Determination of Time-Dependent Coefficients in Time-Fractional Diffusion Equations by Variational Iteration Method

Mimi Li,Xianzheng Jia,Gongsheng Li,Zhiyuan Li

doi:10.5539/jmr.v12n1p82

Abstract

Inverse problems of determining time-dependent coefficients in partial differential equations are difficult to deal with in general cases. The variational iteration method is introduced to determine the time-dependent coefficient in the fractional diffusion equation as well as the solution of the forward problem. By utilizing the additional condition and the property of the fractional derivative, an expression of the unknown is derived by which a nonlinear dynamical differential equation is obtained. The variational iteration method is applied to solve the nonlinear system and the time-dependent coefficient can be reconstructed in a semi-analytical form. Such method can give explicit expression of the solution in the meaning of approximation, or exact solution to the inverse problem in some cases. Several examples are presented to demonstrate feasibility and effectiveness of the proposed method for inverse time-dependent coefficient problems in the fractional diffusion equations.

Highlights

The fractional differential equations and their applications have attracted much attention during the last two decades
If the usual integer-order derivative in a classical diffusion equation is replaced with a fractional derivative of any real number in time or space, a time or space fractional diffusion equation is established, which has more and more applications in the fields of physics, mechanics, environmental science and hydrology, etc
As for inverse problems arising in the time/space fractional diffusion equations, we refer to Chen, Liu, Jiang, Turner and Burrage (2016), Cheng, Nakagawa, Yamamoto and Yamazaki (2009), Chi, Li and Jia (2011), Jin and Rundell (2015), Li, Zhang, Jia and Yamamoto (2013), Li, Imanuvilov and Yamamoto(2016), Liu and Yamamoto(2010), Liu, Rundell and Yamamoto (2016), Luchko, Rundell and Yamamoto(2013), Miller and Yamamoto (2013), Sakamoto and Yamamoto (2011), Wei, L

Summary

Introduction

The fractional differential equations and their applications have attracted much attention during the last two decades. As we know the variational iteration method is a semi-analytical method to get a solution of a linear/nonlinear algebra/differential equation arising from mathematical physics and engineering science, see the original work by He (1997, 2000, 2007), etc. By the variational iteration method, we consider the inverse problem of determining the time-dependent diffusion coefficient a(t) in the time-fractional diffusion equation. The variational iteration method has been utilized to solve nonlinear problems for classical integer-order diffusion equations and for fractional diffusion equations, there are few researches on inverse coefficient problems by the method. One purpose of this work is try to apply the variational iteration method to solve some inverse coefficient problems in the fractional diffusion equations, especially for determination of a time-dependent coefficient which is difficult to cope with in general cases.

The Variational Iteration Method

Inversion for Diffusion Coefficient in Time-FDE

Transformation of the Inverse Coefficient Problem

Inversion for the Diffusion Coefficient

Conclusion