Application of Rothe’s method to fractional differential equations

Abstract

In this paper we consider an initial value problem for a fractional differential equation formulated in a Banach space X where the fractional derivative is Riemann-Liouville type of order 0 < α < 1. We establish the existence and uniqueness of a strong solution of the problem by applying the method of semi-discretization in time, also known as the method of lines or more popularly as Rothe's method. The dual space X * of X is assumed to be uniformly convex. In the final section, we illustrate the applicability of the theoretical results with the help of an example.

Existence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe method

“…One such method that has drawn the attention of many scholars is the Rothe method. Researchers have exploited and enhanced this technique for study of various differential fractional equations [18][19][20]. Yang [21] presented a difference scheme for a kind of linear space-time fractional convection-diffusion equation using a finite difference method.…”

Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation

“…For instance, Tang [8] studied the 246 Convergence Analysis of Space Discretization of Time Fractional Telegraph Equation convergence and superconvergence for the time fractional optimal control problems via a fully discrete finite element scheme. Sontakke and Pandit [9] studied the convergence of nonlinear fractional partial differential equations via the fractional Adomian decomposition method. An [10] investigated the superconvergence of a time-space discretized scheme for a time-fractional diffusion problem.…”