In this article, fractional exponential operator is considered as a general approach for solving partial fractional differential equations. An integral representation for this operator is derived from the Bromwich integral for the inverse Mellin transform. Also, effectiveness of this operator for obtaining the formal solution of system of diffusion equations is discussed. Copyright © 2011 John Wiley & Sons, Ltd.

Solution to system of partial fractional differential equations using the fractional exponential operators

In this article, we derive the coefficient set {H
 m
 (x,y)}
 =1
 ∞
 using the generating function e
 xt+yϕ(t) When the complex function ϕ(t) is entire, using the inverse Mellin transform, and when ϕ(t) has singular points, using the inverse Laplace transform, the coefficient set is obtained Also, bi-orthogonality of this set with its associated functions and its applications in the explicit solutions of partial fractional differential equations is discussed

On the generating function e xt+yϕ(t) and its fractional calculus

In this paper, the Bromwich integral for the inverse Mellin transform is used for finding an integral representation for a fractional exponential operator. This operator can be considered as an approach for solving partial fractional differential equations. Also, application of this operator for obtaining a formal solution of the time-fractional telegraph equation is discussed.

/pdf/fractional-exponential-operators-and-time-fractional-2j8t407chi.pdf

Fractional exponential operators and time-fractional telegraph equation

Abstract In this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

A class of integral transforms which satisfies certain assumptions is considered. Several examples, including the Fourier, Laplace and Mellin transforms, are in this class as well as transforms whi...

A general class of integral transforms and an expression for their convolution formulas

In this paper, the authors apply complex inversion formula for the $\mathcal{L}_2$-transform and also show some applications of the $\mathcal{L}_2$-transform for solving of singular integral equation with trigonometric kernel and system of partial fractional differential equations.

SOLUTION TO SYSTEM OF PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS USING THE $\mathcal{L}_2$-TRANSFORM

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SOLUTION TO SYSTEM OF PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS USING THE $\mathcal{L}_2$-TRANSFORM