SOLVING PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS USING THE $\mathcal{L}_A $-TRANSFORM

Abstract

In this article, the authors introduce the [Formula: see text]-transform and derive the complex inversion formula and a convolution theorem for the transform. Furthermore, the fundamental solution of the Cauchy type fractional diffusion equation on fractals is given by means of the [Formula: see text]-transform in terms of the Wright functions. Also, the Cauchy fractional disturbance equation with continuous or discrete distribution of time fractional derivative is introduced and by using the [Formula: see text]-transform, its solution is expressed in terms of the Laplace type integral.

Numerical Inversion Technique for the One and Two-Dimensional L2-Transform Using the Fourier Series and Its Application to Fractional Partial Differential Equations

“…It is obvious that the sum of the right-hand side of (3)(4)(5)(6)(7)(8) can be an approximation value of f (t 1 , t 2 ) as…”

“…To control of the truncation error E t , we can use the epsilon algorithm (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) to accelerate the convergence of the series and evaluate the valuesf p+1 (t 1 , t 2 ), f p+ p 4 (t 1 , t 2 ) in (3-9) until the difference between them be small.…”

“…The Laplace-type integral transform called the L 2 -transform was introduced by Yurekli and Sadek [16] and is denoted as follows Authors [1][2][3][4] for real numbers α 1 , α 2 and positive constant M ). Also, the inversion integral formulas for the one and two-dimensional L 2 -transform in terms of the Bromwich's integral can be presented [1] as follows ( Some contributions and applications of the one the L 2 -transform are shown by Yurekli [17] and authors [1][2][3][4].…”

“…Also, the inversion integral formulas for the one and two-dimensional L 2 -transform in terms of the Bromwich's integral can be presented [1] as follows ( Some contributions and applications of the one the L 2 -transform are shown by Yurekli [17] and authors [1][2][3][4]. Yurekli showed the Parseval-Goldstein theorems involving the L 2 -transform and the Laplace transform may be used to yield identities involving several well-known integral transforms and infinite integrals of elementary and special functions.…”

“…Yurekli showed the Parseval-Goldstein theorems involving the L 2 -transform and the Laplace transform may be used to yield identities involving several well-known integral transforms and infinite integrals of elementary and special functions. Moreover, authors [1][2][3][4] presented roles of the one and two-dimensional L 2 -transform in linear partial differential equations and system of differential and integral equations and utilized this transform as a useful and supplementary tool for analyzing the systems which the Laplace transform can not easily or never solve them. Our aim in this article is based on the inversion of the functions which the Laplace transform of them does not exist and can be expressed as functions with damping motions near the zero point.…”