Approximate solutions of a time-fractional diffusion equation with a source term using the variational iteration method

TL;DR

This paper develops approximate solutions for a time-fractional diffusion equation with a source term using the Variational Iteration Method, demonstrating rapid convergence and ease of implementation for different source cases.

Contribution

The paper introduces a novel application of the Variational Iteration Method to solve time-fractional diffusion equations with source terms, providing explicit series solutions with fast convergence.

Findings

Series solutions converge exponentially fast as terms increase.

VIM solutions require only a few terms for accurate approximation.

Method is easily implementable on symbolic computation platforms.

Abstract

We consider a time fractional differential equation of order $\alpha$, $0<\alpha<1$, $$ \frac{\partial c(x,t)}{\partial t}={}^C_0\mathcal{D}_t^{\alpha}[(Ac)(x,t)]+q(x,t) ,\quad x > 0, t > 0, \quad c(x,0)=f(x). $$ where ${}^C_0\mathcal{D}_t^{\alpha}$ is the Caputo fractional derivative of order $\alpha$, $A$ is a linear differential operator, $q(x,t)$ is a source term, and $f(x)$ is the inital condition. Approximate (truncated) series solutions are obtained by means of the Variational Iteration Method (VIM). We find the series solutions for different cases of the source term, in a form that is readily implementable on the computer where symbolic computation platform is available. The error in truncated solution $c_n$ diminishes exponentially fast for a given $\alpha$ as the number of terms in the series increases. VIM has several advantages over other methods that produce solutions in the series form. The truncated VIM solutions often converge rapidly requiring only a few terms for fast and accurate approximations.