Identifying the space source term problem for time-space-fractional diffusion equation

In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.


Introduction
Let T be a given positive number, a bounded domain in R n (n ≥ 1) with a smooth boundary ∂ . In this work, we consider the inverse source problem of the time-fractional diffusion equation as follows: where D β t u(x, t) is the Caputo fractional derivative of order β defined as [1] in the following form: where (·) is the Gamma function. In fact (g, , ϕ) is noised by observation data (g ε , ε , ϕ ε ) where the order of ε is the noise level. We have © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
In all functions g(x), (x), and ϕ(t) are given data. It is well known that, if ε is small enough, the sought solution f (x) may have a large error. It is known that the inverse source problem mentioned above is ill-posed. In general, the definition of the ill-posed problem was introduced in [2]. Therefore, regularization is needed.
As is well known, in the last few decades, the fractional calculation is a concept that has a great influence on the mathematical background and its application in modeling real problems. Fractional calculus has many applications in mechanics, physics and engineering science, etc. We present to the reader much of the published work on these issues, such as [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references cited therein. This makes it attractive to study this model.
The space source term problem for the time-fractional partial differential equations has attracted a lot of attention, and much work has been completed to study many aspects of this problem, specifically as follows.
In 2019, the authors Yan, Xiong and Wei proposed a conjugate gradient algorithm to solve the Tikhonov regularization problem for the case γ = 1.
In the case f (x) = 1, in 2014, Fan Yang and his group considered the Fourier transform and the quasi-reversibility regularization method; see [24]. Recently, the simple source problem, i.e, ϕ(t) = 1 and γ = 1 in Eq. (1.1) has been considered by Fan Yang, Zhang and Li, see [20,21,[25][26][27]; the authors used the Landweber iterative regularization, Truncation regularization and Tikhonov regularization methods solve this problem and achieved the results of convergence results to the order of p p+1 for 0 < p < 2 and 1 2 for p > 2, respectively.
The problem (1.1) with discrete random noise has been studied by Tuan et al, they used the filter regularization and trigonometric methods to solve this problem (1.1); see [28][29][30]. According to our searching, the results about applying the quasi-reversibility regularization method to solve the inverse source problem for the time-space-fractional diffusion equation is still limited. To the best of our knowledge, this is one of the first results of this type of problem. In particular, one addressed the case where L γ and the right-hand side ϕ(t)f (x) are represented in a general form. Motivated by all the above reasons, we consider the quasi-reversibility regularization method to solve the problem (1.1). The present paper aims to use the quasi-reversibility regularization method (QR method) to solve the problem (1.1).
The outline of the paper is as follows. In Sect. 2, we show some basic concepts, the function setting, the definitions, and the ill-posed problem are presented in Sect.

Preliminary results
The eigenvalues of the operator L γ is introduced in [31]. Let us recall that the spectral problem admits a family of eigenvalues where β > 0 and γ ∈ R are arbitrary constant.

Theorem 2.11
Let g, f ∈ L 2 ( ) and ϕ ∈ L ∞ (0, T), then there exists a unique weak solution u ∈ C([0, T]; L 2 ( )) ∪ C([0, T]; D ζ ( )) for (1.1) given by By a simple transformation we can see that This implies that Proof Denote ϕ L ∞ (0,T) = P(ϕ 0 , ϕ 1 ). A linear operator is defined R : L 2 ( ) → L 2 ( ) as follows: Because of k(x, ξ ) = k(ξ , x), we can see that R is self-adjoint operator. Next, we are going to prove its compactness. Let us define R M as follows: This implies that 24) and the corresponding eigenvectors are e k which is known as an orthonormal basis in L 2 ( ). From (2.19), the inverse source problem can be formulated as an operator equation, and by Kirsch ([2]), we conclude that the problem (1.1) is ill-posed. We present an example. Fix β and choose Because of (2.18) and combining (2.26), the source term f m is If we have input data , g = 0, then the source term f = 0. An error in L 2 ( ) norm between ( , g) and ( m , g m ) is Combining (2.29) and (2.32), we conclude that the inverse source problem is not wellposed.

Quasi-reversibility method
In this section, the quasi-reversibility method is used to investigate problem (1.1), and give information for convergence of the two estimates under a prior parameter choice rule and a posterior parameter choice rule, respectively.

Construction of a regularization method
We employ the QR method to established a regularized problem, namely where g ε , ε are perturbed initial data and final data satisfying and α(ε) is a regularization parameter. We can assert that From now on, for brevity, we denote and

A prior parameter choice
Afterwards, f (·)f ε,α(ε) (·) L 2 ( ) is shown under a suitable choice for the regularization parameter. To do this, we introduced the following lemma.
Proof (1) If j ≥ 1 then from s ≥ λ 1 , we get (2) If 0 < j < 1 then it can be seen that . Solving G (s) = 0, we can see that This is precisely the assertion of the lemma.
Theorem 3.2 Let f be as (2.18) and the noise assumption (2.13) hold. We obtain the following two cases. (3.8) Proof By the triangle inequality, we know The proof falls naturally into two steps.

A posterior parameter choice
In this subsection, a posterior regularization parameter choice rule is considered. By the Morozov discrepancy principle here we find ζ such that see [2], where ζ > 1 is a constant. We know there exists an unique solution for

29)
which gives the required results.
Here, put Therefore, combining (3.33) to (3.34), we know that From (3.35), it is very easy to see that (3.36) Therefore, we conclude that which gives the required results.
• Next, the relative error estimation is defined by (4.7) In Fig. 1, we show the convergent estimate between exact solution and its approximation by the quasi-reversibility method under a prior parameter choice rule and under a posterior parameter choice rule. In Fig. 2, we show the convergent estimate between the sought solution and its approximation by QRM and the corresponding errors with ε = 0.2. Similarly, in Fig. 3 and in Fig. 4, we show the comparison in the cases ε = 0.02 and ε = 0.0125. While drawing these figures, we choose values β = 0.5, γ = 0.5 and j = 1. In the tables of errors that we calculated in this numerical example, we present the error estimation for both a prior and a posterior parameter choice rule, respectively. In Table 1, we give the comparison of the convergent rate between the sought solution and the regularized solutions. Next, in Table 2, we fixed ε = 0.034. In the first column, with β p+1 = β p + 0.11, p = 1, 8 with β 1 = 0.11. Using Eq. (4.7), we show the error estimate between the sought solution and its approximation with β = 0.3, in the second column and the third column. Similarly    Table 1 The error between the regularized solutions and sought solution at β = 0.5, γ = 0.5  Table 2 The error between the regularized solutions and the sought solution at ε = 0.034

Conclusions
In this work, we use the QR method to regularize the inverse problem to determine an unknown source term of a space-time-fractional diffusion equation. We showed that the problem (1.1) is ill-posed in the sense of Hadamard. Next, we give the results for the convergent estimate between the regularized solution and the sought solution under a prior and a posterior parameter choice rule. We illustrate our theoretical results by a numerical example. In future work, we will be interested in the case of the source function being a function of the general form f (x, t), and this is still an open problem and will show more difficulty.