A Double-Parameter Regularization Scheme for the Backward Diffusion Problem with a Time-Fractional Derivative

Abstract

In this paper, we investigate the regularization of the backward problem for a diffusion process with a time-fractional derivative. We propose a novel double-parameter regularization scheme that integrates the quasi-reversibility method for the governing equation with the quasi-boundary method. Theoretical analysis establishes the regularity and the convergence analysis of the regularized solution, along with a convergence rate under an a-priori regularization parameter choice rule in the general-dimensional case. Finally, numerical experiments validate the effectiveness of the proposed scheme.

1. Introduction

As a significant generalization of classical diffusion theory, fractional diffusion equations have attracted considerable attention from researchers due to their distinctive advantages in characterizing anomalous diffusion phenomena. By extending the classical time derivative to a fractional-order derivative, these equations can effectively describe two typical types of anomalous diffusion behavior: superdiffusion (with diffusion rates faster than normal diffusion) and subdiffusion (with diffusion rates slower than normal diffusion) [1,2]. Recent research demonstrates that significant progress has been made in the systematic investigation of forward problems for time-fractional diffusion equations (including initial value problems and initial-boundary value problems). The main advances are reflected in the following aspects: the existence theory of weak solutions based on fixed-point theorems [3], mathematical proofs of maximum principles [4], and studies on numerical solution methods such as finite difference methods [5], finite element method [6], and parallel-in-time algorithms [7], among others.

However, for practical applications [8] where certain parameters such as boundary conditions, initial conditions, diffusion coefficients, source terms, or the order of fractional derivative are unknown, the reconstruction of these unknowns based on additional observational data generates a class of typical inverse problems in the study of fractional diffusion equations. Recently, these problems have received great attention by many researchers. In [9], the source term identification problem was studied. The reconstruction of the Robin coefficient was investigated in [10]. The spatial potential term was recovered from final-time measurement data in [11]. Furthermore, the inverse problems to determine multiple parameters in fractional diffusion equations simultaneously have been considered recently in [12,13,14,15], and the references therein. Here, we consider an inverse problem of the time-fractional diffusion equation for reconstructing the initial distribution.

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ (with $d\in \{1,2,3\}$) be a bounded domain with a sufficiently smooth boundary $\partial \mathrm{\Omega }$. Denote by $L^2\left(\mathrm{\Omega }\right)$ the Hilbert space of square-integrable functions on $\mathrm{\Omega }$, equipped with the standard inner product

$$(f,h)=\underset{\mathrm{\Omega }}{\int }f\left(x\right)h\left(x\right) dx.$$

Here, we consider the following initial-boundary value problem involving a time-fractional derivative:

$$\left\{\begin{array}{cc}{\partial }_t^{\alpha }u(x,t)=\left(\mathcal{L}u\right)(x,t),\hfill & x\in \mathrm{\Omega }, t>0, 0<\alpha \le 1,\hfill \\ u(x,t)=0,\hfill & x\in \partial \mathrm{\Omega }, t>0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in \overline{\mathrm{\Omega }},\hfill \end{array}\right.$$

(1)

where ${\partial }_t^{\alpha }u$ denotes the Caputo fractional derivative of order $\alpha$, defined as

$${\partial }_t^{\alpha }u(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{0}{\overset{t}{\int }}\frac{\partial u(x,s)}{\partial s}·{(t-s)}^{-\alpha } ds,}\hfill & 0<\alpha <1,\hfill \\ {\displaystyle \frac{\partial u(x,t)}{\partial t},}\hfill & \alpha =1,\hfill \end{array}\right.$$

and $\mathrm{\Gamma }(·)$ is the Gamma function. Notably, when $\alpha =1$, problem (1) reduces to the classical parabolic initial-boundary value problem. In this paper, however, we restrict our attention to the case $0<\alpha <1$.

$-\mathcal{L}$ is a symmetric uniformly elliptic operator defined on $D(-\mathcal{L})=H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)$ given by

$$\begin{array}{c}\hfill -\mathcal{L}=-\sum _{i=1}^d\frac{\partial }{\partial x_i}\left(\sum _{j=1}^d{a}_{ij}\left(x\right)\frac{\partial }{\partial x_j}\right)+c\left(x\right), x\in \mathrm{\Omega }\end{array}$$

in which the coefficients satisfy

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{ij}=a_{ji},1\le i,j\le d, a_{ij}\left(x\right)\in C^1\left(\overline{\mathrm{\Omega }}\right)\hfill \\ {\nu }_0{\displaystyle \sum _{i=1}^d}{\xi }_i^2\le {\displaystyle \sum _{i,j=1}^d}{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le {\nu }_1{\displaystyle \sum _{i=1}^d}{\xi }_i^2, x\in \overline{\mathrm{\Omega }},\xi \in {\mathbb{R}}^d\hfill \\ c\left(x\right)\ge 0, x\in \overline{\mathrm{\Omega }},c\left(x\right)\in C\left(\overline{\mathrm{\Omega }}\right).\hfill \end{array}\right.\end{array}$$

with positive constants ${\nu }_0$ and ${\nu }_1$.

It is well-known that the forward problem is well-posed when the initial data $u_0\left(x\right)$ is given. In this work, however, we are concerned with the backward problem of reconstructing the unknown initial distribution $u_0\left(x\right)$ from the final state $u(x,T)=g\left(x\right)$ for $x\in \mathrm{\Omega }$. The backward problem consists of determining the solution $u(x,t)$ for $t\in [0,T)$ based on either the exact final data $g\left(x\right)$ or noisy measurements $g^{\delta }\left(x\right)$ satisfying

$$\parallel g^{\delta }{-g\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le \delta ,$$

(2)

where $\delta >0$ is a known noise level. This inverse (backward) problem is known to be ill-posed in the sense of Hadamard, as small perturbations in the final data may lead to large deviations in the reconstructed solution.

The backward problems of the time-fractional diffusion equations have been extensively investigated in recent years. In 2010, Liu and Yamamoto firstly proposed a quasi-reversibility regularization method in the one-dimensional case [16]. In 2011, Sakamoto and Yamamoto gave the uniqueness based on the theoretical results of the forward problem on [17]. Subsequently, some effective regularization methods have continued to emerge, such as the Tikhonov method [18,19], quasi-boundary value method [20], modified quasi-boundary value method [21], projection method [22], fractional Tikhonov and fractional Landweber method [23,24], exponential Tikhonov regularization method [15], fractional-order quasi-reversibility method [25], and so on. Some of the ideas in this paper come from the above references.

In this work, we propose a new regularization strategy that integrates the fractional quasi-reversibility method for the underlying evolution equation with the quasi-boundary value technique. The combination of these two regularization frameworks is designed to improve the approximation quality to the true solution from both theoretical and numerical perspectives. Concretely, we formulate the following regularized problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }{v}_{\eta ,\mu }=\mathcal{L}{v}_{\eta ,\mu }+\eta {\partial }_t^{\alpha }\mathcal{L}{v}_{\eta ,\mu },\hfill & x\in \mathrm{\Omega }, t>0,\hfill \\ v_{\eta ,\mu }(x,t)=0,\hfill & x\in \partial \mathrm{\Omega }, t\in (0,T],\hfill \\ v_{\eta ,\mu }(x,T)+\mu v_{\eta ,\mu }(x,0)=g\left(x\right),\hfill & x\in \overline{\mathrm{\Omega }},\hfill \end{array}\right.\end{array}$$

(3)

where $\eta$ and $\mu$ serve as two independent regularization parameters. It is worth noting that the model degenerates to the standard quasi-boundary value method (QBVM) when $\eta =0$, relying solely on the parameter $\mu >0$. On the other hand, setting $\mu =0$ retrieves the classical fractional quasi-reversibility method governed by the single parameter $\eta >0$. This type of combined regularization approach has previously been applied to the classical heat equation context as discussed in [26].

The remainder of this paper is structured as follows. Section 2 presents some essential preliminaries and technical lemmas. In Section 3, we analyze the regularity of the regularized solution and establish its convergence properties. Section 4 is devoted to deriving the convergence rate under an a priori parameter selection rule in the general-dimensional case. Finally, Section 5 provides numerical experiments to demonstrate the effectiveness and accuracy of the proposed method.

2. Preliminaries

In this section, we present some basic definitions and preliminary results that will be used throughout the analysis.

Denote $\{{\lambda }_n,{\omega }_n\left(x\right):n=1,2,\cdots \}$ as the eigensystem of operator $-\mathcal{L}$ acting on $D(-\mathcal{L}).$ That is, ${\omega }_n\left(x\right)$ satisfies

$$\left\{\begin{array}{cc}\mathcal{L}{\omega }_n\left(x\right)+{\lambda }_n{\omega }_n\left(x\right)=0,\hfill & x\in \mathrm{\Omega }\hfill \\ {\omega }_n\left(x\right)=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}\right.$$

It is well-known that ${\left\{{\omega }_n\left(x\right)\right\}}_{n=1}^{+\infty }$ constitutes the base of $L^2\left(\mathrm{\Omega }\right)$ and

$$0<{\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_n\le \cdots , \underset{n\to \infty }{\lim }{\lambda }_n=\infty .$$

For $p\ge 0$, introduce the function space

$$D\left({(-\mathcal{L})}^p\right)=\left\{\psi \in L^2\left(\mathrm{\Omega }\right)|\sum _{n=1}^{+\infty }{\lambda }_n^{2p}{\left|(\psi ,{\omega }_n)\right|}^2<+\infty \right\}.$$

Obviously, $D{(-\mathcal{L})}^p$ are Hilbert spaces with the following norms

$${\parallel \psi \parallel }_p:={\left(\sum _{n=1}^{+\infty }{\lambda }_n^{2p}{\left|(\psi ,{\omega }_n)\right|}^2\right)}^{\frac{1}{2}}, \psi \in D\left({(-\mathcal{L})}^p\right).$$

Moreover, we need some preliminary results on the Mittag–Leffler function [27] which is defined by

$$E_{\alpha ,\gamma }\left(z\right):=\sum _{k=0}^{+\infty }\frac{z^k}{\mathrm{\Gamma }(\alpha k+\gamma )}, z\in C, \alpha >0, \gamma \ge 0,$$

The Mittag–Leffler function $E_{\alpha ,\gamma }\left(z\right)$ generalizes the exponential function $e^z$ in that $E_{1,1}\left(z\right)=e^z$.

Lemma 1

(See [16]). Let $0<{\alpha }_0<{\alpha }_1<1$. Then for $x<0$, there exist positive constants $C_1^{+}, C_1^{-}$ and $C_2^{+}$ depending on ${\alpha }_0$ and ${\alpha }_1$ such that

$$\frac{C_1^{-}}{\mathrm{\Gamma }(1-\alpha )}\frac{1}{1-x}\le E_{\alpha ,1}\left(x\right)\le \frac{C_1^{+}}{\mathrm{\Gamma }(1-\alpha )}\frac{1}{1-x},$$

and

$$E_{\alpha ,0}\left(x\right)\le \frac{C_2^{+}}{\mathrm{\Gamma }(1-\alpha )}\frac{1}{1-x}.$$

These estimates are uniform for all $\alpha \in [{\alpha }_0,{\alpha }_1]$.

Lemma 2

(See [21,28]). For any ${\lambda }_n$ satisfying ${\lambda }_n\ge {\lambda }_1>0$, there exist positive constants $\underline{C}$, $\overline{C}$ depending on α, T, ${\lambda }_1$ such that

$$\frac{\underline{C}}{{\lambda }_n}\le E_{\alpha ,1}(-{\lambda }_n{T}^{\alpha })\le \frac{\overline{C}}{{\lambda }_n}.$$

Lemma 3

(See [28,29,30,31]). For any $\alpha \in (0,1)$, $t>0$, we have $0<E_{\alpha ,1}(-t)<1$. Moreover, the function $E_{\alpha ,1}(-t)$ is completely monotonic, that is

$${(-1)}^n\frac{d^n}{d{t}^n}{E}_{\alpha ,1}(-t)\ge 0, \forall n\in N.$$

Lemma 4

(See [16,27]). For any $\xi \in \mathbb{R}$ and $\xi \ne 0$, we have

$$\frac{d}{d\xi }{E}_{\alpha ,1}\left(\xi t^{\alpha }\right)=\frac{1}{\alpha \xi }{E}_{\alpha ,0}\left(\xi t^{\alpha }\right), t>0, \alpha \in (0,1).$$

3. Convergence Analysis and Error Estimate

In this section, we present the main theoretical results: existence and uniqueness of the solution (3) and convergence analysis on the regularizing solution.

Assume that ${\left\{{\omega }_n\left(x\right)\right\}}_{n=1}^{\infty }$ forms an orthonormal basis in $L^2\left(\mathrm{\Omega }\right)$. Then any function $g\left(x\right)\in L^2\left(\mathrm{\Omega }\right)$ can be represented by the following series expansion:

$$g\left(x\right)=\sum _{n=1}^{+\infty }{g}_n {\omega }_n\left(x\right),$$

(4)

where the Fourier coefficients are given by $g_n={(g,{\omega }_n)}_{L^2\left(\mathrm{\Omega }\right)}$.

For a given initial function $u_0\in L^2\left(\mathrm{\Omega }\right)$, it has been established in Theorem 2.1 of [17] that the forward problem (1) admits a unique solution $u(x,t)$. By employing the method of separation of variables, the formal solution to (1) can be expressed as

$$u(x,t)=\sum _{n=1}^{\infty }\frac{E_{\alpha ,1}(-{\lambda }_n{t}^{\alpha })}{E_{\alpha ,1}(-{\lambda }_n{T}^{\alpha })}{g}_n {\omega }_n\left(x\right),$$

where $\left\{{\lambda }_n\right\}$ and $\left\{{\omega }_n\left(x\right)\right\}$ denote the eigenvalues and corresponding eigenfunctions of the spatial operator $-\mathcal{L}$, and $E_{\alpha ,1}(·)$ is the Mittag–Leffler function.

First, we consider the existence and uniqueness of the solution to the regularized problem (3). In fact, we can establish the following theorem.

Theorem 1.

Assume that $g\in D(-\mathcal{L})$ and $\eta \in (0,+\infty )$, $\mu \in (0,+\infty )$. Then the initial-boundary value problem (3) has a unique solution

$$\begin{array}{c}\hfill v_{\eta ,\mu }(x,t)\in C\left([0,T];L^2\left(\mathrm{\Omega }\right)\right)\cap C\left((0,T];H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)\right).\end{array}$$

Proof of Theorem 1.

Our proof process can be seen as two steps.

Step 1: Existence and regularity of the solution.

We begin by analyzing the existence and regularity properties of the solution to the initial-boundary value problem (3). To this end, we apply the method of separation of variables to construct the solution in series form.

Let $u(x,t)=X\left(x\right)T\left(t\right)$ be a separable solution to the regularized problem (3). Substituting this form into the equation yields the following identity:

$$\frac{{\partial }_t^{\alpha }T\left(t\right)}{T\left(t\right)}=\frac{\mathcal{L}X\left(x\right)}{X\left(x\right)-\eta \mathcal{L}X\left(x\right)}=-\beta ,$$

where $-\beta$ is the separation constant, which is to be determined. Therefore, we deduce an eigenvalue problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathcal{L}X\left(x\right)-\beta \eta \mathcal{L}X\left(x\right)=-\beta X\left(x\right),\hfill & x\in \mathrm{\Omega }\hfill \\ X\left(x\right)=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}\right.\end{array}$$

(5)

and an Ordinary Differential Equation (ODE) with fractional derivative

$$\begin{array}{c}\hfill {\partial }_t^{\alpha }T\left(t\right)+\beta T\left(t\right)=0, t>0.\end{array}$$

(6)

We know that the eigensystem $\{{\lambda }_n,{\omega }_n\left(x\right)\}$ satisfies $\mathcal{L}{\omega }_n\left(x\right)=-{\lambda }_n{\omega }_n\left(x\right)$ and ${\omega }_n\left(x\right){|}_{\partial \mathrm{\Omega }}=0$. So eigenvalue problem (5) yields

$$\begin{array}{cc}\hfill {\beta }_n=\frac{{\lambda }_n}{1+\eta {\lambda }_n}, X_n\left(x\right)={\omega }_n\left(x\right),& n=1,2,\cdots .\end{array}$$

For fixed $\beta ={\beta }_n$, solving the corresponding ODE (6) yields

$$\begin{array}{cc}\hfill T_n\left(t\right)=E_{\alpha ,1}(-{\beta }_n{t}^{\alpha }),& n=1,2,\cdots .\end{array}$$

By combining the above results, the last equation of (3) and (4), we obtain

$$v_{\eta ,\mu }(x,t)=\sum _{n=1}^{\infty }\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }{g}_n{\omega }_n\left(x\right).$$

Now we verify the regularity of $v_{\eta ,\mu }(x,t)$. For any $t>0,$ we obtain from Lemma 1

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{\left[\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }{\lambda }_n{g}_n\right]}^2& =& \sum _{n=1}^{\infty }{\left[\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right]}^2{\lambda }_n^2{g}_n^2\hfill \\ & \le & \frac{{\left(C_1^{+}\right)}^2}{{[C_1^{-}+\mu \mathrm{\Gamma }(1-\alpha )]}^2}{\left(1+\frac{T^{\alpha }}{\eta }\right)}^2\sum _{n=1}^{\infty }{\lambda }_n^2{g}_n^2,\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{\left[\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }{g}_n\right]}^2& \le & \sum _{n=1}^{\infty }{\left[\frac{1}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right]}^2{g}_n^2\hfill \\ & \le & {\left[\frac{C_1^{-}}{\mathrm{\Gamma }(1-\alpha )}+\mu \right]}^{-2}{\left(1+\frac{T^{\alpha }}{\eta }\right)}^2\sum _{n=1}^{\infty }{g}_n^2.\hfill \end{array}$$

(8)

Since $g\in D(-\mathcal{L})$, it follows that the series ${\displaystyle \sum _{n=1}^{\infty }}{\lambda }_n^2{g}_n^2$ is convergent. Consequently, both series in (7) and (8) converge uniformly. So we have

$$\begin{array}{c}\hfill v_{\eta ,\mu }(x,t)\in C\left([0,T];L^2\left(\mathrm{\Omega }\right)\right)\cap C\left((0,T];H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)\right).\end{array}$$

Step 2: Uniqueness of the solution.

In this step, we establish the uniqueness of the solution to the regularized problem (3). Let $v(x,t)$ be the solution to (3) for $g\left(x\right)=0$. Then

$$v(x,t)=\sum _{n=1}^{\infty }{v}_n\left(t\right){\omega }_n\left(x\right),$$

where the coefficients $v_n\left(t\right)=(v(·,t),{\omega }_n)$ and ${\partial }_t^{\alpha }{v}_n\left(t\right)=-{\beta }_n{v}_n\left(t\right).$ On the other hand, the condition $v(x,T)+\mu v(x,0)=0$ yields $v_n\left(T\right)+\mu v_n\left(0\right)=0$. The the existence and uniqueness of the fractional ordinary equation differential yields $v_n\left(t\right)\equiv 0$ in $t\in [0,T]$, which leads to $v(x,t)\equiv 0$ because ${\left\{{\omega }_n\left(x\right)\right\}}_{n=1}^{+\infty }$ is a complete orthogonal system in $L^2\left(\mathrm{\Omega }\right)$. □

Theorem 2.

Assume that $g\in D(-\mathcal{L})$. Then for every $\alpha \in [{\alpha }_0,{\alpha }_1]$,

$$v_{\eta ,\mu }(·,t)\to u(·,t)$$

in $C((0,T];L^2\left(\mathrm{\Omega }\right))$ as $\eta ,\mu \to 0$. Moreover,

$$\parallel v_{\eta ,\mu }{(·,0)-u(·,0)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\to 0,$$

as $\eta ,\mu \to 0$.

Proof of Theorem 2.

A direct calculation yields

$$\begin{array}{ccc}\hfill v_{\eta ,\mu }(x,t)-u(x,t)& =& \sum _{n=1}^{\infty }\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }{g}_n{\omega }_n\left(x\right)-\sum _{n=1}^{\infty }\frac{E_{\alpha ,1}(-{\lambda }_n{t}^{\alpha })}{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{g}_n{\omega }_n\left(x\right)\hfill \\ & =:& \sum _{n=1}^{\infty }{h}_{n,\alpha ,T}\left(t\right)g_n{\omega }_n\left(x\right)+\sum _{n=1}^{\infty }{r}_{n,\alpha ,T}\left(t\right)g_n{\omega }_n\left(x\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill h_{n,\alpha ,T}\left(t\right)& =& \frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }-\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)},\hfill \\ \hfill r_{n,\alpha ,T}\left(t\right)& =& \frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}-\frac{E_{\alpha ,1}(-{\lambda }_n{t}^{\alpha })}{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}, t\in (0,T].\hfill \end{array}$$

Indeed, according to Lemmas 2 and 3, we obtain

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{h}_{n,\alpha ,T}^2\left(t\right)g_n^2=\left(\sum _{n=1}^N+\sum _{n=N+1}^{\infty }\right)\frac{E_{\alpha ,1}^2\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}^2\left(-{\beta }_n{T}^{\alpha }\right)}{\left(\frac{-\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2{g}_n^2\end{array}$$

$$\begin{array}{ccc}& \le & \sum _{n=1}^N\frac{g_n^2}{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}{\left(\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2+\sum _{n=N+1}^{\infty }\frac{g_n^2}{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\left[\sum _{n=1}^N{\left(\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2{\lambda }_n^2{g}_n^2+\sum _{n=N+1}^{\infty }{\lambda }_n^2{g}_n^2\right],\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{r}_{n,\alpha ,T}^2\left(t\right)g_n^2& =& \left(\sum _{n=1}^N+\sum _{n=N+1}^{\infty }\right)\frac{E_{\alpha ,1}^2\left(-{\lambda }_n{t}^{\alpha }\right)}{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}{\left(\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)}-1\right)}^2{g}_n^2\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\sum _{n=1}^N{\left(\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)}-1\right)}^2{\lambda }_n^2{g}_n^2\hfill \\ & +& \frac{2}{{\underline{C}}^2}\sum _{n=N+1}^{\infty }\left(\frac{E_{\alpha ,1}^2\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}^2\left(-{\beta }_n{T}^{\alpha }\right)}\frac{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}^2\left(-{\lambda }_n{t}^{\alpha }\right)}+1\right){\lambda }_n^2{g}_n^2\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\sum _{n=1}^N{\left(\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)}-1\right)}^2{\lambda }_n^2{g}_n^2\hfill \\ & +& \frac{2}{{\underline{C}}^2}\left[{\left(\frac{C_1^{+}}{C_1^{-}}\right)}^2{\left(\frac{T}{t}\right)}^{2\alpha }+1\right]\sum _{n=N+1}^{\infty }{\lambda }_n^2{g}_n^2,\hfill \end{array}$$

(10)

Since $g\in D(-\mathcal{L})$, the series ${\displaystyle \sum _{n=1}^{+\infty }}{\lambda }_n^2{g}_n^2$ converges. Hence, for any $ϵ>0$, there exists an integer $N=N\left(ϵ\right)$ such that

$$\begin{array}{c}\hfill \sum _{n=N\left(ϵ\right)+1}^{+\infty }{\lambda }_n^2{g}_n^2\le \frac{ϵ}{2}.\end{array}$$

(11)

By means of the continuity of $\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }$, ${\beta }_n$ and $E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)$, we can choose the appropriate $\eta \left(ϵ\right), \mu \left(ϵ\right)$ such that

$$\begin{array}{c}\hfill {\left(\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2\le \frac{ϵ}{2}, {\left(\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)}-1\right)}^2\le \frac{ϵ}{2},\end{array}$$

(12)

for all $N=1,2,\cdots ,N\left(ϵ\right)$. Combining (9)–(12), we have

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{h}_{n,\alpha ,T}^2\left(t\right)g_n^2& \le & \frac{1}{{\underline{C}}^2}{(\parallel g\parallel }_1^2+1)\frac{ϵ}{2},\hfill \\ \hfill \sum _{n=1}^{\infty }{r}_{n,\alpha ,T}^2\left(t\right)g_n^2& \le & \frac{1}{{\underline{C}}^2}\left\{{\parallel g\parallel }_1^2+2\left[{\left(\frac{C_1^{+}}{C_1^{-}}\right)}^2{\left(\frac{T}{t}\right)}^{2\alpha }+1\right]\right\}\frac{ϵ}{2}.\hfill \end{array}$$

(13)

As $ϵ>0$ has been arbitrary chosen, it follows that

$$\sum _{n=1}^{\infty }{h}_{n,\alpha ,T}^2\left(t\right)g_n^2\to 0, \sum _{n=1}^{\infty }{r}_{n,\alpha ,T}^2\left(t\right)g_n^2\to 0, as ϵ\to 0.$$

This means that ${\parallel v_{\eta ,\mu }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\to 0$ as $\eta ,\mu \to 0$ holds for all $t\in (0,T]$. We note that the constant in (13) depends on the variable $t>0$, thus requiring special consideration of the convergence at $t=0$. For the exact case $t=0$, we have

$$\begin{array}{ccc}\hfill h_{n,\alpha ,T}\left(0\right)& =& \frac{1}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }-\frac{1}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)},\hfill \\ \hfill r_{n,\alpha ,T}\left(0\right)& =& \frac{1}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}-\frac{1}{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}.\hfill \end{array}$$

We can easily get the results below

$$\begin{array}{ccc}\hfill v_{\eta ,\mu }(x,0)-u(x,0)& =& \sum _{n=1}^{\infty }{h}_{n,\alpha ,T}\left(0\right)g_n{\omega }_n\left(x\right)+\sum _{n=1}^{\infty }{r}_{n,\alpha ,T}\left(0\right)g_n{\omega }_n\left(x\right).\hfill \end{array}$$

On the other hand, we have from Lemma 1 that

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{h}_{n,\alpha ,T}^2\left(0\right)g_n^2& =& \left(\sum _{n=1}^N+\sum _{n=N+1}^{\infty }\right)\frac{g_n^2}{E_{\alpha ,1}^2(-{\beta }_n{T}^{\alpha })}{\left(\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\left[\sum _{n=1}^N{\left(\frac{\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2{\lambda }_n^2{g}_n^2+\sum _{n=N+1}^{\infty }{\lambda }_n^2{g}_n^2\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{r}_{n,\alpha ,T}^2\left(0\right)g_n^2& =& \left(\sum _{n=1}^N+\sum _{n=N+1}^{\infty }\right)\frac{g_n^2}{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}{\left(\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}-1\right)}^2\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\sum _{n=1}^N{\left(\frac{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}-1\right)}^2{\lambda }_n^2{g}_n^2\hfill \\ & +& \frac{2}{{\underline{C}}^2}\left[{\left(\frac{C_1^{+}}{C_1^{-}}\right)}^2+1\right]\sum _{n=N+1}^{\infty }{\lambda }_n^2{g}_n^2.\hfill \end{array}$$

Applying an analogous procedure to the case $t>0$ yields

$${\parallel v_{\eta ,\mu }(·,0)-u(·,0)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\to 0,$$

as $\eta ,\mu \to 0$. □

However, for problems encountered in real life, we only know additional noisy data. So we consider the noisy input data $g^{\delta }\left(x\right)$ of $g\left(x\right)$ as our input data. Let $v_{\eta ,\mu }^{\delta }$ be the solution of (3) with $g^{\delta }\left(x\right),$ that is

$$v_{\eta ,\mu }^{\delta }(x,t)=\sum _{n=1}^{\infty }\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }{g}_n^{\delta }{\omega }_n\left(x\right),$$

where $g_n^{\delta }=(g^{\delta },{\omega }_n), n=1,2,\cdots$.

Theorem 3.

Let $\alpha \in [{\alpha }_0,{\alpha }_1]$ and $g^{\delta }\left(x\right)$ denote the noisy data which satisfies (2). Then

$${\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\to 0, t\in [0,T)$$

as $\delta \to 0$, where the regularization parameters $\eta =\eta \left(\delta \right)$ and $\mu =\mu \left(\delta \right)$ depend on the noise level δ and satisfy

$$\begin{array}{c}\hfill \eta \left(\delta \right)\to 0, \mu \left(\delta \right)\to 0, \frac{\delta }{\frac{\eta }{\eta +1}+\mu }\to 0, as \delta \to 0.\end{array}$$

(14)

Proof of Theorem 3.

Note that $t\in [0,T)$,

$$\begin{array}{ccc}\hfill {\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}& \le & {\parallel v_{\eta ,\mu }^{\delta }(·,t)-v_{\eta ,\mu }(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}+{\parallel v_{\eta ,\mu }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

By Theorem 1 and the condition in (14), the last term on the above right-hand side tends to zero as $\delta \to 0$. Therefore, it remains to estimate the first term, namely,

$$\begin{array}{ccc}& & {\parallel v_{\eta ,\mu }^{\delta }(·,t)-v_{\eta ,\mu }(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2=\sum _{n=1}^{\infty }{\left[\frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right]}^2{(g_n^{\delta }-g_n)}^2\hfill \\ & \le & \sum _{n=1}^{\infty }\frac{1}{{\left[E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu \right]}^2}{(g_n^{\delta }-g_n)}^2\le \frac{1}{{\left[\frac{C_1^{-}}{\mathrm{\Gamma }(1-\alpha )}\frac{\eta }{\eta +T^{\alpha }}+\mu \right]}^2}\sum _{n=1}^{\infty }{(g_n^{\delta }-g_n)}^2\hfill \\ & \le & \frac{1}{{\left[\frac{C_1^{-}}{\mathrm{\Gamma }(1-\alpha )\max \{1,T^{\alpha }\}}\frac{\eta }{\eta +1}+\mu \right]}^2}\sum _{n=1}^{\infty }{(g_n^{\delta }-g_n)}^2\le C_1^2(\alpha ,T){\left(\frac{\delta }{\frac{\eta }{\eta +1}+\mu }\right)}^2,\hfill \end{array}$$

where $C_1(\alpha ,T)=\frac{1}{\min \left\{\frac{C_1^{-}}{\mathrm{\Gamma }(1-\alpha )\max \{1,T^{\alpha }\}},1\right\}}.$

From the above discussion, we obtain the corresponding results. □

4. The Convergence Rate of the Corresponding Solution

In addition, if some additional conditions can be given to the initial data, the convergence rate of the corresponding solution can be explicitly characterized, leading to the following provable result.

Theorem 4.

Suppose that $g\in D\left({(-\mathcal{L})}^{\frac{3}{2}}\right)$ with ${\parallel g\parallel }_{\frac{3}{2}}\le M$. If the regularization parameters are chosen as $\eta =\mathcal{O}\left({\delta }^{\frac{2}{3}}\right)$ and $\mu =\mathcal{O}\left({\delta }^{\frac{2}{3}}\right)$, then there exists a constant $C>0$, depending only on M, T, ${\alpha }_1$, ${\alpha }_2$, and ${\lambda }_1$, such that for any $\eta ,\mu >0$, the following estimate holds:

$${\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C{\delta }^{\frac{1}{3}}, \forall t\in [0,T).$$

Proof of Theorem 4.

According to Lemmas 2 and 3, we obtain

$$\begin{array}{ccc}\hfill {\parallel v_{\eta ,\mu }(·,t)-v_{\eta ,0}(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }\frac{E_{\alpha ,1}^2\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}^2\left(-{\beta }_n{T}^{\alpha }\right)}{\left(\frac{-\mu }{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)+\mu }\right)}^2{g}_n^2\hfill \\ & \le & \sum _{n=1}^{\infty }\frac{1}{E_{\alpha ,1}^2\left(-{\lambda }_n{T}^{\alpha }\right)}{\left(\frac{\mu }{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)+\mu }\right)}^2{g}_n^2\hfill \\ & \le & \frac{1}{{\underline{C}}^2}\sum _{n=1}^{+\infty }{\left(\frac{\mu {\lambda }_n^{\frac{1}{2}}}{\mu {\lambda }_n+\underline{C}}\right)}^2{\lambda }_n^3{g}_n^2\le \frac{\mu }{4{\underline{C}}^3}\sum _{n=1}^{+\infty }{\lambda }_n^3{g}_n^2\hfill \end{array}$$

(15)

Taking advantage of the previous discussion, we have

$$\begin{array}{ccc}\hfill {\parallel v_{\eta ,0}(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{r}_{n,\alpha ,T}^2\left(t\right)g_n^2,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill r_{n,\alpha ,T}\left(t\right)& =& \frac{E_{\alpha ,1}\left(-{\beta }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\beta }_n{T}^{\alpha }\right)}-\frac{E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-{\lambda }_n{T}^{\alpha }\right)}\hfill \\ & =& G^{\prime }\left({\xi }_n\right)({\lambda }_n-{\beta }_n)=G^{\prime }\left({\xi }_n\right)\frac{\eta {\lambda }_n^2}{1+\eta {\lambda }_n},\hfill \end{array}$$

(16)

where $G\left(\xi \right)=\frac{E_{\alpha ,1}\left(\xi t^{\alpha }\right)}{E_{\alpha ,1}\left(\xi T^{\alpha }\right)}$ and ${\xi }_n\in (-{\lambda }_n,-{\beta }_n)$.

For $\xi \ne 0$, it follows from Lemmas 1 and 4

$$\begin{array}{ccc}\hfill \left|G^{\prime }\left(\xi \right)\right|& =& \frac{1}{\alpha \left|\xi \right|}\frac{|E_{\alpha ,0}\left(\xi t^{\alpha }\right)E_{\alpha ,1}\left(\xi T^{\alpha }\right)-E_{\alpha ,1}\left(\xi t^{\alpha }\right)E_{\alpha ,0}\left(\xi T^{\alpha }\right)|}{E_{\alpha ,1}^2\left(\xi T^{\alpha }\right)}\hfill \\ & \le & \frac{1}{\alpha \left|\xi \right|}\left(\frac{\left|E_{\alpha ,0}\left(\xi t^{\alpha }\right)\right|}{E_{\alpha ,1}\left(\xi T^{\alpha }\right)}+\frac{\left|E_{\alpha ,1}\left(\xi t^{\alpha }\right)E_{\alpha ,0}\left(\xi T^{\alpha }\right)\right|}{E_{\alpha ,1}^2\left(\xi T^{\alpha }\right)}\right)\hfill \\ & \le & \frac{2}{\alpha \left|\xi \right|}\frac{C_1^{+}·C_2^{+}}{{\left(C_1^{-}\right)}^2}\frac{1-\xi T^{\alpha }}{1-\xi t^{\alpha }}\le \frac{2}{\alpha }\frac{C_1^{+}·C_2^{+}}{{\left(C_1^{-}\right)}^2}\left(\frac{1}{\left|\xi \right|}+T^{\alpha }\right)\hfill \\ & \to & \frac{2C_1^{+}·C_2^{+}}{{\left(C_1^{-}\right)}^2}\frac{T^{\alpha }}{\alpha }, \xi \to -\infty .\hfill \end{array}$$

Then there exists a constant $\tilde{C}=\tilde{C}\left(\alpha \right)$ such that

$$\begin{array}{c}\hfill \left|G^{\prime }\left(\xi \right)\right|\le \tilde{C}\left(\alpha \right)\frac{T^{\alpha }}{\alpha }.\end{array}$$

(17)

As for $\xi =0$, we obtain

$$\begin{array}{ccc}\hfill G^{\prime }\left(0\right)& =& \underset{\xi \to 0}{\lim }\frac{\frac{E_{\alpha ,1}\left(\xi t^{\alpha }\right)}{E_{\alpha ,1}\left(\xi T^{\alpha }\right)}-1}{\xi }\hfill \\ & =& \underset{\xi \to 0}{\lim }\frac{E_{\alpha ,1}\left(\xi t^{\alpha }\right)-E_{\alpha ,1}\left(\xi T^{\alpha }\right)}{\xi E_{\alpha ,1}\left(\xi T^{\alpha }\right)}=\frac{t^{\alpha }-T^{\alpha }}{\mathrm{\Gamma }(1+\alpha )}.\hfill \end{array}$$

(18)

With the aid of (17), (18) and the triangle inequality, we obtain

$$|G^{\prime }\left(\xi \right)|\le C_2\left(\alpha \right)\max \left\{\frac{T^{\alpha }}{\alpha },1,2T^{\alpha }\right\}, \xi \in [0,+\infty ),$$

where $t\in [0,T)$ and $C_2\left(\alpha \right)=\max \left\{\frac{1}{\mathrm{\Gamma }(1+\alpha )},\tilde{C}\left(\alpha \right)\right\}$.

We can obtain from (16)

$$\begin{array}{ccc}\hfill {\parallel v_{\eta ,0}(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }|G^{\prime }\left({\xi }_n\right){|}^2{\left(\frac{\eta {\lambda }_n^2}{1+\eta {\lambda }_n}\right)}^2{g}_n^2\le \sum _{n=1}^{\infty }\frac{\eta }{4}|G^{\prime }\left({\xi }_n\right){|}^2{\lambda }_n^3{g}_n^2\hfill \\ & \le & C_3^2(\alpha ,T)\eta \sum _{n=1}^{+\infty }{\lambda }_n^3{g}_n^2,\hfill \end{array}$$

(19)

where $C_3(\alpha ,T)=\frac{C_2\left(\alpha \right)}{2}\max \left\{\frac{T^{\alpha }}{\alpha },1,2T^{\alpha }\right\}$.

Since ${\parallel g\parallel }_{\frac{3}{2}}\le M$, we have ${\displaystyle \sum _{n=1}^{\infty }}{\lambda }_n^3{g}_n^2\le M^2.$ Combining the result in Theorem 2, (15) and (19), we obtain

$$\begin{array}{c}\hfill {\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\le C_4^2(\alpha ,T,M,{\lambda }_1)\left({\left(\frac{\delta }{\frac{\eta }{\eta +1}+\mu }\right)}^2+\eta +\mu \right),\end{array}$$

where $C_4(\alpha ,T,M,{\lambda }_1)=\max \left\{C_1(\alpha ,T),M{C}_3(\alpha ,T),\frac{M}{2{\underline{C}}^{\frac{3}{2}}}\right\}$. Therefore, we have

$$\begin{array}{c}\hfill {\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C_4(\alpha ,T,M,{\lambda }_1)\left(\frac{\delta }{\frac{\eta }{\eta +1}+\mu }+{\mu }^{\frac{1}{2}}+{\eta }^{\frac{1}{2}}\right).\end{array}$$

If we choose $\eta =\mathcal{O}({\delta }^{\frac{2}{3}}),\mu =\mathcal{O}({\delta }^{\frac{2}{3}}),$ then the regularizing solution $v_{\eta ,\mu }^{\delta }$ with respect to $g^{\delta }$ has the following convergence rate:

$$\begin{array}{c}\hfill {\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C_4(\alpha ,T,M,{\lambda }_1){\delta }^{\frac{1}{3}},\end{array}$$

for any fixed $\alpha \in [{\alpha }_0,{\alpha }_1]$ and all $t\in [0,T)$. □

5. Numerical Examples

In this section, we present three numerical examples to demonstrate the effectiveness of the proposed method. To assess the accuracy of the numerical solution, we introduce the relative error defined by

$$R_{\mathrm{error}}\left(t\right)=\frac{{\parallel v_{\eta ,\mu }^{\delta }(·,t)-u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}}{{\parallel u(·,t)\parallel }_{L^2\left(\mathrm{\Omega }\right)}}.$$

All numerical experiments were performed using MATLAB R2024a on an Intel Core Ultra 7 155H-based workstation (32 GB RAM). And the Mittag–Leffler function was computed via the mlf( ) routine from the MATLAB Central File Exchange, which implements the algorithm by Podlubny et al. in [32].

Example 1.

Consider the one-dimensional problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u(x,t)=\frac{{\partial }^{2}u}{\partial x^2},\hfill & x\in (0,\pi ),t\in (0,T],\hfill \\ u(0,t)=u(\pi ,t)=0,\hfill & t\in (0,T),\hfill \\ u(x,0)=u_0\left(x\right)=\sin x+\sin 2x,\hfill & x\in [0,\pi ].\hfill \end{array}\right.\end{array}$$

The measurement data is given at the final time $T=1$ with the fractional order $\alpha =\frac{1}{2}$.

For this simple model, the eigenvalues and eigenfunctions can be explicitly expressed as

$${\lambda }_n=n^2, {\omega }_n\left(x\right)=\sqrt{\frac{2}{\pi }}\sin \left(nx\right), n=1,2,\dots .$$

The corresponding exact solution takes the form

$$u(x,t)=E_{\frac{1}{2},1}(-t^{\frac{1}{2}})\sin x+E_{\frac{1}{2},1}(-4t^{\frac{1}{2}})\sin 2x.$$

To simulate noisy final-time data, we introduce random perturbations to the exact data $g\left(x\right)$, yielding

$$g^{\delta }\left(x\right)=g\left(x\right)+\sqrt{\frac{2}{\pi }}·\delta ·\mathrm{randn}\left(x\right),$$

(20)

where $\mathrm{randn}\left(x\right)$ is a normal distribution with mean 0 and standard deviation 1 uniformly distributed in $[-1,1],$ $\delta$ is the error level which can be calculated by $\delta :={\parallel g^{\delta }(·)-g(·)\parallel }_{L^2(0,\pi )}$ from (20). Correspondingly, the regularized solutions are given by

$$v_{\eta ,\mu }^{\delta }(x,t)\approx \sum _{n=1}^{10}\frac{E_{\frac{1}{2},1}\left(-\frac{{\lambda }_n}{1+\eta {\lambda }_n}{t}^{\frac{1}{2}}\right)}{E_{\frac{1}{2},1}\left(-\frac{{\lambda }_n}{1+\eta {\lambda }_n}{T}^{\frac{1}{2}}\right)+\mu }{g}_n^{\delta }\sin nx,$$

with the coefficient $g_n^{\delta }=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}{g}^{\delta }\left(x\right)\sin nxdx$.

In Figure 1, we present the exact data $u(x,0),u(x,T)$ and inversion results of $u(x,0)$ for different noise level $\delta$ with the reasonable regularization parameters $\eta =\mu ={(0.05\times \delta )}^{\frac{2}{3}}$. It can be seen that the inversion results are satisfactory. In

Table 1. Error results for Example 1 with different $\kappa$ with $\eta =\mu ={(0.05\times \delta )}^{\kappa }$.

$\mathit{\delta }$	${\mathit{R}}_{\mathit{error}}(\mathit{t}=0)$
	$\mathbf{\kappa }=\mathbf{1}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{2}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{3}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{4}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{5}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{6}/\mathbf{6}$	$\mathbf{\kappa }=\mathbf{7}/\mathbf{6}$
$\delta =0.1$	0.6864	0.5124	0.3402	0.2577	0.3428	0.5060	0.6513
$\delta =0.01$	0.6186	0.3552	0.1482	0.0661	0.0685	0.0776	0.0812
$\delta =0.001$	0.5410	0.2159	0.0557	0.0347	0.0418	0.0440	0.0444

, we show the numerical errors of Example 1 for different index $\kappa$ with $\eta =\mu ={(0.05\times \delta )}^{\kappa }$ in case of $\delta =0.001,0.01,0.1$. The numerical results clearly validate the proposed regularization parameter selection strategy, in accordance with the theoretical estimate established in Theorem 4.

Example 2.

We also consider a numerical result for the same backward problem 1 in case of a non-smooth unknown initial state. Let the exact initial data be

$$\begin{array}{c}\hfill u(x,0)=\left\{\begin{array}{cc}x, \hfill & x\in [0,\frac{\pi }{2}],\hfill \\ \pi -x, \hfill & x\in (\frac{\pi }{2},\pi ].\hfill \end{array}\right.\end{array}$$

The measurement data is given at the final time $T=2$.

The solutions of Example 2 can be expressed by

$$\begin{array}{c}\hfill u(x,t)=\sum _{n=1}^{\infty }\frac{4}{\pi n^2}\sin \frac{n\pi }{2}{E}_{\alpha ,1}(-n^2{t}^{\alpha })\sin nx.\end{array}$$

We approximate this infinite series by its first 50 terms, then the noisy data for $g^{\delta }\left(x\right)$ is given by

$$\begin{array}{c}\hfill g^{\delta }\left(x\right)\approx \sum _{n=1}^{50}\frac{4}{\pi n^2}\sin \frac{n\pi }{2}{E}_{\alpha ,1}(-n^2{t}^{\alpha })\sin nx+\sqrt{\frac{2}{\pi }}·\delta ·\mathrm{randn}\left(x\right).\end{array}$$

Following [25], we approximate the regularized solution by

$$v_{\eta ,\mu }^{\delta }(x,t)\approx \sum _{n=1}^4\frac{E_{\alpha ,1}\left(-\frac{n^2}{1+\eta n^2}{t}^{\alpha }\right)}{E_{\alpha ,1}\left(-\frac{n^2}{1+\eta n^2}{T}^{\alpha }\right)+\mu }{g}_n^{\delta }\sin nx,$$

with the coefficient $g_n^{\delta }=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}{g}^{\delta }\left(x\right)\sin nxdx$.

The numerical results under our double-parameter regularization scheme are presented for different noise levels $\delta$ with the regularization parameters $\eta =\mu ={(0.05\times \delta )}^{\frac{2}{3}}$ in Figure 2 and Figure 3 in which $\alpha$ is $0.2$ and $0.6$ respectively. On the other hand, the error results at different time and noise levels for $\alpha =0.6$ are shown in

Table 2. Error results for Example 2 with different time and noise levels for $\alpha =0.6$.

$\mathit{\delta }$	${\mathit{R}}_{\mathit{error}}\left(\mathit{t}\right)$
	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{0.4}$	$\mathit{t}=\mathbf{0.8}$	$\mathit{t}=\mathbf{1.2}$	$\mathit{t}=\mathbf{1.6}$	$\mathit{t}=\mathbf{2}$
$\delta =0.1$	0.2215	0.0811	0.0719	0.0697	0.0686	0.0680
$\delta =0.01$	0.0639	0.0190	0.0184	0.0182	0.0181	0.0181
$\delta =0.001$	0.0485	0.0049	0.0047	0.0046	0.0046	0.0045

. It can be found that the reconstruction is still efficient for a non-smooth unknown initial data.

Example 3.

Consider the two-dimensional problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_t^{\alpha }u(x,t)=▵u,\hfill & x\in \mathrm{\Omega }=(0,\pi )\times (0,\pi ),t\in (0,T],\hfill \\ u(x,t)=0,\hfill & x\in \partial \mathrm{\Omega },t\in (0,T],\hfill \\ u(x,0)=x_1(\pi -x_1)x_2(\pi -x_2),\hfill & x\in (0,\pi )\times (0,\pi ).\hfill \end{array}\right.\end{array}$$

The measurement data is given at the final time $T=1$ with the fractional order $\alpha =\frac{1}{2}$.

In this numerical process, the eigenvalues and the eigenfunctions are

$${\lambda }_{mn}=m^2+n^2, {\omega }_{mn}\left(x\right)=\frac{2}{\pi }\sin m{x}_1\sin n{x}_2, m,n=1,2,\cdots .$$

The exact solution has the representation

$$u(x,t)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{d}_{mn}{E}_{\alpha ,1}[-(m^2+n^2)t^{\alpha }]\sin m{x}_1\sin n{x}_2,$$

where

$$d_{mn}=\frac{16}{{\pi }^2{m}^3{n}^3}[1-{(-1)}^m][1-{(-1)}^n], m,n=1,2,\cdots .$$

We approximate this infinite series by its first 100-terms, then the noisy data of $g\left(x\right)$ is simulated by

$$g^{\delta }\left(x\right)=\sum _{m=1}^{10}\sum _{n=1}^{10}{d}_{mn}{E}_{\alpha ,1}\left[-(m^2+n^2)T^{\alpha }\right]\sin m{x}_1\sin n{x}_2+\frac{2}{\pi }·\delta ·\mathrm{randn}\left(x\right).$$

(21)

The regularized solution is approximated by

$$v_{\eta ,\mu }^{\delta }(x,t)\approx \sum _{m=1}^{10}\sum _{n=1}^{10}\frac{E_{\alpha ,1}\left[-\frac{m^2+n^2}{1+\eta (m^2+n^2)}{t}^{\alpha }\right]}{E_{\alpha ,1}\left[-\frac{m^2+n^2}{1+\eta (m^2+n^2)}{T}^{\alpha }\right]+\mu }{g}_{mn}^{\delta }\sin m{x}_1\sin n{x}_2,$$

with the coefficient $g_{mn}^{\delta }=\frac{2}{\pi }\underset{\mathrm{\Omega }}{\int }{g}^{\delta }\left(x\right){\omega }_{mn}\left(x\right)dx$.

The exact $u(x,0)$ and its inversions in case of $\alpha =0.2,0.6$ with $\delta =0.01$ are presented in Figure 4, Figure 5 and Figure 6 respectively.

Table 3. Error results for Example 3 with different time and noise levels for $\alpha =0.6$.

$\mathit{\delta }$	${\mathit{R}}_{\mathit{error}}\left(\mathit{t}\right)$
	$\mathit{t}=\mathbf{0}$	$\mathit{t}=\mathbf{0.2}$	$\mathit{t}=\mathbf{0.4}$	$\mathit{t}=\mathbf{0.6}$	$\mathit{t}=\mathbf{0.8}$	$\mathit{t}=\mathbf{1}$
$\delta =0.1$	0.1579	0.1232	0.1153	0.1109	0.1082	0.1063
$\delta =0.01$	0.0401	0.029	0.0280	0.0270	0.0264	0.0259
$\delta =0.001$	0.0091	0.0066	0.0063	0.0060	0.0058	0.0057

demonstrates the influence of $\delta$ for the error results at different time and noise levels for $\alpha =0.6$. We can find the same conclusion as that for the one-dimensional case.

From Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, the comparison between the exact and reconstructed initial data demonstrates that the proposed method yields highly accurate approximations under moderate noise levels. Additionally, the plots of absolute errors versus noise levels and different fractional orders further confirm the method’s stability with respect to data perturbations.

6. Conclusions

This work proposes a novel regularization scheme for time-fractional diffusion backward problems by combining the fractional quasi-reversibility method with the quasi-boundary value approach. The integration of these two regularization techniques yields complementary effects, offering not only improved stability compared to each method individually but also more accurate numerical approximations to the exact solution. It is worth emphasizing that the proposed method is also applicable to time-fractional wave equations, where the order of the fractional derivative lies in the interval (1, 2). Future research will focus on two main directions: (1) conducting convergence analysis and error estimates of the regularized solutions under a posteriori parameter selection strategies, and (2) extending and applying the proposed method to a broader class of superdiffusion equations.