International Journal of Applied Mathematics Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 7, Issue 4, Jun - Jul 2018; 19-32 © IASET
GENERALIZED LAPLACE TRANSFORM AND FRACTIONAL EQUATIONS OF DISTRIBUTED ORDERS A. K. Thakur, Geeta Kewat & Hetram Suryavanshi Dr. C. V. Raman University, Kota, Bilaspur, Chhattisgarh, India
ABSTRACT In this article, we introduce a generalized Laplace transform and fractional equations of distributed orders and evaluate the results of the complex inversion formula for the exponential Mellin transform, the exponential Laplace transform of
-Derivatives.
KEYWORDS: Mellin Transform, Fractional Derivatives, Fractional Diffusion Equation
Article History Received: 14 May 2018 | Revised: 23 May 2018 | Accepted: 25 May 2018
1. INTRODUCTION This section is devoted to a presentation of some basic facts from the theory of the Mellin integral transform that are used in the further discussions. For more information regarding the Mellin integral transform including its properties and particular cases we refer the interested reader to e.g. [2], [3], [5]. The Mellin integral transform of a sufficiently well-behaved function f is defined as M{f(t); s} =
∗
(s) =
∞
( )
dt
(2.1)
and the inverse Mellin integral transform as f(t) = M-1{f(t);s} =
∗
∞ ∞
(s) =
∗(
)t ds, t > 0,
= ℜ(s)
(2.2)
Where the integral is understood in the sense of the Cauchy principal value. It is worth mentioning that the Mellin integral transform can be obtained from the Fourier-integral transform by the variables substitution t= M{f(t); s} =
∗
(s) =
and by rotation of the complex plane by a right angle: ∞
( )
dt =
∞ ∞
(
= ξ{f
);
},
)
(
)
= ξ{ (
)
(
)
Where ξ {f(x); κ} denotes the Fourier transform of the function f at the point κ. Accordingly, the inverse Mellin transforms and the convolution for the Mellin transform can be obtained by the same substitutions from the inverse Fourier
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