 developed develop VERB american_scientific_journal-4152 12 35 a a DET american_scientific_journal-4152 12 36 new new ADJ american_scientific_journal-4152 12 37 form form NOUN american_scientific_journal-4152 12 38 of of ADP american_scientific_journal-4152 12 39 fractional fractional ADJ american_scientific_journal-4152 12 40 laplace laplace NOUN american_scientific_journal-4152 12 41 transform transform NOUN american_scientific_journal-4152 12 42 with with ADP american_scientific_journal-4152 12 43 the the DET american_scientific_journal-4152 12 44 mittage mittage NOUN american_scientific_journal-4152 12 45 leffler leffl ADJ american_scientific_journal-4152 12 46 function function NOUN american_scientific_journal-4152 12 47 for for ADP american_scientific_journal-4152 12 48 the the DET american_scientific_journal-4152 12 49 entire entire ADJ american_scientific_journal-4152 12 50 class class NOUN american_scientific_journal-4152 12 51 of of ADP american_scientific_journal-4152 12 52 functions function NOUN american_scientific_journal-4152 12 53 that that PRON american_scientific_journal-4152 12 54 are be AUX american_scientific_journal-4152 12 55 fractional fractional ADV american_scientific_journal-4152 12 56 differentiable differentiable ADJ american_scientific_journal-4152 12 57 [ [ X american_scientific_journal-4152 12 58 4 4 NUM american_scientific_journal-4152 12 59 ] ] PUNCT american_scientific_journal-4152 12 60 . . PUNCT american_scientific_journal-4152 13 1 in in ADP american_scientific_journal-4152 13 2 2010 2010 NUM american_scientific_journal-4152 13 3 in in ADP american_scientific_journal-4152 13 4 k. k. PROPN american_scientific_journal-4152 13 5 k. k. PROPN american_scientific_journal-4152 13 6 sharma sharma PROPN american_scientific_journal-4152 13 7 defined define VERB american_scientific_journal-4152 13 8 fractional fractional ADJ american_scientific_journal-4152 13 9 laplace laplace NOUN american_scientific_journal-4152 13 10 transform transform NOUN american_scientific_journal-4152 13 11 as as ADP american_scientific_journal-4152 13 12 a a DET american_scientific_journal-4152 13 13 special special ADJ american_scientific_journal-4152 13 14 case case NOUN american_scientific_journal-4152 13 15 of of ADP american_scientific_journal-4152 13 16 linear linear ADJ american_scientific_journal-4152 13 17 canonical canonical ADJ american_scientific_journal-4152 13 18 transform transform NOUN american_scientific_journal-4152 13 19 with with ADP american_scientific_journal-4152 13 20 representative representative ADJ american_scientific_journal-4152 13 21 matrix matrix NOUN american_scientific_journal-4152 13 22 and and CCONJ american_scientific_journal-4152 13 23 used use VERB american_scientific_journal-4152 13 24 it it PRON american_scientific_journal-4152 13 25 in in ADP american_scientific_journal-4152 13 26 problems problem NOUN american_scientific_journal-4152 13 27 [ [ X american_scientific_journal-4152 13 28 5 5 NUM american_scientific_journal-4152 13 29 ] ] PUNCT american_scientific_journal-4152 13 30 . . PUNCT american_scientific_journal-4152 14 1 the the DET american_scientific_journal-4152 14 2 convolution convolution NOUN american_scientific_journal-4152 14 3 structure structure NOUN american_scientific_journal-4152 14 4 for for ADP american_scientific_journal-4152 14 5 the the DET american_scientific_journal-4152 14 6 two two NUM american_scientific_journal-4152 14 7 versions version NOUN american_scientific_journal-4152 14 8 of of ADP american_scientific_journal-4152 14 9 fractional fractional ADJ american_scientific_journal-4152 14 10 laplace laplace NOUN american_scientific_journal-4152 14 11 transform transform NOUN american_scientific_journal-4152 14 12 was be AUX american_scientific_journal-4152 14 13 developed develop VERB american_scientific_journal-4152 14 14 [ [ PUNCT american_scientific_journal-4152 14 15 6 6 NUM american_scientific_journal-4152 14 16 ] ] PUNCT american_scientific_journal-4152 14 17 . . PUNCT american_scientific_journal-4152 15 1 various various ADJ american_scientific_journal-4152 15 2 of of ADP american_scientific_journal-4152 15 3 fractional fractional ADJ american_scientific_journal-4152 15 4 laplace laplace NOUN american_scientific_journal-4152 15 5 transform transform NOUN american_scientific_journal-4152 15 6 properties property NOUN american_scientific_journal-4152 15 7 are be AUX american_scientific_journal-4152 15 8 discussed discuss VERB american_scientific_journal-4152 15 9 which which PRON american_scientific_journal-4152 15 10 are be AUX american_scientific_journal-4152 15 11 useful useful ADJ american_scientific_journal-4152 15 12 in in ADP american_scientific_journal-4152 15 13 application application NOUN american_scientific_journal-4152 15 14 to to PART american_scientific_journal-4152 15 15 differential differential VERB american_scientific_journal-4152 15 16 and and CCONJ american_scientific_journal-4152 15 17 integral integral ADJ american_scientific_journal-4152 15 18 equations equation NOUN american_scientific_journal-4152 15 19 or or CCONJ american_scientific_journal-4152 15 20 problems problem NOUN american_scientific_journal-4152 15 21 in in ADP american_scientific_journal-4152 15 22 non non ADJ american_scientific_journal-4152 15 23 - - ADJ american_scientific_journal-4152 15 24 extensive extensive ADJ american_scientific_journal-4152 15 25 statistical statistical ADJ american_scientific_journal-4152 15 26 mechanics mechanic NOUN american_scientific_journal-4152 15 27 [ [ X american_scientific_journal-4152 15 28 7 7 NUM american_scientific_journal-4152 15 29 ] ] PUNCT american_scientific_journal-4152 15 30 . . PUNCT american_scientific_journal-4152 16 1 in in ADP american_scientific_journal-4152 16 2 this this DET american_scientific_journal-4152 16 3 paper paper NOUN american_scientific_journal-4152 16 4 we we PRON american_scientific_journal-4152 16 5 have have AUX american_scientific_journal-4152 16 6 established establish VERB american_scientific_journal-4152 16 7 the the DET american_scientific_journal-4152 16 8 relationship relationship NOUN american_scientific_journal-4152 16 9 of of ADP american_scientific_journal-4152 16 10 fractional fractional ADJ american_scientific_journal-4152 16 11 laplace laplace NOUN american_scientific_journal-4152 16 12 transform transform NOUN american_scientific_journal-4152 16 13 with with ADP american_scientific_journal-4152 16 14 other other ADJ american_scientific_journal-4152 16 15 transforms transform NOUN american_scientific_journal-4152 16 16 . . PUNCT american_scientific_journal-4152 17 1 the the DET american_scientific_journal-4152 17 2 fractional fractional ADJ american_scientific_journal-4152 17 3 laplace laplace NOUN american_scientific_journal-4152 17 4 transform transform NOUN american_scientific_journal-4152 17 5 is be AUX american_scientific_journal-4152 17 6 defined define VERB american_scientific_journal-4152 17 7 in in ADP american_scientific_journal-4152 17 8 [ [ PUNCT american_scientific_journal-4152 17 9 5 5 NUM american_scientific_journal-4152 17 10 ] ] PUNCT american_scientific_journal-4152 17 11 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 17 12 ) ) PUNCT american_scientific_journal-4152 17 13 } } PUNCT american_scientific_journal-4152 18 1 = = SYM american_scientific_journal-4152 18 2 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 18 3 οΏ½ οΏ½ NOUN american_scientific_journal-4152 18 4 𝛼𝛼(𝑠𝑠 𝛼𝛼(𝑠𝑠 NOUN american_scientific_journal-4152 18 5 ) ) PUNCT american_scientific_journal-4152 18 6 = = SYM american_scientific_journal-4152 18 7 οΏ½ οΏ½ PROPN american_scientific_journal-4152 18 8 πœ™πœ™(𝑑𝑑)π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 πœ™πœ™(𝑑𝑑)π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 PROPN american_scientific_journal-4152 18 9 ) ) PUNCT american_scientific_journal-4152 18 10 ∞ ∞ PROPN american_scientific_journal-4152 19 1 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 19 2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4152 19 3 ( ( PUNCT american_scientific_journal-4152 19 4 1 1 NUM american_scientific_journal-4152 19 5 ) ) PUNCT american_scientific_journal-4152 19 6 where where SCONJ american_scientific_journal-4152 19 7 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 NOUN american_scientific_journal-4152 19 8 ) ) PUNCT american_scientific_journal-4152 19 9 is be AUX american_scientific_journal-4152 19 10 called call VERB american_scientific_journal-4152 19 11 kernel kernel PROPN american_scientific_journal-4152 20 1 and and CCONJ american_scientific_journal-4152 20 2 it it PRON american_scientific_journal-4152 20 3 is be AUX american_scientific_journal-4152 20 4 defined define VERB american_scientific_journal-4152 20 5 as as ADP american_scientific_journal-4152 20 6 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 PROPN american_scientific_journal-4152 20 7 ) ) PUNCT american_scientific_journal-4152 21 1 = = SYM american_scientific_journal-4152 21 2 οΏ½ οΏ½ PROPN american_scientific_journal-4152 21 3 1 1 NUM american_scientific_journal-4152 21 4 βˆ’ βˆ’ NOUN american_scientific_journal-4152 21 5 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-4152 21 6 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 21 7 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 21 8 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-4152 21 9 cot cot VERB american_scientific_journal-4152 21 10 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 21 11 2 2 NUM american_scientific_journal-4152 21 12 οΏ½ οΏ½ PROPN american_scientific_journal-4152 21 13 𝑑𝑑2βˆ’π‘ π‘ 2 𝑑𝑑2βˆ’π‘ π‘ 2 PROPN american_scientific_journal-4152 21 14 + + ADJ american_scientific_journal-4152 21 15 2𝑖𝑖𝑠𝑠𝑑𝑑 2𝑖𝑖𝑠𝑠𝑑𝑑 NUM american_scientific_journal-4152 21 16 sec𝛼𝛼 sec𝛼𝛼 ADJ american_scientific_journal-4152 21 17 οΏ½ οΏ½ PROPN american_scientific_journal-4152 21 18 ( ( PUNCT american_scientific_journal-4152 21 19 2 2 NUM american_scientific_journal-4152 21 20 ) ) PUNCT american_scientific_journal-4152 21 21 this this PRON american_scientific_journal-4152 21 22 reduces reduce VERB american_scientific_journal-4152 21 23 into into ADP american_scientific_journal-4152 21 24 classical classical ADJ american_scientific_journal-4152 21 25 laplace laplace NOUN american_scientific_journal-4152 21 26 transform transform NOUN american_scientific_journal-4152 21 27 if if SCONJ american_scientific_journal-4152 21 28 we we PRON american_scientific_journal-4152 21 29 put put VERB american_scientific_journal-4152 21 30 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 21 31 = = SYM american_scientific_journal-4152 21 32 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4152 21 33 2 2 NUM american_scientific_journal-4152 21 34 . . PUNCT american_scientific_journal-4152 22 1 the the DET american_scientific_journal-4152 22 2 inversion inversion NOUN american_scientific_journal-4152 22 3 formula formula NOUN american_scientific_journal-4152 22 4 for for ADP american_scientific_journal-4152 22 5 fractional fractional ADJ american_scientific_journal-4152 22 6 laplace laplace NOUN american_scientific_journal-4152 22 7 transform transform NOUN american_scientific_journal-4152 22 8 can can AUX american_scientific_journal-4152 22 9 be be AUX american_scientific_journal-4152 22 10 obtained obtain VERB american_scientific_journal-4152 22 11 by by ADP american_scientific_journal-4152 22 12 replacing replace VERB american_scientific_journal-4152 22 13 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 22 14 by by ADP american_scientific_journal-4152 22 15 βˆ’π›Όπ›Ό βˆ’π›Όπ›Ό X american_scientific_journal-4152 22 16 in in ADP american_scientific_journal-4152 22 17 equations equation NOUN american_scientific_journal-4152 22 18 ( ( PUNCT american_scientific_journal-4152 22 19 1 1 NUM american_scientific_journal-4152 22 20 ) ) PUNCT american_scientific_journal-4152 22 21 and and CCONJ american_scientific_journal-4152 22 22 ( ( PUNCT american_scientific_journal-4152 22 23 2 2 X american_scientific_journal-4152 22 24 ) ) PUNCT american_scientific_journal-4152 22 25 properties property NOUN american_scientific_journal-4152 22 26 of of ADP american_scientific_journal-4152 22 27 kernel kernel PROPN american_scientific_journal-4152 22 28 function function NOUN american_scientific_journal-4152 22 29 of of ADP american_scientific_journal-4152 22 30 laplace laplace NOUN american_scientific_journal-4152 22 31 transform transform NOUN american_scientific_journal-4152 22 32 stated state VERB american_scientific_journal-4152 22 33 in in ADP american_scientific_journal-4152 22 34 [ [ X american_scientific_journal-4152 22 35 3 3 X american_scientific_journal-4152 22 36 ] ] PUNCT american_scientific_journal-4152 22 37 by by ADP american_scientific_journal-4152 22 38 gudadhe gudadhe NOUN american_scientific_journal-4152 22 39 are be AUX american_scientific_journal-4152 22 40 given give VERB american_scientific_journal-4152 22 41 as as SCONJ american_scientific_journal-4152 22 42 follows follow VERB american_scientific_journal-4152 22 43 ⎩ ⎩ PROPN american_scientific_journal-4152 22 44 βŽͺ βŽͺ NOUN american_scientific_journal-4152 22 45 βŽͺ βŽͺ NOUN american_scientific_journal-4152 22 46 ⎨ ⎨ PROPN american_scientific_journal-4152 22 47 βŽͺ βŽͺ NOUN american_scientific_journal-4152 22 48 βŽͺ βŽͺ NOUN american_scientific_journal-4152 22 49 ⎧ ⎧ PROPN american_scientific_journal-4152 22 50 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 PROPN american_scientific_journal-4152 22 51 ) ) PUNCT american_scientific_journal-4152 23 1 = = SYM american_scientific_journal-4152 23 2 π‘˜π‘˜π›Όπ›Ό(𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑒𝑒 PROPN american_scientific_journal-4152 23 3 , , PUNCT american_scientific_journal-4152 23 4 𝑑𝑑 𝑑𝑑 ADJ american_scientific_journal-4152 23 5 ) ) PUNCT american_scientific_journal-4152 23 6 π‘˜π‘˜βˆ’π›Όπ›Ό(𝑑𝑑,𝑒𝑒 π‘˜π‘˜βˆ’π›Όπ›Ό(𝑑𝑑,𝑒𝑒 NUM american_scientific_journal-4152 23 7 ) ) PUNCT american_scientific_journal-4152 23 8 = = SYM american_scientific_journal-4152 23 9 π‘˜π‘˜π›Όπ›Ό π‘˜π‘˜π›Όπ›Ό NOUN american_scientific_journal-4152 23 10 βˆ—(𝑑𝑑,𝑒𝑒 βˆ—(𝑑𝑑,𝑒𝑒 NOUN american_scientific_journal-4152 23 11 ) ) PUNCT american_scientific_journal-4152 23 12 π‘˜π‘˜π›Όπ›Ό(βˆ’π‘‘π‘‘,𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(βˆ’π‘‘π‘‘,𝑒𝑒 NOUN american_scientific_journal-4152 23 13 ) ) PUNCT american_scientific_journal-4152 24 1 = = SYM american_scientific_journal-4152 24 2 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,βˆ’π‘’π‘’ π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,βˆ’π‘’π‘’ PROPN american_scientific_journal-4152 24 3 ) ) PUNCT american_scientific_journal-4152 24 4 οΏ½ οΏ½ PROPN american_scientific_journal-4152 24 5 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒)π‘˜π‘˜π›½π›½(𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒)π‘˜π‘˜π›½π›½(𝑒𝑒 PROPN american_scientific_journal-4152 24 6 , , PUNCT american_scientific_journal-4152 24 7 𝑧𝑧)𝑑𝑑𝑒𝑒 𝑧𝑧)𝑑𝑑𝑒𝑒 NOUN american_scientific_journal-4152 24 8 ∞ ∞ PROPN american_scientific_journal-4152 24 9 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 24 10 = = SYM american_scientific_journal-4152 24 11 π‘˜π‘˜π›Όπ›Ό+𝛽𝛽(𝑑𝑑 π‘˜π‘˜π›Όπ›Ό+𝛽𝛽(𝑑𝑑 PROPN american_scientific_journal-4152 24 12 , , PUNCT american_scientific_journal-4152 24 13 𝑧𝑧 𝑧𝑧 NOUN american_scientific_journal-4152 24 14 ) ) PUNCT american_scientific_journal-4152 24 15 ( ( PUNCT american_scientific_journal-4152 24 16 3 3 X american_scientific_journal-4152 24 17 ) ) PUNCT american_scientific_journal-4152 24 18 οΏ½ οΏ½ PROPN american_scientific_journal-4152 24 19 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 π‘˜π‘˜π›Όπ›Ό(𝑑𝑑,𝑒𝑒 PROPN american_scientific_journal-4152 24 20 ) ) PUNCT american_scientific_journal-4152 24 21 ∞ ∞ PROPN american_scientific_journal-4152 25 1 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 25 2 π‘˜π‘˜π›Όπ›Ό π‘˜π‘˜π›Όπ›Ό NOUN american_scientific_journal-4152 25 3 βˆ—(𝑑𝑑,𝑒𝑒)𝑑𝑑𝑑𝑑 βˆ—(𝑑𝑑,𝑒𝑒)𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4152 25 4 = = SYM american_scientific_journal-4152 25 5 𝛿𝛿(𝑒𝑒 𝛿𝛿(𝑒𝑒 PROPN american_scientific_journal-4152 25 6 βˆ’ βˆ’ NOUN american_scientific_journal-4152 25 7 𝑒𝑒′ 𝑒𝑒′ NOUN american_scientific_journal-4152 25 8 ) ) PUNCT american_scientific_journal-4152 25 9 the the DET american_scientific_journal-4152 25 10 kernel kernel NOUN american_scientific_journal-4152 25 11 which which PRON american_scientific_journal-4152 25 12 is be AUX american_scientific_journal-4152 25 13 defined define VERB american_scientific_journal-4152 25 14 in in ADP american_scientific_journal-4152 25 15 eq eq NOUN american_scientific_journal-4152 25 16 ( ( PUNCT american_scientific_journal-4152 25 17 3 3 NUM american_scientific_journal-4152 25 18 ) ) PUNCT american_scientific_journal-4152 25 19 has have VERB american_scientific_journal-4152 25 20 the the DET american_scientific_journal-4152 25 21 similar similar ADJ american_scientific_journal-4152 25 22 expressions expression NOUN american_scientific_journal-4152 25 23 as as SCONJ american_scientific_journal-4152 25 24 almeida almeida PROPN american_scientific_journal-4152 25 25 defined define VERB american_scientific_journal-4152 25 26 [ [ PUNCT american_scientific_journal-4152 25 27 8 8 NUM american_scientific_journal-4152 25 28 ] ] SYM american_scientific_journal-4152 25 29 . . PUNCT american_scientific_journal-4152 26 1 2 2 X american_scientific_journal-4152 26 2 . . NOUN american_scientific_journal-4152 26 3 results result NOUN american_scientific_journal-4152 26 4 and and CCONJ american_scientific_journal-4152 26 5 discussions discussion NOUN american_scientific_journal-4152 26 6 in in ADP american_scientific_journal-4152 26 7 this this DET american_scientific_journal-4152 26 8 section section NOUN american_scientific_journal-4152 26 9 we we PRON american_scientific_journal-4152 26 10 are be AUX american_scientific_journal-4152 26 11 presenting present VERB american_scientific_journal-4152 26 12 some some DET american_scientific_journal-4152 26 13 important important ADJ american_scientific_journal-4152 26 14 relationship relationship NOUN american_scientific_journal-4152 26 15 of of ADP american_scientific_journal-4152 26 16 fractional fractional ADJ american_scientific_journal-4152 26 17 laplace laplace NOUN american_scientific_journal-4152 26 18 transform transform NOUN american_scientific_journal-4152 26 19 with with ADP american_scientific_journal-4152 26 20 other other ADJ american_scientific_journal-4152 26 21 transform transform NOUN american_scientific_journal-4152 26 22 like like ADP american_scientific_journal-4152 26 23 fractional fractional ADJ american_scientific_journal-4152 26 24 fourier fourier NOUN american_scientific_journal-4152 26 25 transform transform NOUN american_scientific_journal-4152 26 26 , , PUNCT american_scientific_journal-4152 26 27 fractional fractional ADJ american_scientific_journal-4152 26 28 mellin mellin NOUN american_scientific_journal-4152 26 29 transform transform NOUN american_scientific_journal-4152 26 30 and and CCONJ american_scientific_journal-4152 26 31 fractional fractional ADJ american_scientific_journal-4152 26 32 sumudu sumudu NOUN american_scientific_journal-4152 26 33 transform transform NOUN american_scientific_journal-4152 26 34 which which PRON american_scientific_journal-4152 26 35 can can AUX american_scientific_journal-4152 26 36 play play VERB american_scientific_journal-4152 26 37 a a DET american_scientific_journal-4152 26 38 significant significant ADJ american_scientific_journal-4152 26 39 role role NOUN american_scientific_journal-4152 26 40 in in ADP american_scientific_journal-4152 26 41 signal signal ADJ american_scientific_journal-4152 26 42 processing processing NOUN american_scientific_journal-4152 26 43 and and CCONJ american_scientific_journal-4152 26 44 other other ADJ american_scientific_journal-4152 26 45 fields field NOUN american_scientific_journal-4152 26 46 of of ADP american_scientific_journal-4152 26 47 applied applied ADJ american_scientific_journal-4152 26 48 mathematics mathematic NOUN american_scientific_journal-4152 26 49 2.1 2.1 NUM american_scientific_journal-4152 26 50 . . PUNCT american_scientific_journal-4152 27 1 the the DET american_scientific_journal-4152 27 2 relationship relationship NOUN american_scientific_journal-4152 27 3 between between ADP american_scientific_journal-4152 27 4 fractional fractional ADJ american_scientific_journal-4152 27 5 laplace laplace NOUN american_scientific_journal-4152 27 6 transform transform NOUN american_scientific_journal-4152 27 7 with with ADP american_scientific_journal-4152 27 8 fractional fractional ADJ american_scientific_journal-4152 27 9 fourier fourier NOUN american_scientific_journal-4152 27 10 transform transform NOUN american_scientific_journal-4152 27 11 since since SCONJ american_scientific_journal-4152 27 12 fractional fractional ADJ american_scientific_journal-4152 27 13 fourier fourier NOUN american_scientific_journal-4152 27 14 transform transform NOUN american_scientific_journal-4152 27 15 of of ADP american_scientific_journal-4152 27 16 πœ™πœ™(𝑑𝑑 πœ™πœ™(𝑑𝑑 PROPN american_scientific_journal-4152 27 17 ) ) PUNCT american_scientific_journal-4152 27 18 is be AUX american_scientific_journal-4152 27 19 defined define VERB american_scientific_journal-4152 27 20 as as ADP american_scientific_journal-4152 27 21 [ [ X american_scientific_journal-4152 27 22 8 8 NUM american_scientific_journal-4152 27 23 ] ] X american_scientific_journal-4152 27 24 american american ADJ american_scientific_journal-4152 27 25 scientific scientific ADJ american_scientific_journal-4152 27 26 research research PROPN american_scientific_journal-4152 27 27 journal journal NOUN american_scientific_journal-4152 27 28 for for ADP american_scientific_journal-4152 27 29 engineering engineering NOUN american_scientific_journal-4152 27 30 , , PUNCT american_scientific_journal-4152 27 31 technology technology NOUN american_scientific_journal-4152 27 32 , , PUNCT american_scientific_journal-4152 27 33 and and CCONJ american_scientific_journal-4152 27 34 sciences science NOUN american_scientific_journal-4152 27 35 ( ( PUNCT american_scientific_journal-4152 27 36 asrjets asrjet NOUN american_scientific_journal-4152 27 37 ) ) PUNCT american_scientific_journal-4152 27 38 ( ( PUNCT american_scientific_journal-4152 27 39 2019 2019 NUM american_scientific_journal-4152 27 40 ) ) PUNCT american_scientific_journal-4152 27 41 volume volume NOUN american_scientific_journal-4152 27 42 55 55 NUM american_scientific_journal-4152 27 43 , , PUNCT american_scientific_journal-4152 27 44 no no DET american_scientific_journal-4152 27 45 1 1 NUM american_scientific_journal-4152 27 46 , , PUNCT american_scientific_journal-4152 27 47 pp pp ADV american_scientific_journal-4152 27 48 11 11 NUM american_scientific_journal-4152 27 49 - - SYM american_scientific_journal-4152 27 50 16 16 NUM american_scientific_journal-4152 27 51 13 13 NUM american_scientific_journal-4152 27 52 ℱ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℱ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 27 53 ) ) PUNCT american_scientific_journal-4152 27 54 } } PUNCT american_scientific_journal-4152 28 1 = = SYM american_scientific_journal-4152 28 2 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 28 3 οΏ½ οΏ½ NOUN american_scientific_journal-4152 28 4 𝛼𝛼(πœ”πœ” 𝛼𝛼(πœ”πœ” NOUN american_scientific_journal-4152 28 5 ) ) PUNCT american_scientific_journal-4152 28 6 = = SYM american_scientific_journal-4152 28 7 οΏ½ οΏ½ PROPN american_scientific_journal-4152 28 8 1 1 NUM american_scientific_journal-4152 28 9 βˆ’ βˆ’ PROPN american_scientific_journal-4152 28 10 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-4152 28 11 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 28 12 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 29 1 οΏ½ οΏ½ PROPN american_scientific_journal-4152 29 2 1 1 NUM american_scientific_journal-4152 29 3 βˆ’ βˆ’ NOUN american_scientific_journal-4152 29 4 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 29 5 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 29 6 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 29 7 οΏ½ οΏ½ PROPN american_scientific_journal-4152 29 8 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 29 9 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 29 10 cot cot NOUN american_scientific_journal-4152 29 11 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 29 12 2 2 NUM american_scientific_journal-4152 29 13 οΏ½ οΏ½ PROPN american_scientific_journal-4152 29 14 𝑑𝑑2+πœ”πœ”2 𝑑𝑑2+πœ”πœ”2 PUNCT american_scientific_journal-4152 29 15 οΏ½ οΏ½ PROPN american_scientific_journal-4152 29 16 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 PROPN american_scientific_journal-4152 29 17 csc csc PROPN american_scientific_journal-4152 29 18 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4152 29 19 , , PUNCT american_scientific_journal-4152 29 20 ∞ ∞ PROPN american_scientific_journal-4152 29 21 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 29 22 when when SCONJ american_scientific_journal-4152 29 23 𝛼𝛼 𝛼𝛼 ADV american_scientific_journal-4152 29 24 is be AUX american_scientific_journal-4152 29 25 not not PART american_scientific_journal-4152 29 26 a a DET american_scientific_journal-4152 29 27 multiple multiple NOUN american_scientific_journal-4152 29 28 of of ADP american_scientific_journal-4152 29 29 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4152 29 30 ( ( PUNCT american_scientific_journal-4152 29 31 5 5 NUM american_scientific_journal-4152 29 32 ) ) PUNCT american_scientific_journal-4152 29 33 the the DET american_scientific_journal-4152 29 34 fractional fractional ADJ american_scientific_journal-4152 29 35 laplace laplace NOUN american_scientific_journal-4152 29 36 transform transform NOUN american_scientific_journal-4152 29 37 is be AUX american_scientific_journal-4152 29 38 defined define VERB american_scientific_journal-4152 29 39 as as ADP american_scientific_journal-4152 29 40 after after ADP american_scientific_journal-4152 29 41 combining combine VERB american_scientific_journal-4152 29 42 equations equation NOUN american_scientific_journal-4152 29 43 ( ( PUNCT american_scientific_journal-4152 29 44 1 1 NUM american_scientific_journal-4152 29 45 ) ) PUNCT american_scientific_journal-4152 29 46 and and CCONJ american_scientific_journal-4152 29 47 ( ( PUNCT american_scientific_journal-4152 29 48 2 2 X american_scientific_journal-4152 29 49 ) ) PUNCT american_scientific_journal-4152 29 50 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 29 51 ) ) PUNCT american_scientific_journal-4152 29 52 } } PUNCT american_scientific_journal-4152 30 1 = = SYM american_scientific_journal-4152 30 2 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 30 3 οΏ½ οΏ½ NOUN american_scientific_journal-4152 30 4 𝛼𝛼(𝑠𝑠 𝛼𝛼(𝑠𝑠 NOUN american_scientific_journal-4152 30 5 ) ) PUNCT american_scientific_journal-4152 30 6 = = PUNCT american_scientific_journal-4152 30 7 οΏ½ οΏ½ PROPN american_scientific_journal-4152 30 8 1 1 NUM american_scientific_journal-4152 30 9 βˆ’ βˆ’ NOUN american_scientific_journal-4152 30 10 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 30 11 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 30 12 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 30 13 οΏ½ οΏ½ PROPN american_scientific_journal-4152 30 14 πœ™πœ™(𝑑𝑑)𝑒𝑒𝑖𝑖 πœ™πœ™(𝑑𝑑)𝑒𝑒𝑖𝑖 PART american_scientific_journal-4152 30 15 cot cot NOUN american_scientific_journal-4152 30 16 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 30 17 2 2 NUM american_scientific_journal-4152 30 18 οΏ½ οΏ½ NOUN american_scientific_journal-4152 30 19 𝑑𝑑2βˆ’π‘ π‘ 2 𝑑𝑑2βˆ’π‘ π‘ 2 PROPN american_scientific_journal-4152 30 20 + + PROPN american_scientific_journal-4152 30 21 2𝑖𝑖𝑠𝑠𝑑𝑑 2𝑖𝑖𝑠𝑠𝑑𝑑 PROPN american_scientific_journal-4152 30 22 sec sec PROPN american_scientific_journal-4152 30 23 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 30 24 οΏ½ οΏ½ PROPN american_scientific_journal-4152 30 25 ∞ ∞ PROPN american_scientific_journal-4152 30 26 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 30 27 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4152 30 28 , , PUNCT american_scientific_journal-4152 30 29 when when SCONJ american_scientific_journal-4152 30 30 𝛼𝛼 𝛼𝛼 ADV american_scientific_journal-4152 30 31 is be AUX american_scientific_journal-4152 30 32 not not PART american_scientific_journal-4152 30 33 a a DET american_scientific_journal-4152 30 34 multiple multiple NOUN american_scientific_journal-4152 30 35 of of ADP american_scientific_journal-4152 30 36 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4152 30 37 ( ( PUNCT american_scientific_journal-4152 30 38 6 6 NUM american_scientific_journal-4152 30 39 ) ) PUNCT american_scientific_journal-4152 30 40 since since SCONJ american_scientific_journal-4152 30 41 𝑠𝑠 𝑠𝑠 ADV american_scientific_journal-4152 30 42 is be AUX american_scientific_journal-4152 30 43 complex complex ADJ american_scientific_journal-4152 30 44 therefore therefore ADV american_scientific_journal-4152 30 45 substituting substitute VERB american_scientific_journal-4152 30 46 𝑠𝑠 𝑠𝑠 ADV american_scientific_journal-4152 30 47 = = SYM american_scientific_journal-4152 30 48 𝜎𝜎 𝜎𝜎 PROPN american_scientific_journal-4152 30 49 + + NUM american_scientific_journal-4152 30 50 π‘–π‘–πœ”πœ” π‘–π‘–πœ”πœ” NOUN american_scientific_journal-4152 30 51 in in ADP american_scientific_journal-4152 30 52 in in ADP american_scientific_journal-4152 30 53 equation equation NOUN american_scientific_journal-4152 30 54 ( ( PUNCT american_scientific_journal-4152 30 55 6 6 NUM american_scientific_journal-4152 30 56 ) ) PUNCT american_scientific_journal-4152 30 57 we we PRON american_scientific_journal-4152 30 58 get get VERB american_scientific_journal-4152 30 59 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 30 60 ) ) PUNCT american_scientific_journal-4152 30 61 } } PUNCT american_scientific_journal-4152 31 1 = = SYM american_scientific_journal-4152 31 2 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 31 3 οΏ½ οΏ½ NOUN american_scientific_journal-4152 31 4 𝛼𝛼(𝜎𝜎 𝛼𝛼(𝜎𝜎 NOUN american_scientific_journal-4152 31 5 + + CCONJ american_scientific_journal-4152 31 6 π‘–π‘–πœ”πœ” π‘–π‘–πœ”πœ” NOUN american_scientific_journal-4152 31 7 ) ) PUNCT american_scientific_journal-4152 31 8 = = SYM american_scientific_journal-4152 31 9 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 10 1 1 NUM american_scientific_journal-4152 31 11 βˆ’ βˆ’ NOUN american_scientific_journal-4152 31 12 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 31 13 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 31 14 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 31 15 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 16 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 31 17 𝑖𝑖 𝑖𝑖 X american_scientific_journal-4152 31 18 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 31 19 2 2 NUM american_scientific_journal-4152 31 20 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 21 𝑑𝑑2βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)2 𝑑𝑑2βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)2 PROPN american_scientific_journal-4152 31 22 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 23 βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 PRON american_scientific_journal-4152 31 24 csc csc PROPN american_scientific_journal-4152 31 25 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4152 31 26 ∞ ∞ PROPN american_scientific_journal-4152 31 27 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 31 28 = = SYM american_scientific_journal-4152 31 29 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 30 1 1 NUM american_scientific_journal-4152 31 31 βˆ’ βˆ’ PROPN american_scientific_journal-4152 31 32 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 31 33 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 31 34 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 31 35 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 36 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 31 37 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 31 38 cot cot NOUN american_scientific_journal-4152 31 39 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 31 40 2 2 NUM american_scientific_journal-4152 31 41 οΏ½ οΏ½ NOUN american_scientific_journal-4152 31 42 𝑑𝑑2βˆ’πœŽπœŽ2+πœ”πœ”2βˆ’2π‘–π‘–πœŽπœŽπœ”πœ” 𝑑𝑑2βˆ’πœŽπœŽ2+πœ”πœ”2βˆ’2π‘–π‘–πœŽπœŽπœ”πœ” NOUN american_scientific_journal-4152 31 43 οΏ½ οΏ½ PROPN american_scientific_journal-4152 31 44 βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 PRON american_scientific_journal-4152 31 45 csc csc PROPN american_scientific_journal-4152 31 46 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4152 31 47 ∞ ∞ PROPN american_scientific_journal-4152 32 1 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 32 2 = = SYM american_scientific_journal-4152 32 3 π‘’π‘’βˆ’ π‘’π‘’βˆ’ NUM american_scientific_journal-4152 32 4 𝑖𝑖 𝑖𝑖 X american_scientific_journal-4152 32 5 cot𝛼𝛼 cot𝛼𝛼 X american_scientific_journal-4152 32 6 2 2 NUM american_scientific_journal-4152 32 7 οΏ½ οΏ½ PROPN american_scientific_journal-4152 32 8 𝜎𝜎2 𝜎𝜎2 PROPN american_scientific_journal-4152 32 9 + + PROPN american_scientific_journal-4152 32 10 2π‘–π‘–πœŽπœŽπœ”πœ” 2π‘–π‘–πœŽπœŽπœ”πœ” PROPN american_scientific_journal-4152 32 11 οΏ½ οΏ½ NOUN american_scientific_journal-4152 32 12 οΏ½ οΏ½ PROPN american_scientific_journal-4152 32 13 1 1 NUM american_scientific_journal-4152 32 14 βˆ’ βˆ’ NOUN american_scientific_journal-4152 32 15 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 32 16 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 32 17 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 32 18 οΏ½ οΏ½ PROPN american_scientific_journal-4152 32 19 πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ PRON american_scientific_journal-4152 32 20 csc csc PROPN american_scientific_journal-4152 32 21 𝛼𝛼𝑒𝑒 𝛼𝛼𝑒𝑒 PROPN american_scientific_journal-4152 32 22 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-4152 32 23 cot cot NOUN american_scientific_journal-4152 32 24 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 32 25 2 2 NUM american_scientific_journal-4152 32 26 οΏ½ οΏ½ PROPN american_scientific_journal-4152 32 27 𝑑𝑑2+πœ”πœ”2 𝑑𝑑2+πœ”πœ”2 PUNCT american_scientific_journal-4152 32 28 οΏ½ οΏ½ PROPN american_scientific_journal-4152 32 29 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 PROPN american_scientific_journal-4152 32 30 csc csc PROPN american_scientific_journal-4152 32 31 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4152 32 32 ∞ ∞ PROPN american_scientific_journal-4152 33 1 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 33 2 = = SYM american_scientific_journal-4152 33 3 π‘’π‘’βˆ’ π‘’π‘’βˆ’ PUNCT american_scientific_journal-4152 33 4 𝑖𝑖 𝑖𝑖 PRON american_scientific_journal-4152 33 5 cot cot NOUN american_scientific_journal-4152 33 6 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 33 7 2 2 NUM american_scientific_journal-4152 33 8 οΏ½ οΏ½ NOUN american_scientific_journal-4152 33 9 𝜎𝜎2 𝜎𝜎2 PROPN american_scientific_journal-4152 33 10 + + PROPN american_scientific_journal-4152 33 11 2π‘–π‘–πœŽπœŽπœ”πœ” 2π‘–π‘–πœŽπœŽπœ”πœ” PROPN american_scientific_journal-4152 33 12 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 13 οΏ½ οΏ½ SYM american_scientific_journal-4152 33 14 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 15 1 1 NUM american_scientific_journal-4152 33 16 βˆ’ βˆ’ NOUN american_scientific_journal-4152 33 17 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 33 18 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 33 19 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 33 20 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 21 ( ( PUNCT american_scientific_journal-4152 33 22 πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ X american_scientific_journal-4152 33 23 csc csc PROPN american_scientific_journal-4152 33 24 𝛼𝛼)𝑒𝑒 𝛼𝛼)𝑒𝑒 PROPN american_scientific_journal-4152 33 25 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 33 26 cot cot NOUN american_scientific_journal-4152 33 27 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 33 28 2 2 NUM american_scientific_journal-4152 33 29 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 30 𝑑𝑑2+πœ”πœ”2 𝑑𝑑2+πœ”πœ”2 PUNCT american_scientific_journal-4152 33 31 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 32 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 PROPN american_scientific_journal-4152 33 33 csc csc PROPN american_scientific_journal-4152 33 34 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4152 33 35 ∞ ∞ PROPN american_scientific_journal-4152 33 36 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 33 37 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 38 ( ( PUNCT american_scientific_journal-4152 33 39 7 7 NUM american_scientific_journal-4152 33 40 ) ) PUNCT american_scientific_journal-4152 33 41 from from ADP american_scientific_journal-4152 33 42 ( ( PUNCT american_scientific_journal-4152 33 43 5 5 NUM american_scientific_journal-4152 33 44 ) ) PUNCT american_scientific_journal-4152 33 45 and and CCONJ american_scientific_journal-4152 33 46 ( ( PUNCT american_scientific_journal-4152 33 47 7 7 X american_scientific_journal-4152 33 48 ) ) PUNCT american_scientific_journal-4152 33 49 we we PRON american_scientific_journal-4152 33 50 get get VERB american_scientific_journal-4152 33 51 the the DET american_scientific_journal-4152 33 52 result result NOUN american_scientific_journal-4152 33 53 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 33 54 ) ) PUNCT american_scientific_journal-4152 33 55 } } PUNCT american_scientific_journal-4152 33 56 = = SYM american_scientific_journal-4152 33 57 π‘’π‘’βˆ’ π‘’π‘’βˆ’ NUM american_scientific_journal-4152 33 58 𝑖𝑖 𝑖𝑖 X american_scientific_journal-4152 33 59 cot𝛼𝛼 cot𝛼𝛼 X american_scientific_journal-4152 33 60 2 2 NUM american_scientific_journal-4152 33 61 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 62 𝜎𝜎2 𝜎𝜎2 PROPN american_scientific_journal-4152 33 63 + + PROPN american_scientific_journal-4152 33 64 2π‘–π‘–πœŽπœŽπœ”πœ” 2π‘–π‘–πœŽπœŽπœ”πœ” PROPN american_scientific_journal-4152 33 65 οΏ½ οΏ½ PROPN american_scientific_journal-4152 33 66 ℱ𝛼𝛼{πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ ℱ𝛼𝛼{πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ PROPN american_scientific_journal-4152 33 67 csc csc PROPN american_scientific_journal-4152 33 68 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 33 69 } } PUNCT american_scientific_journal-4152 33 70 ( ( PUNCT american_scientific_journal-4152 33 71 8) 8) NUM american_scientific_journal-4152 33 72 eq eq NOUN american_scientific_journal-4152 33 73 ( ( PUNCT american_scientific_journal-4152 33 74 8) 8) NUM american_scientific_journal-4152 33 75 shows show VERB american_scientific_journal-4152 33 76 the the DET american_scientific_journal-4152 33 77 relationship relationship NOUN american_scientific_journal-4152 33 78 of of ADP american_scientific_journal-4152 33 79 fractional fractional ADJ american_scientific_journal-4152 33 80 laplace laplace NOUN american_scientific_journal-4152 33 81 transform transform NOUN american_scientific_journal-4152 33 82 with with ADP american_scientific_journal-4152 33 83 fractional fractional ADJ american_scientific_journal-4152 33 84 fourier fourier NOUN american_scientific_journal-4152 33 85 transform transform NOUN american_scientific_journal-4152 33 86 and and CCONJ american_scientific_journal-4152 33 87 this this DET american_scientific_journal-4152 33 88 relation relation NOUN american_scientific_journal-4152 33 89 will will AUX american_scientific_journal-4152 33 90 reduces reduce VERB american_scientific_journal-4152 33 91 to to ADP american_scientific_journal-4152 33 92 classical classical ADJ american_scientific_journal-4152 33 93 relation relation NOUN american_scientific_journal-4152 33 94 of of ADP american_scientific_journal-4152 33 95 laplace laplace NOUN american_scientific_journal-4152 33 96 transform transform NOUN american_scientific_journal-4152 33 97 and and CCONJ american_scientific_journal-4152 33 98 fourier fourier NOUN american_scientific_journal-4152 33 99 transform transform NOUN american_scientific_journal-4152 33 100 if if SCONJ american_scientific_journal-4152 33 101 we we PRON american_scientific_journal-4152 33 102 put put VERB american_scientific_journal-4152 33 103 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 33 104 = = SYM american_scientific_journal-4152 33 105 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4152 33 106 2 2 NUM american_scientific_journal-4152 33 107 . . PUNCT american_scientific_journal-4152 34 1 2.2 2.2 NUM american_scientific_journal-4152 34 2 . . PUNCT american_scientific_journal-4152 34 3 relation relation NOUN american_scientific_journal-4152 34 4 between between ADP american_scientific_journal-4152 34 5 fractional fractional ADJ american_scientific_journal-4152 34 6 laplace laplace NOUN american_scientific_journal-4152 34 7 transform transform NOUN american_scientific_journal-4152 34 8 and and CCONJ american_scientific_journal-4152 34 9 fractional fractional ADJ american_scientific_journal-4152 34 10 mellin mellin PROPN american_scientific_journal-4152 34 11 transform transform NOUN american_scientific_journal-4152 34 12 in in ADP american_scientific_journal-4152 34 13 order order NOUN american_scientific_journal-4152 34 14 to to PART american_scientific_journal-4152 34 15 develop develop VERB american_scientific_journal-4152 34 16 a a DET american_scientific_journal-4152 34 17 relation relation NOUN american_scientific_journal-4152 34 18 of of ADP american_scientific_journal-4152 34 19 frlt frlt NOUN american_scientific_journal-4152 34 20 with with ADP american_scientific_journal-4152 34 21 frmt frmt NOUN american_scientific_journal-4152 34 22 let let VERB american_scientific_journal-4152 34 23 us we PRON american_scientific_journal-4152 34 24 consider consider VERB american_scientific_journal-4152 34 25 the the DET american_scientific_journal-4152 34 26 change change NOUN american_scientific_journal-4152 34 27 of of ADP american_scientific_journal-4152 34 28 variable variable NOUN american_scientific_journal-4152 34 29 which which PRON american_scientific_journal-4152 34 30 is be AUX american_scientific_journal-4152 34 31 defined define VERB american_scientific_journal-4152 34 32 by by ADP american_scientific_journal-4152 34 33 π‘’π‘’βˆ’π‘‘π‘‘ π‘’π‘’βˆ’π‘‘π‘‘ NOUN american_scientific_journal-4152 34 34 = = PUNCT american_scientific_journal-4152 34 35 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-4152 34 36 replacing replace VERB american_scientific_journal-4152 34 37 𝑑𝑑 𝑑𝑑 NOUN american_scientific_journal-4152 34 38 by by ADP american_scientific_journal-4152 34 39 π‘’π‘’βˆ’π‘‘π‘‘ π‘’π‘’βˆ’π‘‘π‘‘ NOUN american_scientific_journal-4152 34 40 in in ADP american_scientific_journal-4152 34 41 equation equation NOUN american_scientific_journal-4152 34 42 ( ( PUNCT american_scientific_journal-4152 34 43 6 6 NUM american_scientific_journal-4152 34 44 ) ) PUNCT american_scientific_journal-4152 34 45 american american ADJ american_scientific_journal-4152 34 46 scientific scientific ADJ american_scientific_journal-4152 34 47 research research NOUN american_scientific_journal-4152 34 48 journal journal NOUN american_scientific_journal-4152 34 49 for for ADP american_scientific_journal-4152 34 50 engineering engineering NOUN american_scientific_journal-4152 34 51 , , PUNCT american_scientific_journal-4152 34 52 technology technology NOUN american_scientific_journal-4152 34 53 , , PUNCT american_scientific_journal-4152 34 54 and and CCONJ american_scientific_journal-4152 34 55 sciences science NOUN american_scientific_journal-4152 34 56 ( ( PUNCT american_scientific_journal-4152 34 57 asrjets asrjet NOUN american_scientific_journal-4152 34 58 ) ) PUNCT american_scientific_journal-4152 34 59 ( ( PUNCT american_scientific_journal-4152 34 60 2019 2019 NUM american_scientific_journal-4152 34 61 ) ) PUNCT american_scientific_journal-4152 34 62 volume volume NOUN american_scientific_journal-4152 34 63 55 55 NUM american_scientific_journal-4152 34 64 , , PUNCT american_scientific_journal-4152 34 65 no no DET american_scientific_journal-4152 34 66 1 1 NUM american_scientific_journal-4152 34 67 , , PUNCT american_scientific_journal-4152 34 68 pp pp ADV american_scientific_journal-4152 34 69 11 11 NUM american_scientific_journal-4152 34 70 - - SYM american_scientific_journal-4152 34 71 16 16 NUM american_scientific_journal-4152 34 72 14 14 NUM american_scientific_journal-4152 34 73 we we PRON american_scientific_journal-4152 34 74 get get VERB american_scientific_journal-4152 34 75 ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)}(𝑠𝑠 ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)}(𝑠𝑠 NOUN american_scientific_journal-4152 34 76 ) ) PUNCT american_scientific_journal-4152 34 77 = = SYM american_scientific_journal-4152 34 78 οΏ½ οΏ½ PROPN american_scientific_journal-4152 34 79 1βˆ’π‘–π‘– 1βˆ’π‘–π‘– NUM american_scientific_journal-4152 34 80 cot cot NOUN american_scientific_journal-4152 34 81 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 34 82 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 34 83 ∫ ∫ PROPN american_scientific_journal-4152 34 84 πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)𝑒𝑒 πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘)𝑒𝑒 PROPN american_scientific_journal-4152 34 85 𝑖𝑖cot𝛼𝛼 𝑖𝑖cot𝛼𝛼 VERB american_scientific_journal-4152 34 86 2 2 NUM american_scientific_journal-4152 34 87 οΏ½ οΏ½ PROPN american_scientific_journal-4152 34 88 𝑑𝑑2βˆ’π‘ π‘ 2 𝑑𝑑2βˆ’π‘ π‘ 2 PROPN american_scientific_journal-4152 34 89 οΏ½ οΏ½ NOUN american_scientific_journal-4152 34 90 βˆ’π‘ π‘ π‘‘π‘‘ βˆ’π‘ π‘ π‘‘π‘‘ ADJ american_scientific_journal-4152 34 91 cscπ›Όπ›Όπ‘‘π‘‘π‘‘π‘‘βˆž cscπ›Όπ›Όπ‘‘π‘‘π‘‘π‘‘βˆž NOUN american_scientific_journal-4152 34 92 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 34 93 ( ( PUNCT american_scientific_journal-4152 34 94 9 9 X american_scientific_journal-4152 34 95 ) ) PUNCT american_scientific_journal-4152 34 96 substituting substitute VERB american_scientific_journal-4152 34 97 π‘’π‘’βˆ’π‘‘π‘‘ π‘’π‘’βˆ’π‘‘π‘‘ NOUN american_scientific_journal-4152 34 98 = = SYM american_scientific_journal-4152 34 99 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4152 34 100 ⟹ ⟹ ADV american_scientific_journal-4152 34 101 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 NOUN american_scientific_journal-4152 35 1 = = SYM american_scientific_journal-4152 36 1 βˆ’π‘‘π‘‘π‘‘π‘‘ βˆ’π‘‘π‘‘π‘‘π‘‘ NOUN american_scientific_journal-4152 37 1 𝑑𝑑 𝑑𝑑 VERB american_scientific_journal-4152 37 2 when when SCONJ american_scientific_journal-4152 37 3 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4152 37 4 β†’ β†’ SYM american_scientific_journal-4152 37 5 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 37 6 then then ADV american_scientific_journal-4152 37 7 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4152 37 8 β†’ β†’ SYM american_scientific_journal-4152 37 9 ∞ ∞ PROPN american_scientific_journal-4152 37 10 , , PUNCT american_scientific_journal-4152 37 11 when when SCONJ american_scientific_journal-4152 37 12 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4152 37 13 β†’ β†’ SYM american_scientific_journal-4152 37 14 ∞ ∞ PROPN american_scientific_journal-4152 37 15 then then ADV american_scientific_journal-4152 37 16 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4152 37 17 β†’ β†’ SYM american_scientific_journal-4152 37 18 0 0 NUM american_scientific_journal-4152 37 19 in in ADP american_scientific_journal-4152 37 20 equation equation NOUN american_scientific_journal-4152 37 21 ( ( PUNCT american_scientific_journal-4152 37 22 6 6 NUM american_scientific_journal-4152 37 23 ) ) PUNCT american_scientific_journal-4152 37 24 we we PRON american_scientific_journal-4152 37 25 get get VERB american_scientific_journal-4152 37 26 = = PUNCT american_scientific_journal-4152 38 1 βˆ’ βˆ’ NOUN american_scientific_journal-4152 38 2 οΏ½ οΏ½ PROPN american_scientific_journal-4152 38 3 1 1 NUM american_scientific_journal-4152 38 4 βˆ’ βˆ’ NOUN american_scientific_journal-4152 38 5 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 38 6 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 38 7 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 38 8 οΏ½ οΏ½ PROPN american_scientific_journal-4152 39 1 πœ™πœ™(𝑦𝑦)𝑒𝑒 πœ™πœ™(𝑦𝑦)𝑒𝑒 PROPN american_scientific_journal-4152 39 2 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 39 3 cot cot NOUN american_scientific_journal-4152 39 4 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 39 5 2 2 NUM american_scientific_journal-4152 39 6 οΏ½ οΏ½ PROPN american_scientific_journal-4152 39 7 ln𝑑𝑑2βˆ’π‘ π‘ 2 ln𝑑𝑑2βˆ’π‘ π‘ 2 NOUN american_scientific_journal-4152 39 8 οΏ½ οΏ½ PROPN american_scientific_journal-4152 39 9 +𝑠𝑠 +𝑠𝑠 X american_scientific_journal-4152 39 10 𝑐𝑐𝑠𝑠𝑐𝑐 𝑐𝑐𝑠𝑠𝑐𝑐 NOUN american_scientific_journal-4152 39 11 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 39 12 ln ln PROPN american_scientific_journal-4152 39 13 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-4152 39 14 𝑑𝑑𝑦𝑦 𝑑𝑑𝑦𝑦 NOUN american_scientific_journal-4152 39 15 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-4152 39 16 0 0 NUM american_scientific_journal-4152 39 17 ∞ ∞ NUM american_scientific_journal-4152 39 18 = = SYM american_scientific_journal-4152 39 19 οΏ½ οΏ½ PROPN american_scientific_journal-4152 39 20 1 1 NUM american_scientific_journal-4152 39 21 βˆ’ βˆ’ PROPN american_scientific_journal-4152 39 22 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 39 23 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 39 24 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 39 25 οΏ½ οΏ½ PROPN american_scientific_journal-4152 40 1 πœ™πœ™(𝑦𝑦)𝑒𝑒 πœ™πœ™(𝑦𝑦)𝑒𝑒 PROPN american_scientific_journal-4152 40 2 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 40 3 cot cot NOUN american_scientific_journal-4152 40 4 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 40 5 2 2 NUM american_scientific_journal-4152 40 6 οΏ½ οΏ½ PROPN american_scientific_journal-4152 40 7 ln𝑑𝑑2βˆ’π‘ π‘ 2 ln𝑑𝑑2βˆ’π‘ π‘ 2 NOUN american_scientific_journal-4152 40 8 οΏ½ οΏ½ PROPN american_scientific_journal-4152 40 9 +ln +ln NOUN american_scientific_journal-4152 40 10 𝑑𝑑𝑠𝑠csc𝛼𝛼 𝑑𝑑𝑠𝑠csc𝛼𝛼 NOUN american_scientific_journal-4152 40 11 𝑑𝑑𝑦𝑦 𝑑𝑑𝑦𝑦 NOUN american_scientific_journal-4152 40 12 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-4152 40 13 ∞ ∞ PROPN american_scientific_journal-4152 40 14 0 0 PUNCT american_scientific_journal-4152 41 1 = = SYM american_scientific_journal-4152 41 2 οΏ½ οΏ½ PROPN american_scientific_journal-4152 41 3 1 1 NUM american_scientific_journal-4152 41 4 βˆ’ βˆ’ NOUN american_scientific_journal-4152 41 5 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 41 6 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 41 7 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 41 8 οΏ½ οΏ½ PROPN american_scientific_journal-4152 42 1 πœ™πœ™(𝑦𝑦)𝑒𝑒 πœ™πœ™(𝑦𝑦)𝑒𝑒 PROPN american_scientific_journal-4152 42 2 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 42 3 cot cot NOUN american_scientific_journal-4152 42 4 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 42 5 2 2 NUM american_scientific_journal-4152 42 6 οΏ½ οΏ½ PROPN american_scientific_journal-4152 42 7 ln𝑑𝑑2βˆ’π‘ π‘ 2 ln𝑑𝑑2βˆ’π‘ π‘ 2 PROPN american_scientific_journal-4152 42 8 οΏ½ οΏ½ PROPN american_scientific_journal-4152 42 9 𝑦𝑦𝑠𝑠 𝑦𝑦𝑠𝑠 PROPN american_scientific_journal-4152 42 10 csc csc PROPN american_scientific_journal-4152 42 11 π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 PROPN american_scientific_journal-4152 42 12 ( ( PUNCT american_scientific_journal-4152 42 13 10 10 NUM american_scientific_journal-4152 42 14 ) ) PUNCT american_scientific_journal-4152 42 15 ∞ ∞ NOUN american_scientific_journal-4152 42 16 0 0 NUM american_scientific_journal-4152 43 1 in in ADP american_scientific_journal-4152 43 2 [ [ X american_scientific_journal-4152 43 3 3 3 NUM american_scientific_journal-4152 43 4 ] ] X american_scientific_journal-4152 43 5 fractional fractional ADJ american_scientific_journal-4152 43 6 mellin mellin PROPN american_scientific_journal-4152 43 7 transform transform NOUN american_scientific_journal-4152 43 8 is be AUX american_scientific_journal-4152 43 9 defined define VERB american_scientific_journal-4152 43 10 as as ADP american_scientific_journal-4152 43 11 ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦 ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦 ADJ american_scientific_journal-4152 43 12 ) ) PUNCT american_scientific_journal-4152 43 13 } } PUNCT american_scientific_journal-4152 43 14 = = SYM american_scientific_journal-4152 43 15 οΏ½ οΏ½ PROPN american_scientific_journal-4152 43 16 1 1 NUM american_scientific_journal-4152 43 17 βˆ’ βˆ’ NOUN american_scientific_journal-4152 43 18 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 43 19 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 43 20 2πœ‹πœ‹ 2πœ‹πœ‹ VERB american_scientific_journal-4152 43 21 οΏ½ οΏ½ PROPN american_scientific_journal-4152 44 1 πœ™πœ™(𝑦𝑦)𝑒𝑒 πœ™πœ™(𝑦𝑦)𝑒𝑒 PROPN american_scientific_journal-4152 44 2 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 44 3 cot cot NOUN american_scientific_journal-4152 44 4 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 44 5 2 2 NUM american_scientific_journal-4152 44 6 οΏ½ οΏ½ PROPN american_scientific_journal-4152 44 7 ln ln NOUN american_scientific_journal-4152 44 8 𝑑𝑑2βˆ’π‘ π‘ 2 𝑑𝑑2βˆ’π‘ π‘ 2 NOUN american_scientific_journal-4152 44 9 οΏ½ οΏ½ PROPN american_scientific_journal-4152 44 10 𝑦𝑦𝑠𝑠 𝑦𝑦𝑠𝑠 PROPN american_scientific_journal-4152 44 11 csc csc PROPN american_scientific_journal-4152 44 12 π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 π›Όπ›Όβˆ’1𝑑𝑑𝑦𝑦 PROPN american_scientific_journal-4152 44 13 ∞ ∞ PROPN american_scientific_journal-4152 44 14 0 0 NUM american_scientific_journal-4152 45 1 ( ( PUNCT american_scientific_journal-4152 45 2 11 11 NUM american_scientific_journal-4152 45 3 ) ) PUNCT american_scientific_journal-4152 45 4 from from ADP american_scientific_journal-4152 45 5 ( ( PUNCT american_scientific_journal-4152 45 6 10 10 NUM american_scientific_journal-4152 45 7 ) ) PUNCT american_scientific_journal-4152 45 8 and and CCONJ american_scientific_journal-4152 45 9 ( ( PUNCT american_scientific_journal-4152 45 10 11 11 X american_scientific_journal-4152 45 11 ) ) PUNCT american_scientific_journal-4152 45 12 we we PRON american_scientific_journal-4152 45 13 get get VERB american_scientific_journal-4152 45 14 the the DET american_scientific_journal-4152 45 15 relation relation NOUN american_scientific_journal-4152 45 16 between between ADP american_scientific_journal-4152 45 17 fractional fractional ADJ american_scientific_journal-4152 45 18 laplace laplace NOUN american_scientific_journal-4152 45 19 transform transform NOUN american_scientific_journal-4152 45 20 and and CCONJ american_scientific_journal-4152 45 21 fractional fractional ADJ american_scientific_journal-4152 45 22 mellin mellin PROPN american_scientific_journal-4152 45 23 transform transform NOUN american_scientific_journal-4152 45 24 ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘ ℒ𝛼𝛼{πœ™πœ™(π‘’π‘’βˆ’π‘‘π‘‘ PROPN american_scientific_journal-4152 45 25 ) ) PUNCT american_scientific_journal-4152 45 26 } } PUNCT american_scientific_journal-4152 46 1 = = SYM american_scientific_journal-4152 46 2 ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦 ℳ𝛼𝛼{πœ™πœ™(𝑦𝑦 ADJ american_scientific_journal-4152 46 3 ) ) PUNCT american_scientific_journal-4152 46 4 } } PUNCT american_scientific_journal-4152 46 5 this this PRON american_scientific_journal-4152 46 6 reduces reduce VERB american_scientific_journal-4152 46 7 to to ADP american_scientific_journal-4152 46 8 classical classical ADJ american_scientific_journal-4152 46 9 relation relation NOUN american_scientific_journal-4152 46 10 between between ADP american_scientific_journal-4152 46 11 laplace laplace NOUN american_scientific_journal-4152 46 12 transform transform VERB american_scientific_journal-4152 46 13 mellin mellin NOUN american_scientific_journal-4152 46 14 transform transform NOUN american_scientific_journal-4152 46 15 for for ADP american_scientific_journal-4152 46 16 𝛼𝛼 𝛼𝛼 NOUN american_scientific_journal-4152 46 17 = = SYM american_scientific_journal-4152 46 18 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-4152 46 19 2 2 NUM american_scientific_journal-4152 46 20 . . PUNCT american_scientific_journal-4152 47 1 2.3 2.3 NUM american_scientific_journal-4152 47 2 . . PUNCT american_scientific_journal-4152 47 3 relation relation NOUN american_scientific_journal-4152 47 4 between between ADP american_scientific_journal-4152 47 5 fractional fractional ADJ american_scientific_journal-4152 47 6 laplace laplace NOUN american_scientific_journal-4152 47 7 transform transform NOUN american_scientific_journal-4152 47 8 and and CCONJ american_scientific_journal-4152 47 9 fractional fractional ADJ american_scientific_journal-4152 47 10 sumudu sumudu NOUN american_scientific_journal-4152 47 11 transform transform NOUN american_scientific_journal-4152 47 12 in in ADP american_scientific_journal-4152 47 13 [ [ X american_scientific_journal-4152 47 14 10 10 NUM american_scientific_journal-4152 47 15 ] ] PUNCT american_scientific_journal-4152 47 16 the the DET american_scientific_journal-4152 47 17 duality duality NOUN american_scientific_journal-4152 47 18 relation relation NOUN american_scientific_journal-4152 47 19 between between ADP american_scientific_journal-4152 47 20 two two NUM american_scientific_journal-4152 47 21 sided sided ADJ american_scientific_journal-4152 47 22 laplace laplace NOUN american_scientific_journal-4152 47 23 transform transform NOUN american_scientific_journal-4152 47 24 and and CCONJ american_scientific_journal-4152 47 25 two two NUM american_scientific_journal-4152 47 26 sided sided ADJ american_scientific_journal-4152 47 27 sumudu sumudu NOUN american_scientific_journal-4152 47 28 transform transform NOUN american_scientific_journal-4152 47 29 is be AUX american_scientific_journal-4152 47 30 given give VERB american_scientific_journal-4152 47 31 by by ADP american_scientific_journal-4152 47 32 𝐺𝐺(𝑒𝑒 𝐺𝐺(𝑒𝑒 PROPN american_scientific_journal-4152 47 33 ) ) PUNCT american_scientific_journal-4152 47 34 = = SYM american_scientific_journal-4152 48 1 1 1 NUM american_scientific_journal-4152 48 2 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-4152 48 3 𝐹𝐹(𝑠𝑠 𝐹𝐹(𝑠𝑠 NOUN american_scientific_journal-4152 48 4 ) ) PUNCT american_scientific_journal-4152 48 5 οΏ½ οΏ½ PROPN american_scientific_journal-4152 48 6 𝑠𝑠=1𝑒𝑒 𝑠𝑠=1𝑒𝑒 PROPN american_scientific_journal-4152 48 7 ( ( PUNCT american_scientific_journal-4152 48 8 12 12 NUM american_scientific_journal-4152 48 9 ) ) PUNCT american_scientific_journal-4152 48 10 or or CCONJ american_scientific_journal-4152 48 11 𝐹𝐹(𝑠𝑠 𝐹𝐹(𝑠𝑠 NOUN american_scientific_journal-4152 48 12 ) ) PUNCT american_scientific_journal-4152 48 13 = = SYM american_scientific_journal-4152 49 1 1 1 NUM american_scientific_journal-4152 49 2 𝑠𝑠 𝑠𝑠 ADV american_scientific_journal-4152 49 3 𝐺𝐺(𝑒𝑒)| 𝐺𝐺(𝑒𝑒)| NUM american_scientific_journal-4152 49 4 𝑒𝑒=1𝑠𝑠 𝑒𝑒=1𝑠𝑠 X american_scientific_journal-4152 49 5 ( ( PUNCT american_scientific_journal-4152 49 6 13 13 NUM american_scientific_journal-4152 49 7 ) ) PUNCT american_scientific_journal-4152 49 8 where where SCONJ american_scientific_journal-4152 49 9 𝐺𝐺(𝑒𝑒 𝐺𝐺(𝑒𝑒 VERB american_scientific_journal-4152 49 10 ) ) PUNCT american_scientific_journal-4152 49 11 = = SYM american_scientific_journal-4152 49 12 𝑆𝑆{𝑓𝑓(𝑑𝑑 𝑆𝑆{𝑓𝑓(𝑑𝑑 NOUN american_scientific_journal-4152 49 13 ) ) PUNCT american_scientific_journal-4152 49 14 } } PUNCT american_scientific_journal-4152 49 15 and and CCONJ american_scientific_journal-4152 49 16 𝐹𝐹(𝑠𝑠 𝐹𝐹(𝑠𝑠 NOUN american_scientific_journal-4152 49 17 ) ) PUNCT american_scientific_journal-4152 49 18 = = SYM american_scientific_journal-4152 49 19 β„’{𝑓𝑓(𝑑𝑑 β„’{𝑓𝑓(𝑑𝑑 NOUN american_scientific_journal-4152 49 20 ) ) PUNCT american_scientific_journal-4152 49 21 } } PUNCT american_scientific_journal-4152 49 22 are be AUX american_scientific_journal-4152 49 23 the the DET american_scientific_journal-4152 49 24 sumudu sumudu NOUN american_scientific_journal-4152 49 25 and and CCONJ american_scientific_journal-4152 49 26 laplace laplace NOUN american_scientific_journal-4152 49 27 transform transform VERB american_scientific_journal-4152 49 28 respectively respectively ADV american_scientific_journal-4152 49 29 american american ADJ american_scientific_journal-4152 49 30 scientific scientific ADJ american_scientific_journal-4152 49 31 research research NOUN american_scientific_journal-4152 49 32 journal journal NOUN american_scientific_journal-4152 49 33 for for ADP american_scientific_journal-4152 49 34 engineering engineering NOUN american_scientific_journal-4152 49 35 , , PUNCT american_scientific_journal-4152 49 36 technology technology NOUN american_scientific_journal-4152 49 37 , , PUNCT american_scientific_journal-4152 49 38 and and CCONJ american_scientific_journal-4152 49 39 sciences science NOUN american_scientific_journal-4152 49 40 ( ( PUNCT american_scientific_journal-4152 49 41 asrjets asrjet NOUN american_scientific_journal-4152 49 42 ) ) PUNCT american_scientific_journal-4152 49 43 ( ( PUNCT american_scientific_journal-4152 49 44 2019 2019 NUM american_scientific_journal-4152 49 45 ) ) PUNCT american_scientific_journal-4152 49 46 volume volume NOUN american_scientific_journal-4152 49 47 55 55 NUM american_scientific_journal-4152 49 48 , , PUNCT american_scientific_journal-4152 49 49 no no DET american_scientific_journal-4152 49 50 1 1 NUM american_scientific_journal-4152 49 51 , , PUNCT american_scientific_journal-4152 49 52 pp pp ADV american_scientific_journal-4152 49 53 11 11 NUM american_scientific_journal-4152 49 54 - - SYM american_scientific_journal-4152 49 55 16 16 NUM american_scientific_journal-4152 49 56 15 15 NUM american_scientific_journal-4152 49 57 the the DET american_scientific_journal-4152 49 58 generalized generalize VERB american_scientific_journal-4152 49 59 fractional fractional ADJ american_scientific_journal-4152 49 60 sumudu sumudu NOUN american_scientific_journal-4152 49 61 transform transform NOUN american_scientific_journal-4152 49 62 can can AUX american_scientific_journal-4152 49 63 be be AUX american_scientific_journal-4152 49 64 established establish VERB american_scientific_journal-4152 49 65 by by ADP american_scientific_journal-4152 49 66 using use VERB american_scientific_journal-4152 49 67 the the DET american_scientific_journal-4152 49 68 duality duality NOUN american_scientific_journal-4152 49 69 relation relation NOUN american_scientific_journal-4152 49 70 between between ADP american_scientific_journal-4152 49 71 laplace laplace NOUN american_scientific_journal-4152 49 72 transform transform NOUN american_scientific_journal-4152 49 73 and and CCONJ american_scientific_journal-4152 49 74 sumudu sumudu NOUN american_scientific_journal-4152 49 75 transform transform NOUN american_scientific_journal-4152 49 76 as as ADV american_scientific_journal-4152 49 77 defined define VERB american_scientific_journal-4152 49 78 in in ADP american_scientific_journal-4152 49 79 a a DET american_scientific_journal-4152 49 80 similar similar ADJ american_scientific_journal-4152 49 81 way way NOUN american_scientific_journal-4152 49 82 as as SCONJ american_scientific_journal-4152 49 83 defined define VERB american_scientific_journal-4152 49 84 in in ADP american_scientific_journal-4152 49 85 equations equation NOUN american_scientific_journal-4152 49 86 ( ( PUNCT american_scientific_journal-4152 49 87 12 12 NUM american_scientific_journal-4152 49 88 ) ) PUNCT american_scientific_journal-4152 49 89 and and CCONJ american_scientific_journal-4152 49 90 ( ( PUNCT american_scientific_journal-4152 49 91 13)[11 13)[11 X american_scientific_journal-4152 49 92 ] ] X american_scientific_journal-4152 49 93 let let VERB american_scientific_journal-4152 49 94 πœ™πœ™(𝑑𝑑 πœ™πœ™(𝑑𝑑 PROPN american_scientific_journal-4152 49 95 ) ) PUNCT american_scientific_journal-4152 49 96 be be AUX american_scientific_journal-4152 49 97 of of ADP american_scientific_journal-4152 49 98 exponential exponential ADJ american_scientific_journal-4152 49 99 order order NOUN american_scientific_journal-4152 49 100 and and CCONJ american_scientific_journal-4152 49 101 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑 NOUN american_scientific_journal-4152 49 102 ) ) PUNCT american_scientific_journal-4152 49 103 } } PUNCT american_scientific_journal-4152 50 1 = = SYM american_scientific_journal-4152 50 2 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 50 3 οΏ½ οΏ½ NOUN american_scientific_journal-4152 50 4 𝛼𝛼(𝑠𝑠 𝛼𝛼(𝑠𝑠 NOUN american_scientific_journal-4152 50 5 ) ) PUNCT american_scientific_journal-4152 50 6 and and CCONJ american_scientific_journal-4152 50 7 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 PROPN american_scientific_journal-4152 50 8 ) ) PUNCT american_scientific_journal-4152 50 9 } } PUNCT american_scientific_journal-4152 51 1 = = SYM american_scientific_journal-4152 51 2 𝐺𝐺𝛼𝛼(𝑒𝑒 𝐺𝐺𝛼𝛼(𝑒𝑒 PROPN american_scientific_journal-4152 51 3 ) ) PUNCT american_scientific_journal-4152 51 4 then then ADV american_scientific_journal-4152 51 5 𝐺𝐺𝛼𝛼(𝑒𝑒 𝐺𝐺𝛼𝛼(𝑒𝑒 ADP american_scientific_journal-4152 51 6 ) ) PUNCT american_scientific_journal-4152 51 7 = = SYM american_scientific_journal-4152 52 1 1 1 NUM american_scientific_journal-4152 52 2 𝑒𝑒 𝑒𝑒 INTJ american_scientific_journal-4152 52 3 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 52 4 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 5 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 52 6 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 7 1 1 NUM american_scientific_journal-4152 52 8 𝑒𝑒 𝑒𝑒 PROPN american_scientific_journal-4152 52 9 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 10 = = SYM american_scientific_journal-4152 52 11 1 1 NUM american_scientific_journal-4152 52 12 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-4152 52 13 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 14 1 1 NUM american_scientific_journal-4152 52 15 βˆ’ βˆ’ PROPN american_scientific_journal-4152 52 16 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 52 17 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 52 18 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 52 19 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 20 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 52 21 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 52 22 cot cot NOUN american_scientific_journal-4152 52 23 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 52 24 2 2 NUM american_scientific_journal-4152 52 25 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 26 𝑑𝑑2βˆ’(1𝑒𝑒)2 𝑑𝑑2βˆ’(1𝑒𝑒)2 PROPN american_scientific_journal-4152 52 27 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 28 βˆ’π‘‘π‘‘π‘’π‘’ βˆ’π‘‘π‘‘π‘’π‘’ PROPN american_scientific_journal-4152 52 29 csc csc PROPN american_scientific_journal-4152 52 30 𝛼𝛼𝑑𝑑𝑑𝑑 𝛼𝛼𝑑𝑑𝑑𝑑 PROPN american_scientific_journal-4152 52 31 ∞ ∞ PROPN american_scientific_journal-4152 52 32 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 52 33 ( ( PUNCT american_scientific_journal-4152 52 34 14 14 NUM american_scientific_journal-4152 52 35 ) ) PUNCT american_scientific_journal-4152 52 36 and and CCONJ american_scientific_journal-4152 52 37 𝐺𝐺𝛼𝛼(𝑒𝑒 𝐺𝐺𝛼𝛼(𝑒𝑒 NOUN american_scientific_journal-4152 52 38 ) ) PUNCT american_scientific_journal-4152 52 39 = = SYM american_scientific_journal-4152 52 40 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 PROPN american_scientific_journal-4152 52 41 ) ) PUNCT american_scientific_journal-4152 52 42 } } PUNCT american_scientific_journal-4152 52 43 = = SYM american_scientific_journal-4152 52 44 1 1 NUM american_scientific_journal-4152 52 45 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-4152 52 46 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 47 1 1 NUM american_scientific_journal-4152 52 48 βˆ’ βˆ’ PROPN american_scientific_journal-4152 52 49 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 52 50 cot𝛼𝛼 cot𝛼𝛼 NOUN american_scientific_journal-4152 52 51 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 52 52 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 53 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 52 54 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-4152 52 55 cot cot NOUN american_scientific_journal-4152 52 56 𝛼𝛼 𝛼𝛼 ADP american_scientific_journal-4152 52 57 2 2 NUM american_scientific_journal-4152 52 58 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 59 𝑑𝑑2βˆ’(1𝑒𝑒)2 𝑑𝑑2βˆ’(1𝑒𝑒)2 PROPN american_scientific_journal-4152 52 60 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 61 βˆ’π‘‘π‘‘π‘’π‘’ βˆ’π‘‘π‘‘π‘’π‘’ PUNCT american_scientific_journal-4152 52 62 csc𝛼𝛼𝑑𝑑𝑑𝑑 csc𝛼𝛼𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4152 52 63 ∞ ∞ PROPN american_scientific_journal-4152 52 64 βˆ’βˆž βˆ’βˆž X american_scientific_journal-4152 52 65 ( ( PUNCT american_scientific_journal-4152 52 66 15 15 NUM american_scientific_journal-4152 52 67 ) ) PUNCT american_scientific_journal-4152 52 68 substituting substitute VERB american_scientific_journal-4152 52 69 𝑒𝑒 𝑒𝑒 NOUN american_scientific_journal-4152 52 70 = = SYM american_scientific_journal-4152 52 71 1 1 NUM american_scientific_journal-4152 52 72 𝑠𝑠 𝑠𝑠 ADV american_scientific_journal-4152 52 73 we we PRON american_scientific_journal-4152 52 74 get get VERB american_scientific_journal-4152 52 75 𝐺𝐺𝛼𝛼 𝐺𝐺𝛼𝛼 VERB american_scientific_journal-4152 52 76 οΏ½ οΏ½ NOUN american_scientific_journal-4152 52 77 1 1 NUM american_scientific_journal-4152 52 78 𝑠𝑠 𝑠𝑠 PROPN american_scientific_journal-4152 52 79 οΏ½ οΏ½ PROPN american_scientific_journal-4152 52 80 = = SYM american_scientific_journal-4152 52 81 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 𝑆𝑆𝛼𝛼{πœ™πœ™(𝑑𝑑 PROPN american_scientific_journal-4152 52 82 ) ) PUNCT american_scientific_journal-4152 52 83 } } PUNCT american_scientific_journal-4152 53 1 = = PUNCT american_scientific_journal-4152 53 2 𝑠𝑠 𝑠𝑠 PROPN american_scientific_journal-4152 53 3 οΏ½ οΏ½ PROPN american_scientific_journal-4152 53 4 1 1 NUM american_scientific_journal-4152 53 5 βˆ’ βˆ’ NOUN american_scientific_journal-4152 53 6 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-4152 53 7 cot𝛼𝛼 cot𝛼𝛼 VERB american_scientific_journal-4152 53 8 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-4152 53 9 οΏ½ οΏ½ PROPN american_scientific_journal-4152 53 10 πœ™πœ™(𝑑𝑑)𝑒𝑒 πœ™πœ™(𝑑𝑑)𝑒𝑒 PROPN american_scientific_journal-4152 53 11 𝑖𝑖 𝑖𝑖 X american_scientific_journal-4152 53 12 cot𝛼𝛼 cot𝛼𝛼 PROPN american_scientific_journal-4152 53 13 2 2 NUM american_scientific_journal-4152 53 14 οΏ½ οΏ½ PROPN american_scientific_journal-4152 53 15 𝑑𝑑2βˆ’(𝑠𝑠)2 𝑑𝑑2βˆ’(𝑠𝑠)2 PROPN american_scientific_journal-4152 53 16 οΏ½ οΏ½ PROPN american_scientific_journal-4152 53 17 βˆ’π‘ π‘ π‘‘π‘‘ βˆ’π‘ π‘ π‘‘π‘‘ PROPN american_scientific_journal-4152 53 18 csc𝛼𝛼𝑑𝑑𝑑𝑑 csc𝛼𝛼𝑑𝑑𝑑𝑑 VERB american_scientific_journal-4152 53 19 ∞ ∞ PROPN american_scientific_journal-4152 53 20 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-4152 53 21 β†’ β†’ SYM american_scientific_journal-4152 53 22 ( ( PUNCT american_scientific_journal-4152 53 23 16 16 NUM american_scientific_journal-4152 53 24 ) ) PUNCT american_scientific_journal-4152 53 25 from from ADP american_scientific_journal-4152 53 26 ( ( PUNCT american_scientific_journal-4152 53 27 16 16 NUM american_scientific_journal-4152 53 28 ) ) PUNCT american_scientific_journal-4152 53 29 and and CCONJ american_scientific_journal-4152 53 30 ( ( PUNCT american_scientific_journal-4152 53 31 1 1 X american_scientific_journal-4152 53 32 ) ) PUNCT american_scientific_journal-4152 53 33 we we PRON american_scientific_journal-4152 53 34 get get VERB american_scientific_journal-4152 53 35 the the DET american_scientific_journal-4152 53 36 relation relation NOUN american_scientific_journal-4152 53 37 of of ADP american_scientific_journal-4152 53 38 fractional fractional ADJ american_scientific_journal-4152 53 39 laplace laplace NOUN american_scientific_journal-4152 53 40 transform transform NOUN american_scientific_journal-4152 53 41 with with ADP american_scientific_journal-4152 53 42 fractional fractional ADJ american_scientific_journal-4152 53 43 sumudu sumudu NOUN american_scientific_journal-4152 53 44 transform transform VERB american_scientific_journal-4152 53 45 1 1 NUM american_scientific_journal-4152 53 46 𝑠𝑠 𝑠𝑠 ADV american_scientific_journal-4152 53 47 𝐺𝐺𝛼𝛼 𝐺𝐺𝛼𝛼 VERB american_scientific_journal-4152 53 48 οΏ½ οΏ½ NOUN american_scientific_journal-4152 53 49 1 1 NUM american_scientific_journal-4152 53 50 𝑠𝑠 𝑠𝑠 PROPN american_scientific_journal-4152 53 51 οΏ½ οΏ½ PROPN american_scientific_journal-4152 53 52 = = SYM american_scientific_journal-4152 53 53 πœ™πœ™ πœ™πœ™ NOUN american_scientific_journal-4152 53 54 οΏ½ οΏ½ NOUN american_scientific_journal-4152 53 55 𝛼𝛼(𝑠𝑠 𝛼𝛼(𝑠𝑠 NOUN american_scientific_journal-4152 53 56 ) ) PUNCT american_scientific_journal-4152 53 57 ( ( PUNCT american_scientific_journal-4152 53 58 17 17 NUM american_scientific_journal-4152 53 59 ) ) SYM american_scientific_journal-4152 53 60 3 3 NUM american_scientific_journal-4152 53 61 . . X american_scientific_journal-4152 54 1 conclusion conclusion NOUN american_scientific_journal-4152 54 2 we we PRON american_scientific_journal-4152 54 3 have have AUX american_scientific_journal-4152 54 4 established establish VERB american_scientific_journal-4152 54 5 mathematically mathematically ADV american_scientific_journal-4152 54 6 the the DET american_scientific_journal-4152 54 7 relationship relationship NOUN american_scientific_journal-4152 54 8 of of ADP american_scientific_journal-4152 54 9 the the DET american_scientific_journal-4152 54 10 fractional fractional ADJ american_scientific_journal-4152 54 11 laplace laplace NOUN american_scientific_journal-4152 54 12 transform transform NOUN american_scientific_journal-4152 54 13 with with ADP american_scientific_journal-4152 54 14 fractional fractional ADJ american_scientific_journal-4152 54 15 mellin mellin NOUN american_scientific_journal-4152 54 16 transform transform NOUN american_scientific_journal-4152 54 17 and and CCONJ american_scientific_journal-4152 54 18 fractional fractional ADJ american_scientific_journal-4152 54 19 fourier fourier NOUN american_scientific_journal-4152 54 20 transform transform NOUN american_scientific_journal-4152 54 21 which which PRON american_scientific_journal-4152 54 22 will will AUX american_scientific_journal-4152 54 23 play play VERB american_scientific_journal-4152 54 24 a a DET american_scientific_journal-4152 54 25 significant significant ADJ american_scientific_journal-4152 54 26 role role NOUN american_scientific_journal-4152 54 27 in in ADP american_scientific_journal-4152 54 28 signal signal ADJ american_scientific_journal-4152 54 29 processing processing NOUN american_scientific_journal-4152 54 30 , , PUNCT american_scientific_journal-4152 54 31 optics optic NOUN american_scientific_journal-4152 54 32 and and CCONJ american_scientific_journal-4152 54 33 other other ADJ american_scientific_journal-4152 54 34 field field NOUN american_scientific_journal-4152 54 35 of of ADP american_scientific_journal-4152 54 36 applied applied ADJ american_scientific_journal-4152 54 37 mathematics mathematic NOUN american_scientific_journal-4152 54 38 , , PUNCT american_scientific_journal-4152 54 39 physics physics NOUN american_scientific_journal-4152 54 40 and and CCONJ american_scientific_journal-4152 54 41 engineering engineering NOUN american_scientific_journal-4152 54 42 references reference NOUN american_scientific_journal-4152 54 43 [ [ X american_scientific_journal-4152 54 44 1 1 X american_scientific_journal-4152 54 45 ] ] PUNCT american_scientific_journal-4152 54 46 v. v. ADP american_scientific_journal-4152 54 47 namias namias PROPN american_scientific_journal-4152 54 48 , , PUNCT american_scientific_journal-4152 54 49 β€œ " PUNCT american_scientific_journal-4152 54 50 the the DET american_scientific_journal-4152 54 51 fractional fractional ADJ american_scientific_journal-4152 54 52 order order NOUN american_scientific_journal-4152 54 53 fourier fourier NOUN american_scientific_journal-4152 54 54 transform transform NOUN american_scientific_journal-4152 54 55 and and CCONJ american_scientific_journal-4152 54 56 its its PRON american_scientific_journal-4152 54 57 application application NOUN american_scientific_journal-4152 54 58 to to ADP american_scientific_journal-4152 54 59 quantum quantum ADJ american_scientific_journal-4152 54 60 mechanics mechanic NOUN american_scientific_journal-4152 54 61 ” " PUNCT american_scientific_journal-4152 54 62 , , PUNCT american_scientific_journal-4152 54 63 i i PRON american_scientific_journal-4152 54 64 m m VERB american_scientific_journal-4152 54 65 a a DET american_scientific_journal-4152 54 66 journal journal NOUN american_scientific_journal-4152 54 67 of of ADP american_scientific_journal-4152 54 68 applied apply VERB american_scientific_journal-4152 54 69 mathematics mathematic NOUN american_scientific_journal-4152 54 70 , , PUNCT american_scientific_journal-4152 54 71 25(3 25(3 NUM american_scientific_journal-4152 54 72 ) ) PUNCT american_scientific_journal-4152 54 73 ( ( PUNCT american_scientific_journal-4152 54 74 1980 1980 NUM american_scientific_journal-4152 54 75 ) ) PUNCT american_scientific_journal-4152 54 76 241 241 NUM american_scientific_journal-4152 54 77 - - SYM american_scientific_journal-4152 54 78 265 265 NUM american_scientific_journal-4152 54 79 [ [ X american_scientific_journal-4152 54 80 2 2 NUM american_scientific_journal-4152 54 81 ] ] PUNCT american_scientific_journal-4152 54 82 a. a. NOUN american_scientific_journal-4152 54 83 torre torre PROPN american_scientific_journal-4152 54 84 . . PUNCT american_scientific_journal-4152 55 1 ( ( PUNCT american_scientific_journal-4152 55 2 2003 2003 NUM american_scientific_journal-4152 55 3 ) ) PUNCT american_scientific_journal-4152 55 4 . . PUNCT american_scientific_journal-4152 56 1 linear linear ADJ american_scientific_journal-4152 56 2 and and CCONJ american_scientific_journal-4152 56 3 radial radial ADJ american_scientific_journal-4152 56 4 canonical canonical ADJ american_scientific_journal-4152 56 5 transforms transform NOUN american_scientific_journal-4152 56 6 of of ADP american_scientific_journal-4152 56 7 fractional fractional ADJ american_scientific_journal-4152 56 8 order order NOUN american_scientific_journal-4152 56 9 . . PUNCT american_scientific_journal-4152 57 1 computational computational ADJ american_scientific_journal-4152 57 2 and and CCONJ american_scientific_journal-4152 57 3 applied applied ADJ american_scientific_journal-4152 57 4 mathematics mathematic NOUN american_scientific_journal-4152 57 5 , , PUNCT american_scientific_journal-4152 57 6 477 477 NUM american_scientific_journal-4152 57 7 - - SYM american_scientific_journal-4152 57 8 486 486 NUM american_scientific_journal-4152 57 9 . . PUNCT american_scientific_journal-4152 58 1 [ [ X american_scientific_journal-4152 58 2 3 3 X american_scientific_journal-4152 58 3 ] ] PUNCT american_scientific_journal-4152 58 4 p. p. NOUN american_scientific_journal-4152 58 5 r.gudadhe r.gudadhe NOUN american_scientific_journal-4152 58 6 , , PUNCT american_scientific_journal-4152 58 7 analytical analytical ADJ american_scientific_journal-4152 58 8 study study NOUN american_scientific_journal-4152 58 9 of of ADP american_scientific_journal-4152 58 10 a a DET american_scientific_journal-4152 58 11 special special ADJ american_scientific_journal-4152 58 12 case case NOUN american_scientific_journal-4152 58 13 of of ADP american_scientific_journal-4152 58 14 complex complex ADJ american_scientific_journal-4152 58 15 canonical canonical ADJ american_scientific_journal-4152 58 16 transform transform NOUN american_scientific_journal-4152 58 17 . . PUNCT american_scientific_journal-4152 59 1 global global ADJ american_scientific_journal-4152 59 2 journal journal PROPN american_scientific_journal-4152 59 3 of of ADP american_scientific_journal-4152 59 4 mathematical mathematical ADJ american_scientific_journal-4152 59 5 sciences science NOUN american_scientific_journal-4152 59 6 , , PUNCT american_scientific_journal-4152 59 7 ( ( PUNCT american_scientific_journal-4152 59 8 2010 2010 NUM american_scientific_journal-4152 59 9 ) ) PUNCT american_scientific_journal-4152 59 10 293 293 NUM american_scientific_journal-4152 59 11 - - SYM american_scientific_journal-4152 59 12 302 302 NUM american_scientific_journal-4152 60 1 [ [ X american_scientific_journal-4152 60 2 4 4 NUM american_scientific_journal-4152 60 3 ] ] PUNCT american_scientific_journal-4152 60 4 g.jumarie g.jumarie NOUN american_scientific_journal-4152 60 5 , , PUNCT american_scientific_journal-4152 60 6 fractional fractional ADJ american_scientific_journal-4152 60 7 laplace laplace NOUN american_scientific_journal-4152 60 8 transform transform NOUN american_scientific_journal-4152 60 9 with with ADP american_scientific_journal-4152 60 10 mittage mittage NOUN american_scientific_journal-4152 60 11 leffler leffl ADJ american_scientific_journal-4152 60 12 function function NOUN american_scientific_journal-4152 60 13 . . PUNCT american_scientific_journal-4152 61 1 applied apply VERB american_scientific_journal-4152 61 2 mathematics mathematic NOUN american_scientific_journal-4152 61 3 , , PUNCT american_scientific_journal-4152 61 4 ( ( PUNCT american_scientific_journal-4152 61 5 2009 2009 NUM american_scientific_journal-4152 61 6 ) ) PUNCT american_scientific_journal-4152 61 7 . . PUNCT american_scientific_journal-4152 62 1 american american PROPN american_scientific_journal-4152 62 2 scientific scientific ADJ american_scientific_journal-4152 62 3 research research PROPN american_scientific_journal-4152 62 4 journal journal NOUN american_scientific_journal-4152 62 5 for for ADP american_scientific_journal-4152 62 6 engineering engineering NOUN american_scientific_journal-4152 62 7 , , PUNCT american_scientific_journal-4152 62 8 technology technology NOUN american_scientific_journal-4152 62 9 , , PUNCT american_scientific_journal-4152 62 10 and and CCONJ american_scientific_journal-4152 62 11 sciences science NOUN american_scientific_journal-4152 62 12 ( ( PUNCT american_scientific_journal-4152 62 13 asrjets asrjet NOUN american_scientific_journal-4152 62 14 ) ) PUNCT american_scientific_journal-4152 62 15 ( ( PUNCT american_scientific_journal-4152 62 16 2019 2019 NUM american_scientific_journal-4152 62 17 ) ) PUNCT american_scientific_journal-4152 62 18 volume volume NOUN american_scientific_journal-4152 62 19 55 55 NUM american_scientific_journal-4152 62 20 , , PUNCT american_scientific_journal-4152 62 21 no no DET american_scientific_journal-4152 62 22 1 1 NUM american_scientific_journal-4152 62 23 , , PUNCT american_scientific_journal-4152 62 24 pp pp ADV american_scientific_journal-4152 62 25 11 11 NUM american_scientific_journal-4152 62 26 - - SYM american_scientific_journal-4152 62 27 16 16 NUM american_scientific_journal-4152 62 28 16 16 NUM american_scientific_journal-4152 63 1 [ [ X american_scientific_journal-4152 63 2 5 5 NUM american_scientific_journal-4152 63 3 ] ] X american_scientific_journal-4152 63 4 k.k k.k PROPN american_scientific_journal-4152 63 5 sharma sharma PROPN american_scientific_journal-4152 63 6 . . PUNCT american_scientific_journal-4152 64 1 fractional fractional ADJ american_scientific_journal-4152 64 2 laplace laplace NOUN american_scientific_journal-4152 64 3 transform transform NOUN american_scientific_journal-4152 64 4 , , PUNCT american_scientific_journal-4152 64 5 springer springer NOUN american_scientific_journal-4152 64 6 - - PUNCT american_scientific_journal-4152 64 7 verlag verlag PROPN american_scientific_journal-4152 64 8 london london PROPN american_scientific_journal-4152 64 9 limited limit VERB american_scientific_journal-4152 64 10 , , PUNCT american_scientific_journal-4152 64 11 4 4 NUM american_scientific_journal-4152 64 12 , , PUNCT american_scientific_journal-4152 64 13 ( ( PUNCT american_scientific_journal-4152 64 14 2010 2010 NUM american_scientific_journal-4152 64 15 ) ) PUNCT american_scientific_journal-4152 64 16 377 377 NUM american_scientific_journal-4152 64 17 - - SYM american_scientific_journal-4152 64 18 379 379 NUM american_scientific_journal-4152 65 1 [ [ X american_scientific_journal-4152 65 2 6 6 NUM american_scientific_journal-4152 65 3 ] ] PUNCT american_scientific_journal-4152 65 4 r.prabhakar r.prabhakar NOUN american_scientific_journal-4152 65 5 , , PUNCT american_scientific_journal-4152 65 6 deshmukhi deshmukhi PROPN american_scientific_journal-4152 65 7 , , PUNCT american_scientific_journal-4152 65 8 s.d s.d PROPN american_scientific_journal-4152 65 9 . . PROPN american_scientific_journal-4152 65 10 alka alka PROPN american_scientific_journal-4152 65 11 , , PUNCT american_scientific_journal-4152 65 12 convolution convolution NOUN american_scientific_journal-4152 65 13 structure structure NOUN american_scientific_journal-4152 65 14 for for ADP american_scientific_journal-4152 65 15 two two NUM american_scientific_journal-4152 65 16 version version NOUN american_scientific_journal-4152 65 17 of of ADP american_scientific_journal-4152 65 18 fractional fractional ADJ american_scientific_journal-4152 65 19 laplace laplace NOUN american_scientific_journal-4152 65 20 transform transform NOUN american_scientific_journal-4152 65 21 , , PUNCT american_scientific_journal-4152 65 22 journal journal NOUN american_scientific_journal-4152 65 23 of of ADP american_scientific_journal-4152 65 24 science science NOUN american_scientific_journal-4152 65 25 and and CCONJ american_scientific_journal-4152 65 26 arts art NOUN american_scientific_journal-4152 65 27 , , PUNCT american_scientific_journal-4152 65 28 2(15),(2011 2(15),(2011 NUM american_scientific_journal-4152 65 29 ) ) PUNCT american_scientific_journal-4152 65 30 143 143 NUM american_scientific_journal-4152 65 31 - - SYM american_scientific_journal-4152 65 32 150 150 NUM american_scientific_journal-4152 65 33 . . PUNCT american_scientific_journal-4152 66 1 [ [ X american_scientific_journal-4152 66 2 7 7 X american_scientific_journal-4152 66 3 ] ] X american_scientific_journal-4152 66 4 r. r. PROPN american_scientific_journal-4152 66 5 a. a. NOUN american_scientific_journal-4152 66 6 treumann treumann PROPN american_scientific_journal-4152 66 7 , , PUNCT american_scientific_journal-4152 66 8 w. w. PROPN american_scientific_journal-4152 66 9 baumjohann baumjohann PROPN american_scientific_journal-4152 66 10 , , PUNCT american_scientific_journal-4152 66 11 fractional fractional ADJ american_scientific_journal-4152 66 12 laplace laplace NOUN american_scientific_journal-4152 66 13 transform transform VERB american_scientific_journal-4152 66 14 - - PUNCT american_scientific_journal-4152 66 15 a a DET american_scientific_journal-4152 66 16 perspective perspective NOUN american_scientific_journal-4152 66 17 , , PUNCT american_scientific_journal-4152 66 18 frontier frontier NOUN american_scientific_journal-4152 66 19 physics physics NOUN american_scientific_journal-4152 66 20 , , PUNCT american_scientific_journal-4152 66 21 2,(2014 2,(2014 NUM american_scientific_journal-4152 66 22 ) ) PUNCT american_scientific_journal-4152 66 23 . . PUNCT american_scientific_journal-4152 67 1 1 1 NUM american_scientific_journal-4152 67 2 - - SYM american_scientific_journal-4152 67 3 4 4 NUM american_scientific_journal-4152 67 4 . . X american_scientific_journal-4152 67 5 institute institute NOUN american_scientific_journal-4152 67 6 for for ADP american_scientific_journal-4152 67 7 nonlinear nonlinear ADJ american_scientific_journal-4152 67 8 science science NOUN american_scientific_journal-4152 67 9 . . PUNCT american_scientific_journal-4152 68 1 springer springer NOUN american_scientific_journal-4152 68 2 , , PUNCT american_scientific_journal-4152 68 3 new new PROPN american_scientific_journal-4152 68 4 york york PROPN american_scientific_journal-4152 68 5 , , PUNCT american_scientific_journal-4152 68 6 ny ny PROPN american_scientific_journal-4152 68 7 ( ( PUNCT american_scientific_journal-4152 68 8 2003 2003 NUM american_scientific_journal-4152 68 9 ) ) PUNCT american_scientific_journal-4152 69 1 [ [ X american_scientific_journal-4152 69 2 8 8 NUM american_scientific_journal-4152 69 3 ] ] X american_scientific_journal-4152 69 4 l.b l.b PROPN american_scientific_journal-4152 69 5 . . PROPN american_scientific_journal-4152 69 6 almeida almeida PROPN american_scientific_journal-4152 69 7 , , PUNCT american_scientific_journal-4152 69 8 β€œ " PUNCT american_scientific_journal-4152 69 9 the the DET american_scientific_journal-4152 69 10 fractional fractional ADJ american_scientific_journal-4152 69 11 fourier fourier NOUN american_scientific_journal-4152 69 12 transform transform NOUN american_scientific_journal-4152 69 13 and and CCONJ american_scientific_journal-4152 69 14 time time NOUN american_scientific_journal-4152 69 15 - - PUNCT american_scientific_journal-4152 69 16 frequency frequency NOUN american_scientific_journal-4152 69 17 representations representation NOUN american_scientific_journal-4152 69 18 ” " PUNCT american_scientific_journal-4152 69 19 , , PUNCT american_scientific_journal-4152 69 20 ieee ieee NOUN american_scientific_journal-4152 69 21 transactions transaction NOUN american_scientific_journal-4152 69 22 on on ADP american_scientific_journal-4152 69 23 signal signal ADJ american_scientific_journal-4152 69 24 processing processing NOUN american_scientific_journal-4152 69 25 , , PUNCT american_scientific_journal-4152 69 26 42(11 42(11 NUM american_scientific_journal-4152 69 27 ) ) PUNCT american_scientific_journal-4152 69 28 ( ( PUNCT american_scientific_journal-4152 69 29 1994 1994 NUM american_scientific_journal-4152 69 30 ) ) PUNCT american_scientific_journal-4152 69 31 3084–3091 3084–3091 NUM american_scientific_journal-4152 69 32 [ [ X american_scientific_journal-4152 69 33 9 9 NUM american_scientific_journal-4152 69 34 ] ] PUNCT american_scientific_journal-4152 69 35 a. a. NOUN american_scientific_journal-4152 69 36 peng peng PROPN american_scientific_journal-4152 69 37 , , PUNCT american_scientific_journal-4152 69 38 y. y. PROPN american_scientific_journal-4152 69 39 wang wang PROPN american_scientific_journal-4152 69 40 , , PUNCT american_scientific_journal-4152 69 41 x. x. PROPN american_scientific_journal-4152 69 42 zuo zuo PROPN american_scientific_journal-4152 69 43 , , PUNCT american_scientific_journal-4152 69 44 l. l. PROPN american_scientific_journal-4152 69 45 gong gong PROPN american_scientific_journal-4152 69 46 , , PUNCT american_scientific_journal-4152 69 47 properties property NOUN american_scientific_journal-4152 69 48 of of ADP american_scientific_journal-4152 69 49 fractional fractional ADJ american_scientific_journal-4152 69 50 mellin mellin PROPN american_scientific_journal-4152 69 51 transform transform NOUN american_scientific_journal-4152 69 52 , , PUNCT american_scientific_journal-4152 69 53 advances advance NOUN american_scientific_journal-4152 69 54 in in ADP american_scientific_journal-4152 69 55 information information NOUN american_scientific_journal-4152 69 56 sciences sciences PROPN american_scientific_journal-4152 69 57 and and CCONJ american_scientific_journal-4152 69 58 services service NOUN american_scientific_journal-4152 69 59 sciences science NOUN american_scientific_journal-4152 69 60 5(11 5(11 NUM american_scientific_journal-4152 69 61 ) ) PUNCT american_scientific_journal-4152 69 62 ( ( PUNCT american_scientific_journal-4152 69 63 2013 2013 NUM american_scientific_journal-4152 69 64 ) ) PUNCT american_scientific_journal-4152 69 65 90 90 NUM american_scientific_journal-4152 69 66 - - SYM american_scientific_journal-4152 69 67 96 96 NUM american_scientific_journal-4152 69 68 . . PUNCT american_scientific_journal-4152 70 1 [ [ X american_scientific_journal-4152 70 2 10 10 NUM american_scientific_journal-4152 70 3 ] ] X american_scientific_journal-4152 70 4 keji keji PROPN american_scientific_journal-4152 70 5 li li PROPN american_scientific_journal-4152 70 6 , , PUNCT american_scientific_journal-4152 70 7 y. y. PROPN american_scientific_journal-4152 70 8 x. x. PROPN american_scientific_journal-4152 70 9 ( ( PUNCT american_scientific_journal-4152 70 10 2011 2011 NUM american_scientific_journal-4152 70 11 ) ) PUNCT american_scientific_journal-4152 70 12 . . PUNCT american_scientific_journal-4152 71 1 a a DET american_scientific_journal-4152 71 2 brief brief ADJ american_scientific_journal-4152 71 3 introduction introduction NOUN american_scientific_journal-4152 71 4 of of ADP american_scientific_journal-4152 71 5 sumudu sumudu NOUN american_scientific_journal-4152 71 6 transform transform NOUN american_scientific_journal-4152 71 7 and and CCONJ american_scientific_journal-4152 71 8 comparison comparison NOUN american_scientific_journal-4152 71 9 with with ADP american_scientific_journal-4152 71 10 other other ADJ american_scientific_journal-4152 71 11 integral integral ADJ american_scientific_journal-4152 71 12 transforms transform NOUN american_scientific_journal-4152 71 13 . . PUNCT american_scientific_journal-4152 72 1 ieee ieee NOUN american_scientific_journal-4152 72 2 , , PUNCT american_scientific_journal-4152 72 3 285 285 NUM american_scientific_journal-4152 72 4 - - SYM american_scientific_journal-4152 72 5 287 287 NUM american_scientific_journal-4152 72 6 . . PUNCT american_scientific_journal-4152 73 1 [ [ X american_scientific_journal-4152 73 2 11 11 NUM american_scientific_journal-4152 73 3 ] ] X american_scientific_journal-4152 73 4 gupta gupta PROPN american_scientific_journal-4152 73 5 , , PUNCT american_scientific_journal-4152 73 6 v.j v.j PROPN american_scientific_journal-4152 73 7 et et PROPN american_scientific_journal-4152 73 8 el el PROPN american_scientific_journal-4152 73 9 . . PROPN american_scientific_journal-4152 73 10 ( ( PUNCT american_scientific_journal-4152 73 11 2010 2010 NUM american_scientific_journal-4152 73 12 ) ) PUNCT american_scientific_journal-4152 73 13 . . PUNCT american_scientific_journal-4152 74 1 a a DET american_scientific_journal-4152 74 2 note note NOUN american_scientific_journal-4152 74 3 of of ADP american_scientific_journal-4152 74 4 fractional fractional ADJ american_scientific_journal-4152 74 5 sumudu sumudu NOUN american_scientific_journal-4152 74 6 transform transform NOUN american_scientific_journal-4152 74 7 , , PUNCT american_scientific_journal-4152 74 8 journal journal NOUN american_scientific_journal-4152 74 9 of of ADP american_scientific_journal-4152 74 10 applied apply VERB american_scientific_journal-4152 74 11 mathematics mathematics NOUN american_scientific_journal-4152 74 12 research research NOUN american_scientific_journal-4152 74 13 article article NOUN american_scientific_journal-4152 74 14 ( ( PUNCT american_scientific_journal-4152 74 15 9 9 NUM american_scientific_journal-4152 74 16 pages page NOUN american_scientific_journal-4152 74 17 ) ) PUNCT american_scientific_journal-4152 74 18 , , PUNCT american_scientific_journal-4152 74 19 article article NOUN american_scientific_journal-4152 74 20 i i PROPN american_scientific_journal-4152 74 21 d d PROPN american_scientific_journal-4152 74 22 154189 154189 NUM american_scientific_journal-4152 74 23 , , PUNCT american_scientific_journal-4152 74 24 volume volume NOUN american_scientific_journal-4152 74 25 2010 2010 NUM american_scientific_journal-4152 74 26 . . PUNCT american_scientific_journal-4152 75 1 https://arxiv.org/find/physics/1/au:+treumann_r/0/1/0/all/0/1 https://arxiv.org/find/physics/1/au:+treumann_r/0/1/0/all/0/1 PROPN american_scientific_journal-4152 75 2 https://arxiv.org/find/physics/1/au:+baumjohann_w/0/1/0/all/0/1 https://arxiv.org/find/physics/1/au:+baumjohann_w/0/1/0/all/0/1 PROPN