On a final value problem for the time-fractional diffusion equation with inhomogeneous source

Abstract

In this paper, we consider an inverse problem for the time-fractional diffusion equation with inhomogeneous source to determine an initial data from the observation data provided at a later time. In general, this problem is ill-posed, therefore we construct a regularizing solution using the quasi-boundary value method. We also proposed both parameter choice rule methods, the a-priori and the a-posteriori methods, to estimate the convergence rate of the regularized methods. In addition, the proposed regularized methods have been verified by numerical experiments, and a comparison of the convergence rate between the a-priori and the a-posteriori choice rule methods is also given.

Keywords:

• Diffusion process
• fractional derivative
• backward problem
• regularization
• inhomogeneous source

AMS Subject Classifications:

• 35K05
• 47A52
• 62G08

1. Introduction

The aim of this paper is to design the methods to recover the initial data from a final value problem for an inhomogeneous time-fractional diffusion equation. Diffusive transport is one of the most important transport mechanisms found in nature. At a microscopic level, the diffusion is the result of the random motion of individual particles, and the use of the Laplacian operator to model this motion relies on the key assumption that this random motion is a stochastic Gaussian process. However, a growing number of studies on diffusive phenomena have shown the prevalence of anomalous diffusion, in which the mean square variance grows faster (in the case of superdiffusion) or slower (in the case of subdiffusion) than in a Gaussian diffusion process. Nowadays, anomalous diffusion becomes ’normal’ in spatially disordered systems, porous media, fractal media,[1–14] turbulent fluids and plasmas [15,16], biological media with traps, binding sites or macro-molecular crowding,[17,18] stock price movements.[19,20] As we know, the study on the inverse problems for fractional-order systems are still not well-covered. Since the fractional derivative is nonlocal, the backward problems for such models become much more difficult due to the following two reasons: The first reason, the backward problems for diffusion processes are ill posed, corresponding to the irreversibility of time; the second reason, the strong smoothing effect of the forward diffusion process makes it very difficult to reconstruct the possible discontinuities arising in the initial status.

In this work, we consider the problem of recovering the distribution u(x, 0) from a final value problem for the following inhomogeneous time-fractional diffusion equation:(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1)

where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^d$ with sufficient smooth boundary $\mathit{\partial }\mathrm{\Omega }$, and $T>0$ is a given number; the source function $F\in L^{\infty }(0,T;L^2(\mathrm{\Omega }))$ and the final data $g\in L^2(\mathrm{\Omega })$ are given. Here $\mathit{\alpha }\in (0,1)$ is the fractional order of $\frac{{\mathit{\partial }}^{\mathit{\alpha }}u}{\mathit{\partial }{t}^{\mathit{\alpha }}}$ referred to the Caputo fractional derivative with respect to t of the order $\mathit{\alpha }$ defined by$${\mathit{\partial }}_t^{\mathit{\alpha }}u=\frac{{\mathit{\partial }}^{\mathit{\alpha }}u}{\mathit{\partial }{t}^{\mathit{\alpha }}}:=\frac{1}{\mathrm{\Gamma }(1-\mathit{\alpha })}{\int }_0^t{(t-s)}^{-\mathit{\alpha }}{u}^{\prime }(s)\mathrm{d}s,$$

and $\mathrm{\Gamma }(x)$ denotes the standard Gamma function. The operator A is a symmetric uniformly elliptic operator. Note that if the fractional order $\mathit{\alpha }$ tends to unity, the fractional derivative ${\mathit{\partial }}_t^{\mathit{\alpha }}u$ becomes the first-order derivative ${\mathit{\partial }}_{t}u$, and thus problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) reproduces the standard parabolic equation. The objective of this study is to obtain the initial status of a physical field from its measurement data at the present time. Therefore, our backward problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) is to find the approximate solution u(x, t) for $t\in [0,T)$ corresponding to (g, F) from the noisy data $(g^{\mathit{ϵ}},F^{\mathit{ϵ}})$ such that(1.2) $$\begin{array}{c}\hfill \Vert g^{\mathit{ϵ}}{-g\Vert }_{L^2(\mathrm{\Omega })}\le \mathit{ϵ};{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}\le \mathit{ϵ},\end{array}$$(1.2)

for some known error level $\mathit{ϵ}>0$ .

As it is well known, that the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) is ill-posed, i.e. the solutions do not always exist, and in the case of existence, the solutions do not depend continuously on the given data. In fact, from small noise of physical measurement data, the corresponding solutions may have the large errors. It makes difficult to a numerical calculation. Hence, a regularization method is required.

When $\mathit{\alpha }=1$, the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) is a classical problem and has been particularly active in the past 30 years using many regularization methods, for example, see [8] and the references therein. However, there are not many studies on the fractional backward problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) for $\mathit{\alpha }\in (0,1)$. For the homogeneous case of backward problem, i.e. $F=0$ in (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ), there are very few works. For example, Sakamoto and Yamamoto [6] have proved that there exists a unique weak solution for this backward problem; Liu et al. [21] used a quasi-reversibility method to solve the homogeneous backward problem in one-dimensional case for special coefficients; and very recently, it has been considered by some other authors, such as [10–12,22–24].

Up-to-date, to the best of our knowledge, we could not find any result on backward problem for inhomogeneous time-fractional diffusion equation. Motivated by this reason, in this paper, we mainly propose a priori and an a posteriori regularization parameter choice rule using the quasi boundary value method. The quasi boundary value method, also called non-local boundary value problem method, is a regularization technique by replacing the final condition by a new approximate condition in such a way that the new problem is well-posed. In comparison with previous studies [3,11,21,22] on solving a time-fractional backward diffusion problem, our method shows an improvement in dealing with the time-fractional inverse diffusion problems with inhomogeneous source.

The manuscript is organized as follows. In Section 2, we introduce some well-known results on a forward problem. The conditional stability are described in Section 3. In Section 4, we propose a quasi-boundary value (QBV) method and provide two convergence estimates under an a priori assumption for the exact solution, and two regularization parameter choice rules. Eventually, two numerical examples are verified in Section 5.

2. The forward time-fractional diffusion problem

In order to solve the backward problem, we first recall in this section some well-known results from the forward problem for inhomogeneous time-fractional diffusion. The forward problem reads as: Given $u_0$ and F, find u satisfying(2.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega }.\hfill \end{array}\right.\end{array}$$(2.1)

Here, A is a symmetric uniformly elliptic operator defined by$$Av(x)=\sum _{i=1}^d\frac{\mathit{\partial }}{\mathit{\partial }{x}_i}\left(\sum _{j=1}^d{A}_{ij}(x)\frac{\mathit{\partial }}{\mathit{\partial }{x}_j}v(x)\right)+C(x)v(x),x\in \mathrm{\Omega },$$

where the coefficient functions $A_{ij}\in C^1(\overline{\mathrm{\Omega }})$ (for $i,j=1,\dots ,d$) and $C\in C(\overline{\mathrm{\Omega }})$ satisfy$$A_{ij}=A_{ji},\mathit{\nu }\sum _{i=1}^d{\mathit{\xi }}_i^2\le \sum _{i=1}^d\sum _{j=1}^d{A}_{ij}(x){\mathit{\xi }}_i{\mathit{\xi }}_j\text{and}C(x)\le 0,x\in \overline{\mathrm{\Omega }},\mathit{\xi }\in {\mathbb{R}}^d.$$

Here, $\mathit{\nu }$ is a positive constant independent of x and $\mathit{\xi }$.

Since A is a linear densely defined self-adjoint and positive definite elliptic operator on the connected bounded domain $\mathrm{\Omega }$ with zero Dirichlet boundary condition, the eigenvalues of A satisfy$$0<{\mathit{\lambda }}_1\le {\mathit{\lambda }}_2\le {\mathit{\lambda }}_3\le \cdots \le {\mathit{\lambda }}_n\le \cdots$$

with ${\mathit{\lambda }}_n\to \infty$ as $n\to \infty$; see [25]. The corresponding eigenfunctions are denoted, respectively, by ${\mathit{\phi }}_n\in H_0^1(\mathrm{\Omega })$. Thus the eigenpair $({\mathit{\lambda }}_n,{\mathit{\phi }}_n)$, $n=1,2,...$, satisfies$$\left\{\begin{array}{cc}A{\mathit{\phi }}_n(x)=-{\mathit{\lambda }}_n{\mathit{\phi }}_n(x),\hfill & x\in \mathrm{\Omega }\hfill \\ {\mathit{\phi }}_n(x)=0,\hfill & x\in \mathit{\partial }\mathrm{\Omega }.\hfill \end{array}\right.$$

The functions ${\mathit{\phi }}_n$ are normalized so that ${\{{\mathit{\phi }}_n\}}_{n=1}^{\infty }$ is an orthonormal basis of $L^2(\mathrm{\Omega })$.

Defining$${\mathbf{H}}^k(\mathrm{\Omega })=\{v\in L^2(\mathrm{\Omega }):\sum _{n=1}^{\infty }{\mathit{\lambda }}_n^{2k}{|〈v,{\mathit{\phi }}_n〉|}^2<+\infty \},$$

where $〈·,·〉$ is the inner product in $L^2(\mathrm{\Omega })$, then ${\mathbf{H}}^k(\mathrm{\Omega })$ is a Hilbert space equipped with norm$${\Vert v\Vert }_{{\mathbf{H}}^k(\mathrm{\Omega })}={\left(\sum _{n=1}^{\infty }{\mathit{\lambda }}_n^{2k}{|〈v,{\mathit{\phi }}_n〉|}^2\right)}^{1/2}.$$

The exact solution u of (2.1(2.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega }.\hfill \end{array}\right.\end{array}$$(2.1) ) can be represented by its Fourier series as follows (see [26])(2.2) $$\begin{array}{c}\hfill u(x,t)=\sum _{n=1}^{\infty }\left(E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})u_{0,n}+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right){\mathit{\phi }}_n(x).\end{array}$$(2.2)

where $E_{\mathit{\alpha },\mathit{\beta }}$ the Mittag–Leffler function which plays an important role in time-fractional PDEs and is defined, for $\mathit{\alpha }>0$ and $\mathit{\beta }\in \mathbb{R}$, by$$E_{\mathit{\alpha },\mathit{\beta }}(z)=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\mathit{\alpha }k+\mathit{\beta })},z\in \mathbb{C}.$$

And$$G_n(\mathit{\alpha },t):=t^{\mathit{\alpha }-1}{E}_{\mathit{\alpha },\mathit{\alpha }}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }}).$$

An existence and uniqueness result for the forward problem (2.1(2.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega }.\hfill \end{array}\right.\end{array}$$(2.1) ) has been obtained in Sakamoto and Yamamoto [6]. Numerical methods for this problem have been studied; see for example Jin et al. [27], McLean [28], Mustapha and McLean [29]. We finish this section by recalling some asymptotic results for the Mittag–Leffler function which can be found in [21] or in [2, Chapter 1]. More general results can be found in [2].

Lemma 2.1:

Let $0<{\mathit{\alpha }}_0<{\mathit{\alpha }}_1<1.$ Then there exists positive constants $C_1^{-}$, $C_1^{+}$, $C_2^{-}$, and $C_2^{+}$ such that for all $\mathit{\alpha }\in [{\mathit{\alpha }}_0,{\mathit{\alpha }}_1]$ there hold$$\begin{array}{cc}\hfill \frac{C_1^{-}}{\mathit{\alpha }}{e}^{x^{1/\mathit{\alpha }}}& \le E_{\mathit{\alpha },1}(x)\le \frac{C_1^{+}}{\mathit{\alpha }}{e}^{x^{1/\mathit{\alpha }}}\forall x\ge 0,\hfill \\ \hfill \frac{C_2^{-}}{\mathrm{\Gamma }(1-\mathit{\alpha })}\frac{1}{1-x}& \le E_{\mathit{\alpha },1}(x)\le \frac{C_2^{+}}{\mathrm{\Gamma }(1-\mathit{\alpha })}\frac{1}{1-x}\forall x\le 0.\hfill \end{array}$$

Lemma 2.2:

Assume that $\mathit{\alpha }\in (0,1)$. Then the Mittag–Leffler function have the asymptotic$$\begin{array}{ccc}\hfill E_{\mathit{\alpha },1}(x)& =\frac{1}{\mathit{\alpha }}{e}^{x^{1/\mathit{\alpha }}}-\frac{1}{x\mathrm{\Gamma }(1-\mathit{\alpha })}+O\left(\frac{1}{x^2}\right),\hfill & \hfill x\to +\infty ,\\ \hfill E_{\mathit{\alpha },1}(x)& =-\frac{1}{x\mathrm{\Gamma }(1-\mathit{\alpha })}+O\left(\frac{1}{x^2}\right),\hfill & \hfill x\to -\infty ,\\ \hfill E_{\mathit{\alpha },0}(x)& =\frac{1}{\mathit{\alpha }}{x}^{1/\mathit{\alpha }}{e}^{x^{1/\mathit{\alpha }}}-\frac{1}{x\mathrm{\Gamma }(-\mathit{\alpha })}+O\left(\frac{1}{x^2}\right),\hfill & \hfill x\to +\infty ,\\ \hfill E_{\mathit{\alpha },0}(x)& =-\frac{1}{x\mathrm{\Gamma }(-\mathit{\alpha })}+O\left(\frac{1}{x^2}\right),\hfill & \hfill x\to -\infty .\end{array}$$

In next section, we consider a backward problem for inhomogeneous time fractional diffusion equation by using regularization methods.

3. The backward time-fractional diffusion problem

3.1. The existence and regularity of solutions

Before proceeding to derive the existence and regularity of the solution of the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ), we introduce a useful lemma.

Lemma 3.1:

 

(a)

Let $F\in L^{\infty }(0,T;L^2(\mathrm{\Omega }))$. If $0<d<4$ then there exists a positive constant $M_1$ such that(3.1) $$\begin{array}{c}\hfill I_1:=\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\end{array}$$(3.1) If $d>0$ and $\mathit{\gamma }>\frac{d}{4}-1$, then there exists a positive constant $M_2$ such that$$I_2:=\sum _{n=1}^{\infty }\frac{1}{{\mathit{\lambda }}_n^{2\mathit{\gamma }}}|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le M_2{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.$$

(b)

Suppose further that $F\in L^{\infty }(0,T;H^2(\mathrm{\Omega }))$. Then for $0<d<4$ there holds$$I_3:=\sum _{n=1}^{\infty }{\mathit{\lambda }}_n^2|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le M_1{\Vert F\Vert }_{L^{\infty }(0,T;H^2(\mathrm{\Omega }))}^2·$$ where $M_1$ is the constant in (3.1(3.1) $$\begin{array}{c}\hfill I_1:=\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\end{array}$$(3.1) ).

Part (a): First, we note that for $0\le t\le T$ there holds(3.2) $$\begin{array}{c}\hfill |F_n{(t)|}^2\le \sum _{k=1}^{\infty }|〈F(·,t),{\mathit{\phi }}_k〉{|}^2\le {\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\end{array}$$(3.2)

Moreover, it follows from [6, Lemma 3.2] that $\frac{\mathrm{d}}{\mathrm{d}t}{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})=-{\mathit{\lambda }}_n{G}_n(\mathit{\alpha },t),t>0,$ which together with $E_{\mathit{\alpha },1}(0)=1$ implies$${\int }_0^t{G}_n(\mathit{\alpha },\mathit{\tau })\mathrm{d}\mathit{\tau }=\frac{1}{{\mathit{\lambda }}_n}(1-E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})).$$

On the other hand, [6, Lemma 3.3] gives $E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})\ge 0$ so that $G_n(\mathit{\alpha },t)\ge 0$ for all $t\ge 0$. Hence(3.3) $$\begin{array}{c}\hfill 0\le {\int }_0^t{G}_n(\mathit{\alpha },\mathit{\tau })\mathrm{d}\mathit{\tau }\le \frac{1}{{\mathit{\lambda }}_n},t\ge 0.\end{array}$$(3.3)

Combining (3.2(3.2) $$\begin{array}{c}\hfill |F_n{(t)|}^2\le \sum _{k=1}^{\infty }|〈F(·,t),{\mathit{\phi }}_k〉{|}^2\le {\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\end{array}$$(3.2) ) and (3.3(3.3) $$\begin{array}{c}\hfill 0\le {\int }_0^t{G}_n(\mathit{\alpha },\mathit{\tau })\mathrm{d}\mathit{\tau }\le \frac{1}{{\mathit{\lambda }}_n},t\ge 0.\end{array}$$(3.3) ) yields$$\begin{array}{cc}\hfill I_1& =\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le {\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\sum _{n=1}^{\infty }{\left|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })\mathrm{d}\mathit{\tau }\right|}^2\hfill \\ \hfill & \le {\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\sum _{n=1}^{\infty }\frac{1}{{\mathit{\lambda }}_n^2}.\hfill \end{array}$$

It is known that ${\mathit{\lambda }}_n\ge C{n}^{2/d}$ for $n\in \mathbb{N}$, where C is a positive constant independent of n; see e.g. Courant and Hilbert [30]. Hence for $0<d<4$ by letting $M_1:=\frac{1}{C^2}{\sum }_{n=1}^{\infty }\frac{1}{n^{4/d}},$ we prove (3.1(3.1) $$\begin{array}{c}\hfill I_1:=\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\le M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\end{array}$$(3.1) ). Similarly, we have$$I_2\le {\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\sum _{n=1}^{\infty }\frac{1}{{\mathit{\lambda }}_n^{2\mathit{\gamma }+2}}.$$

If $d>0$ and $\mathit{\gamma }>\frac{d}{4}-1$ (implying $4(\mathit{\gamma }+1)/d>1$), then by letting$$M_2:=\frac{1}{C^{2\mathit{\gamma }+2}}\sum _{n=1}^{\infty }\frac{1}{n^{4(\mathit{\gamma }+1)/d}},$$

we complete the proof Part (a).

Part (b): For $0\le t\le T$, we have$${\mathit{\lambda }}_n^2|F_n{(t)|}^2\le \sum _{k=1}^{\infty }{\mathit{\lambda }}_k^2|〈F(·,t),{\mathit{\phi }}_k〉{|}^2\le {\Vert F\Vert }_{L^{\infty }(0,T;H^2(\mathrm{\Omega }))}^2,$$

and the boundedness of $I_3$ follows as that of $I_1$.

The following theorem gives the necessary and sufficient conditions for the existence of the solution of (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ).

Theorem 3.2:

Assume that $g\in L^2(\mathrm{\Omega })$ and $F\in L^{\infty }(0,T;L^2(\mathrm{\Omega }))$.

(a)	If the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a unique solution, then(3.4) $$\begin{array}{c}\hfill g_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0)+{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }·\end{array}$$(3.4) where we recall that $g_n=〈g,{\mathit{\phi }}_n〉$ and $F_n(\mathit{\tau })=〈F(·,\mathit{\tau }),{\mathit{\phi }}_n〉$. Moreover,(3.5) $$\begin{array}{cc}\hfill {\Vert u(·,t)\Vert }_{L^2(\mathrm{\Omega })}& \le \sqrt{3}\frac{C_2^{+}{T}^{\mathit{\alpha }}}{C_2^{-}{t}^{\mathit{\alpha }}}\left({\Vert g\Vert }_{L^2(\mathrm{\Omega })}\right.\left.+\sqrt{M_1}{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}\right)\hfill \\ \hfill & +\sqrt{3}\sqrt{M_1}{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}·\hfill \end{array}$$(3.5) where $C_2^{+}$, $C_2^{-}$, and $M_1$ are the constants given in Lemmas 2.1 and 3.1.
(b)	The problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a unique solution u if and only if(3.6) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\left[\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2<\infty ·\end{array}$$(3.6)
(c)	If g, F satisfies $g\in H^2(\mathrm{\Omega })$ and $F\in L^{\infty }(0,T;H^2(\mathrm{\Omega }))$ then the condition (3.6(3.6) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\left[\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2<\infty ·\end{array}$$(3.6) ) holds when $d<4$.

Part (a): Suppose Problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a unique solution u. Then u is the solution of the forward problem (2.1(2.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega }.\hfill \end{array}\right.\end{array}$$(2.1) ) with $u_0(x)={\sum }_{n=1}^{\infty }{u}_n(0){\mathit{\phi }}_n(x).$ It follows from (2.2(2.2) $$\begin{array}{c}\hfill u(x,t)=\sum _{n=1}^{\infty }\left(E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})u_{0,n}+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right){\mathit{\phi }}_n(x).\end{array}$$(2.2) ) that$$g(x)=u(x,T)=\sum _{n=1}^{\infty }\left(E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0)+{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right){\mathit{\phi }}_n(x),$$

which implies (3.4(3.4) $$\begin{array}{c}\hfill g_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0)+{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }·\end{array}$$(3.4) ). Solving (3.4(3.4) $$\begin{array}{c}\hfill g_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0)+{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }·\end{array}$$(3.4) ) for $u_n(0)$ we obtain(3.7) $$\begin{array}{c}\hfill u_n(0)=\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}·\end{array}$$(3.7)

Therefore(3.8) $$\begin{array}{cc}\hfill {\Vert u(·,t)\Vert }_{L^2(\mathrm{\Omega })}^2& =\sum _{n=1}^{\infty }\left[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left(g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right)\right.\hfill \\ \hfill & {\left.+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2·\hfill \end{array}$$(3.8)

Lemma 2.1 gives$$0\le \frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\le \frac{C_2^{+}}{1+{\mathit{\lambda }}_n{t}^{\mathit{\alpha }}}\frac{1+{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}}{C_2^{-}}=\frac{C_2^{+}{T}^{\mathit{\alpha }}({\mathit{\lambda }}_n+1/T^{\mathit{\alpha }})}{C_2^{-}{t}^{\mathit{\alpha }}({\mathit{\lambda }}_n+1/t^{\mathit{\alpha }})}\le \frac{C_2^{+}{T}^{\mathit{\alpha }}}{C_2^{-}{t}^{\mathit{\alpha }}}·$$

Using the inequality ${(a+b+c)}^2\le 3(a^2+b^2+c^2)$ and Lemma 3.1 we deduce$$\begin{array}{cc}\hfill {\Vert u(·,t)\Vert }_{L^2(\mathrm{\Omega })}^2& \le 3\sum _{n=1}^{\infty }{\left(\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2({(g_n)}^2+|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2)\hfill \\ \hfill & +3\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le 3{\left(\frac{C_2^{+}{T}^{\mathit{\alpha }}}{C_2^{-}{t}^{\mathit{\alpha }}}\right)}^2\left[{\Vert g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\right]+3M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2·\hfill \end{array}$$

Thus (3.5(3.5) $$\begin{array}{cc}\hfill {\Vert u(·,t)\Vert }_{L^2(\mathrm{\Omega })}& \le \sqrt{3}\frac{C_2^{+}{T}^{\mathit{\alpha }}}{C_2^{-}{t}^{\mathit{\alpha }}}\left({\Vert g\Vert }_{L^2(\mathrm{\Omega })}\right.\left.+\sqrt{M_1}{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}\right)\hfill \\ \hfill & +\sqrt{3}\sqrt{M_1}{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}·\hfill \end{array}$$(3.5) ) is followed.

Part (b): Now assume that (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a unique solution. Then (3.7(3.7) $$\begin{array}{c}\hfill u_n(0)=\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}·\end{array}$$(3.7) ) gives$$\sum _{n=1}^{\infty }{\left[\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2=\sum _{n=1}^{\infty }{\left|u_n(0)\right|}^2={\Vert u(·,0)\Vert }_{L^2(\mathrm{\Omega })}^2<\infty ·$$

Conversely, if (3.6(3.6) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\left[\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2<\infty ·\end{array}$$(3.6) ) holds, then if $u_0$ is defined by(3.9) $$\begin{array}{c}\hfill u_0(x)=\sum _{n=1}^{\infty }\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{\mathit{\phi }}_n(x),\end{array}$$(3.9)

then clearly $u_0\in L^2(\mathrm{\Omega })$. Now, we consider the problem of finding u satisfying the forward time-fractional diffusion problem$$\left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.$$

Since $u_0\in L^2(\mathrm{\Omega })$ and $F\in L^{\infty }(0,T;L^2(\mathrm{\Omega }))$, the above problem has a unique solution u given by (2.2(2.2) $$\begin{array}{c}\hfill u(x,t)=\sum _{n=1}^{\infty }\left(E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})u_{0,n}+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right){\mathit{\phi }}_n(x).\end{array}$$(2.2) ). Combining (2.2(2.2) $$\begin{array}{c}\hfill u(x,t)=\sum _{n=1}^{\infty }\left(E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})u_{0,n}+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right){\mathit{\phi }}_n(x).\end{array}$$(2.2) ) and (3.9(3.9) $$\begin{array}{c}\hfill u_0(x)=\sum _{n=1}^{\infty }\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{\mathit{\phi }}_n(x),\end{array}$$(3.9) ), we obtain $u(x,T)={\sum }_{n=1}^{\infty }{g}_n{\mathit{\phi }}_n(x)=g(x),$ so that u is a solution of the backward problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ).

Part (c): Finally, we prove that if g and F satisfy $g\in H^2(\mathrm{\Omega })$ and $F\in L^{\infty }(0,T;H^2(\mathrm{\Omega }))$ then the condition (3.6(3.6) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }{\left[\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2<\infty ·\end{array}$$(3.6) ) holds. In fact, we have(3.10) $$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{\left[\frac{{\displaystyle g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2& \le 2\sum _{n=1}^{\infty }\frac{|g_n{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & +2\sum _{n=1}^{\infty }\frac{|〈{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau },{\mathit{\phi }}_n〉{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}·\hfill \end{array}$$(3.10)

Using the inequality (see Lemma 2.1) $\frac{1}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\le \frac{1+{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}}{C_2^{-}}\le \frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}{\mathit{\lambda }}_n,$ we obtain(3.11) $$\begin{array}{c}\hfill \sum _{n=1}^{\infty }\frac{|g_n{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\le \sum _{n=1}^{\infty }{\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2{\mathit{\lambda }}_n^2|g_n{|}^2={\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2{\Vert g\Vert }_{H^2(\mathrm{\Omega })}^2\end{array}$$(3.11)

and(3.12) $$\begin{array}{cc}& \sum _{n=1}^{\infty }\frac{|〈{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau },{\mathit{\phi }}_n〉{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & \le {\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2\sum _{n=1}^{\infty }{\mathit{\lambda }}_n^2|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le M_1{\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2{\Vert F\Vert }_{L^{\infty }(0,T;H^2(\mathrm{\Omega }))}^2,\hfill \end{array}$$(3.12)

where in the last step we used Lemma 3.1(b). Combining (3.10(3.10) $$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }{\left[\frac{{\displaystyle g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2& \le 2\sum _{n=1}^{\infty }\frac{|g_n{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & +2\sum _{n=1}^{\infty }\frac{|〈{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau },{\mathit{\phi }}_n〉{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}·\hfill \end{array}$$(3.10) )–(3.12(3.12) $$\begin{array}{cc}& \sum _{n=1}^{\infty }\frac{|〈{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau },{\mathit{\phi }}_n〉{|}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & \le {\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2\sum _{n=1}^{\infty }{\mathit{\lambda }}_n^2|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le M_1{\left(\frac{T^{\mathit{\alpha }}+1/{\mathit{\lambda }}_1}{C_2^{-}}\right)}^2{\Vert F\Vert }_{L^{\infty }(0,T;H^2(\mathrm{\Omega }))}^2,\hfill \end{array}$$(3.12) ), we complete the proof of Part (c).

Theorem 3.3:

Assume that (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a solution u which satisfies $u(·,0)\in {\mathbf{H}}^k(\mathrm{\Omega })$ for some $k>0$. Let M be a positive constant which satisfies(3.13) $$\begin{array}{c}\hfill {\Vert u(·,0)\Vert }_{H^k(\mathrm{\Omega })}\le M·\end{array}$$(3.13)

Then we have(3.14) $$\begin{array}{c}\hfill {\Vert u(·,0)\Vert }_{L^2(\mathrm{\Omega })}\le P(k,g,F)M^{\frac{1}{k+1}}·\end{array}$$(3.14)

where$$\begin{array}{c}\hfill P(k,g,F)=(\frac{\sqrt{2}}{C_2^{-}}{)}^{\frac{k}{k+1}}{\left({\Vert g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\right)}^{\frac{k}{2(k+1)}},\end{array}$$

where $M_1$ is the constant given by Lemma 3.1.

It follows from (3.8(3.8) $$\begin{array}{cc}\hfill {\Vert u(·,t)\Vert }_{L^2(\mathrm{\Omega })}^2& =\sum _{n=1}^{\infty }\left[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left(g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right)\right.\hfill \\ \hfill & {\left.+{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2·\hfill \end{array}$$(3.8) ) that(3.15) $$\begin{array}{cc}& {\Vert u(·,0)\Vert }_{L^2(\mathrm{\Omega })}^2\hfill \\ \hfill & =\sum _{n=1}^{\infty }\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & =\sum _{n=1}^{\infty }\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^{\frac{2}{k+1}}{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^{\frac{2k}{k+1}}}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & \le A_1^{\frac{1}{k+1}}{A}_2^{\frac{k}{k+1}},\hfill \end{array}$$(3.15)

where$$\begin{array}{cc}\hfill A_1& =\sum _{n=1}^{\infty }\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^{2k+2}}\text{and}\hfill \\ \hfill A_2& =\sum _{n=1}^{\infty }{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2·\hfill \end{array}$$

The term $A_1$ can be estimated by using Lemma 2.1 and (3.7(3.7) $$\begin{array}{c}\hfill u_n(0)=\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}·\end{array}$$(3.7) ) as follows(3.16) $$\begin{array}{cc}\hfill A_1& =\sum _{n=1}^{\infty }\frac{1}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^{2k}}\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & \le \sum _{n=1}^{\infty }\frac{{\mathit{\lambda }}_n^{2k}}{{(C_2^{-})}^{2k}}|u_n{(0)|}^2=\frac{1}{{(C_2^{-})}^{2k}}{\Vert u(·,0)\Vert }_{H^k(\mathrm{\Omega })}^2·\hfill \end{array}$$(3.16)

The term $A_2$ can be easily estimated by using Lemma 3.1(3.17) $$\begin{array}{cc}\hfill A_2& \le 2\sum _{n=1}^{\infty }{(g_n)}^2+2\sum _{n=1}^{\infty }|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le 2\left({\Vert g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\right)·\hfill \end{array}$$(3.17)

Combining (3.15(3.15) $$\begin{array}{cc}& {\Vert u(·,0)\Vert }_{L^2(\mathrm{\Omega })}^2\hfill \\ \hfill & =\sum _{n=1}^{\infty }\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^2}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & =\sum _{n=1}^{\infty }\frac{{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^{\frac{2}{k+1}}{\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]}^{\frac{2k}{k+1}}}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^2}\hfill \\ \hfill & \le A_1^{\frac{1}{k+1}}{A}_2^{\frac{k}{k+1}},\hfill \end{array}$$(3.15) )–(3.17(3.17) $$\begin{array}{cc}\hfill A_2& \le 2\sum _{n=1}^{\infty }{(g_n)}^2+2\sum _{n=1}^{\infty }|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le 2\left({\Vert g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\right)·\hfill \end{array}$$(3.17) ), we complete the proof of the Theorem.

4. QBV regularization method and error estimate

In this section, we shall regularize the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) with pertubed given data $F^{\mathit{ϵ}}$ and $g^{\mathit{ϵ}}$ satisfying (1.2(1.2) $$\begin{array}{c}\hfill \Vert g^{\mathit{ϵ}}{-g\Vert }_{L^2(\mathrm{\Omega })}\le \mathit{ϵ};{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}\le \mathit{ϵ},\end{array}$$(1.2) ). The regularized problem reads as: Find $u_{\mathit{\beta }}^{\mathit{ϵ}}$ satisfying(4.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}{u}_{\mathit{\beta }}^{\mathit{ϵ}}}{\mathit{\partial }{t}^{\mathit{\alpha }}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.1)

where $\mathit{\beta }$ is a regularizing parameter. The basic idea of the above regularization, which was initially developed for parabolic final value problems,[31,32] is to add an appropriate ‘corrector’, $\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}$, into the final value condition. This regularization is called the QBV method which is applied for some other ill-posed problem, for example [4,9,33,35].

The condition $u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)$ gives(4.2) $$\begin{array}{c}\hfill 〈u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0),{\mathit{\phi }}_n(x)〉=\frac{g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}·\end{array}$$(4.2)

Therefore, (4.1(4.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}{u}_{\mathit{\beta }}^{\mathit{ϵ}}}{\mathit{\partial }{t}^{\mathit{\alpha }}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.1) ) can be rewritten as(4.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}}{\mathit{\partial }{t}^{\mathit{\alpha }}}{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=h^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.3)

where(4.4) $$\begin{array}{c}\hfill h^{\mathit{ϵ}}(x)=\sum _{n=1}^{\infty }\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\end{array}$$(4.4)  

4.1. The existence, uniqueness of a solution of the problem (4.1(4.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}{u}_{\mathit{\beta }}^{\mathit{ϵ}}}{\mathit{\partial }{t}^{\mathit{\alpha }}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.1) )

In the next theorem, we shall study the existence, uniqueness and stability of (weak) solution of the problem (4.1(4.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}{u}_{\mathit{\beta }}^{\mathit{ϵ}}}{\mathit{\partial }{t}^{\mathit{\alpha }}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.1) ).

Theorem 4.1:

The problem (4.1(4.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}{u}_{\mathit{\beta }}^{\mathit{ϵ}}}{\mathit{\partial }{t}^{\mathit{\alpha }}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,T)+\mathit{\beta }{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=g^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.1) ) has uniquely a solution $u_{\mathit{\beta }}^{\mathit{ϵ}}\in L^{\infty }([0,T];L^2(\mathrm{\Omega }))$-pagination satisfying(4.5) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.5)

Moreover, the solution depends continuously on $g\in L^2(\mathrm{\Omega })$.

According the results of the forward problem, it is easy to show that the solution of Problem (4.3(4.3) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}}{\mathit{\partial }{t}^{\mathit{\alpha }}}{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=A{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)+F^{\mathit{ϵ}}(x,t)(x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)=0(x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)=h^{\mathit{ϵ}}(x)x\in \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(4.3) ) satisfies (4.5(4.5) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.5) ). We omit it here. In the following section, we will provide two convergence estimates for $\Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}$ by using an a priori and posteriori choice rules for the regularization parameter. To do so, we introduce the function $v_{\mathit{\beta }}$ as follows:(4.6) $$\begin{array}{cc}\hfill v_{\mathit{\beta }}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.6)

We show the smoothness of the regularization $u_{\mathit{\beta }}^{\mathit{ϵ}}$. Indeed, using the inequality $\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}$$\le \frac{1}{\mathit{\beta }(\mathit{ϵ})}$ and Lemma 3.1, part a, we get for all $t\in [0,T]$(4.7) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)\Vert }_{L^2(\mathrm{\Omega })}^2& =\sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}(g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau })\hfill \\ \hfill & +{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }{]}^2\hfill \\ \hfill & \le \frac{3}{{[\mathit{\beta }(\mathit{ϵ})]}^2}\sum _{n=1}^{\infty }({(g_n^{\mathit{ϵ}})}^2+|{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2)\hfill \\ \hfill & +3\sum _{n=1}^{\infty }|{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & \le \frac{3}{{[\mathit{\beta }(\mathit{ϵ})]}^2}[\Vert g^{\mathit{ϵ}}{\Vert }_{L^2(\mathrm{\Omega })}^2+M_1\Vert F^{\mathit{ϵ}}{\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2]+3M_1{\Vert F^{\mathit{ϵ}}\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2.\hfill \end{array}$$(4.7)

This implies that $u_{\mathit{\beta }}^{\mathit{ϵ}}\in L^{\infty }([0,T];L^2(\mathrm{\Omega }))$.

4.2. Convergence estimate under an a priori regularization parameter choice rule

In the following, even though we will choose $\mathit{\beta }$ depending on $\mathit{ϵ}$, we still clearly show the dependence of the solution of (4.1) on $\mathit{\beta }$, namely we will still denote the solution by $u_{\mathit{\beta }}^{\mathit{ϵ}}$, because the different choices of $\mathit{\beta }$ give rise to different error estimates. We first introduce the following elementary lemma.

Lemma 4.2:

Let $k,\mathit{\mu },{\mathit{\lambda }}_1$ be positive constant. Let G be a function defined by(4.8) $$\begin{array}{c}\hfill G(z)=\frac{\mathit{\mu }{z}^{1-k}}{\mathit{\mu }z+C_2^{-}},z>0,\end{array}$$(4.8)

where $C_2^{-}$ is defined in Lemma 2.1. Then(4.9) $$\begin{array}{c}\hfill G(z)\le \left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-}){\mathit{\mu }}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\mu },\text{if}k\ge 1,z\ge {\mathit{\lambda }}_1.\hfill \end{array}\right.\end{array}$$(4.9)

where(4.10) $$\begin{array}{c}\hfill {\tilde{B}}_1(k,C_2^{-})={\left[\frac{k}{C_2^{-}}\right]}^k[(1-k){]}^{1-k},{\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)=\frac{1}{C_2^{-}{\mathit{\lambda }}_1^{k-1}}·\end{array}$$(4.10)  

(1)	If $k\ge 1$, then from $z\ge {\mathit{\lambda }}_1$, we have(4.11) $$\begin{array}{c}\hfill G(z)=\frac{\mathit{\mu }{z}^{1-k}}{\mathit{\mu }z+C_2^{-}}=\frac{\mathit{\mu }}{(\mathit{\mu }z+C_2^{-})z^{k-1}}\le \frac{\mathit{\mu }}{C_2^{-}{z}^{k-1}}\le \frac{\mathit{\mu }}{C_2^{-}{\mathit{\lambda }}_1^{k-1}}·\end{array}$$(4.11)
(2)	If $0<k<1$, then it is easy to see that ${lim}_{z\to 0}G(z)={lim}_{z\to +\infty }G(z)=0·$ By taking the derivative of G with respect to z, we have(4.12) $$\begin{array}{c}\hfill G^{\prime }(z)=\frac{\mathit{\mu }{z}^{-k}[(1-k)C_2^{-}-\mathit{\mu }kz]}{{(\mathit{\mu }z+C_2^{-})}^2}·\end{array}$$(4.12)

G(z) attains maximum value at $z=z_0$ which satisfies $G^{\prime }(z)=0$. Solving $G^{\prime }(z_0)=0$, we get $z_0=\frac{(1-k)C_2^{-}}{k\mathit{\mu }}$.

Hence,(4.13) $$\begin{array}{c}\hfill G(z)\le G(z_0)=G(\frac{(1-k)C_2^{-}}{k\mathit{\mu }})={\left[\frac{k}{C_2^{-}}\right]}^k[(1-k){]}^{1-k}{\mathit{\mu }}^k·\end{array}$$(4.13)

We are now ready to prove the main result with a priori choices of the regularization parameter $\mathit{\beta }$.

Theorem 4.3:

Let us choose the regularizing parameter $\mathit{\beta }=\mathit{\beta }(\mathit{ϵ})$ satisfying(4.14) $$\begin{array}{c}\hfill \underset{\mathit{ϵ}\to 0}{lim}\mathit{\beta }(\mathit{ϵ})=\underset{\mathit{ϵ}\to 0}{lim}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}=0·\end{array}$$(4.14) (1) Assuming that (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) has a solution u satisfying $u(.,0)\in L^2(\mathrm{\Omega })$. Then for $t\in (0,T]$, the following estimate holds(4.15) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \sqrt{6(M_1+1)}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+\mathit{\beta }(\mathit{ϵ})\frac{C_2^{+}}{C_2^{-}}{\left(\frac{T}{t}\right)}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}·\end{array}$$(4.15) (2) Suppose the a priori condition (3.13(3.13) $$\begin{array}{c}\hfill {\Vert u(·,0)\Vert }_{H^k(\mathrm{\Omega })}\le M·\end{array}$$(3.13) ) holds for some $k>0$ and $M>0$. Then we have(4.16) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}& \le \sqrt{6(M_1+1)}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+\left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})M|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)M\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}\right.\hfill \end{array}$$(4.16)

Here $M_1$ and $C_2^{-}$ are recalled from Lemmas 3.1 and 2.1, respectively

Remark 4.4:

(1)

Let us choose $\mathit{\beta }(\mathit{ϵ})={\mathit{ϵ}}^k,0<k<1$, then(4.17) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \sqrt{6(M_1+1)}{\mathit{ϵ}}^{1-k}+{\mathit{ϵ}}^k\frac{C_2^{+}}{C_2^{-}}(\frac{T}{t}{)}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$(4.17)

(2)

(a) If $0<k<1$ then by choosing $\mathit{\beta }(\mathit{ϵ})={\left(\frac{\mathit{ϵ}}{M}\right)}^{\frac{1}{k+1}}$, the convergence estimate $\Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}$ is of order ${\mathit{ϵ}}^{\frac{k}{k+1}}$. (b) If $k\ge 1$ then by choosing $\mathit{\beta }(\mathit{ϵ})={\left(\frac{\mathit{ϵ}}{M}\right)}^{\frac{1}{2}}$, the convergence estimate $\Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}$ is of order ${\mathit{ϵ}}^{\frac{1}{2}}$.

By the triangle inequality, we know(4.18) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,t)-v_{\mathit{\beta }}{(.,t)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert v_{\mathit{\beta }}(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$(4.18) Proof Part 1: For $t>0$, we first give an estimate for the first term. In fact, from (4.5(4.5) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.5) ), (4.6(4.6) $$\begin{array}{cc}\hfill v_{\mathit{\beta }}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.6) ), and using the inequality ${(a+b+c)}^2\le 3(a^2+b^2+c^2)$ , we obtain(4.19) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,t)-v_{\mathit{\beta }}{(.,t)\Vert }_{L^2(\mathrm{\Omega })}^2& \le 3\sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{]}^2(g_n^{\mathit{ϵ}}-g_n{)}^2\hfill \\ \hfill & +3\sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{]}^2\hfill \\ \hfill & |{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })(F_n^{\mathit{ϵ}}(\mathit{\tau })-F_n(\mathit{\tau }))\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & +3\sum _{n=1}^{\infty }[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })(F_n^{\mathit{ϵ}}(\mathit{\tau })-F_n(\mathit{\tau }))\mathrm{d}\mathit{\tau }{]}^2\hfill \\ \hfill & \le \frac{3}{{[\mathit{\beta }(\mathit{ϵ})]}^2}(\Vert g^{\mathit{ϵ}}{-g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2)\hfill \\ \hfill & +3M_1{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\hfill \\ \hfill & \le 3{\mathit{ϵ}}^2[M_1+\frac{M_1+1}{{[\mathit{\beta }(\mathit{ϵ})]}^2}]\le 6(M_1+1)\frac{{\mathit{ϵ}}^2}{{[\mathit{\beta }(\mathit{ϵ})]}^2}·\hfill \end{array}$$(4.19)

where we used that $M_1+1\le \frac{M_1+1}{\mathit{\beta }(\mathit{ϵ})}.$ Next, we continue to estimate the second term as follows:(4.20) $$\begin{array}{cc}& \Vert v_{\mathit{\beta }}(x,t)-u(x,t){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}-\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}]\hfill \\ \hfill & [g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }[\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}]{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}<u(x,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \mathit{\beta }(\mathit{ϵ})\underset{n\in \mathbb{N}}{sup}\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\sqrt{\sum _{n=1}^{\infty }{|<u(.,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x)|}^2}\hfill \\ \hfill & \le \mathit{\beta }(\mathit{ϵ})\frac{C_2^{+}}{C_2^{-}}{(\frac{T}{t})}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}·\hfill \end{array}$$(4.20) Combining (4.17(4.17) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \sqrt{6(M_1+1)}{\mathit{ϵ}}^{1-k}+{\mathit{ϵ}}^k\frac{C_2^{+}}{C_2^{-}}(\frac{T}{t}{)}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$(4.17) ) and (4.18(4.18) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,t)-v_{\mathit{\beta }}{(.,t)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert v_{\mathit{\beta }}(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$(4.18) ), we obtain (4.14(4.14) $$\begin{array}{c}\hfill \underset{\mathit{ϵ}\to 0}{lim}\mathit{\beta }(\mathit{ϵ})=\underset{\mathit{ϵ}\to 0}{lim}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}=0·\end{array}$$(4.14) ).

Proof Part 2:

Now, we return to the proof of Part 2. We have the following estimate:(4.21) $$\begin{array}{cc}\hfill \Vert v_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}& =\sqrt{{\sum }_{n=1}^{\infty }{\left[\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2|<u(x,0),{\mathit{\phi }}_n(x)>{|}^2}\hfill \\ \hfill & \le \sqrt{{\sum }_{n=1}^{\infty }{\left[\frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{-k}}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right]}^2{\mathit{\lambda }}_n^{2k}|<u(x,0),{\mathit{\phi }}_n(x)>{|}^2}\hfill \\ \hfill & \le \underset{n\in \mathbb{N}}{sup}[\frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{-k}}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}]\Vert u(.,0){\Vert }_{H^k(\mathrm{\Omega })}.\hfill \end{array}$$(4.21)

On other hand, using the inequality $E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})\ge \frac{C_2^{-}}{{\mathit{\lambda }}_n}$ and Lemma 4.1, we have(4.22) $$\begin{array}{c}\hfill \frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{-k}}{\mathit{\beta }(\mathit{ϵ})+\frac{C_2^{-}}{{\mathit{\lambda }}_n}}=\mathit{\beta }(\mathit{ϵ})G({\mathit{\lambda }}_n)\le \left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}·\right.\end{array}$$(4.22)

Combining (4.17(4.17) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,t)-u(.,t)\Vert }_{L^2(\mathrm{\Omega })}\le \sqrt{6(M_1+1)}{\mathit{ϵ}}^{1-k}+{\mathit{ϵ}}^k\frac{C_2^{+}}{C_2^{-}}(\frac{T}{t}{)}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$(4.17) ), (4.19(4.19) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,t)-v_{\mathit{\beta }}{(.,t)\Vert }_{L^2(\mathrm{\Omega })}^2& \le 3\sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{]}^2(g_n^{\mathit{ϵ}}-g_n{)}^2\hfill \\ \hfill & +3\sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{]}^2\hfill \\ \hfill & |{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })(F_n^{\mathit{ϵ}}(\mathit{\tau })-F_n(\mathit{\tau }))\mathrm{d}\mathit{\tau }{|}^2\hfill \\ \hfill & +3\sum _{n=1}^{\infty }[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })(F_n^{\mathit{ϵ}}(\mathit{\tau })-F_n(\mathit{\tau }))\mathrm{d}\mathit{\tau }{]}^2\hfill \\ \hfill & \le \frac{3}{{[\mathit{\beta }(\mathit{ϵ})]}^2}(\Vert g^{\mathit{ϵ}}{-g\Vert }_{L^2(\mathrm{\Omega })}^2+M_1{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2)\hfill \\ \hfill & +3M_1{\Vert F^{\mathit{ϵ}}-F\Vert }_{L^{\infty }(0,T;L^2(\mathrm{\Omega }))}^2\hfill \\ \hfill & \le 3{\mathit{ϵ}}^2[M_1+\frac{M_1+1}{{[\mathit{\beta }(\mathit{ϵ})]}^2}]\le 6(M_1+1)\frac{{\mathit{ϵ}}^2}{{[\mathit{\beta }(\mathit{ϵ})]}^2}·\hfill \end{array}$$(4.19) ), (4.20(4.20) $$\begin{array}{cc}& \Vert v_{\mathit{\beta }}(x,t)-u(x,t){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }[\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}-\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}]\hfill \\ \hfill & [g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }[\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}]{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}<u(x,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \mathit{\beta }(\mathit{ϵ})\underset{n\in \mathbb{N}}{sup}\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\sqrt{\sum _{n=1}^{\infty }{|<u(.,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x)|}^2}\hfill \\ \hfill & \le \mathit{\beta }(\mathit{ϵ})\frac{C_2^{+}}{C_2^{-}}{(\frac{T}{t})}^{\mathit{\alpha }}{\Vert u(.,0)\Vert }_{L^2(\mathrm{\Omega })}·\hfill \end{array}$$(4.20) ), we obtain(4.23) $$\begin{array}{cc}& \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)-v_{\mathit{\beta }}{(x,0)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert v_{\mathit{\beta }}(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill \\ \hfill & \le \sqrt{2(M_1+1)}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+\left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})M|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)M\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}·\right.\hfill \end{array}$$(4.23)

The proof is completed.

4.3. Convergence estimate under an a posteriori regularization parameter choice rule

First, we recall $u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)$ as in (4.5(4.5) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,t)& =\sum _{n=1}^{\infty }\frac{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{t}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\left[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x)\hfill \\ \hfill & +\sum _{n=1}^{\infty }\left[{\int }_0^t{G}_n(\mathit{\alpha },t-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }\right]{\mathit{\phi }}_n(x).\hfill \end{array}$$(4.5) ) as follows(4.24) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)& =\sum _{n=1}^{\infty }\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x),\hfill \\ \hfill u_{\mathit{\beta }}(x,0)& =\sum _{n=1}^{\infty }\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x)·\hfill \\ \hfill \end{array}$$(4.24)

Denoting(4.25) $$\begin{array}{cc}\hfill h^{\mathit{ϵ}}(x)& =\sum _{n=1}^{\infty }[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x),\hfill \\ \hfill h(x)& =\sum _{n=1}^{\infty }[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x)·\hfill \\ \hfill \end{array}$$(4.25)

In this section, we introduce the convergence estimate for $\Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}$ by using an a posteriori choice rule for the regularization parameter.

To find u(x, 0), we just need to solve the following integral equation(4.26) $$\begin{array}{c}\hfill Ku(x,0)={\int }_{\mathrm{\Omega }}k(x,\mathit{\xi })u(\mathit{\xi },0)\mathrm{d}\mathit{\xi }·\end{array}$$(4.26)

where the kernel is(4.27) $$\begin{array}{c}\hfill k(x,\mathit{\xi })=\sum _{n=1}^{\infty }{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\phi }}_n(x){\mathit{\phi }}_n(\mathit{\xi })·\end{array}$$(4.27)

Next, we know that ${\{{\mathit{\phi }}_n\}}_{n=1}^{\infty }$ is an orthogonal basic in $L^2(\mathrm{\Omega })$ and (4.24(4.24) $$\begin{array}{cc}\hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)& =\sum _{n=1}^{\infty }\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}[g_n^{\mathit{ϵ}}-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x),\hfill \\ \hfill u_{\mathit{\beta }}(x,0)& =\sum _{n=1}^{\infty }\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}[g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }]{\mathit{\phi }}_n(x)·\hfill \\ \hfill \end{array}$$(4.24) ), we know that the singular values ${\{{\mathit{\upsilon }}_n\}}_{n=1}^{\infty }$ for the linear self-adjoint compact operator K are(4.28) $$\begin{array}{c}\hfill {\mathit{\upsilon }}_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})·\end{array}$$(4.28)

and corresponding eigenvectors is ${\mathit{\phi }}_n$.

Motivated by the remarks at the end of Section 4.4 in [3], we apply a modified discrepancy principle in the following form(4.29) $$\begin{array}{c}\hfill \Vert \mathit{\beta }{(\mathit{\beta }+K)}^{-1}(K{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)-h^{\mathit{ϵ}}(x)){\Vert }_{L^2(\mathrm{\Omega })}=m\mathit{ϵ}·\end{array}$$(4.29)

where $m>1$ is a constant. According to the following lemma, there exists a unique solution for (4.27(4.27) $$\begin{array}{c}\hfill k(x,\mathit{\xi })=\sum _{n=1}^{\infty }{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\phi }}_n(x){\mathit{\phi }}_n(\mathit{\xi })·\end{array}$$(4.27) ) if $\Vert h^{\mathit{ϵ}}{\Vert }_{L^2(\mathrm{\Omega })}>m\mathit{ϵ}>0$.

Lemma 4.5:

Set $P(\mathit{\beta })=\Vert \mathit{\beta }{(\mathit{\beta }+K)}^{-1}(K{u}_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)-h^{\mathit{ϵ}}(x)){\Vert }_{L^2(\mathrm{\Omega })}$. If $\Vert h^{\mathit{ϵ}}{\Vert }_{L^2(\mathrm{\Omega })}>\mathit{ϵ}>0$ then the following results are obtained:

(a)	$\mathit{\rho }(\mathit{\beta })$ is a continuous function.
(b)	${lim}_{\mathit{\beta }\to 0}$$\mathit{\rho }(\mathit{\beta })$ = 0.
(c)	${lim}_{\mathit{\beta }\to +\infty }$$\mathit{\rho }(\mathit{\beta })=\Vert g^{\mathit{ϵ}}{(x)\Vert }_{L^2(\mathrm{\Omega })}$.
(d)	$\mathit{\rho }(\mathit{\beta })$ is a strictly increasing function over $(0,+\infty )$.

The proofs of (a), (b), (c) and (d) are straightforward results by the expression of(4.30) $$\begin{array}{c}\hfill P(\mathit{\beta })=\sqrt{{\sum }_{n=1}^{\infty }(\frac{\mathit{\beta }}{\mathit{\beta }+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^4|〈h^{\mathit{ϵ}}(x),{\mathit{\phi }}_n(x)〉{|}^2}·\end{array}$$(4.30)

Theorem 4.6:

Suppose that the a priori condition ${\Vert u(.,0)\Vert }_{{\mathbf{H}}^k(\mathrm{\Omega })}\le M$, $k>0$ and assumption (4.28(4.28) $$\begin{array}{c}\hfill {\mathit{\upsilon }}_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})·\end{array}$$(4.28) ) holds. Let $m>B$ such that $\Vert h^{\mathit{ϵ}}{\Vert }_{L^2(\mathrm{\Omega })}>m\mathit{ϵ}>0$ with B defined in Lemma 4.3. The regularization parameter $\mathit{\beta }(\mathit{ϵ})$ is chosen by the modified discrepancy principle (4.27(4.27) $$\begin{array}{c}\hfill k(x,\mathit{\xi })=\sum _{n=1}^{\infty }{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\phi }}_n(x){\mathit{\phi }}_n(\mathit{\xi })·\end{array}$$(4.27) ). Then we obtain:

(1) If $0<k<1$, we have a convergence estimate(4.31) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}& \le {\mathit{ϵ}}^{\frac{k}{k+1}}{M}^{\frac{1}{k+1}}[H(m,k)+Q(m,k,{\mathit{\lambda }}_n)]·\hfill \end{array}$$(4.31)

whereby(4.32) $$\begin{array}{c}\hfill H(m,k)=(\frac{m+B}{C_2^{-}}{)}^{\frac{k}{k+1}},Q(m,k)=\frac{B(k+1)}{C_2^{-}}[\frac{C_2^{+}{(1-k)}^{(1-k)}}{4(m-B)}{]}^{\frac{1}{1+k}}·\end{array}$$(4.32) (2) If $k\ge 1$, we have a convergence estimate(4.33) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}& \le {\mathit{ϵ}}^{\frac{1}{2}}{M}^{\frac{1}{2}}[R(m,k)+S(m,k)]·\hfill \end{array}$$(4.33)

whereby(4.34) $$\begin{array}{c}\hfill R(m,k)={\left(\frac{a(B+m)}{C_2^{-}}\right)}^{\frac{1}{2}},S(m,k)=\sqrt{\frac{B}{4C_2^{-}{\mathit{\lambda }}_1^{\frac{k-1}{2}}(m-B)}}·\end{array}$$(4.34)

Before proving Theorem 4.3, we introduce the following Lemma.

Lemma 4.7:

For constants $k>0,\mathit{\mu }>0,s\ge {\mathit{\lambda }}_1>0$, we have:(4.35) $$\begin{array}{c}\hfill H(z)=\frac{\mathit{\mu }{z}^{\frac{1-k}{2}}}{\mathit{\mu }z+C_2^{-}}\le \left\{\begin{array}{c}{\tilde{B}}_3(k,C_2^{-}){\mathit{\mu }}^{\frac{1+k}{2}},0<k<1,\\ {\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\mu },k\ge 1.\end{array}\right.\end{array}$$(4.35)

where ${\tilde{B}}_3(k,C_2^{-})=({\displaystyle \frac{1-k}{1+k}}{)}^{\frac{1-k}{2}}\frac{1}{2{(C_2^{-})}^{\frac{k+1}{2}}}$ and ${\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1)=\frac{1}{C_2^{-}{\mathit{\lambda }}_1^{\frac{k-1}{2}}}·$

Proof:

 

(1)

If $k\ge 1$, then for $z\ge {\mathit{\lambda }}_1$, we have.(4.36) $$\begin{array}{c}\hfill H(z)\le \frac{\mathit{\mu }}{\mathit{\mu }z+C_2^{-}}\frac{1}{z^{\frac{k-1}{2}}}\le \frac{\mathit{\mu }}{C_2^{-}}\frac{1}{{\mathit{\lambda }}_1^{\frac{k-1}{2}}}=\frac{1}{C_2^{-}{\mathit{\lambda }}_1^{\frac{k-1}{2}}}\mathit{\mu }·\end{array}$$(4.36)

(2)

If $0<k<1$, then we have ${lim}_{z\to 0}H(z)={lim}_{z\to \infty }H(z)=0$, thus we know.(4.37) $$\begin{array}{c}\hfill H(z)\le \underset{z\in (0,+\infty )}{sup}H(z)\le H(z_0)\end{array}$$(4.37) where $z_0\in (0,+\infty )$ such that $H^{\prime }(z_0)=0.$ Notice that $z_0=\frac{(1-k)C_2^{-}}{(1+k)\mathit{\mu }}>0$, then we have:(4.38) $$\begin{array}{c}\hfill H(z)\le H(z_0)=H\left(\frac{(1-k)C_2^{-}}{(1+k)\mathit{\mu }}\right)={\left({\displaystyle \frac{1-k}{1+k}}\right)}^{\frac{1-k}{2}}\frac{1}{2{(C_2^{-})}^{\frac{k+1}{2}}}{\mathit{\mu }}^{\frac{1+k}{2}}·\end{array}$$(4.38)

From (4.27(4.27) $$\begin{array}{c}\hfill k(x,\mathit{\xi })=\sum _{n=1}^{\infty }{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\phi }}_n(x){\mathit{\phi }}_n(\mathit{\xi })·\end{array}$$(4.27) ), we obtain(4.39) $$\begin{array}{cc}\hfill m\mathit{ϵ}& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{h}_n^{\mathit{ϵ}}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2(h_n^{\mathit{ϵ}}-h_n){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & +\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{h}_n{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}.\hfill \end{array}$$(4.39)

Using the inequality $\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}<1$, we get(4.40) $$\begin{array}{c}\hfill \Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2\left(h_n^{\mathit{ϵ}}-h_n\right){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\le {\Vert h^{\mathit{ϵ}}-h\Vert }_{L^2(\mathit{\omega })}\le B\mathit{ϵ}.\end{array}$$(4.40)

Since $h_n=E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0)$, we have the following estimate(4.41) $$\begin{array}{cc}& \Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{h}_n{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\lambda }}_n^k{\mathit{\lambda }}_n^{-k}{u}_n(0){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \underset{n\in \mathbb{N}}{sup}[{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\lambda }}_n^{-k}]\Vert u(.,0){\Vert }_{H^k(\mathrm{\Omega })}\hfill \\ \hfill & \le C_2^{+}\underset{n\in \mathbb{N}}{sup}(\frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{\frac{1-k}{2}}}{C_2^{-}+{\mathit{\lambda }}_n\mathit{\beta }(\mathit{ϵ})}{)}^{2}M\le \left\{\begin{array}{c}M{C}_2^{+}|{\tilde{B}}_3(k,C_2^{-}){|}^2{|\mathit{\beta }(\mathit{ϵ})|}^{1+k},0<k<1,\\ M{C}_2^{+}|{\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1){|}^2|\mathit{\beta }(\mathit{ϵ}){|}^2,k\ge 1.\end{array}\right.\hfill \end{array}$$(4.41)

Therefore(4.42) $$\begin{array}{c}\hfill m\mathit{ϵ}\le \left\{\begin{array}{c}B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_3(k,C_2^{-}){|}^2{|\mathit{\beta }(\mathit{ϵ})|}^{1+k},0<k<1,\hfill \\ B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1){|}^2|\mathit{\beta }(\mathit{ϵ}){|}^2,k\ge 1.\hfill \end{array}\right.\end{array}$$(4.42)

Now we begin to prove Theorem 4.3.

Proof part 1:  $\text{For}(0<k<1)$, by the triangle inequality, we have(4.43) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert u_{\mathit{\beta }}(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}·\end{array}$$(4.43) Step 1: We first estimate the first term on the right-hand side of (4.36(4.36) $$\begin{array}{c}\hfill H(z)\le \frac{\mathit{\mu }}{\mathit{\mu }z+C_2^{-}}\frac{1}{z^{\frac{k-1}{2}}}\le \frac{\mathit{\mu }}{C_2^{-}}\frac{1}{{\mathit{\lambda }}_1^{\frac{k-1}{2}}}=\frac{1}{C_2^{-}{\mathit{\lambda }}_1^{\frac{k-1}{2}}}\mathit{\mu }·\end{array}$$(4.36) ), $\Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}.$(4.44) $$\begin{array}{cc}& \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}=\sqrt{{\sum }_{n=1}^{\infty }(\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{|<h^{\mathit{ϵ}}(x)-h(x),{\mathit{\phi }}_n(x)>|}^2}\hfill \\ \hfill & \le \frac{1}{[\mathit{\beta }(\mathit{ϵ})]}\sqrt{{\sum }_{n=1}^{\infty }|<h^{\mathit{ϵ}}(x)-h(x),{\mathit{\phi }}_n(x)>{|}^2}\le \frac{1}{[\mathit{\beta }(\mathit{ϵ})]}{\Vert h^{\mathit{ϵ}}-h\Vert }_{L^2(\mathrm{\Omega })}\le \frac{B\mathit{ϵ}}{[\mathit{\beta }(\mathit{ϵ})]}.\hfill \end{array}$$(4.44) Step 2: Continuing to estimate the second term on the right-hand side of (4.36(4.36) $$\begin{array}{c}\hfill H(z)\le \frac{\mathit{\mu }}{\mathit{\mu }z+C_2^{-}}\frac{1}{z^{\frac{k-1}{2}}}\le \frac{\mathit{\mu }}{C_2^{-}}\frac{1}{{\mathit{\lambda }}_1^{\frac{k-1}{2}}}=\frac{1}{C_2^{-}{\mathit{\lambda }}_1^{\frac{k-1}{2}}}\mathit{\mu }·\end{array}$$(4.36) ), $\Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}$. From (4.22(4.22) $$\begin{array}{c}\hfill \frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{-k}}{\mathit{\beta }(\mathit{ϵ})+\frac{C_2^{-}}{{\mathit{\lambda }}_n}}=\mathit{\beta }(\mathit{ϵ})G({\mathit{\lambda }}_n)\le \left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}·\right.\end{array}$$(4.22) ) and (4.23(4.23) $$\begin{array}{cc}& \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(x,0)-v_{\mathit{\beta }}{(x,0)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert v_{\mathit{\beta }}(x,0)-u(x,0)\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill \\ \hfill & \le \sqrt{2(M_1+1)}\frac{\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+\left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})M|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)M\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}·\right.\hfill \end{array}$$(4.23) ), and the a priori bounded condition of u(x, 0), we have:(4.45) $$\begin{array}{cc}& \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}=\Vert \sum _{n=1}^{\infty }\frac{-\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{u}_n(0){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^k(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^{1-k}\hfill \\ \hfill & \frac{u_n(0)}{|E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){|}^k}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}.\hfill \end{array}$$(4.45)

Using the Hölder’s inequality, we obtain(4.46) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\le J_1^{\frac{k}{k+1}}{J}_2^{\frac{1}{k+1}}.\end{array}$$(4.46)

where(4.47) $$\begin{array}{cc}\hfill J_1& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^{k+1}(\frac{\mathit{\beta }(\mathit{ϵ})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})+\mathit{\beta }(\mathit{ϵ})}{)}^{1-k}\hfill \\ \hfill & \frac{u_n(0)}{{[E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})]}^k}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \end{array}$$(4.47) (4.48) $$\begin{array}{cc}\hfill J_2& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^{1-k}\frac{u_n(0)}{{[E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})]}^k}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}.\hfill \end{array}$$(4.48)

The first term $J_1$ is estimated as follows(4.49) $$\begin{array}{cc}\hfill J_1& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})u_n(0){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2<h(x)-h^{\mathit{ϵ}}(x),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & +\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2<h^{\mathit{ϵ}}(x),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le B\mathit{ϵ}+\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2<h^{\mathit{ϵ}}(x),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}=(B+m)\mathit{ϵ}.\hfill \end{array}$$(4.49)

The second term $J_2$ can be estimated(4.50) $$\begin{array}{cc}\hfill J_2& \le \Vert \sum _{n=1}^{\infty }\frac{<u(.,0),{\mathit{\phi }}_n(x)>}{{[E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})]}^k}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert \sum _{n=1}^{\infty }\frac{1}{{[C_2^{-}]}^k}{\mathit{\lambda }}_n^k<u(.,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \frac{1}{{[C_2^{-}]}^k}\Vert \sum _{n=1}^{\infty }{\mathit{\lambda }}_n^k<u(.,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\le \frac{1}{{[C_2^{-}]}^k}{\Vert u(.,0)\Vert }_{H^k(\mathrm{\Omega })}\le \frac{M}{{[C_2^{-}]}^k}·\hfill \end{array}$$(4.50)

Combining (4.42(4.42) $$\begin{array}{c}\hfill m\mathit{ϵ}\le \left\{\begin{array}{c}B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_3(k,C_2^{-}){|}^2{|\mathit{\beta }(\mathit{ϵ})|}^{1+k},0<k<1,\hfill \\ B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1){|}^2|\mathit{\beta }(\mathit{ϵ}){|}^2,k\ge 1.\hfill \end{array}\right.\end{array}$$(4.42) ) and (4.43(4.43) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert u_{\mathit{\beta }}(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}·\end{array}$$(4.43) ) into (4.39(4.39) $$\begin{array}{cc}\hfill m\mathit{ϵ}& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{h}_n^{\mathit{ϵ}}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2(h_n^{\mathit{ϵ}}-h_n){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & +\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^2{h}_n{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}.\hfill \end{array}$$(4.39) ), we obtain:(4.51) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}& \le \frac{B\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+{(m+B)}^{\frac{k}{k+1}}\frac{M^{\frac{1}{k+1}}}{{[C_2^{-}]}^{\frac{k}{k+1}}}{\mathit{ϵ}}^{\frac{k}{k+1}}\hfill \\ \hfill & \le {\mathit{ϵ}}^{\frac{k}{k+1}}{M}^{\frac{1}{k+1}}[H(m,k)+Q(m,k)]·\hfill \end{array}$$(4.51)

where we used (4.35(4.35) $$\begin{array}{c}\hfill H(z)=\frac{\mathit{\mu }{z}^{\frac{1-k}{2}}}{\mathit{\mu }z+C_2^{-}}\le \left\{\begin{array}{c}{\tilde{B}}_3(k,C_2^{-}){\mathit{\mu }}^{\frac{1+k}{2}},0<k<1,\\ {\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\mu },k\ge 1.\end{array}\right.\end{array}$$(4.35) ).

Proof part 2: For $k\ge 1$, The proof in this case is similar to above proof in part 1.

First, we have(4.52) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}^2& =\Vert \sum _{n=1}^{\infty }\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{u}_n(0){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}^2·\hfill \end{array}$$(4.52)

Using the Hölder inequality, we obtain that(4.53) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\le J_1^{\frac{1}{2}}{J}_2^{\frac{1}{2}}·\end{array}$$(4.53)

where $J_1$ and $J_2$ are defined in (4.40(4.40) $$\begin{array}{c}\hfill \Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2\left(h_n^{\mathit{ϵ}}-h_n\right){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\le {\Vert h^{\mathit{ϵ}}-h\Vert }_{L^2(\mathit{\omega })}\le B\mathit{ϵ}.\end{array}$$(4.40) ) and (4.41(4.41) $$\begin{array}{cc}& \Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{h}_n{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & =\Vert \sum _{n=1}^{\infty }{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\lambda }}_n^k{\mathit{\lambda }}_n^{-k}{u}_n(0){\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \underset{n\in \mathbb{N}}{sup}[{\left(\frac{\mathit{\beta }(\mathit{ϵ})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}\right)}^2{E}_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }}){\mathit{\lambda }}_n^{-k}]\Vert u(.,0){\Vert }_{H^k(\mathrm{\Omega })}\hfill \\ \hfill & \le C_2^{+}\underset{n\in \mathbb{N}}{sup}(\frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{\frac{1-k}{2}}}{C_2^{-}+{\mathit{\lambda }}_n\mathit{\beta }(\mathit{ϵ})}{)}^{2}M\le \left\{\begin{array}{c}M{C}_2^{+}|{\tilde{B}}_3(k,C_2^{-}){|}^2{|\mathit{\beta }(\mathit{ϵ})|}^{1+k},0<k<1,\\ M{C}_2^{+}|{\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1){|}^2|\mathit{\beta }(\mathit{ϵ}){|}^2,k\ge 1.\end{array}\right.\hfill \end{array}$$(4.41) ). Now we need to estimate $J_2$. Since $k\ge 1$, there exists a positive number a such that ${\Vert u(.,0)\Vert }_{H^1(\mathrm{\Omega })}\le a{\Vert u(.,0)\Vert }_{H^k(\mathrm{\Omega })}\le aM$. This leads to(4.54) $$\begin{array}{cc}\hfill J_2& \le \frac{1}{C_2^{-}}\Vert \sum _{n=1}^{\infty }{\mathit{\lambda }}_n<u(.,0),{\mathit{\phi }}_n(x)>{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\le \frac{1}{C_2^{-}}{\Vert u(.,0)\Vert }_{H^1(\mathrm{\Omega })}\le \frac{aM}{C_2^{-}}·\hfill \end{array}$$(4.54)

Combining (4.42(4.42) $$\begin{array}{c}\hfill m\mathit{ϵ}\le \left\{\begin{array}{c}B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_3(k,C_2^{-}){|}^2{|\mathit{\beta }(\mathit{ϵ})|}^{1+k},0<k<1,\hfill \\ B\mathit{ϵ}+M{C}_2^{+}|{\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1){|}^2|\mathit{\beta }(\mathit{ϵ}){|}^2,k\ge 1.\hfill \end{array}\right.\end{array}$$(4.42) ), (4.46(4.46) $$\begin{array}{c}\hfill \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\le J_1^{\frac{k}{k+1}}{J}_2^{\frac{1}{k+1}}.\end{array}$$(4.46) ), (4.47(4.47) $$\begin{array}{cc}\hfill J_1& =\Vert \sum _{n=1}^{\infty }(\frac{\mathit{\beta }(\mathit{ϵ})E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}{)}^{k+1}(\frac{\mathit{\beta }(\mathit{ϵ})}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})+\mathit{\beta }(\mathit{ϵ})}{)}^{1-k}\hfill \\ \hfill & \frac{u_n(0)}{{[E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})]}^k}{\mathit{\phi }}_n(x){\Vert }_{L^2(\mathrm{\Omega })}\hfill \end{array}$$(4.47) ) gives(4.55) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}& \le M^{\frac{1}{2}}{\left(\frac{a(B+m)}{C_2^{-}}\right)}^{\frac{1}{2}}{\mathit{ϵ}}^{\frac{1}{2}}.\hfill \end{array}$$(4.55)

Using the inequality and noting (4.35(4.35) $$\begin{array}{c}\hfill H(z)=\frac{\mathit{\mu }{z}^{\frac{1-k}{2}}}{\mathit{\mu }z+C_2^{-}}\le \left\{\begin{array}{c}{\tilde{B}}_3(k,C_2^{-}){\mathit{\mu }}^{\frac{1+k}{2}},0<k<1,\\ {\tilde{B}}_4(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\mu },k\ge 1.\end{array}\right.\end{array}$$(4.35) ) for $k\ge 1$, we conclude that(4.56) $$\begin{array}{cc}\hfill \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}{(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}& \le \Vert u_{\mathit{\beta }}^{\mathit{ϵ}}(.,0)-u_{\mathit{\beta }}{(.,0)\Vert }_{L^2(\mathrm{\Omega })}+{\Vert u_{\mathit{\beta }}(.,0)-u(.,0)\Vert }_{L^2(\mathrm{\Omega })}\hfill \\ \hfill & \le \frac{B\mathit{ϵ}}{\mathit{\beta }(\mathit{ϵ})}+M^{\frac{1}{2}}{\left(\frac{a(B+m)}{C_2^{-}}\right)}^{\frac{1}{2}}{\mathit{ϵ}}^{\frac{1}{2}}\hfill \\ \hfill & \le M^{\frac{1}{2}}[R(m,k)+S(m,k)]{\mathit{ϵ}}^{\frac{1}{2}}·\hfill \end{array}$$(4.56)

5. Numerical experiments

To verify our proposal methods, we carry out numerical experiments for above regularization methods. Two different numerical examples corresponding to $T=1$, and $\mathit{\alpha }=0.2$ and 0.6 are described in this section. In order to illustrated the sensitivity of the computational accuracy to the noise of the measurement data, we use the random function to generate the noisy data. The perturbation was defined as $\mathit{ϵ}$ rand(size()), where rand(size()) is a random number, and $\mathit{ϵ}$ plays as an amplitude of the errors. For computing the Mittag-Leffler function, we applied an algorithm in [3].

In this example, we consider a two-dimensional case of Problem (1.1) as follows:(5.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{{\mathit{\partial }}^{\mathit{\alpha }}u(x,y,t)}{\mathit{\partial }{t}^{\mathit{\alpha }}}=u_{xx}(x,y,t)+u_{yy}(x,y,t)+F(x,y,t),(x,y)\in (0,\mathit{\pi })\times (0,\mathit{\pi }),t\in (0,1]\hfill \\ u(x,y,t)=0,(x,y)\in (0,\mathit{\pi })\times (0,\mathit{\pi }),t\in (0,1]\hfill \\ u(x,y,T)=g(x,y),(x,y)\in (0,\mathit{\pi })\times (0,\mathit{\pi })·\hfill \end{array}\right.\end{array}$$(5.1)

whereby: $Au(x,t)=u_{xx}(x,y,t)+u_{yy}(x,y,t)·$(5.2) $$\begin{array}{cc}\hfill F(x,y,t)& ={10}^{-1}\frac{t^{1-\mathit{\alpha }}}{\mathrm{\Gamma }(2-\mathit{\alpha })}{e}^t[e^{{(x+y)}^{\mathit{\alpha }}}-1]\hfill \\ \hfill & sin(x(\mathit{\pi }-x))sin(y(\mathit{\pi }-y))-({10}^{-1}[e^t+1]sin(x(\mathit{\pi }-x))sin(y(\mathit{\pi }-y))\hfill \\ \hfill & (2\mathit{\alpha }(\mathit{\alpha }-1){(x+y)}^{\mathit{\alpha }-2}{e}^{{(x+y)}^{\mathit{\alpha }}}+{[\mathit{\alpha }{(x+y)}^{\mathit{\alpha }-1}]}^2{e}^{{(x+y)}^{\mathit{\alpha }}}\hfill \\ \hfill & -(e^{{(x+y)}^{\mathit{\alpha }}}-1){(\mathit{\pi }-2x)}^2-(e^{{(x+y)}^{\mathit{\alpha }}}-1)\hfill \\ \hfill & {(\mathit{\pi }-2y)}^2)+5^{-1}[e^t+1]cos(x(\mathit{\pi }-x))sin(y(\mathit{\pi }-y))\hfill \\ \hfill & (\mathit{\alpha }{(x+y)}^{\mathit{\alpha }-1}{e}^{{(x+y)}^{\mathit{\alpha }}}(\mathit{\pi }-2x)\hfill \\ \hfill & +(e^{{(x+y)}^{\mathit{\alpha }}}-1))+5^{-1}[e^t+1]sin(x(\mathit{\pi }-x))cos(y(\mathit{\pi }-y))\hfill \\ \hfill & (\mathit{\alpha }{(x+y)}^{\mathit{\alpha }-1}{e}^{{(x+y)}^{\mathit{\alpha }}}(\mathit{\pi }-2y)\hfill \\ \hfill & +(e^{{(x+y)}^{\mathit{\alpha }}}-1))),\hfill \\ \hfill g(x,y)& =5^{-1}{e}^1[e^{{(x+y)}^{\mathit{\alpha }}}-1]sin(x(\mathit{\pi }-x))sin(y(\mathit{\pi }-y))·\hfill \end{array}$$(5.2)

By choosing ${\mathit{\lambda }}_{p,q}=p^2+q^2$, and from (3.7(3.7) $$\begin{array}{c}\hfill u_n(0)=\frac{g_n-{\int }_0^T{G}_n(\mathit{\alpha },T-\mathit{\tau })F_n(\mathit{\tau })\mathrm{d}\mathit{\tau }}{E_{\mathit{\alpha },1}(-{\mathit{\lambda }}_n{T}^{\mathit{\alpha }})}·\end{array}$$(3.7) ) and (4.22(4.22) $$\begin{array}{c}\hfill \frac{\mathit{\beta }(\mathit{ϵ}){\mathit{\lambda }}_n^{-k}}{\mathit{\beta }(\mathit{ϵ})+\frac{C_2^{-}}{{\mathit{\lambda }}_n}}=\mathit{\beta }(\mathit{ϵ})G({\mathit{\lambda }}_n)\le \left\{\begin{array}{c}{\tilde{B}}_1(k,C_2^{-})|\mathit{\beta }(\mathit{ϵ}){|}^k\text{if}0<k<1\hfill \\ {\tilde{B}}_2(k,C_2^{-},{\mathit{\lambda }}_1)\mathit{\beta }(\mathit{ϵ}),\text{if}k\ge 1\hfill \end{array}·\right.\end{array}$$(4.22) ) we can deduce the exact and its regularized solutions as follows:(5.3) $$\begin{array}{cc}\hfill u(x,y,0)& =\sum _{p=1}^I\sum _{q=1}^J\frac{1}{E_{\mathit{\alpha },1}(-(p^2+q^2))}\hfill \\ \hfill & \times (g_{p,q}(x,y)-{\int }_0^1{G}_{p,q}(\mathit{\alpha },1-\mathit{\tau })F_{pq}(\mathit{\tau })\mathrm{d}\mathit{\tau }){\mathit{\phi }}_{pq}(x,y),\hfill \\ \hfill u_{\mathit{\beta }}^{\mathit{ϵ}}(x,y,0)& =\sum _{p=1}^I\sum _{q=1}^J\frac{1}{\mathit{\beta }(\mathit{ϵ})+E_{\mathit{\alpha },1}(-(p^2+q^2))}\hfill \\ \hfill & \times (g_{p,q}^{\mathit{ϵ}}(x,y)-{\int }_0^1{G}_{p,q}(\mathit{\alpha },1-\mathit{\tau })F_{pq}^{\mathit{ϵ}}(\mathit{\tau })\mathrm{d}\mathit{\tau }){\mathit{\phi }}_{pq}(x,y)·\hfill \\ \hfill \end{array}$$(5.3)

The whole numerical procedure is proceeded in the following steps:

Step 1. Choosing I,J and L to generate spatial and temporal discretizations as follows:(5.4) $$\begin{array}{cc}\hfill x_i& =i\mathrm{\Delta }x,\mathrm{\Delta }x=\frac{\mathit{\pi }}{I},i=\overline{0,I},\hfill \\ \hfill y_j& =j\mathrm{\Delta }y,\mathrm{\Delta }y=\frac{\mathit{\pi }}{J},j=\overline{0,J},\hfill \\ \hfill t_l& =l\mathrm{\Delta }t,\mathrm{\Delta }t=\frac{1}{L},l=\overline{0,L}·\hfill \end{array}$$(5.4)

Of course, the higher value of I, J and L will provide more stable and accurate numerical calculation, however in the following examples $I=J=L=101$ are satisfied.

Step 2. We choose couple $(g^{\mathit{ϵ}},F^{\mathit{ϵ}})$ as observed data-set including the noise in such a way that:(5.5) $$\begin{array}{cc}\hfill g^{\mathit{ϵ}}& =g(.,.)+\mathit{ϵ}rand(size(g(.,.))),\hfill \\ \hfill F^{\mathit{ϵ}}& =F(.,.,.)+\mathit{ϵ}(rand(size(F(.,.,.))))·\hfill \end{array}$$(5.5) Step 3. We put $u{(.,.,0)}_{\mathit{\beta }(\mathit{ϵ})}^{\mathit{ϵ}}\left(x_i,y_j\right)=u{(.,.,0)}_{\mathit{\beta }(\mathit{ϵ}),i,j}^{\mathit{ϵ}}$ and $u(.,.,0)\left(x_i,y_j\right)=u_{i,j}$, constructing two matrices contained all discrete values of $u{(.,.,0)}_{\mathit{\beta }}^{\mathit{ϵ}}$ and u(., ., 0) denoted by ${\mathrm{\Lambda }}_{\mathit{\beta }}^{\mathit{ϵ}}$ and $\mathrm{\Psi }$, respectively.(5.6) $$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_{\mathit{\beta }(\mathit{ϵ})}^{\mathit{ϵ}}& =\left[\begin{array}{ccccc}{u}_{(0,0,0)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,0,1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& ...& u_{(0,0,I-1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,0,I)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}\\ u_{(0,1,0)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,1,1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& ...& u_{(0,1,I-1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,1,I)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}\\ u_{(0,2,0)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,2,1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& ...& u_{(0,2,I-1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,2,I)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}\\ ...& ...& \ddots & ...& ...\\ u_{(0,J,0)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,J,1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& ...& u_{(0,J,I-1)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}& u_{(0,J,I)}^{\mathit{\beta }(\mathit{ϵ}),\mathit{ϵ}}\end{array}\right]\in {\mathbb{R}}^{I+1}\times {\mathbb{R}}^{J+1},\hfill \\ \hfill \mathrm{\Psi }& =\left[\begin{array}{ccccc}{u}_{(0,0,0)}& u_{(0,0,1)}& ...& u_{(0,0,I-1)}& u_{(0,0,I)}\\ u_{(0,1,0)}& u_{(0,1,1)}& ...& u_{(0,1,I-1)}& u_{(0,1,I)}\\ u_{(0,2,0)}& u_{(0,2,1)}& ...& u_{(0,2,I-1)}& u_{(0,2,I)}\\ ...& ...& \ddots & ...& ...\\ u_{(0,J,0)}& u_{(0,J,1)}& ...& u_{(0,J,I-1)}& u_{(0,J,I)}\end{array}\right]\in {\mathbb{R}}^{I+1}\times {\mathbb{R}}^{J+1}·\hfill \end{array}$$(5.6) Step 4. The error estimation is followed by

Relative error estimation:(5.7) $$\begin{array}{cc}\hfill E_1& =\frac{\sqrt{{\sum }_{i=0}^I{\sum }_{j=0}^J{|u_{\mathit{\beta }}^{\mathit{ϵ}}(x_i,y_j,0)-u(x_i,y_j,0)|}_{L^2(0,\mathit{\pi })\times L^2(0,\mathit{\pi })}^2}}{\sqrt{{\sum }_{i=0}^J{\sum }_{j=0}^I{|u(x_i,y_j,0)|}_{L^2(0,\mathit{\pi })}^2}}·\hfill \end{array}$$(5.7)

Absolute error estimation:(5.8) $$\begin{array}{cc}\hfill E_2& =\sqrt{\frac{1}{(I+1)(J+1)}{\sum }_{i=0}^J{\sum }_{j=0}^I{|u_{\mathit{\beta }}^{\mathit{ϵ}}(x_i,y_j,0)-u(x_i,y_j,0)|}_{L^2(0,\mathit{\pi })\times L^2(0,\mathit{\pi })}^2}·\hfill \end{array}$$(5.8)

For the a priori choice rule method we choose ${\mathit{\beta }}_{pri}={(\frac{\mathit{ϵ}}{M})}^{\frac{1}{2}}$, then from the exact solution we have ${\Vert u(.,0)\Vert }_{H^2(\mathrm{\Omega })}\le M$ with $M=1168$ in the case of $\mathit{\alpha }=0.2$; and with $M=1685$ in the case of $\mathit{\alpha }=0.6$. For the a posteriori parameter choice rule method, we choose ${\mathit{\beta }}_{pos}=\frac{{\mathit{ϵ}}^{\frac{1}{2}}}{M_{m,\mathit{\alpha }}}$ based on $m=3.6$, with $M_{m,\mathit{\alpha }}=113546$ at $\mathit{\alpha }=0.2$, and $M_{m,\mathit{\alpha }}=171653$ at $\mathit{\alpha }=0.6$, respectively.

Table 1. Error estimation between the exact and its regularized solutions for both, the a priori and the a posteriori parameter choice rule methods in Example 2 with $\mathit{\alpha }=0.2$.

$\mathit{ϵ}$	$E_1^{{\mathit{\beta }}_{pri}}$	$E_2^{{\mathit{\beta }}_{pri}}$	$E_1^{{\mathit{\beta }}_{pos}}$	$E_2^{{\mathit{\beta }}_{pos}}$
$1\text{E}-01$	$6.06\text{E}-01$	$4.64\text{E}-01$	$5.09\text{E}-02$	$4.18\text{E}-02$
$1\text{E}-02$	$5.16\text{E}-02$	$3.18\text{E}-02$	$5.31\text{E}-03$	$4.05\text{E}-03$
$1\text{E}-03$	$7.81\text{E}-03$	$4.65\text{E}-03$	$4.86\text{E}-04$	$1.18\text{E}-04$
$1\text{E}-04$	$2.37\text{E}-04$	$2.01\text{E}-04$	$4.29\text{E}-05$	$3.74\text{E}-05$

Table 2. Error estimation between the exact and its regularized solutions for both, the a priori and the a posteriori parameter choice rule methods in Example 2 with $\mathit{\alpha }=0.6$.

$\mathit{ϵ}$	$E_1^{{\mathit{\beta }}_{pri}}$	$E_2^{{\mathit{\beta }}_{pri}}$	$E_1^{{\mathit{\beta }}_{pos}}$	$E_2^{{\mathit{\beta }}_{pos}}$
$1\text{E}-01$	$6.25\text{E}-01$	$3.56\text{E}-01$	$3.94\text{E}-02$	$2.56\text{E}-02$
$1\text{E}-02$	$5.44\text{E}-02$	$1.12\text{E}-02$	$3.20\text{E}-03$	$2.03\text{E}-03$
$1\text{E}-03$	$5.81\text{E}-03$	$3.55\text{E}-03$	$2.56\text{E}-04$	$1.50\text{E}-04$
$1\text{E}-04$	$5.02\text{E}-04$	$1.12\text{E}-04$	$4.77\text{E}-05$	$3.58\text{E}-05$

Figure 1. A comparison between the exact and its regularized solutions for the a priori parameter choice rule in Example with $\mathit{\alpha }=0.2$.

Figure 2. A comparison between the exact and its regularized solutions for the a posteriori parameter choice rule in Example with $\mathit{\alpha }=0.2$.

Figure 3. A comparison between the exact and its regularized solutions for the a priori parameter choice rule in Example with $\mathit{\alpha }=0.6$.

Figure 4. A comparison between the exact and its regularized solutions for the a posteriori parameter choice rule in Example with $\mathit{\alpha }=0.6$.

Figures 1 and 2 show a comparison between the exact and its regularized solutions for both parameter choice rule methods, the a priori and the a posteriori, with $\mathit{\alpha }=0.2$. Figures 3 and 4 show a comparison between the exact and its regularized solutions for both parameter choice rule methods, the a priori and the a posteriori, with $\mathit{\alpha }=0.6$. Tables 1 and 2 show the relative and absolute error estimates between the exact and its regularized solutions for both, the a priori and the a posteriori parameter choice rule methods with $\mathit{\alpha }=0.2$ and $\mathit{\alpha }=0.6$, respectively. It shows in the Figures 1–4 that when $\mathit{ϵ}\le {10}^{-1}$ in comparison with the posteriori parameter choice rule method, the numerical solution of the priori parameter choice rule method is more oscillated around the exact solution. However once $\mathit{ϵ}$ tends to zero, both methods have been converged to the exact solution very well. The numerical results obtained from $\mathit{\alpha }=0.2$ and $\mathit{\alpha }=0.6$ show the same tendency. At the beginning of the calculation, when the error estimation between the exact and its regularized solutions is still large (about ${10}^{-1}-{10}^{-2}$), the solutions obtained from the a priori parameter choice rule are oscillated around the exact solution more than the results obtained the a posteriori parameter choice rule. As shown in Tables 1 and 2, in general the a posteriori choice rule method is converged to the exact solution faster than the a priori parameter choice rule method by about one order of ${10}^{-1}$.

6. Conclusion

In this paper, we investigate the backward problem for the time-fractional diffusion equation with inhomogeneous source. We applied the QBV method to regularize the problem. In the theoretical results, we obtain the error estimate for both proposed methods, the a priori and the a posteriori parameter choice rule methods, based on the priori stability condition. From numerical results, we try to test with very complicated source term functions, it shows that our proposed regularization methods are converged very well to the exact solutions. Furthermore, it also shows that the posteriori choice rule method is converged to the exact solution better than the priori choice rule method. Since the time-fractional diffusion is more focused on the subdiffusion phenomena, which are characterized by a heavy-tailed waiting time distribution in diverging temporal moments and a non-Markovian dynamics, in the future work, we plan to study on a backward problem for space-time fractional diffusion problems, which can deal with both subdiffusive and superdiffusive problems appeared in a wide range of practical problems in surface and subsurface hydrology, biology and epidemiology. Furthermore, in the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ) we still have not taken in account the convection term, if the diffusion process is also driven by a hydrodynamic problem, it absolutely needs to add this term into the problem (1.1(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{\partial }}_t^{\mathit{\alpha }}u(x,t)=Au(x,t)+F(x,t),\hfill & (x,t)\in \mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0,\hfill & (x,t)\in \mathit{\partial }\mathrm{\Omega }\times (0,T),\hfill \\ u(x,T)=g(x),\hfill & x\in \mathrm{\Omega }·\hfill \end{array}\right.\end{array}$$(1.1) ), the backward problem would be more challenging, and it is on demand in a future work as well.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments leading to the improvement of our manuscript.

Additional information

Funding

This work is supported by Vietnam National University-Ho Chi Minh City (VNU-HCM) under Grant named ‘Some ill-posed problems for partial differential equation’.

Notes

No potential conflict of interest was reported by the authors.