Mathematical Problems in Engineering

Volume 2019, Article ID 2562580, 13 pages

https://doi.org/10.1155/2019/2562580

## A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255049, China

Correspondence should be addressed to Gongsheng Li; nc.ude.tuds@sgil

Received 20 September 2018; Revised 9 December 2018; Accepted 30 December 2018; Published 13 January 2019

Academic Editor: Mariano Torrisi

Copyright © 2019 Shengnan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the basis functions of the approximate space. The inversion solutions with noisy data which give good approximations to the exact solution demonstrate effectiveness of the inversion algorithm for the simultaneous inversion problem.

#### 1. Introduction

The fractional calculus and 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, we get a time or space fractional diffusion equation, which has more and more applications in the fields of physics, mechanics, viscoelastic materials, environmental science, hydrology, nanofluid, etc. On the attempting to describe some real processes and complex dynamical systems using the fractional diffusion models, several researches confronted with the situation that the fractional derivative did not remain constant and changed; say, in the interval from 0 to 1, from 1 to 2 or even from 0 to 2. To study such phenomena in mathematics, an effective way is to employ the fractional derivative of the variable order; i.e., the order of the fractional derivative can change with the time or/and space variables. The variable-order fractional models and the corresponding theory of variable-order operators give a new mathematical framework for dealing with complex dynamical systems, which becomes a new direction in the research of fractional differential equations (see, e.g., [1, 2]). For the concept of variable fractional differentials and integrals and the connection between the mathematical concepts of fractional order and physical process, see [3–5] and references therein.

There are quite a few of researches on numerical methods for solving the variable-order time/space fractional diffusion equation models. We refer to some work given by F. Liu and his group, see [6–9] for instance, also see the work given by Sun et al. [10], and the master degree thesis given by Zhang [11] for numerical solutions to the variable-order time fractional diffusion models with variable coefficients. Recently, we refer to the work by Bhrawy and Zaky et al. [12–16] for numerical methods for multidimensional space/time variable-order fractional differential equations, where approximate solutions of the forward variable-order fractional differential equations and spectral techniques are discussed based on orthogonal polynomials. On concrete applications of the fractional calculus models, we always encounter with inverse problems which need to determine some unknowns in the model using suitable additional data. As for inverse problems arising in the time/space fractional diffusion equations with constant fractional orders, we refer to [17–29], and recently see [30–32], etc. However, there are few studies on inverse problems associated with variable-order fractional differential equations to our knowledge. Jia and Peng gave a uniqueness result for the inverse source problem in the variable-order space fractional diffusion equation using a Carleman type estimate [33], and Liu et al. [34] gave numerical inversions for determining the variable fractional order in the variable-order time fractional diffusion equation using the homotopy regularization algorithm.

Since the diffusion coefficient denotes the characteristics of the medium, the variable time fractional order is a key index denoting the correlations in the model, and they are always unknown in real-life problems, it is of much importance to determine these two variable-dependent parameters utilizing the inverse problem method. Although the uniqueness and stability of such a simultaneous inverse problem are very difficult to obtain at present, it is still meaningful to study its numerical inversions from numerics. In this paper, we deal with the simultaneous inversion problem of determining the space-dependent fractional order and diffusion coefficient in the variable-order time fractional diffusion equation utilizing the homotopy regularization algorithm. Such an inversion problem is more difficult than those discussed in the previous work by the following reasons:(i)The model is complicated, which involves the variable-dependent fractional order and diffusion coefficient, and the computational complexity is high.(ii)The inverse problem is to determine two kinds of parameters simultaneously, and both of the diffusion coefficient and the fractional order are spatially dependent functions such that the ill-posedness of the numerical inversion is much severe.(iii)The approximate space for the unknowns is different. We employ the Legendre polynomials as the basis functions instead of the ordinary polynomials to perform the inversion algorithm.

Therefore it is much challenging to deal with the simultaneous inversion problem arising in the variable-order fractional diffusion equation. The most contribution of this paper is devoted to give effective numerical inversions for the simultaneous inversion problem utilizing the homotopy regularization in the framework of Legendre polynomials approximations.

The rest of the paper is organized as follows. In Section 2, a finite difference scheme for solving the forward problem is given. In Section 3, the homotopy regularization algorithm is introduced for solving the inverse problem of determining the fractional order and the diffusion coefficient simultaneously. In Section 4, numerical inversions with noisy data are performed in Legendre polynomial approximate space, and concluding remarks are given in Section 5.

#### 2. The Forward Problem and Finite Difference Scheme

##### 2.1. The Forward Problem

Denote and for and . Consider the variable-order time fractional diffusion equation with a space-dependent diffusion coefficient:where is the state variable, is the fractional differential order depending upon the space and time variables and , is the diffusion coefficient, and is a source term. Here denotes the variable-order fractional derivative in the sense of Caputo, which is defined bywhere is the Gamma function, and there exist two positive constants and such that Given the initial value conditionand the homogeneous boundary conditiona forward problem is composed by (1) with the initial boundary conditions (4)-(5).

##### 2.2. The Finite Difference Scheme

In this subsection, an implicit finite difference scheme is set forth to solve the forward problem (1), (4)-(5) numerically. Although a similar difference scheme has been discussed in [11], we still give it here for the completeness of this paper.

By discretizing the space domain by and the time domain by and the variable-order fractional derivative by the difference discretization, we havewhere and is the space mesh step and is the time mesh step.

Next, denote as the midpoint of the neighboring nodes and . The integer-order diffusion term in (1) is discretized by

Denoting , , , and , substituting (6) and (7) into (1), and omitting high-order terms, we getThe initial boundary value conditions are discretized as follows:and

Finally, denote ; (8) is arranged asfor . In the case of , there isand we have for Denote , and letfor . We rewrite (13) in a matrix form:wherefor , and for and .

Lemma 1. *The coefficient matrix () defined by (16) is strictly diagonally dominant, and the difference scheme (15) has only one solution.*

*Proof. *For any , there is . Noting there holdswhich means thatThis shows that the matrix () is strictly diagonally dominant, and the difference scheme (15) has unique solution.

Furthermore, it is not difficult to prove that the scheme (15) is of unconditional stability and convergence with a similar method as used in [34].

Lemma 2 (see [11, 34]). *The difference scheme (15) is unconditional stable and convergent to the exact solution of the forward problem for any finite time .*

##### 2.3. Numerical Testification

In this subsection, we give a numerical computation for the forward problem utilizing the difference scheme (15). The analytic solution is chosen asand the model parameters are taken as follows:and the source term

The numerical results with different time step and space step are listed in Tables 1 and 2 at given , respectively. The relative error in the solutions is expressed bywhere and denote the exact and numerical solutions, respectively. Moreover, the numerical and exact solutions are plotted in Figure 1.