Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation

Abstract

The diffusion equation with a general convolutional derivative in time is considered on a bounded domain, as one of the boundary conditions is nonlocal. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. To find the source term and the solution, we resort to generalized eigenfunction expansion, using a bi-orthogonal pair of bases. Estimates for the time-dependent components in the spectral expansions are established and applied to prove uniqueness and existence in the classical sense. Analytical and numerical examples are provided.

Introduction

Anomalous diffusion processes are modeled by employing different types of fractional partial differential and integro-differential equations (see [1] and the references cited there). In particular, the power-law dependence on time of the mean squared displacement can be captured by time-fractional diffusion equations. However, most of the anomalous diffusion phenomena in complex systems do not show a mono-scaling behavior. Instead, transitions between different diffusion regimes in course of time are observed. One way to capture such multi-scaling behavior is by replacing the relatively simple operators of fractional derivatives by more general operators with specific memory kernels [2], [3], [4].

A large number of different generalizations of the classical fractional calculus operators have been proposed and extensively discussed recently, see e.g. [5], [6] for some general unifying models of fractional calculus.

In this work we adopt the definition of generalized fractional derivative of Caputo type introduced in [7] (see also [8]) in the form

$$\left({\mathbb{D}}_t^{\left(k\right)}f\right)\left(t\right)=\frac{\text{d}}{\text{d}t}{\int }_0^{t}k(t-\tau )f\left(\tau \right)\text{d}\tau -k\left(t\right)f\left(0\right),t>0,$$

where $k\left(t\right)$ is a nonnegative locally integrable kernel. The exact assumptions imposed on the kernel will be specified later. Here we list some basic particular cases.

The Caputo fractional derivative of order $\alpha \in (0,1)$ is recovered from (1) for the power-law memory kernel $k\left(t\right)={\omega }_{1-\alpha }\left(t\right)$, where

$${\omega }_{\beta }\left(t\right)=\frac{t^{\beta -1}}{\Gamma \left(\beta \right)},\beta >0,t>0,$$

with $\Gamma \left(\cdot \right)$ being the Gamma function. Other basic examples of memory kernels are the multi-term power-law memory kernel

$$k\left(t\right)=\sum _{j=1}^m{q}_j{\omega }_{1-{\alpha }_j}\left(t\right),0<{\alpha }_j<1,q_j>0,j=1,\dots ,m,$$

the distributed-order memory kernel

$$k\left(t\right)={\int }_0^1{\omega }_{1-\alpha }\left(t\right)p\left(\alpha \right)\text{d}\alpha ,$$

where $p\left(\cdot \right)$ is a nonnegative weight function, and the truncated power-law memory kernel

$$k\left(t\right)=e^{-\gamma t}{\omega }_{1-\alpha }\left(t\right),\gamma >0,0<\alpha <1.$$

The time-fractional diffusion equation with the general integro-differential operator (1) is discussed in [2], [3], where its relevance for describing a broad class of anomalous nonscaling patterns is pointed out. The Cauchy problem for the general diffusion equation on an unbounded space domain is studied in detail in [7]. In [9] some uniqueness and existence results, as well as a maximum principle, are established for the initial–boundary-value problem. Optimal estimates for the decay in time of solutions to the general time-fractional diffusion equations on a bounded domain subject to homogeneous Dirichlet boundary condition are deduced in [10], where it is shown that the different kernels can have very different kinds of decay, e.g. exponential, algebraic, or logarithmic. An initial value problem for a semilinear differential equation with the general fractional derivative (1) is studied in [11], where by using the Schauder fixed point theorem, the uniqueness and the local/global existence of solution are established.

Inverse problems for various types of fractional evolution equations are extensively studied recently, see e.g. [12], [13], [14]. Different kinds of inverse problems for the diffusion equations with the Caputo time-derivative are considered in [15], [16], [17], [18], [19], [20]. For a comprehensive tutorial on inverse problems for anomalous diffusion processes we refer to [21]. Inverse problems for subdiffusion equations with a more general time-derivative, such as multi-term and distributed-order time-fractional equations, are also considered. Uniqueness for two kinds of inverse problems of identifying the orders of fractional derivatives in multi-term time-fractional diffusion equation is established in [22]. Uniqueness results for the recovering of weight function in distributed-order diffusion equations from one interior point observation of the solution are obtained in [23], [24]. An inverse problem for the general fractional derivative is studied in [25] and the results are applied to determine time- and space-dependent sources in general time-fractional diffusion and wave equations. Existence, uniqueness and stability for the inverse source problem with final overdetermination for a generalized subdiffusion equation are established in [26].

In this work, we are concerned with the problem of determining a space-dependent source $h\left(x\right)$ and the solution $u(x,t)$ to the following nonlocal boundary-value problem with final overdetermination

$${\mathbb{D}}_t^{\left(k\right)}u(x,t)=u_{xx}(x,t)+h\left(x\right),x\in (0,1),t\in (0,T),$$$$u(1,t)=0,u_x(0,t)=u_x(1,t),t\in (0,T],$$$$u(x,0)=0,u(x,T)=g\left(x\right),x\in [0,1],$$

where the operator ${\mathbb{D}}_t^{\left(k\right)}$ acting with respect to the time variable is defined in (1), $g\left(x\right)$ is a known square integrable function and $T>0$ is the final time.

Due to the nonlocal character of the second boundary condition in (4), the corresponding spatial differential operator is non-selfadjoint and standard eigenfunction expansion technique does not apply. Non-selfadjoint operators appear e.g. in the modeling of processes with dissipation [27]. In many cases a nonlocal condition is more realistic in treating physical problems than the classical local conditions, which motivates the study of nonlocal boundary-value problems. Direct problems for diffusion equations with nonlocal boundary conditions are considered e.g. in [28], [29], [30], [31]. Inverse source problems with nonlocal boundary conditions are studied e.g. in [32], [33], [34], [35], [36]. The papers [32] and [33] are concerned with particular cases of the inverse source problem (3)–(5) with ${\mathbb{D}}_t^{\left(k\right)}=\frac{\text{d}}{\text{d}t}$ and ${\mathbb{D}}_t^{\left(k\right)}={\mathbb{D}}_t^{\alpha }$, the Caputo time-fractional derivative of order $\alpha \in (0,1)$, respectively. Inverse source problems for diffusion equations with other types of time-fractional operators and boundary conditions (4) are studied in [34], [35].

The aim of this work is to construct spectral expansions for the source function $h\left(x\right)$ and solution $u(x,t)$ of problem (3)–(5) and to prove that these expansions provide a unique solution in the classical sense, based on estimates for the impulse-response solution of the ${\mathbb{D}}_t^{\left(k\right)}$-relaxation equation. Explicit expressions for the time-dependent components in the spectral expansions are derived for the basic memory kernels, with the main emphasis on the multi-term and the truncated power-law memory kernels. To this end we propose a Prabhakar-type generalization of the multinomial Mittag-Leffler function, introduced in [37]. To illustrate the analytical findings, numerical results are presented.

The rest of this paper is organized as follows. Section 2 contains preliminaries on Bernstein functions and functions of Mittag-Leffler type. In Section 3 the assumptions on the memory kernel $k$ in the definition of the general fractional derivative are formulated and basic examples are given. The impulse-response solution to the general fractional relaxation equation is studied in Section 4. Formal spectral expansions for the solution of problem (3)–(5) and the source term are obtained in Section 5. In Section 6 estimates for the time-dependent components in the spectral expansions are established and used to prove the main result concerning uniqueness and existence in the classical sense. Explicit representations for the time-dependent components in the cases of the basic kernels are derived in Section 7. Section 8 contains analytical and numerical examples. Concluding remarks are given in Section 9.