Certain results of Aleph-Function based on natural transform of fractional order >
ISSN: 2689-7636
Annals of Mathematics and Physics
Research Article       Open Access      Peer-Reviewed

Certain results of Aleph-Function based on natural transform of fractional order

Aarti Pathak1, Rajshree Mishra2, DK Jain3, Farooq Ahmad4 and Altaf Ahmad Bhat5*

1School of Mathematics and Allied Sciences, Jiwaji University, Gwalior, MP, India
2Department of Mathematics, Government Model Science College, Gwalior, MP India
3Department of Engineering Mathematics Computing, MITS, Gwalior, MP India
4Department of Mathematics, Govt. College for Women Nawakadal, J&K, India
5Department of General Requirements, University of Technology and Applied Sciences, Salalah, Oman
*Corresponding authors: Altaf Ahmad Bhat, Department of General Requirements, University of Technology and Applied Sciences, Salalah, Oman, Email: altaf.sal@cas.edu.om
Received: 13 April, 2023 | Accepted: 24 April, 2023 | Published: 25 April, 2023
Keywords: N-transform of fractional order; L-transform of fractional order; S-transform of fractional order & Aleph-function

Cite this as

Pathak A, Mishra R, Jain DK, Ahmad F, Bhat AA (2023) Certain results of Aleph-Function based on natural transform of fractional order. Ann Math Phys 6(1): 052-057. DOI: 10.17352/amp.000078

Copyright Licence

© 2023 Pathak A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

In this research article, a new type of fractional integral transform namely the N-transform of fractional order is proposed, and derived a number of useful results of a more generalized function (Aleph-function) of fractional calculus by making use of the N-transform of fractional order. Further, the relation between it and other fractional transforms is given and some special cases have also been discussed.

Introduction

Our translation of real-world problems to mathematical expressions relies on calculus, which in turn relies on the differentiation and integration operations of arbitrary order with a sort of misnomer fractional calculus which is also a natural generalization of calculus and its mathematical history is equally long. It plays a significant role in a number of fields such as physics, rheology, quantitative biology, electro-chemistry, scattering theory, diffusion, transport theory, probability, elasticity, control theory, engineering mathematics, and many others. Fractional calculus like many other mathematical disciplines and ideas has its origin in the quest of researchers to expand its applications to new fields. This freedom of order opens new dimensions and many problems of applied sciences can be tackled in a more efficient way by means of fractional calculus.

Laplace and Sumudu transformations are closely linked to natural transformation. The Natural transform, also known as the N-transform, was initially introduced by Khan and Khan (2008) [1]; Al-Omari (2013) [2]; Belgacem and Silambarasan [3] explored its features (2012b). Maxwell's equations were solved using the Natural transform in Belgacem and Silambarasan (2011, 2012a) [4]. Belgacem and Silambarasan's (2011, 2012c) works on Natural transformation can be found here [5] for more information. If we assume that the function is fractional derivative and continuous, the Natural transform often works with continuous and continuously differentiable functions. The Natural transform, like the Laplace and Sumudu transforms, does not work since the function is not derivative. In a similar vein, we must establish a new term that we will call fractional Natural transform [6-8].

The purpose of this research article is to calculate the fractional order natural transform of the Aleph function.

Definitions and preliminaries used in this paper

Classical Laplace transform: The Laplace transform is very useful in analysis and design for systems that are linear and time-invariant (LTI). Beginning in about 1910, transform techniques were applied to signal processing at Bell Labs for signal filtering and telephone long-line communication by H. Bode and others. Transform theory subsequently provided the backbone of Classical Control Theory as practiced during the World Wars and up to about 1960, when State Variable techniques began to be used for control design. Pierre Simon Laplace was a French mathematician who lived from 1749-1827, during the Age of Enlightenment characterized by the French Revolution, Rousseau, Voltaire and Napoleon Bonaparte.

Suppose f(t) is a real-valued function defined over the interval (0,∞). The Laplace transform of f(t) is defined by

L[ f( t ) ]= 0 e st f( t )d(t) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaKqbaoaadmaakeaajugibiaadAgajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaaaiaawUfacaGLDbaajugibiabg2da9KqbaoaapehakeaajugibiaadwgajuaGdaahaaWcbeqaaKqzGeGaeyOeI0Iaam4CaiaadshaaaaaleaajugibiaaicdaaSqaaKqzGeGaeyOhIukacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaajugibiaadsgacaGGOaGaamiDaiaacMcaaaa@54E5@

Or f( s )= 0 e st f( t )d(t) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaKqbaoaabmaakeaajugibiaadohaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbsacqGHsislcaWGZbGaamiDaaaaaSqaaKqzGeGaaGimaaWcbaqcLbsacqGHEisPaiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadshaaOGaayjkaiaawMcaaKqzGeGaamizaiaacIcacaWG0bGaaiykaaaa@50F9@

The Laplace transform is said to exist if the above integral is convergent for some values of s.

The Inverse Laplace Transform can be described as the transformation into a function of time. In the Laplace inverse formula, f(s) is the Transform of f(t), while in the Inverse Transform f(t) is the Inverse Laplace Transform of f(s). Therefore, we can write this Inverse Laplace transform formula as follows:

f( t ) =  L -1 { f }( t )= 1 2πi Lim L γiL γ+iL e st f(s)ds MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@719F@

Natural transform: In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan[1] 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence of the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of both transforms. Most recently, F. B. M. Belgacem [2] has renamed it the natural transform and has proposed a detailed theory and applications. The natural transform of a function f(t), defined for all real numbers t ≥ 0, is the function R(u, s), defined by:

R( u, s ) =N[ f( t ) ]= 0 e st f( ut )dt,Re(S)>0,u( τ 1 , τ 2 )       (1) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbGcdaqadaqaaKqzGeGaamyDaiaacYcacaGGGcGaam4CaaGccaGLOaGaayzkaaqcLbsacaqGGaGaeyypa0deaaaaaaaaa8qacaWGobGcdaWadaWdaeaajugib8qacaWGMbGcdaqadaWdaeaajugib8qacaWG0baakiaawIcacaGLPaaaaiaawUfacaGLDbaajugibiabg2da9OWaaybCaeqal8aabaqcLbsapeGaaGimaaWcpaqaaKqzGeWdbiabg6HiLcqdpaqaaKqzGeWdbiabgUIiYdaacaWGLbGcpaWaaWbaaSqabeaajugib8qacqGHsislcaWGZbGaamiDaaaacaWGMbGcdaqadaWdaeaajugib8qacaWG1bGaamiDaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiDaiaacYcacaWGsbGaamyzaiaacIcacaWGtbGaaiykaiabg6da+iaaicdacaGGSaGaamyDaiaacIcacqGHsislcqaHepaDk8aadaWgaaWcbaqcLbsapeGaaGymaaWcpaqabaqcLbsapeGaaiilaiabes8a0PWdamaaBaaaleaajugib8qacaaIYaaal8aabeaajugib8qacaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeykaaaa@77B7@

Provided the function f (t) ∈ R2 is defined in the set

A= { f( t ) |M, τ 1 , τ 2 >0.|f(t)|<M   e |t| τ j MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6391@

Khan [1] showed that the above integral converges to the Laplace transform when u = 1 and into Sumudu transform for s = 1.

Laplace transform of error function:

L{erf(t)} = 1 s exp( s 2 4 )erfc( s 2 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGmbGaai4EaiaabwgacaqGYbGaaeOzaiaacIcacaqG0bGaaiykaiaac2hacaqGGaGaeyypa0JcqaaaaaaaaaWdbmaalaaapaqaaKqzGeWdbiaaigdaaOWdaeaajugib8qacaWGZbaaaiGacwgacaGG4bGaaiiCaOWaaeWaa8aabaWdbmaalaaapaqaaKqzGeWdbiaadohak8aadaahaaWcbeqaaKqzGeWdbiaaikdaaaaak8aabaqcLbsapeGaaGinaaaaaOGaayjkaiaawMcaaKqzGeGaamyzaiaadkhacaWGMbGaam4yaOWaaeWaa8aabaWdbmaalaaapaqaaKqzGeWdbiaadohaaOWdaeaajugib8qacaaIYaaaaaGccaGLOaGaayzkaaaaaa@5802@

Where: L{f} denotes the Laplace transform of the function f.

erf denotes the error function.

erfc denotes the complementary error function.

Fractional natural transform of order α :

R α + (u, s)= N α + [ f( x ) ]= 0 E α ( s α x α )f( ux ) (dx) α ,0<α1. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7823@

Or

R α + (u, s)= Lim M 0 M E α ( s α x α )f( ux ) (dx) α        (2) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@71F1@

Where s, u ∈ C and Eα(x) is the Mittag–Leffler function, E α ()= n=0 x n αn! MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaabweak8aadaWgaaWcbaqcLbsapeGaaeySdaWcpaqabaqcLbsapeGaaiikaiaabQhacaqGGcGaaiykaiabg2da9OWaaybCaeqal8aabaqcLbsapeGaamOBaiabg2da9iaaicdaaSWdaeaajugib8qacqGHEisPa0Wdaeaajugib8qacqGHris5aaGcdaWcaaWdaeaajugib8qacaWG4bGcpaWaaWbaaSqabeaajugib8qacaWGUbaaaaGcpaqaaKqzGeWdbiabeg7aHjaad6gacaGGHaaaaaaa@50E8@

Fractional Laplace transform reported in Jumarie (2009a) [9]:

From the above definition, when u = 1

L α + (1, s) L α + [ f( x ) ]= 0 E α ( s α x α )f( x ) (dx) α , 0<α1. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7678@

Or

L α + (1, s)== Lim M 0 M E α ( s α x α )f( x ) (dx) α       (3) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@712E@

Where s ∈ C and Eα (x) is the Mittag–Leffler function, E α ()=  n=0 x n αn! MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaabweak8aadaWgaaWcbaqcLbsapeGaaeySdaWcpaqabaqcLbsapeGaaiikaiaabQhacaqGGcGaaiykaiabg2da9iaabckakmaawahabeWcpaqaaKqzGeWdbiaad6gacqGH9aqpcaaIWaaal8aabaqcLbsapeGaeyOhIukan8aabaqcLbsapeGaeyyeIuoaaOWaaSaaa8aabaqcLbsapeGaamiEaOWdamaaCaaaleqabaqcLbsapeGaamOBaaaaaOWdaeaajugib8qacqaHXoqycaWGUbGaaiyiaaaaaaa@520B@

Fractional Sumudu transform which is proposed by Gupta, Sharma and Kiliçman (2010):

From the above definition, when S = 1

S α + (u, 1)= S α + [ f( x ) ]= 0 E α ( x α )f( ux ) (dx) α ,0<α1. MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7469@

Or

S α + (u, 1)== Lim M 0 M E α ( x α )f( ux ) (dx) α        (4) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6F59@

Where u ∈ C and Eα (x) is the Mittag–Leffler function, E α (z)= n=0 x n αn! MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaabweak8aadaWgaaWcbaqcLbsapeGaaeySdaWcpaqabaqcLbsapeGaaiikaiaabQhacaGGPaGaeyypa0JcdaGfWbqabSWdaeaajugib8qacaWGUbGaeyypa0JaaGimaaWcpaqaaKqzGeWdbiabg6HiLcqdpaqaaKqzGeWdbiabggHiLdaakmaalaaapaqaaKqzGeWdbiaadIhak8aadaahaaWcbeqaaKqzGeWdbiaad6gaaaaak8aabaqcLbsapeGaeqySdeMaamOBaiaacgcaaaaaaa@4FC5@

Aleph-function:

The Aleph function is defined in terms of the Mellin-Barnes type integral in the following manner [10]:

p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A059@

= 1 2πi L j=1 m ( b j B j s ) j=1 n ( 1 a j + A j s ) i=1 r τ i j=m+1 q i (1 b ji + B ji s) j=n+1 p i [ a ji A ji s] z s ds      (5) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@BFB2@

Lemma-I: For instance the fractional natural transform of the f( x )= x nα , n N MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadAgakmaabmaapaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaKqzGeWdaiabg2da98qacaWG4bGcpaWaaWbaaSqabeaajugib8qacaWGUbGaeqySdegaa8aacaGGSaGaaeiiaiaad6gacaGGGcWdbiabgIGiolaad6eaaaa@48EF@ then

N α + [ x nα ]= 0 E α ( s α x α ) (ux) nα (dx) α = u nα 0 E α ( s α x α ) (x) nα (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8C9C@

We put t=xs. we get

N α + [ x nα ]= u nα s (n+1)α 0 E α ( t α ) (t) nα (dt) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FC5@

Or

N α + [ x nα ]= (α!) u nα s (n+1)α Γ α ( n+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F8D@

Note: Γ α ( n )=  1 (α!) 0 E α ( x α ) (x) (n1)α (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66F7@

Lemma-II: For instance the fractional Laplace transform of the f( x )= x nα , n N MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadAgakmaabmaapaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaKqzGeWdaiabg2da98qacaWG4bGcpaWaaWbaaSqabeaajugib8qacaWGUbGaeqySdegaa8aacaGGSaGaaeiiaiaad6gacaGGGcWdbiabgIGiolaad6eaaaa@48EF@ then

L α + [ x nα ]= 0 E α ( s α x α ) (x) nα (dx) α = 0 E α ( s α x α ) (x) nα (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@868F@

We put t = xs. we get

L α + [ x nα ]= 1 s (n+1)α 0 E α ( t α ) (t) nα (dt) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6C0D@

Or

L α + [ x nα ]= (α!) s (n+1)α Γ α ( n+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadYeak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaamWaa8aabaqcLbsapeGaamiEaOWdamaaCaaaleqabaqcLbsapeGaamOBaiabeg7aHbaaaOGaay5waiaaw2faaKqzGeGaeyypa0JcdaWcaaWdaeaajugib8qacaGGOaGaeqySdeMaaiyiaiaacMcaaOWdaeaajugib8qacaWGZbGcpaWaaWbaaSqabeaajugib8qacaGGOaGaamOBaiabgUcaRiaaigdacaGGPaGaeqySdegaaaaacqqHtoWrk8aadaWgaaWcbaqcLbsapeGaaeySdaWcpaqabaGcpeWaaeWaa8aabaqcLbsapeGaaeOBaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaaa@5B69@

Note: Γ α ( n )=  1 (α!) 0 E α ( x α ) (x) (n1)α (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66F7@

Lemma-II: For instance the fractional Laplace transform of the f( x )= x nα , n N MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadAgakmaabmaapaqaaKqzGeWdbiaadIhaaOGaayjkaiaawMcaaKqzGeWdaiabg2da98qacaWG4bGcpaWaaWbaaSqabeaajugib8qacaWGUbGaeqySdegaa8aacaGGSaGaaeiiaiaad6gacaGGGcWdbiabgIGiolaad6eaaaa@48EF@ then

S α + [ x nα ]= 0 E α ( x α ) (ux) nα (dx) α = u nα 0 E α ( x α ) (x) nα (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@85A9@

We put t = x, we get

S α + [ x nα ]= u nα 0 E α ( t α ) (t) nα (dt) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66E5@

Or

S α + [ x nα ]=(α!) u nα Γ α ( n+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadofak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaamWaa8aabaqcLbsapeGaamiEaOWdamaaCaaaleqabaqcLbsapeGaamOBaiabeg7aHbaaaOGaay5waiaaw2faaKqzGeGaeyypa0Jaaiikaiabeg7aHjaacgcacaGGPaGaamyDaOWdamaaCaaaleqabaqcLbsapeGaamOBaiabeg7aHbaacaqGtoGcpaWaaSbaaSqaaKqzGeWdbiaabg7aaSWdaeqaaOWdbmaabmaapaqaaKqzGeWdbiaab6gacqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaa@56AD@

Note: Γ α ( n )=  1 (α!) 0 E α ( x α ) (x) (n1)α (dx) α MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66F7@

Main results

Fractional natural transform of order α

In this section, we derived the fractional natural transform of order α in relationship with the known generalized function of fractional calculus known as the Aleph function.

Theorem (1): Let, N α + [ f( x ) ],0<α1 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaad6eak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaamWaa8aabaqcLbsapeGaamOzaOWaaeWaa8aabaqcLbsapeGaamiEaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaqcLbsacaGGSaGaaGimaiabgYda8iabeg7aHjabgsMiJkaaigdaaaa@4B8F@ be the fractional natural transform of order α associated with Aleph-function. Then there holds the following relationship

N α + { { p i , q i; τ i ;r m,n [ z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AC39@

1 s p i , q i; τ i ;r m,n+1 [ u s | ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i (0,1) ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AA97@

Provided the function f (t) ∈ R2

Proof: By using the definition of the generalized function of fractional Aleph -function and fractional natural transform of order α we get

N α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaad6eak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaaiWaa8aabaqcLbsapeGaeyynHaScpaWaa0baaSqaaKqzGeWdbiaadchak8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaaiilaiaadghak8aadaWgaaadbaqcLbsapeGaamyAaiaacUdacqaHepaDk8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaai4oaiaadkhaaWWdaeqaaaWcbaqcLbsapeGaamyBaiaacYcacaWGUbaaaOWaamWaa8aabaWdbmaaeiaapaqaaKqzGeWdbiaadQfaaOGaayjcSdWdamaaDaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadkgak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGcbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad2gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamOyaOWdamaaBaaameaajugib8qacaWGQbGaaiiOaiaacYcaaWWdaeqaaKqzGeWdbiaadkeak8aadaWgaaadbaqcLbsapeGaamOAaiaadMgaaWWdaeqaaaWcpeGaayjkaiaawMcaaaGaay5waiaaw2faaOWdamaaBaaameaajugib8qacaWGTbGaey4kaSIaaGymaiaacYcacaWGXbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqabaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadggak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad6gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamyyaOWdamaaBaaameaajugib8qacaWGQbGaamyAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgacaWGPbaam8aabeaaaSWdbiaawIcacaGLPaaaaiaawUfacaGLDbaak8aadaWgaaadbaqcLbsapeGaamOBaiabgUcaRiaaigdacaGGSaGaamiCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaaeqaaaaaaOWdbiaawUfacaGLDbaaaiaawUhacaGL9baajugibiabg2da9aaa@A9C9@

N α + { 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] z k dk};  Re( α ) > 0  MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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XqjcaGGOaGaaGymaiabgkHiTiaadkgak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiabgUcaRiaadkeak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiaadUgacaGGPaGcdaqfWaqabSWdaeaajugib8qacaWGQbGaeyypa0JaamOBaiabgUcaRiaaigdaaSWdaeaajugib8qacaWGWbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqdbaqcLbsapeGaey4dIunaaiaa=XqjcaGGBbGaamyyaOWdamaaBaaaleaajugib8qacaWGQbGaamyAaaWcpaqabaqcLbsapeGaeyOeI0IaamyqaOWdamaaBaaaleaajugib8qacaWGQbGaamyAaaWcpaqabaqcLbsapeGaam4Aaiaac2faaaGaamOEaOWdamaaCaaaleqabaqcLbsapeGaam4AaaaapaGaamizaiaadUgacaGG9bGaai4oaiaabccacaGGGcGaaeOuaiaabwgakmaabmaabaqcLbsacqaHXoqyaOGaayjkaiaawMcaaKqzGeGaaeiiaiabg6da+iaabccacaaIWaGaaiiOaaaa@CC71@

N α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaad6eak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaaiWaa8aabaqcLbsapeGaeyynHaScpaWaa0baaSqaaKqzGeWdbiaadchak8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaaiilaiaadghak8aadaWgaaadbaqcLbsapeGaamyAaiaacUdacqaHepaDk8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaai4oaiaadkhaaWWdaeqaaaWcbaqcLbsapeGaamyBaiaacYcacaWGUbaaaOWaamWaa8aabaWdbmaaeiaapaqaaKqzGeWdbiaadQfaaOGaayjcSdWdamaaDaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadkgak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGcbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad2gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamOyaOWdamaaBaaameaajugib8qacaWGQbGaaiiOaiaacYcaaWWdaeqaaKqzGeWdbiaadkeak8aadaWgaaadbaqcLbsapeGaamOAaiaadMgaaWWdaeqaaaWcpeGaayjkaiaawMcaaaGaay5waiaaw2faaOWdamaaBaaameaajugib8qacaWGTbGaey4kaSIaaGymaiaacYcacaWGXbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqabaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadggak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad6gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamyyaOWdamaaBaaameaajugib8qacaWGQbGaamyAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgacaWGPbaam8aabeaaaSWdbiaawIcacaGLPaaaaiaawUfacaGLDbaak8aadaWgaaadbaqcLbsapeGaamOBaiabgUcaRiaaigdacaGGSaGaamiCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaaeqaaaaaaOWdbiaawUfacaGLDbaaaiaawUhacaGL9baaaaa@A834@

{ 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk } N α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C348@

N α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9C9@

{ 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk} N α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C12A@

By making use of lemma –I in above equation, we get

N α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9C9@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk u k s (k+1) Γ( k+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@C56B@

Or

N α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A834@

1 s 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k )Γ( 10+k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk u k s k MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C78F@

Or

N α + { p i , q i; τ i ;r m,n [ z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A854@

= 1 s p i , q i; τ i ;r m,n+1 [ u s | ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i (0,1) ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AC36@

This completes the proof of the theorem.

Fractional laplace transform of order α

In this section, we derived the fractional Laplace transform of order α in relationship with the known function of fractional calculus known as the Aleph function.

Theorem (2): Let L α + [ f( x ) ],0<α1 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadYeak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaamWaa8aabaqcLbsapeGaamOzaOWaaeWaa8aabaqcLbsapeGaamiEaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaqcLbsacaGGSaGaaGimaiabgYda8iabeg7aHjabgsMiJkaaigdaaaa@4B8D@ , be the fractional Laplace transform of order α associated with Aleph-function. Then there holds the following relationship

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A832@

1 s p i , q i; τ i ;r m,n+1 [ s 1 | ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i (1,0) ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AB43@

Provided the function f (t) ∈ R2

Proof: By using the definition of the generalized function of fractional ML-function and fractional Laplace transform of order α we get

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadYeak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaaiWaa8aabaqcLbsapeGaeyynHaScpaWaa0baaSqaaKqzGeWdbiaadchak8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaaiilaiaadghak8aadaWgaaadbaqcLbsapeGaamyAaiaacUdacqaHepaDk8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaai4oaiaadkhaaWWdaeqaaaWcbaqcLbsapeGaamyBaiaacYcacaWGUbaaaOWaamWaa8aabaWdbmaaeiaapaqaaKqzGeWdbiaadQfaaOGaayjcSdWdamaaDaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadkgak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGcbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad2gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamOyaOWdamaaBaaameaajugib8qacaWGQbGaaiiOaiaacYcaaWWdaeqaaKqzGeWdbiaadkeak8aadaWgaaadbaqcLbsapeGaamOAaiaadMgaaWWdaeqaaaWcpeGaayjkaiaawMcaaaGaay5waiaaw2faaOWdamaaBaaameaajugib8qacaWGTbGaey4kaSIaaGymaiaacYcacaWGXbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqabaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadggak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad6gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamyyaOWdamaaBaaameaajugib8qacaWGQbGaamyAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgacaWGPbaam8aabeaaaSWdbiaawIcacaGLPaaaaiaawUfacaGLDbaak8aadaWgaaadbaqcLbsapeGaamOBaiabgUcaRiaaigdacaGGSaGaamiCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaaeqaaaaaaOWdbiaawUfacaGLDbaaaiaawUhacaGL9baaaaa@A832@

L α + { 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] z k dk}; Re( α ) > 0 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@CA27@

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9C7@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk  L α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@BFB4@

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A832@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk  L α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@BFB4@

By making use of lemma –II in the above equation, we get

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A832@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk 1 s (k+1) Γ( k+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@C308@

Or

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A832@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k )Γ( k+1 )Γ( 10+k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] 1 s (k+1) dk MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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XqjcaGGOaGaaGymaiabgkHiTiaadkgak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiabgUcaRiaadkeak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiaadUgacaGGPaGcdaqfWaqabSWdaeaajugib8qacaWGQbGaeyypa0JaamOBaiabgUcaRiaaigdaaSWdaeaajugib8qacaWGWbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqdbaqcLbsapeGaey4dIunaaiaa=XqjcaGGBbGaamyyaOWdamaaBaaaleaajugib8qacaWGQbGaamyAaaWcpaqabaqcLbsapeGaeyOeI0IaamyqaOWdamaaBaaaleaajugib8qacaWGQbGaamyAaaWcpaqabaqcLbsapeGaam4Aaiaac2faaaGcdaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaam4CaOWdamaaCaaaleqabaqcLbsapeGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaaaaaGaaeizaiaabUgaaaa@CC5A@

L α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A832@

1 s p i , q i; τ i ;r m,n+1 [ s 1 | ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i (1,0) ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AB43@

This completes the proof of the theorem.

Fractional Sumudu transform of order α

In this section, we derived the fractional Sumudu transform of order α in relationship with the known function of fractional calculus known as ML-function.

Theorem (3): Let S α + [ f( x ) ],0<α1 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadofak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaamWaa8aabaqcLbsapeGaamOzaOWaaeWaa8aabaqcLbsapeGaamiEaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaqcLbsacaGGSaGaaGimaiabgYda8iabeg7aHjabgsMiJkaaigdaaaa@4B94@ , be the fractional Sumudu transform of order α associated with Aleph-function. Then there holds the following relationship

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= 1 s E α 1 ( u s )  MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadofak8aadaqhaaWcbaqcLbsapeGaeqySdegal8aabaqcLbsapeGaey4kaScaaOWaaiWaa8aabaqcLbsapeGaeyynHaScpaWaa0baaSqaaKqzGeWdbiaadchak8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaaiilaiaadghak8aadaWgaaadbaqcLbsapeGaamyAaiaacUdacqaHepaDk8aadaWgaaadbaqcLbsapeGaamyAaaadpaqabaqcLbsapeGaai4oaiaadkhaaWWdaeqaaaWcbaqcLbsapeGaamyBaiaacYcacaWGUbaaaOWaamWaa8aabaWdbmaaeiaapaqaaKqzGeWdbiaadQfaaOGaayjcSdWdamaaDaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadkgak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGcbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad2gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamOyaOWdamaaBaaameaajugib8qacaWGQbGaaiiOaiaacYcaaWWdaeqaaKqzGeWdbiaadkeak8aadaWgaaadbaqcLbsapeGaamOAaiaadMgaaWWdaeqaaaWcpeGaayjkaiaawMcaaaGaay5waiaaw2faaOWdamaaBaaameaajugib8qacaWGTbGaey4kaSIaaGymaiaacYcacaWGXbGcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaaqabaaaleaak8qadaqadaWcpaqaaKqzGeWdbiaadggak8aadaWgaaadbaqcLbsapeGaamOAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgaaWWdaeqaaaWcpeGaayjkaiaawMcaaOWdamaaBaaameaajugib8qacaaIXaGaaiilaiaad6gaaWWdaeqaaOWdbmaadmaal8aabaqcLbsapeGaeqiXdqNcpaWaaSbaaWqaaKqzGeWdbiaadMgaaWWdaeqaaOWdbmaabmaal8aabaqcLbsapeGaamyyaOWdamaaBaaameaajugib8qacaWGQbGaamyAaiaacckacaGGSaaam8aabeaajugib8qacaWGbbGcpaWaaSbaaWqaaKqzGeWdbiaadQgacaWGPbaam8aabeaaaSWdbiaawIcacaGLPaaaaiaawUfacaGLDbaak8aadaWgaaadbaqcLbsapeGaamOBaiabgUcaRiaaigdacaGGSaGaamiCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaaeqaaaaaaOWdbiaawUfacaGLDbaaaiaawUhacaGL9baajugibiabg2da9OWaaSaaa8aabaqcLbsapeGaaGymaaGcpaqaaKqzGeWdbiaadohaaaGaaeyraOWdamaaDaaaleaajugib8qacaqGXoaal8aabaqcLbsapeGaaGymaaaacaGGOaGcdaWcaaWdaeaajugib8qacaqG1baak8aabaqcLbsapeGaae4CaaaacaGGPaGaaeiOaaaa@B743@

Provided the function f (t) ∈ R2

Proof: By using the definition of the generalized function of fractional Aleph -function and fractional Sumudu transform of order α we get

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A839@

S α + 1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] z k dk; Re( α ) > 0  MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C8B1@

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9CE@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk S α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@BF14@

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9CE@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk S α + { z k } MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@BF2F@

By making use of lemma –III in the above equation, we get

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9DD@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] dk u k Γ( k+1 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=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@BE47@

Or

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9CE@

1 2πi L j=1 m ( b j B j k ) j=1 n ( 1 a j + A j k )Γ( 10+k ) i=1 r τ i j=m+1 q i (1 b ji + B ji k) j=n+1 p i [ a ji A ji k] u k dk MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaajugib8qacaaIXaaak8aabaqcLbsapeGaaGOmaiabec8aWjaadMgaaaGcdaGfWbqabSWdaeaajugib8qacaWGmbaal8aabaaaneaajugib8qacqGHRiI8aaGcdaWcaaWdaeaapeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaaigdaaSWdaeaajugib8qacaWGTbaan8aabaqcLbsapeGaey4dIunaamXvP5wqSX2qVrwzqf2zLnharyqtHX2z15gih9gDOL2yaGqbciaa=XqjkmaabmaapaqaaKqzGeWdbiaadkgak8aadaWgaaWcbaqcLbsapeGaamOAaaWcpaqabaqcLbsapeGaeyOeI0IaamOqaOWdamaaBaaaleaajugib8qacaWGQbaal8aabeaajugib8qacaWGRbaakiaawIcacaGLPaaadaqfWaqabSWdaeaajugib8qacaWGQbGaeyypa0JaaGymaaWcpaqaaKqzGeWdbiaad6gaa0Wdaeaajugib8qacqGHpis1aaGaa8hdLOWaaeWaa8aabaqcLbsapeGaaGymaiabgkHiTiaadggak8aadaWgaaWcbaqcLbsapeGaamOAaaWcpaqabaqcLbsapeGaey4kaSIaamyqaOWdamaaBaaaleaajugib8qacaWGQbaal8aabeaajugib8qacaWGRbaakiaawIcacaGLPaaajugibiabfo5ahPWaaeWaa8aabaqcLbsapeGaaGymaiabgkHiTiaaicdacqGHRaWkcaqGRbaakiaawIcacaGLPaaaa8aabaWdbmaavadabeWcpaqaaKqzGeWdbiaadMgacqGH9aqpcaaIXaaal8aabaqcLbsapeGaamOCaaqdpaqaaKqzGeWdbiabggHiLdaacqaHepaDk8aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqabaGcpeWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaad2gacqGHRaWkcaaIXaaal8aabaqcLbsapeGaamyCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaa0qaaKqzGeWdbiabg+GivdaacaWFmuIaaiikaiaaigdacqGHsislcaWGIbGcpaWaaSbaaSqaaKqzGeWdbiaadQgacaWGPbaal8aabeaajugib8qacqGHRaWkcaWGcbGcpaWaaSbaaSqaaKqzGeWdbiaadQgacaWGPbaal8aabeaajugib8qacaWGRbGaaiykaOWaaubmaeqal8aabaqcLbsapeGaamOAaiabg2da9iaad6gacqGHRaWkcaaIXaaal8aabaqcLbsapeGaamiCaOWdamaaBaaameaajugib8qacaWGPbaam8aabeaaa0qaaKqzGeWdbiabg+GivdaacaWFmuIaai4waiaadggak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiabgkHiTiaadgeak8aadaWgaaWcbaqcLbsapeGaamOAaiaadMgaaSWdaeqaaKqzGeWdbiaadUgacaGGDbaaaiaadwhak8aadaahaaWcbeqaaKqzGeWdbiaadUgaaaWdaiaabsgacaqGRbaaaa@C06D@

S α + { p i , q i; τ i ;r m,n [ Z| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] }= MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A9CE@

p i , q i; τ i ;r m,n+1 [ u| ( b j , B j ) 1,m [ τ i ( b j , B ji ) ] m+1, q i (0,1) ( a j , A j ) 1,n [ τ i ( a ji , A ji ) ] n+1, p i ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A61E@

This completes the proof of the theorem.

Special cases

In this section, we discuss some of the important special cases of the main results established discussed above, If we take ∞ = τi =1 in the theorems (1), (2), and (3), we get well-known results of ordinary calculus like the Natural transform of Saxena’s I-function, Laplace transform of Saxena’s I-function, and finally ordinary Sumudu transform of Saxena’s I-function as reported in [11].

Conclusion

We are trying for more specified and detailed results of this transformation. The results proved in this paper give some contributions to the theory of the fractional order transform, believed to be new and are likely to find certain applications to the solution of the fractional differential and integral equations order equations. The importance of this work is to find the half derivative, fractional order Laplace Natural transforms, etc.

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  7. Duffy DG. Transform methods for solving partial differential equations, Second edition, Chapman & Hall/CRC, BocaRaton, FL, 2004.
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