We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1 < α ≤ 2. This operator is defined by a combination of the left and right Riemann–Liouville fractional derivatives. We stu...

In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivati...

In this article, we introduce the fractional differential systems in the sense of the Weber fractional derivatives and study the asymptotic stability of these systems. We present the stability regions and then compare the stability regions of fractional differential systems with the Riemann-Liouville and Weber fractional derivatives.

In this paper, a time-fractional derivative nonlinear Schrödinger equation involving the Riemann–Liouville fractional is investigated. We first perform Lie symmetry analysis of equation, and then derive reduced equations under admitted optimal-symmetry system. Moreover, with invariant subspace method, several exact solutions for their figures are presented. Finally, new conservation theorem app...

In this paper, the ( / ) G G  -expansion method is extended to solve fractional differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order. For illustrating the validity of this method, we apply it to fi...

This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several subdomains, and a fractional derivative (FDs) at a time node point is approximated using a modified Grünwald-Letnikov ap...

In this work, the fractional Lie symmetry method is used to find exact solutions of time-fractional coupled Drinfeld-Sokolov-Wilson equations with Riemann-Liouville derivative. Time-fractional are obtained by replacing first-order time derivative derivatives (FD) order $\alpha$ in classical (DSW) model. Using method, generators obtained. With help generators, FCDSW reduced into ordinary differe...

We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α ∈ (3/2, 2) on the unit interval (0, 1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value pro...