Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations

Abstract

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology.

1. Introduction

With the rapid advancement of science and technology, anomalous diffusion phenomena observed in natural systems have attracted growing attention. It has become increasingly evident that classical differential equation models often fail to accurately capture certain diffusion processes. Over the last few decades, fractional diffusion equations have emerged as powerful modeling tools, finding widespread applications across diverse fields such as porous media, non-Newtonian fluid mechanics, and viscoelastic materials [1]. In practical applications, crucial parameters such as boundary conditions, initial states, diffusion coefficients, and source terms are frequently unknown and thus must be determined using additional information, giving rise to inverse problems for fractional differential equations. Substantial progress has recently been made in the study of such inverse problems, including results related to uniqueness [2,3,4,5,6,7,8] and numerical approaches [9,10].

The time–space fractional diffusion Equation (TSFDE), as a fundamental mathematical tool for characterizing anomalous diffusion, and it can be flexibly tailored to meet specific research objectives as follows: incorporating temporal correlations or memory effects yields time-fractional diffusion equations; accounting for spatial correlations or nonlocal effects leads to space-fractional diffusion equations; and considering both results in time–space fractional diffusion equations. Inverse problems for single-term cases of these equations have been extensively investigated both domestically and abroad. Notable studies have addressed backward problems [11,12,13], source identification [14,15,16,17,18,19,20,21,22], and the determination of the order of fractional derivatives [23,24,25].

On the other hand, ultraslow diffusion characterized by logarithmic growth of mean squared displacement has been observed in various domains, such as polymer physics and particle motion in quenched random force fields. Such phenomena cannot be captured using classical advection–diffusion or conventional fractional diffusion models. To address these cases, distributed-order derivatives have been introduced, leading to distributed-order fractional diffusion equations. When the weight function of a distributed-order derivative is represented as a finite linear combination of Dirac delta functions with positive coefficients, the widely-employed multi-term fractional diffusion equation arises, which more faithfully models diffusion in systems with multiple time scales, coupled mechanisms, or nonlocal memory effects [26].

In this study, we investigate an inverse problem associated with the following multi-term TSFDE:

$$\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}u(x,t)=-{(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}u(x,t)+q\left(x\right)f\left(t\right),x\in \mathsf{\Omega },t\in (0,T],$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^d$ denotes a bounded domain with a sufficiently smooth boundary $\partial \mathsf{\Omega }$, and d is the spatial dimension. For a fixed positive integer m, ${\beta }_j$ and $r_j$$(j=1,2,\dots ,m)$ are positive constants such that $0<{\beta }_m<\dots <{\beta }_1<1$. $q\left(x\right)$ and $f\left(t\right)$ denote the spatially and temporally dependent source terms, respectively. The Caputo fractional left-sided derivative ${\partial }_{0+}^{{\beta }_j}$ is defined by the following:

$${\partial }_{0+}^{{\beta }_j}u(x,t)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-{\beta }_j)}}{\int }_0^t{\displaystyle \frac{\partial u(x,s)}{\partial s}}{\displaystyle \frac{ds}{{(t-s)}^{{\beta }_j}}},0<{\beta }_j<1,0<t\le T,$$

where $\mathsf{\Gamma }$ denotes the Gamma function, and $T>0$ represents the prescribed final time of the process under consideration.

The fractional orders $\mathit{\beta }=({\beta }_1,...,{\beta }_m)$ and the coefficients $\mathit{r}=(r_1,...,r_m)$ are restricted in the admissible sets

$$\begin{array}{ccc}\hfill & & \mathcal{B}:=\{({\beta }_1,...,{\beta }_m)\in {\mathbb{R}}^m;\overline{\beta }\ge {\beta }_1>{\beta }_2>\dots >{\beta }_m\ge \underline{\beta }\},\hfill \\ \hfill & & \mathcal{R}:=\{(r_1,...,r_m)\in {\mathbb{R}}^m;r_1=1,r_j\in [\underline{r},\overline{r}],(j=2,...,m)\},\hfill \end{array}$$

with fixed $0<\underline{\beta }<\overline{\beta }<1$ and $0<\underline{r}<\overline{r}$.

The fractional Laplacian operator, denoted as ${(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}$, is defined for $\alpha$ within the range $1<\alpha \le 2$, based on the spectral decomposition of the classical Laplace operator $-\mathsf{\Delta }$. More specifically, in the context of a Hilbert space, this operator can be rigorously presented as follows:

$$\begin{array}{c}\hfill {\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right):=\left\{u=\sum _{n=1}^{\infty }{a}_n{\psi }_n:{\parallel u\parallel }_{{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)}^2=\sum _{n=1}^{\infty }{a}_n^2{\overline{\lambda }}_n^{\alpha }<\infty \right\},\end{array}$$

and the operator ${(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}$ by

$$\begin{array}{c}\hfill {(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}u:=\sum _{n=1}^{\infty }{a}_n{\psi }_n\mapsto \sum _{n=1}^{\infty }{a}_n{\overline{\lambda }}_n^{\alpha /2}{\psi }_n,\end{array}$$

and it maps ${\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)$ onto $L_2\left(\mathsf{\Omega }\right)$, In this expression, ${\overline{\lambda }}_n$ and ${\psi }_n$ represent the eigenvalues and the corresponding eigenfunctions of the Laplacian operator $-\mathsf{\Delta }$, respectively. Here, the eigenfunctions ${\psi }_n$ satisfy the equation $-\mathsf{\Delta }{\psi }_n={\overline{\lambda }}_n{\psi }_n$, subject to the given boundary conditions as follows:

$$\left\{\begin{array}{ccccc}\hfill & -\mathsf{\Delta }{\psi }_k={\overline{\lambda }}_k{\psi }_k,\hfill & \hfill in& \hfill & \hfill \mathsf{\Omega },\\ \hfill & {\psi }_k=0,\hfill & \hfill on& \hfill & \hfill \partial \mathsf{\Omega }.\end{array}\right.$$

therefore we set

$$\begin{array}{c}\hfill {\parallel u\parallel }_{{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)}={\parallel {(-▵)}^{{\displaystyle \frac{\alpha }{2}}}u\parallel }_{L^2\left(\mathsf{\Omega }\right)}.\end{array}$$

For further discussion on Laplacian operator ${(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}$, please refer to the literature [27].

Suppose the unknown function u is subject to the following initial and boundary conditions:

$$\begin{array}{ccc}\hfill & & u(x,0)=g\left(x\right),x\in \overline{\mathsf{\Omega }},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill & & u(x,t)=0,x\in \partial \mathsf{\Omega },t\in (0,T].\hfill \end{array}$$

(3)

When the functions $q\left(x\right)$, $f\left(t\right)$, and $g\left(x\right)$ are all given, the problem consisting of Equations (1)–(3) is known as the direct or forward problem, where the goal is to compute the solution $u(x,t)$. On the other hand, in the inverse problem, the aim is to recover the unknown time-dependent source function $f\left(t\right)$. This identification is performed with the aid of supplementary measurement data at a fixed location $x_0$ as follows:

$$\begin{array}{c}\hfill u(x_0,t)=\phi \left(t\right),0<t\le T,\end{array}$$

(4)

where $x_0\in \mathsf{\Omega }$ denotes an interior observation point. The objective is to reconstruct $f\left(t\right)$ in the system (1)–(3) using the information provided by $\phi \left(t\right)$.

Moreover, in the particular case where $m=1$, Equation (1) reduces to the following single-term form:

$${\partial }_{0+}^{\beta }u(x,t)=-{(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}u(x,t)+q\left(x\right)f\left(t\right),x\in \mathsf{\Omega },t\in (0,T].$$

(5)

Research into inverse problems for multi-term fractional diffusion equations has been mainly focused on time-fractional variants, yielding significant achievements in several directions as follows: identification of fractional derivative orders [28,29,30]; reconstruction of various source terms [31,32,33]; potential function determination [34,35,36]; and inverse initial data problems [37,38], among others.

To the best of our knowledge, research focused on the inverse problem for multi-term time–space fractional differential Equations (TSFDEs) remains rather limited, Malik et al. [39] investigated the identification of a time-dependent source and diffusion concentration within a TSFRDE characterized by multi-term Hilfer-type time derivatives and Caputo spatial derivatives, subject to homogeneous boundary conditions. Their work established the local Hadamard well-posedness of the inverse source problem, although no numerical algorithms were proposed. In contrast, our study emphasizes the question of uniqueness in recovering a time-dependent source term $f\left(t\right)$ from interior measurement data, thereby generalizing the single-term TSFDE scenario considered by Li et al. [16]. As far as we are aware, our work is the first to tackle the inverse problem specifically for multi-term TSFDEs.

The rest of the paper is structured as follows. In Section 2, we provide preliminary concepts and mathematical tools that serve as the foundation for the subsequent sections. Section 3 details the application of the matrix transfer technique to the direct problem described by Equations (1)–(3), demonstrating the existence and uniqueness of strong solutions as well as developing an implicit finite difference scheme. Section 4 delivers estimates on the stability and uniqueness properties of the inverse source problem. In Section 5, the source identification problem is transformed into a variational problem using the Tikhonov regularization method, and then, an approximate solution to the inverse problem is obtained by employing the optimal perturbation algorithm. In Section 6, numerical findings for six examples in one-dimensional and two-dimensional settings are examined. Finally, we offer a conclusion in Section 7.

2. Preliminary

Throughout this paper, we adopt the following definitions and foundational results, as established in [40].

Definition 1. 

The multinomial Mittag–Leffler function is defined as

$$E_{({\beta }_1,\dots ,{\beta }_m),\beta }(z_1,\dots ,z_m)=\sum _{k=0}^{\infty }\sum _{l_1+\dots +l_m=k}{\displaystyle \frac{(k;l_1,\dots ,l_m){\displaystyle \prod _{j=1}^m}{z}_j^{l_j}}{\mathsf{\Gamma }(\beta +{\displaystyle \sum _{j=1}^m}{\beta }_j{l}_j)}},$$

where $0<\beta <2$, $0<{\beta }_j<1$, and $z_j\in \mathbb{C}$ for $j=1,\dots ,m$. Here, $(k;l_1,\dots ,l_m)$ denotes the multinomial coefficient,

$$\begin{array}{c}\hfill (k;l_1,\dots ,l_m):={\displaystyle \frac{k!}{l_1!\dots l_m!}}\mathrm{with}k=\sum _{j=1}^m{l}_j,\end{array}$$

where each $l_j$$(1\le j\le m)$ is a non-negative integer.

With regard to the multinomial Mittag–Leffler function, the following three important results from [40] will be utilized in our analysis.

Proposition 1 

(See [40]). Let $0<\beta <2$ and $0<{\beta }_m<\dots <{\beta }_1<1$ be arbitrary. Assume that ${\beta }_1\pi /2<\mu <{\beta }_1\pi$ and there exists $K>0$ such that $-K\le z_j<0(j=2,\dots ,m)$. Then, there exists a constant $C=C({\beta }_j,\beta ,\mu ,K)>0(j=2,\dots ,m)$ such that

$$\mid E_{({\beta }_1,{\beta }_1-{\beta }_2,\dots ,{\beta }_1-{\beta }_m),\beta }(z_1,\dots ,z_m)\mid \le {\displaystyle \frac{C}{1+\mid z_1\mid }},\mu \le |arg\left(z_1\right)|\le \pi .$$

For convenience, we will adopt the following abbreviations

$$E_{{\beta }^{\prime },\beta }^{\left(n\right)}\left(t\right):=E_{({\beta }_1,{\beta }_1-{\beta }_2,\dots ,{\beta }_1-{\beta }_m),\beta }(-{\lambda }_n{t}^{{\beta }_1},-r_2{t}^{{\beta }_1-{\beta }_2},\dots ,-r_m{t}^{{\beta }_1-{\beta }_m}),t>0$$

where ${\lambda }_n={{\overline{\lambda }}_n}^{{\displaystyle \frac{\alpha }{2}}}$ is the nth eigenvalue of ${(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}$$(1<\alpha \le 2)$; $0<\beta <2$, ${\beta }_j$, and $r_j$ are the positive constants in (1).

Proposition 2 

(See [40]). Let $0<\beta <2$, $0<{\beta }_j<1(j=1,\dots ,m)$, and $z_j\in \mathbb{c}(j=1,\dots ,m)$ be fixed. Then

$${\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}+\sum _{j=1}^m{z}_j{E}_{({\beta }_1,\dots ,{\beta }_m),\beta +{\beta }_j}(z_1,\dots ,z_m)=E_{({\beta }_1,\dots ,{\beta }_m),\beta }(z_1,\dots ,z_m).$$

Concerning the regularity of $E_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}\left(t\right)$$(t>0)$ in Proposition 1, the estimate

$$\begin{array}{ccc}\hfill E_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}\left(t\right)& =& {\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)}}-{\lambda }_n{t}^{{\beta }_1}{E}_{{\beta }^{\prime },2{\beta }_1}^{\left(n\right)}\left(t\right)-\sum _{j=2}^m{r}_j{t}^{{\beta }_1-{\beta }_j}{E}_{{\beta }^{\prime },2{\beta }_1-{\beta }_j}^{\left(n\right)}\left(t\right)\hfill \\ & <& {\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)}},t>0\hfill \end{array}$$

is essential.

Proposition 3. 

Let $1>{\beta }_1>\dots >{\beta }_m>0$, then we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left\{t^{{\beta }_1}{E}_{{\beta }^{\prime },1+{\beta }_1}^{\left(n\right)}\left(t\right)\right\}=t^{{\beta }_1-1}{E}_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}\left(t\right),t>0.\end{array}$$

Lemma 1. 

Assuming that $f\left(t\right)\in L^{\infty }(0,T)$, $1>{\beta }_1>\dots >{\beta }_m>0$, ${\lambda }_n\ge 0$, denote

$$W_n\left(t\right)={\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,\beta 1}^{\left(n\right)}(t-\tau )d\tau ,t\in (0,T],$$

(6)

and define $W_n\left(0\right)=0$. Then $W_n\left(t\right)\in C[0,T]$. Moreover, we can obtain the following estimates:

$$\begin{array}{ccc}\hfill \left|W_n\left(t\right)\right|& \le & C_1{\displaystyle \frac{{\Vert f\Vert }_{L^{\infty }(0,T)}}{{\lambda }_n}},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \left|W_n\left(t\right)\right|& \le & C_2{\Vert f\Vert }_{L^{\infty }(0,T)}.\hfill \end{array}$$

(8)

where $C_1,C_2>0$ is a constant.

Proof. 

Let us first prove the continuity of $W_n\left(t\right)$. Supposing $h>0$, for any t, $t+h\in (0,T]$, we have

$$\begin{array}{ccc}\hfill \left|W_n(t+h)-W_n\left(t\right)\right|& =& \left|{\int }_0^{t+h}f\left(\tau \right){(t+h-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,\beta 1}^{\left(n\right)}(t+h-\tau )d\tau -{\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau \right|\hfill \\ \hfill & =& |{\int }_0^{t}f\left(\tau \right)[{(t+h-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t+h-\tau )-{(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )]d\tau +\hfill \\ & & {\int }_t^{t+h}f\left(\tau \right){(t+h-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t+h-\tau )d\tau |\hfill \\ & \le & {\parallel f\parallel }_{L^{\infty }(0,T)}\left|{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)-t^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right|.\hfill \end{array}$$

It is clear that we have $\underset{h\to 0^{+}}{lim}{W}_n(t+h)=W_n\left(t\right)$. By a similar deduction, we have $\underset{h\to 0^{-}}{lim}{W}_n(t+h)=W_n\left(t\right)$. Therefore, we can obtain $W_n\left(t\right)\in C[0,T]$. Consequently for each $0\le t\le T$, we deduce

$$\begin{array}{ccc}\hfill \left|W_n\left(t\right)\right|& =& \left|{\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau \right|\hfill \\ \hfill & \le & {\Vert f\Vert }_{L^{\infty }(0,T)}\left|{\int }_0^t{(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau \right|\hfill \\ & \le & {\Vert f\Vert }_{L^{\infty }(0,T)}{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\hfill \\ & \le & {\Vert f\Vert }_{L^{\infty }(0,T)}{\displaystyle \frac{C_1{t}^{{\beta }_1}}{1+{\lambda }_n{t}^{{\beta }_1}}}\hfill \\ & \le & C_1{\displaystyle \frac{{\Vert f\Vert }_{L^{\infty }(0,T)}}{{\lambda }_n}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|W_n\left(t\right)\right|& =& \left|{\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau \right|\hfill \\ \hfill & \le & {\Vert f\Vert }_{L^{\infty }(0,T)}\left|{\int }_0^t{(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau \right|\hfill \\ & \le & {\Vert f\Vert }_{L^{\infty }(0,T)}\left|{\int }_0^t{(t-\tau )}^{{\beta }_1-1}{\displaystyle \frac{C}{1+{\lambda }_n{(t-\tau )}^{{\beta }_1}}}d\tau \right|\hfill \\ & \le & {C\Vert f\Vert }_{L^{\infty }(0,T)}\left|{\int }_0^t{(t-\tau )}^{{\beta }_1-1}d\tau \right|\hfill \\ & \le & C_2{\Vert f\Vert }_{L^{\infty }(0,T)}.\hfill \end{array}$$

□

3. Regularity of the Solution and Difference Scheme for the Direct Problem

Let us denote the eigenvalues of the Laplacian operator $-\mathsf{\Delta }$ under homogeneous Dirichlet boundary conditions by ${\overline{\lambda }}_n$ and the corresponding eigenfunctions by ${\psi }_n\in H^2\left(\mathsf{\Omega }\right)\cap {{\mathcal{H}}_0}^1\left(\mathsf{\Omega }\right)$. Specifically, these eigenfunctions satisfy the spectral problem $-\mathsf{\Delta }{\psi }_n={\overline{\lambda }}_n{\psi }_n$ along with the boundary condition ${\psi }_n{|}_{\partial \mathsf{\Omega }}=0$. Enumerating the eigenvalues according to their multiplicities, we have $0<{\overline{\lambda }}_1\le {\overline{\lambda }}_2\le \dots \le {\overline{\lambda }}_n\le \dots$ , and the family ${\left\{{\psi }_n\right\}}_{n=1}^{\infty }$ forms an orthonormal basis in $L^2\left(\mathsf{\Omega }\right)$.

Next, we introduce the definition of a strong solution for the direct problem (1)–(3), and we subsequently establish the existence and uniqueness of such a solution by following the approach in [3].

Theorem 1. 

Suppose that $g\in {\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)$, $q\in L^2\left(\mathsf{\Omega }\right)$, and $f\in L^{\infty }(0,T)$, and fix $\mathit{\beta }\in \mathcal{B}$, $\mathit{r}\in \mathcal{R}$. Then there exists a unique solution $u(x,t)$ to (1)–(3), given by

$$u(x,t)=\sum _{n=1}^{\infty }\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)\left(g,{\psi }_n\right){\psi }_n\left(x\right)+\sum _{n=1}^{\infty }{W}_n\left(t\right)\left(q,{\psi }_n\right){\psi }_n\left(x\right),$$

(9)

where ${\lambda }_n={\overline{\lambda }n}^{\alpha /2}$ and $W_n\left(t\right)={\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{\beta \prime ,{\beta }_1}^{\left(n\right)}(t-\tau )d\tau$. Moreover, the following estimates hold:

$${\parallel u\parallel }_{C([0,T];L^2\left(\mathsf{\Omega }\right))}\le C_3\left({\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}+{\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right),$$

(10)

$${\parallel u\parallel }_{L^2(0,T;{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right))}\le C_4\left({\parallel g\parallel }_{{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)}+{\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right).$$

(11)

where $C_3$ and $C_4$ are positive constants that depend on ${\beta }_j$$(j=1,2,\dots ,m)$, T, and Ω.

Proof. 

Based on Reference [40] and the method of separation of variables, we can easily obtain the analytical expression 9 for the direct problem (1)–(3). It only remains to prove the regularity and estimates of the solution.

We first verify $u\in C\left([0,T];L^2\left(\mathsf{\Omega }\right)\right)$ and $\underset{t\to 0}{lim}{∥u(·,0)-g(·)∥}_{L^2\left(\mathsf{\Omega }\right)}=0$. Define

$$\begin{array}{ccc}\hfill u_1(x,t)& :=& \sum _{n=1}^{\infty }\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)(g,{\psi }_n)\psi \left(x\right),\hfill \\ \hfill u_2(x,t)& :=& \sum _{n=1}^{\infty }{W}_n\left(t\right)(q,{\psi }_n){\psi }_n\left(x\right),\hfill \end{array}$$

then we have $u(x,t)=u_1(x,t)+u_2(x,t)$. By using the result of Proposition 1, we get

$$\begin{array}{ccc}\hfill {∥u_1(x,t)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)}^2{(g,{\psi }_n)}^2\hfill \\ & \le & 2\left[1+{\left({\displaystyle \frac{C{\lambda }_n{t}^{{\beta }_1}}{1+{\lambda }_n{t}^{{\beta }_1}}}\right)}^2\right]{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & \le & C_5{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

and together with (8), we can obtain

$$\begin{array}{c}\hfill \parallel u_2{(x,t)\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2=\sum _{n=1}^{\infty }{W_n\left(t\right)}^2{(q,{\psi }_n)}^2\le {\left(C_2{∥f∥}_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right)}^2.\end{array}$$

Therefore, we can obtain

$$\begin{array}{ccc}\hfill {\parallel u(x,t)\parallel }_{L^2\left(\mathsf{\Omega }\right)}& \le & \parallel u_1{(x,t)\parallel }_{L^2\left(\mathsf{\Omega }\right)}+{\parallel u_2(x,t)\parallel }_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill & \le & C_3\left({\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}+{\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right),\hfill \end{array}$$

(12)

where $C_3$ represents positive constants depending on ${\beta }_j(j=1,2,\dots ,m)$, T, $\mathsf{\Omega }$.

For t, $t+h\in [0,T]$, we have

$$\begin{array}{ccc}& & u(x,t+h)-u(x,t)\hfill \\ & =& \sum _{n=1}^{\infty }\left[\left(1-{\lambda }_n{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)\right)(g,{\psi }_n)+W_n(t+h)(q,{\psi }_n)\right]{\psi }_n\left(x\right)-\hfill \\ & & \sum _{n=1}^{\infty }\left[\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)(g,{\psi }_n)+W_n\left(t\right)(q,{\psi }_n)\right]{\psi }_n\left(x\right)\hfill \\ \hfill & =& \sum _{n=1}^{\infty }\left[{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)-{\lambda }_n{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)\right](g,{\psi }_n){\psi }_n\left(x\right)+\hfill \\ & & \sum _{n=1}^{\infty }\left(W_n(t+h)-W_n\left(t\right)\right)(q,{\psi }_n){\psi }_n\left(x\right)\hfill \\ & :=& R_1(x,t;h)+R_2(x,t;h).\hfill \end{array}$$

From Proposition 1 and Lemma 1, we derive the following estimate:

$$\begin{array}{ccc}\hfill {∥R_1(x,t;h)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{\left[{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)-{\lambda }_n{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)\right]}^2{(g,{\psi }_n)}^2\hfill \\ & \le & 2\sum _{n=1}^{\infty }\left[{\left({\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)}^2+{\left({\lambda }_n{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)\right)}^2\right]{(g,{\psi }_n)}^2\hfill \\ & \le & C_6{∥g∥}_{L^2\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill {∥R_2(x,t;h)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{(W_n(t+h)-W_n\left(t\right))}^2{(q,{\psi }_n)}^2\hfill \\ & \le & 2\sum _{n=1}^{\infty }\left(W_n^2(t+h)+W_n^2\left(t\right)\right){(q,{\psi }_n)}^2\hfill \\ & \le & C_7{\parallel f\parallel }_{L^{\infty }(0,T)}^2{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

Consequently, we infer that

$$\begin{array}{ccc}\hfill {∥u(x,t+h)-u(x,t)∥}_{L^2\left(\mathsf{\Omega }\right)}& \le & C_8\left({∥g∥}_{L^2\left(\mathsf{\Omega }\right)}+{\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right).\hfill \end{array}$$

Since $\underset{h\to 0}{lim}|{(t+h)}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}(t+h)-t^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)|=0$ for each $n\in \mathbb{N}$, by using the Lebesgue theorem, we can arrive at

$$\underset{h\to 0}{lim}{∥u(x,t+h)-u(x,t)∥}_{L^2\left(\mathsf{\Omega }\right)}=0,$$

(13)

which $u\in C([0,T];L^2\left(\mathsf{\Omega }\right))$.

Utilizing $\underset{t\to 0}{lim}{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)=0$ and (13), we deduce that

$$\underset{t\to 0^{+}}{lim}{∥u(·,t)-g(·)∥}_{L^2\left(\mathsf{\Omega }\right)}=0.$$

Next, we prove that ${(-▵)}^{{\displaystyle \frac{\alpha }{2}}}u\in C\left(\left(\left(0,T\right];L^2\left(\mathsf{\Omega }\right)\right)\cap L^2\left(0,T;L^2\left(\mathsf{\Omega }\right)\right)\right)$ and $u\in L^2$$(0,T;{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right))$. Using (9), we have

$$\begin{array}{ccc}\hfill {(-▵)}^{{\displaystyle \frac{\alpha }{2}}}u(x,t)& =& \sum _{n=1}^{\infty }{\lambda }_n\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)(g,{\psi }_n)\psi \left(x\right)+\sum _{n=1}^{\infty }{\lambda }_n{W}_n\left(t\right)(q,{\psi }_n){\psi }_n\left(x\right)\hfill \\ & :=& w_1(x,t)+w_2(x,t).\hfill \end{array}$$

For $0<t\le T$, by using (7), we derive the estimate

$$\begin{array}{ccc}\hfill {∥w_1(·,t)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{\lambda }_n^2{\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)}^2{(g,{\psi }_n)}^2\le 2\sum _{n=1}^{\infty }{{\lambda }_n}^2\left(1+{\left({\displaystyle \frac{C{\lambda }_n{t}^{{\beta }_1}}{1+{\lambda }_n{t}^{{\beta }_1}}}\right)}^2\right){(g,{\psi }_n)}^2\hfill \\ & \le & C_9{\parallel g\parallel }_{{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {∥w_2(·,t)∥}_{L^2\left(\mathsf{\Omega }\right)}^2=\sum _{n=1}^{\infty }{\lambda }_n^2{(q,{\psi }_n)}^2{W}_n^2\left(t\right)\le C_{10}^{\prime }\sum _{n=1}^{\infty }{\lambda }_n^2{\displaystyle \frac{{\parallel f\parallel }_{L^{\infty }(0,T)}^2}{{\lambda }_n^2}}{(q,{\psi }_n)}^2\le C_{10}{\parallel f\parallel }_{L^{\infty }(0,T)}^2{∥q∥}_{L^2\left(\mathsf{\Omega }\right)}^2.\end{array}$$

Since $w_1(x,t)$ and $w_2(x,t)$ are uniformly convergent in $L^2\left(\mathsf{\Omega }\right)$ with respect to t over any interval $[t_0,T]$ for an arbitrary $t_0>0$, it follows in a manner analogous to the argument in the first part of the proof that ${(-\mathsf{\Delta })}^{{\displaystyle \frac{\alpha }{2}}}u\in C([0,T];L^2\left(\mathsf{\Omega }\right))$. Consequently, we obtain the estimate

$$\begin{array}{c}\hfill {∥u∥}_{L^2(0,T;{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right))}={∥{(-▵)}^{{\displaystyle \frac{\alpha }{2}}}u∥}_{L^2(0,T;L^2\left(\mathsf{\Omega }\right))}\le C_4\left({\parallel g\parallel }_{{\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)}+{\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{L^2\left(\mathsf{\Omega }\right)}\right),\end{array}$$

where $C_4$ is a positive constant depending on the parameters ${\beta }_j$$(j=1,2,\dots ,m)$, the final time T, and the spatial domain $\mathsf{\Omega }$. □

In order to employ inversion techniques for solving the inverse problem at each step, it is first necessary to address the direct problem (1)–(3). In the subsequent analysis, we present an implicit finite difference scheme utilizing the matrix transfer technique, as detailed in [41,42,43], to numerically solve the direct problem (1)–(3). For the sake of simplicity and clarity, our discussion is restricted to the one-dimensional multi-term TSFDE case.

In the finite difference framework, the time and spatial domains are discretized with step sizes $\mathsf{\Delta }t=T/N$ and $\mathsf{\Delta }x=1/M$, respectively. The temporal grid points are given by $t_n=n\mathsf{\Delta }t$ for $n=0,1,\dots ,N$, while the spatial points are $x_i=i\mathsf{\Delta }x$ for $i=0,1,\dots ,M$. The notation $u_i^n\approx u(x_i,t_n)$ represents the numerical approximation of the function u at the grid point $(x_i,t_n)$.

We begin by considering the classical diffusion equation accompanied by the initial and boundary value conditions.

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+q\left(x\right)f\left(t\right),0<x<1,t>0,\hfill \\ u(x,0)=g\left(x\right),\hfill \\ u(0,t)=u(1,t)=0.\hfill \end{array}\right.$$

(14)

Introducing a finite difference approximation, we obtain

$$\left\{\begin{array}{c}{\displaystyle \frac{d{u}_i}{\partial t}}={\displaystyle \frac{1}{h}}(u_{i-1}-2u_i+u_{i+1})+f\left(t\right)q_i,0<t<T,\hfill \\ u_i=g_i,\hfill \\ u_0=u_N=0,\hfill \end{array}\right.$$

where $u_i=u(x_i,t)$, $q_i=q\left(x_i\right)$, $g_i=g\left(x_i\right),i=1,2,\dots M$, and h is the space step defined as $h={\displaystyle \frac{1}{M}}$. The above equations can be expressed as the following system of ODEs:

$${\displaystyle \frac{\partial \mathit{U}}{\partial t}}=-\eta \mathit{B}\mathit{U}+f\left(t\right)Q,$$

(15)

where $\eta ={\displaystyle \frac{1}{h^2}}$ and $\mathit{U},b\in {\mathbb{R}}^{N-1}$, $\mathbf{B}\in {\mathbb{R}}^{N-1\times N-1}$,

$$\begin{array}{cccc}\mathit{U}=\left(\begin{array}{c}{u}_1\\ u_2\\ ⋮\\ u_{N-1}\end{array}\right),& {\mathit{U}}^0=\left(\begin{array}{c}{g}_1\\ g_2\\ ⋮\\ g_{N-1}\end{array}\right),& Q=\left(\begin{array}{c}{q}_1\\ q_2\\ ⋮\\ q_{N-1}\end{array}\right),& \mathit{B}=\left(\begin{array}{cccccc}2& -1& & & & \\ -1& 2& -1& & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & -1& \\ & & & -1& 2& \end{array}\right),\end{array}$$

Let $\mathit{B}$ be a real, non-singular, symmetric matrix of size $(N-1)\times (N-1)$. By virtue of its symmetry and non-singularity, there exists a non-singular matrix $P\in {\mathbb{R}}^{(N-1)\times (N-1)}$ such that $\mathit{B}$ can be diagonalized as

$$\mathit{B}=P\mathsf{\Lambda }{P}^T,$$

where $\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{N-1})$ is a diagonal matrix of the eigenvalues ${\lambda }_i(i=1,2,\dots ,N-1)$ of $\mathit{B}$.

With this preparation, using the methods described in References [41,42], we now reformulate the direct problem (1)–(3) in the following matrix representation:

$$\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}\mathit{U}=-\overline{\eta }{\mathit{B}}^{{\displaystyle \frac{\alpha }{2}}}\mathit{U}+f\left(t\right)Q,$$

(16)

where $\overline{\eta }={\displaystyle \frac{1}{h^{\alpha }}}$, ${\mathit{B}}^{{\displaystyle \frac{\alpha }{2}}}=P{\mathsf{\Lambda }}^{{\displaystyle \frac{\alpha }{2}}}{P}^T$. The time-fractional derivative is approximated by

$${\partial }_{0^{+}}^{{\beta }_j}u(x,t_n)\approx {\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(a_0^{\left({\beta }_j\right)}u(x,t_n)-\sum _{k=1}^{n-1}(a_{n-k-1}^{\left({\beta }_j\right)}-a_{n-k}^{\left({\beta }_j\right)})u(x,t_k)-a_{n-1}^{\left({\beta }_j\right)}u(x,t_0)\right),$$

where $a_l^{\left({\beta }_j\right)}={(l+1)}^{1-{\beta }_j}-l^{1-{\beta }_j},l\ge 0$, this scheme was used in [44]. We have ${\omega }_k^{{\beta }_j}=a_{n-k-1}^{\left({\beta }_j\right)}-a_{n-k}^{\left({\beta }_j\right)}$, and

$$\left\{\begin{array}{c}{\partial }_{0^{+}}^{{\beta }_j}{u}_1^n={\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(a_0^{\left({\beta }_j\right)}{u}_1^n-\sum _{k=1}^{n-1}{\omega }_k^{{\beta }_j}{u}_1^k-a_{n-1}^{\left({\beta }_j\right)}{u}_1^0\right),\hfill \\ {\partial }_{0^{+}}^{{\beta }_j}{u}_2^n={\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(a_0^{\left({\beta }_j\right)}{u}_2^n-\sum _{k=1}^{n-1}{\omega }_k^{{\beta }_j}{u}_2^k-a_{n-1}^{\left({\beta }_j\right)}{u}_2^0\right),\hfill \\ \dots \dots \dots \hfill \\ {\partial }_{0^{+}}^{{\beta }_j}{u}_{N-1}^n={\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(a_0^{\left({\beta }_j\right)}{u}_{N-1}^n-\sum _{k=1}^{n-1}{\omega }_k^{{\beta }_j}{u}_{N-1}^k-a_{n-1}^{\left({\beta }_j\right)}{u}_{N-1}^0\right).\hfill \end{array}\right.$$

(17)

Subsequently, the implicit difference scheme defined by Equations (16) and (17) can be recast in the following matrix form:

$$\left\{\begin{array}{c}\mathit{A}{\mathit{U}}^n=b,\hfill \\ U^0=G,\hfill \end{array}\right.$$

(18)

where

$$U^n=\left(u_1^n,u_2^n,\dots ,u_{N-1}^n\right),$$

$$G^n=\left(g_1,g_2,\dots ,g_{N-1}\right),$$

$$A=\sum _{j=1}^m\left(a_0^{\left({\beta }_j\right)}{\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}{I}_{(N-1)\times (N-1)}\right)+\overline{\eta }{B}^{{\displaystyle \frac{\alpha }{2}}},$$

and

$$\begin{array}{c}b={\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(\begin{array}{cccc}{u}_1^1& u_1^2& \dots & u_1^{n-1}\\ u_2^1& u_2^2& \dots & u_2^{n-1}\\ \dots & \dots & \dots & \dots \\ u_{N-1}^1& u_{N-1}^2& \dots & u_{N-1}^{n-1}\end{array}\right)\left(\begin{array}{c}{\omega }_1^{\left({\beta }_j\right)}\\ {\omega }_2^{\left({\beta }_j\right)}\\ ⋮\\ {\omega }_{n-1}^{\left({\beta }_j\right)}\end{array}\right)+a_{n-1}^{\left({\beta }_j\right)}{\displaystyle \frac{{\tau }^{-{\beta }_j}}{\mathsf{\Gamma }(2-{\beta }_j)}}\left(\begin{array}{c}{g}_1\\ g_2\\ ⋮\\ g_{N-1}\end{array}\right)\right)+f\left(t_n\right)Q.\end{array}$$

The above describes a finite difference discretization of a one-dimensional multi-term space–time fractional diffusion equation. In the two-dimensional case, the approach is analogous; however, the matrix obtained from the spatial fractional-order discretization no longer maintains a symmetric tridiagonal form. Instead, it takes on a symmetric tridiagonal block structure as follows:

$$\begin{array}{c}\hfill B={\left(\begin{array}{ccccc}{B}_1& -I& & & \\ -I& B_1& \ddots & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & B_1& -I\\ & & & -I& B_1\end{array}\right)}_{{(M+1)}^2\times {(M+1)}^2},\end{array}$$

where

$$\begin{array}{c}\hfill B_1={\left(\begin{array}{ccccc}4& -1& & & \\ -1& 4& \ddots & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & 4& -1\\ & & & -1& 4\end{array}\right)}_{(M+1)\times (M+1)}.\end{array}$$

The detailed discretization procedure for the two-dimensional scenario will not be reiterated here.

4. Uniqueness for the Inverse Problem

Theorem 2. 

Let $u_1$ and $u_2$ denote the solutions to the problem corresponding to source terms $f_1,f_2\in L^{\infty }(0,T)$, respectively, with a fixed $q\in {\mathcal{H}}_0^{2\gamma }\left(\mathsf{\Omega }\right)$, where $\gamma >{\displaystyle \frac{d}{2}}$, and $g\in {\mathcal{H}}_0^{\alpha }\left(\mathsf{\Omega }\right)$. Suppose there exists an interior point $x_0\in \mathsf{\Omega }$ such that $q\left(x_0\right)\ne 0$. Then, there exists a positive constant $C_{11}$, depending on the parameters ${\beta }_j$$(j=1,\dots ,m)$, the function q, the domain Ω, and the time T, such that the following stability estimate holds:

$${∥f_1-f_2∥}_{L^{\infty }(0,T)}\le C_{11}{\left|\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}(u_1(x_0,t)-u_2(x_0,t))\right|}_{L^{\infty }(0,T)},$$

(19)

which establishes the uniqueness and stability for retrieving the time-dependent source term.

Proof. 

Let $u=u_1-u_2$, $f=f_1-f_2$, then

$$\begin{array}{ccc}\hfill u(x,t)& =& \sum _{n=1}^{\infty }{\int }_0^{t}f\left(\tau \right){(t-\tau )}^{{\beta }_1-1}{E}_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}(t-\tau )d\tau (q,{\psi }_n){\psi }_n\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& :=& \sum _{n=1}^{\infty }{W}_n\left(t\right)(q,{\psi }_n){\psi }_n.\hfill \end{array}$$

(21)

substitute (20) into Equation (1), and we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cccc}& \sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}{W}_n\left(t\right)+{\lambda }_n{W}_n\left(t\right)=f\left(t\right),\hfill & & t>0,\hfill \\ & W_n\left(0\right)=0,\hfill \end{array}\right.\end{array}$$

and

$$\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}u(x,t)=f\left(t\right)\sum _{n=0}^{\infty }(q,{\psi }_n){\psi }_n\left(x\right)-\sum _{n=0}^{\infty }{\lambda }_n{W}_n\left(t\right)(q,{\psi }_n){\psi }_n\left(x\right).$$

Next, we proved the convergence of the above series. If $m>{\displaystyle \frac{d}{4}}$, there exists a constant $C>0$ such that

$$\parallel {\psi }_n{\parallel }_{C\left(\mathsf{\Omega }\right)}\le C{\parallel {\psi }_n\parallel }_{H^{2m}\left(\mathsf{\Omega }\right)},$$

and

$$\parallel {\psi }_n{\parallel }_{H^{2m}\left(\mathsf{\Omega }\right)}={\parallel {(-\mathsf{\Delta })}^m{\psi }_n\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C{\left({\overline{\lambda }}_n\right)}^m,n=1,2,\dots .$$

It is easy to see that

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{\parallel (q,{\psi }_n){\psi }_n\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}& \le & C\sum _{n=1}^{\infty }\parallel (q,{\psi }_n){\psi }_n{\parallel }_{H^{2m}\left(\mathsf{\Omega }\right)}\hfill \\ & \le & C\sum _{n=1}^{\infty }\left|(q,{\psi }_n)\right|{{\overline{\lambda }}_n}^m\hfill \\ & \le & C{\left(\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{{\overline{\lambda }}_n}^{2v}}}\right)}^{{\displaystyle \frac{1}{2}}}{\left(\sum _{n=1}^{\infty }{\overline{\lambda }}_n^{2(m+v)}{(q,{\psi }_n)}^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Since ${\overline{\lambda }}_n>C{n}^{{\displaystyle \frac{d}{2}}}(n\in N)$, it follows that $v>{\displaystyle \frac{d}{4}}$ and $\gamma >m+v>{\displaystyle \frac{d}{2}}$. By $q\in {\mathcal{H}}_0^{2\gamma }\left(\mathsf{\Omega }\right)$, the series ${\displaystyle \sum _{n=1}^{\infty }}(q,{\psi }_n){\psi }_n\left(x\right)$ uniformly converges on $(x,t)\in \overline{\mathsf{\Omega }}\times [0,T]$ . Furthermore, in view of Lemma 1, we thus deduce

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }\left|{\lambda }_n{W}_n\left(t\right)(q,{\psi }_n){\psi }_n\left(x\right)\right|& \le & \sum _{n=1}^{\infty }|(q,{\psi }_n)|\parallel {\psi }_n{\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}{\parallel W_n\left(t\right)\parallel }_{C[0,T]}\hfill \\ & \le & {C\parallel f\parallel }_{L^{\infty }(0,T)}{\parallel q\parallel }_{{\mathcal{H}}_0^{2\gamma }\left(\mathsf{\Omega }\right)},\hfill \end{array}$$

for some appropriate constant C. Then, the series is uniformly convergent in $\overline{\mathsf{\Omega }}\times [0,T]$.

Hence at the point $x_0$,

$$\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}u(x_0,t)=f\left(t\right)\sum _{n=1}^{\infty }(q,{\psi }_n){\psi }_n\left(x_0\right)-\sum _{n=1}^{\infty }{\lambda }_n{W}_n\left(t\right)(q,{\psi }_n){\psi }_n\left(x_0\right).$$

Since $q\in {\mathcal{H}}_0^{2\gamma }\left(\mathsf{\Omega }\right)$ and ${\mathcal{H}}_0^{2\gamma }\left(\mathsf{\Omega }\right)\hookrightarrow C\left(\overline{\mathsf{\Omega }}\right)$ is a Sobolev space, we discover

$$q\left(x_0\right)=\sum _{n=0}^{\infty }(q,{\psi }_n){\psi }_n\left(x_0\right).$$

Let $Q\left(t\right):={\displaystyle \sum _{n=1}^{\infty }}(-{\lambda }_n)(q,{\psi }_n)E_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}\left(t\right){\psi }_n\left(x_0\right)$. The same approach is used and may find that

$$\sum _{n=1}^{\infty }(-{\lambda }_n)(q,{\psi }_n)E_{{\beta }^{\prime },{\beta }_1}^{\left(n\right)}\left(t\right){\psi }_n\left(x\right)$$

is uniformly convergent on $\overline{\mathsf{\Omega }}\times [0,T]$ as well. Therefore, we have $Q\left(t\right)\in C[0,T]$.

Collecting all the results above, we arrive at

$$\left|f\left(t\right)\right|\le C_{12}\left({∥\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}u(x_0,t)∥}_{L^{\infty }(0,T)}+{\parallel Q\parallel }_{C[0,T]}{\int }_0^t{(t-s)}^{{\beta }_1-1}f\left(s\right)ds\right),t\in (0,T].$$

(22)

Thus, the weakly singular Gronwall-type inequality with kernel ${(t-s)}^{{\beta }_1-1}$ yields the estimate

$$\left|f\left(t\right)\right|\le C_{11}{∥\sum _{j=1}^m{r}_j{\partial }_{0+}^{{\beta }_j}u(x_0,t)∥}_{L^{\infty }(0,T)},t\in (0,T],$$

(23)

where $C_{11}>0$, depending on ${\beta }_j(j=1,\dots ,m)$, where q, $\mathsf{\Omega }$, and T are constants.

□

5. The Inversion Algorithm

In the preceding section, we provided a theoretical justification demonstrating that the source term $f\left(t\right)$ can be uniquely identified from the measurement data collected at a single boundary point. Building upon this result, in the present section, we propose a computational approach to obtain an approximate solution for the source term. This method leverages the additional data specified in (4) to enhance the numerical reconstruction of $f\left(t\right)$.

Through the integration of Equations (4) and (9), a first-kind Volterra integral equation is derived as follows:

$$\mathcal{A}p:={\int }_0^{t}k(t,\tau )f\left(\tau \right)d\tau =y\left(t\right),$$

(24)

where the kernel function takes the form

$$\left\{\begin{array}{ccccc}\hfill & k(t,\tau )={(t-\tau )}^{{\beta }_1-1}\sum _{n=1}^{\infty }{W}_n{\psi }_n\left(x_0\right)E_{\beta \prime ,\beta 1}^{\left(n\right)}(t-\tau ),\hfill & \hfill & \hfill & \hfill t>\tau ,\\ \hfill & k(t,\tau )=0,\hfill & \hfill & \hfill & \hfill t\le \tau .\end{array}\right.$$

and the right-hand side term $y\left(t\right)$ represents the measured data at the interior point $x_0$ as follows:

$$y\left(t\right)=\phi \left(t\right)-\sum _{n=1}^{\infty }\left(1-{\lambda }_n{t}^{{\beta }_1}{E}_{\beta \prime ,{\beta }_1+1}^{\left(n\right)}\left(t\right)\right)\left(g,{\psi }_n\right){\psi }_n\left(x\right),$$

(25)

where $q_n=(q,{\psi }_n)$ and $g_n=(g,{\psi }_n)$.

As established in the seminal works of [16,45], the kernel function $k(t,\tau )$ exhibits weak singularity over the domain $[0,T]\times [0,T]$, and the associated operator $\mathcal{A}$ is compact when mapping from $C[0,T]$ to $C[0,T]$. Consequently, the inverse source problem (1)–(4) is inherently ill-posed. This mathematical characteristic implies that the solution $f\left(t\right)$ lacks continuous dependence on the input data, meaning that even infinitesimal perturbations in the measurement data may lead to significant deviations in the reconstructed solution.

To resolve this challenging inverse problem, we make use of the optimal perturbation algorithm to estimate the time-dependent source term $f\left(t\right)$. Let us consider ${{\varphi }_k\left(t\right)}_{k=1}^{\infty }\subset C^2[0,1]$ as a complete set of basis functions. The source term can then be approximated through the following finite-dimensional expansion:

$$f\left(t\right)\approx f^K\left(t\right)=\sum _{k=1}^K{a}_k{\varphi }_k\left(t\right),$$

(26)

where $f^K\left(t\right)$ represents the K-th order approximation to $f\left(t\right)$, $K\in \mathbf{N}$ denotes the truncation level controlling the approximation accuracy, and $a_k,k=1,2,\dots K$ are the coefficients of expansion.

We define the finite-dimensional approximation space as follows:

$${\mathsf{\Phi }}^K=span\{{\varphi }_1,{\varphi }_2,\dots ,{\varphi }_K\},$$

and associate each approximation $f^K\left(t\right)\in {\mathsf{\Phi }}^K$ with its coefficient vector $\mathit{a}=(a_1,a_2,\dots ,a_K)\in R^K$. This parameterization transforms the infinite-dimensional inverse problem into a finite-dimensional optimization problem.

To handle the intrinsic ill-posedness of the inverse problem and to ensure numerical robustness, the Tikhonov regularization method is employed in solving the following minimization problem as follows:

$$minJ\left(\mathit{a}\right)={\displaystyle \frac{1}{2}}\parallel u(x_0,t;\mathit{a})-\phi \left(t\right){\parallel }_{L^2(0,T)}^2+{\displaystyle \frac{\mu }{2}}{\parallel \mathit{a}\parallel }^2,$$

(27)

where $\mu >0$ is taken as the regularization parameter that controls the trade-off between solution accuracy and stability; $u(x_0,t;\mathit{a})$ is the solution of the direct problem (1)–(3) for any prescribed $f^K\left(t\right)$, given by (26); and $\phi \left(t\right)$ denotes the measured data at the observation point $x_0$. The first term in the functional $J\left(\mathit{a}\right)$ measures the discrepancy between the computed solution and the observed data, while the second term serves as a regularization term that penalizes large values of the coefficient vector $\mathit{a}$, thereby ensuring the stability of the numerical solution.

To numerically solve the inverse problem (27) for reconstructing the source term $f\left(t\right)$, we employ an optimal perturbation algorithm. For any given coefficient vector ${\mathit{a}}_k\in {\mathbb{R}}^K$, we consider the following iterative update scheme:

$${\mathit{a}}_{j+1}={\mathit{a}}_j+\delta {\mathit{a}}_j,j=0,1,\dots ,$$

(28)

where $\delta {\mathit{a}}_j$ represents the perturbation term at the j-th iteration. For notational simplicity, we will hereafter denote ${\mathit{a}}_j$ and $\delta {\mathit{a}}_j$ as $\mathit{a}$ and $\delta \mathit{a}$, respectively.

Applying the first-order Taylor expansion to $u(x_0,t;\mathit{a}+\delta \mathit{a})$ about $\mathit{a}$ while neglecting higher-order terms yields the following linear approximation:

$$\begin{array}{c}\hfill u(x_0,t;\mathit{a}+\delta \mathit{a})\approx u(x_0,t;\mathit{a})+{\nabla }_{\mathit{a}}^{T}u(x_0,t;\mathit{a})·\delta \mathit{a},\end{array}$$

To determine the optimal perturbation, we minimize the following regularized error functional:

$$F\left(\delta \mathit{a}\right)={\displaystyle \frac{1}{2}}\parallel {\nabla }_{\mathit{a}}^{T}u(x_0,t;\mathit{a})·\delta \mathit{a}-[\phi \left(t\right)-u(x_0,t;\mathit{a})]{\parallel }_{L^2(0,T)}^2+{\displaystyle \frac{\mu }{2}}{\parallel \delta \mathit{a}\parallel }^2.$$

(29)

where $\mu$ serves as a regularization parameter with the following adaptive selection strategy:

$$\mu =\mu \left(n\right)={\displaystyle \frac{1}{1+exp\left(\theta (n-n_0)\right)}},$$

where n denotes the iteration count, $n_0$ represents a predetermined threshold (set to 5 in our implementation), and $\theta \ge 0$ (chosen as 0.8) controls the transition rate of this sigmoid-type regularization parameter, following the approach in [46].

We now proceed to discretize the temporal domain $[0,T]$ using a uniform grid, $0=t_1<t_2<\dots <t_S=T$, where S denotes the number of grids. The above $L^2$ norm can then be approximated by the discrete Euclidean norm, yielding the following regularized minimization problem:

$$F\left(\delta \mathit{a}\right)={\displaystyle \frac{1}{2}}{\parallel B\delta \mathit{a}-(\eta -\beta )\parallel }_2^2+{\displaystyle \frac{\mu }{2}}{\parallel \delta \mathit{a}\parallel }_2^2,$$

(30)

where the sensitivity matrix $B\in {\mathbb{R}}^{S\times K}$ is defined component-wise as follows:

$$\begin{array}{ccc}& B& ={\left(b_{sk}\right)}_{S\times K},\hfill \\ & b_{sk}& ={\displaystyle \frac{u(x_0,t_s;a_1,\dots ,a_k+\tau ,\dots ,a_K)-u(x_0,t_k;\mathit{a})}{\tau }},\hfill \\ & & s=1,2,\dots ,S,\hfill \end{array}$$

where $\tau >0$ denotes the finite difference step size. The vectors $\beta$ and $\eta$ represent the computed and measured data, respectively, as follows:

$$\begin{array}{ccc}& \beta & =(u(x_0,t_1;\mathit{a}),u(x_0,t_2;\mathit{a}),\dots ,u(x_0,t_S;\mathit{a})),\hfill \\ & \eta & =(\phi \left(t_1\right),\phi \left(t_2\right),\dots ,\phi \left(t_S\right)).\hfill \end{array}$$

Following standard regularization theory [47], the minimization of (30) leads to the following normal equation:

$$(\mu I+B^{T}B)\delta \mathit{a}=B^T(\eta -\beta ).$$

(31)

The optimal perturbation is consequently obtained through the following regularized solution:

$$\delta \mathit{a}={(\mu I+B^{T}B)}^{-1}{B}^T(\eta -\beta ).$$

(32)

The iterative inversion process continues until either the maximum iteration count is reached or the perturbation satisfies the convergence criterion as follows:

$$\Vert \delta \mathit{a}\Vert \le ϵ,$$

(33)

where $ϵ>0$ is a prescribed tolerance threshold governing the termination of the algorithm.

6. Numerical Experiments

In this part, we prove the effectiveness of the optimal perturbation algorithm with numerical results by using five examples for the one-dimensional scenarios and two-dimensional scenarios. The algorithms convergence and stability are examined.

We set $T=1$ in all our experiments. The accurate data are perturbed to create the noisy data randomly, i.e.,

$${\phi }^{\delta }=\phi +\delta \phi ·(2·rand\left(size\left(\phi \right)\right)-1),$$

with the use of $\epsilon =\parallel {\phi }^{\delta }{-\phi \parallel }_{L^2(\mathsf{\Gamma }\times (0,T))}$, the corresponding noise level can be calculated.

Intending to show the accuracy of the numerical solution, the relative root mean square (RRMS) error is estimated to be

$$\begin{array}{c}\hfill err\left(f\right)={\left({\displaystyle \frac{{\sum }_{i=1}^n{\left(f^K\left(t_i\right)-f\left(t_i\right)\right)}^2}{{\sum }_{i=1}^{n}f{\left(t_i\right)}^2}}\right)}^{1/2},\end{array}$$

(34)

where n denotes the total number of uniformly distributed points on the time interval $[0,1]$, the term $f^K\left(t\right)$ represents the reconstructed source term at the final iteration, while $f\left(t\right)$ corresponds to the exact (analytical) solution.

For generality and simplicity in numerical experiments, unless otherwise specified, we adopt the following parameter settings: the order of $m=3$ and the regularization coefficients $r_j=1(j=1,2,3)$ are all set to 1. The convergence threshold $ϵ$ is chosen as the first iteration ${10}^{-6}$, ensuring high precision in the iterative process. The initial guess for the first iteration is a zero vector (i.e., $\mathit{a}=\mathbf{0}$), and the numerical differentiation step size $\tau$ is assigned a value of 0.01 to balance computational accuracy and efficiency.

6.1. One-Dimensional Case

To maintain generality, we discretize both time intervals $[0,1]$ and $[0,T]$, using 51 uniformly distributed grid points when addressing the direct problem. For the one-dimensional scenario, the spatial domain is specified as $\mathsf{\Omega }=(0,1)$. Throughout our computations, unless otherwise indicated, the observation location is chosen as $x_0=0.5$. The following three examples are explored via numerical experiments to evaluate the effectiveness of the proposed method.

Example 1. 

Suppose the initial condition is $g\left(x\right)=x^2{(1-x)}^2$, and the spatial source function is $q\left(x\right)=e^{-x}$, with fractional orders ${\beta }_1=0.8,{\beta }_2=0.5,{\beta }_3=0.2$, and temporal source component $f\left(t\right)=1+t^2$. To simulate the measured data, we solve the direct problem (1)–(3) using the finite difference method. By setting $K=3$ and choosing the basis functions ${\mathsf{\Phi }}^k=\{1,t,t^2\}$, we generate the required input data at the observation point $x_0$.

The numerical results for different levels of noise, specifically $\delta =0,0.1\%,0.5\%,$ and $1\%$, are illustrated in Figure 1. The results clearly demonstrate that the proposed approach remains both robust and accurate even when the observation data are corrupted by noise up to $1\%$.

We fixed the relative noise level at $\delta =1\%$ and the fractional order at $\alpha =1.2$.

Table 1. RRMS errors for Example 1 corresponding to different values of $\mathit{\beta }$.

β	(0.3, 0.2, 0.1)	(0.5, 0.4, 0.3)	(0.7, 0.6, 0.5)	(0.9, 0.8, 0.7)
$err\left(f\right)$	0.0022	0.0031	0.0045	0.0067

presents the RRMS errors $\epsilon \left(f\right)$ as defined in (34). For Example 1, different values of $\beta$ are presented in Table 1. The data clearly demonstrate that as the components of $\mathit{\beta }$ decrease, the numerical precision is enhanced, indicating the improved accuracy of the method. Additionally, for Example 1 with various settings of $\alpha$, the RRMS errors $\epsilon \left(f\right)$ are reported in

Table 2. RRMS errors for Example 1 corresponding to different values of $\mathit{\alpha }$.

α	1.1	1.3	1.5	1.7	1.9
$err\left(f\right)$	0.0043	0.0039	0.0035	0.0032	0.0029

, where the noise level is maintained at $\delta =1\%$ and the parameters are set to ${\beta }_1=0.8$, ${\beta }_2=0.5$, and ${\beta }_3=0.2$. The results illustrate that higher values of $\alpha$ are associated with increased numerical accuracy. Furthermore, by fixing both the relative noise level at $\delta =1\%$ and the parameters ${\beta }_1=0.8$, ${\beta }_2=0.5$, ${\beta }_3=0.2$, and $\alpha =1.2$, the influence of the observation point $x_0$ on the RRMS error $\epsilon \left(f\right)$ is summarized in

Table 3. RRMS errors for Example 1 corresponding to different values of $x_0$.

x0	0.1	0.3	0.5	0.7	0.9
$err\left(f\right)$	0.0028	0.0036	0.0041	0.0040	0.0033

. These results indicate that the accuracy of the solution exhibits some dependency on the specific choice of $x_0$, suggesting there is a certain sensitivity pertaining to the method to the selection of this parameter.

Figure 2 presents the numerical results for Example 1 under various values of $\alpha$ and for several noise levels, $\delta =0,0.1\%,0.5\%,$ and $1\%$, with the parameters fixed as ${\beta }_1=0.9$; ${\beta }_2=0.8$…; and ${\beta }_9=0.1$. The computed solutions exhibit a strong agreement with the true function $f\left(t\right)$, which demonstrates the high accuracy of the numerical approximation. These findings indicate that the optimal perturbation algorithm is both effective and robust when identifying the source term, even in the presence of noise.

Table 4. RRMS errors for Example 1 at different values of $\alpha$ and noise levels.

		δ	0	0.001	0.005	0.01
	err(f)
α
1.2			$1.16\times {10}^{-9}$	$7.42\times {10}^{-4}$	$3.71\times {10}^{-2}$	$7.43\times {10}^{-2}$
1.8			$1.15\times {10}^{-9}$	$5.90\times {10}^{-4}$	$3.95\times {10}^{-2}$	$5.90\times {10}^{-2}$

displays the RRMS errors $err\left(f\right)$ for Example 1 across different values of $\alpha$, under a fixed relative noise level $\delta =1\%$ and ${\beta }_1=0.9$; ${\beta }_2=0.8$, …; and ${\beta }_9=0.1$. As shown, the RRMS error $err\left(f\right)$ changes only marginally as $\alpha$ varies, indicating that the method maintains stable accuracy with respect to parameter $\alpha$.

Table 5. RRMS errors for Example 1 at different values of $\alpha$ and $x_0$.

		δ	0.1	0.3	0.5	0.7	0.9
	err(f)
x0
1.2			$0.0040$	$0.0062$	$0.0074$	$0.0071$	$0.0049$
1.8			$0.0035$	$0.0049$	$0.0059$	$0.0059$	$0.0047$

reports the RRMS errors $err\left(f\right)$ for Example 1 at different locations of $x_0$, fixing the noise level at $\delta =1\%$, with ${\beta }_1=0.9$; ${\beta }_2=0.8$, …; ${\beta }_9=0.1$; and $\alpha =1.2$. The results show a slight dependence on the value of $x_0$, but overall the reconstruction remains stable, further supporting the robustness of the approach.

Example 2. 

Assume the initial data are given by $g\left(x\right)=sin\left(\pi x\right)$ and let the coefficient and source functions can be defined as $q\left(x\right)=x^2{(1-x)}^2+cos\left(\pi x\right)+1$, with parameters ${\beta }_1=0.8,{\beta }_2=0.5$, and ${\beta }_3=0.2$, $f\left(t\right)=2e^{-2t}sin\left(4\pi t\right)+t$ . To generate the input data at the observation point $x_0$, the direct problem (1)–(3) is solved numerically using the finite difference method. We set $K=10$ and ${\mathsf{\Phi }}^k=\{1,t,t^2\dots t^{10}\}$.

Figure 3 presents numerical results for varying values of $\alpha$ and for different noise levels, namely, $\delta =0,0.1\%,0.5\%,$ and $1\%$. It can be observed that the numerical approximation accurately captures the steady-state boundary for all tested conditions. However, numerical accuracy begins to degrade toward the initial time, specifically at $t=0$.

Example 3. 

We consider a non-smooth scenario characterized by a cusp in the source term, specifically choosing $f\left(t\right)=-|2t-1|+1$. The initial data are set as $g\left(x\right)=x^2{(1-x)}^2$, while the space-dependent coefficient is chosen as $q\left(x\right)=e^x$ , With parameters fixed at ${\beta }_1=0.8$, ${\beta }_2=0.5$, and ${\beta }_3=0.2$, the direct problem (1)–(3) is solved using the finite difference method to obtain the measurement data $u(x_0,t)$ employed in the reconstruction. For the numerical experiments, we select $K=10$ and use the basis set ${\mathsf{\Phi }}^k=\{1,t,t^2\dots t^{10}\}$.

Figure 4 displays the computed results for different values of $\alpha$ with noise levels of $\delta =0,0.1\%,0.5\%,$ and $1\%$. The results show that, overall, the identification quality remains satisfactory, except in the vicinity of $t=0$ and near the location of the non-smooth point, where some loss of accuracy is observed due to the underlying cusp in the source term.

6.2. Two-Dimensional Case

The spatial coordinates are assigned as $(x,y)$. In the case of a two-dimensional case, boundary $x_0=(0.5,0.5)$ is applied and the space domain $\mathsf{\Omega }$ is supposed to be $(0,1)\times (0,1)$. The grid points on $[0,1]\times [0,1]$ are $31\times 31$, and we calculate 50 layers for a given step at time direction when solving the direct problem. We provide the numerical outcomes for three examples to show the precision and consistency of our suggested approach.

Example 4. 

Given a source function $q(x,y)=x^2+e^y$, $f\left(t\right)=e^{-t}$, and the initial data $g(x,y)=sin\left(\pi x\right)sin\left(\pi y\right)$. The measurement data $u(0.5,0.5,t)$ are obtained by solving the direct problem (1)–(3) using the precise input data from the finite difference approach, where we take ${\beta }_1=0.8$, ${\beta }_2=0.5$, and ${\beta }_3=0.2$ and set $K=8$ ,${\mathsf{\Phi }}^k=\{1,t,t^2\dots t^8\}$.

Example 5. 

The time-dependent source term $f\left(t\right)=sin\left(2\pi t\right)$ is taken into consideration. Consider the initial data $g(x,y)=x(1-x)y(1-y)$ and take a source function $q(x,y)=x^2+e^y$. We construct the input data on observation point $x_0$ by solving the direct problem (1)–(3) by using a finite difference method and set $K=8$, ${\mathsf{\Phi }}^k=\{1,t,t^2\dots t^8\}$.

Example 6. 

We take into account a time-dependent source term $f\left(t\right)=\left|2t-1\right|-1$. Let the initial data $g(x,y)=sin\left(\pi x\right)sin\left(\pi y\right)$ and choose a source function $q(x,y)=x^2+y^2$. We create the input data on observation point $x_0$ by utilizing the finite difference approach to solve the direct problems (1)–(3) and set $K=10$, ${\mathsf{\Phi }}^k=\{1,t,t^2\dots t^{10}\}$.

Figure 5, Figure 6 and Figure 7 show us the numerical outcomes for Examples 4–6 at varying noise levels in the situations of ${\beta }_1=0.8,{\beta }_2=0.5,{\beta }_3=0.2$, and $\alpha =1.2,1.8$. It can be seen that our suggested approach works well for solving two-dimensional instances as well.

7. Conclusions

In this study, we explored the identification of time-dependent source terms within multi-term TSFDE. Initially, we rigorously established the existence and uniqueness of solutions to the associated direct problem, ensuring a solid theoretical foundation. Additionally, we derived both the uniqueness and stability estimates for the corresponding inverse problem, which seeks to recover the time-dependent source term using supplementary internal observation data.

From a computational perspective, the direct problem was efficiently solved via an implicit finite difference scheme augmented by the matrix transfer technique, allowing us to address the challenges posed by fractional derivatives. For the inverse problem, we recast the source identification task as a variational problem utilizing Tikhonov regularization, and we employed the optimal perturbation approach to obtain stable approximate solutions. Comprehensive numerical experiments, encompassing five one- and two-dimensional test cases, verified the proposed algorithm’s effectiveness and robustness in identifying time-dependent source terms for multi-term TSFDEs, delivering high accuracy even in the presence of noise.

However, when using the optimal perturbation algorithm to obtain approximate solutions for inverse problems, the parameter K and the basis functions have no fixed selection method and are generally chosen empirically, which poses significant limitations in practical applications. In the future, we plan to explore algorithms such as the conjugate gradient method, trust region methods, or deep learning, which can avoid the need to select basis functions and the number of terms in the expansion, thus making the approach more suitable for practical applications. On the other hand, future research will focus on extending this method to the identification of source terms in multi-term time–space fractional diffusion-wave equations, thereby further broadening its scope of applications.