Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative

This article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.


Introduction
In recent times, the fractional calculus has been extensively studied. There are many definitions for the fractional derivative operator. The Riemann-Liouville fractional derivative and the Caputo fractional derivative are two of the most often used definitions of the fractional derivatives. Recently, in the case the kernel has no singularities, Caputo and Fabrizio have introduced a new definition of the fractional derivatives [1]. Fractional calculus is a subject with a long history and has gained great interest in different fields of applied science, and many authors considered this topic [2][3][4][5][6][7][8][9][10][11][12]. For example, Alrefai and Abdeljawad in [13] studied the analysis for fractional diffusion equations with fractional derivative with non-singular kernel.
In this work, we restore the space source term problem for a fractional diffusion equation with a Caputo-Fabrizio derivative. The fractional diffusion equation is discussed in this paper as follows: u(x, 0) = g(x), in , (1.1) where 0 < α < 1, T > 0 is a fixed value and ∈ R d is a bounded domain with sufficient smooth boundary ∂ . CF 0 D α t is the Caputo-Fabrizio operator for fractional derivatives of order α. From [14], we have the following formula: with * defines the convolution and A(α) is a normalization function such that A(0) = A(1) = 1.
For the spatial-fractional and temporal-fractional derivatives, there are a lot of new definitions to be presented. In [15][16][17][18], we can see more properties and applications of this new fractional derivative. In [19], the authors proved uniqueness of the solution to an initial value problem for linear fractional differential equation with 1 < α ≤ 2.
The problem (1.1) is associated to anomalous diffusion phenomenon in physical motivations. In the case 0 < α < 1, the model (1.1) is used for the super-diffusive model in heterogeneous media. Some physical background is found. For example, as regards the physical application of fractional calculus, the authors in [20] have shown that the fractional calculus is effective to account for the damping controllers which are compared to the classical derivative. The authors in [21] considered the separation and stability of solutions to nonlinear systems involving Caputo-Fabrizio derivatives.
Our purpose in this paper is to find an inversion source problem for (1.1). Assume that the source term F(x, t) of the problem (1.1) is a forward problem, which can be split into F (x)Q(t) where Q(t) is known in advance. Hence, we want to identify the space source term F (x) by using the value of the final time T as follows: In fact, the measurements are noised, instead of receiving the exact data H, we have the approximation data H ε that satisfies where ε > 0 is the noise level. A small error between the given observation H ε and the exact data H ε leads to a large error in the solution in output data. This is a reason why we propose some regularization methods in order to recover stable approximations for the unknown space source function.
In this paper, we apply the fractional Landweber method and modified quasi-boundary value method to restore the unknown space source function F . The idea of the modified quasi-boundary value method mainly comes from Denche and Bessila's paper [22], where they solved the following backward heat conduction problem. For references to this method, the reader can see [23][24][25]. In [26], Klann and Ramlau introduced the fractional Landweber method for a linear ill-posed problem. We give an a posteriori choice of the regularization parameter based on an a priori bound of the exact solution that cannot be known exactly in practice. Here, the regularization parameter by the a priori rule is easier than an a posteriori rule. For references to this method, the reader can see [27]. According to this work, we can compare which method is the more optimal for the space source term problem for the fractional diffusion equation with the Caputo-Fabrizio derivative.
The paper is organized as follows. In Sect. 2, we recall some preliminary results. The exact solution, the ill-posedness of the inverse problem and the conditional stability are also discussed in Sect. 2. In Sects. 3 and 4, we present a modified quasi-boundary value method and the fractional Landweber regularization method. The convergence estimate under an a priori assumption for the exact solution and the a posteriori regularization parameter choice rule are considered in here. In Sect. 5, we show a numerical example to illustrate the proposed method.

The solution and some notations 2.1 Some basic results
In this section, we recall some useful results. Let us consider the operator B on the domain D(-B) := H 1 0 ( ) ∩ H 2 ( ). We assume that -B has eigenvalues λ k with the corresponding eigenfunction w k ∈ D(-B).
Note 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · ≤ λ k ≤ · · · and λ k → ∞ as k → ∞. The negative Laplacian operator -on X is the most popular case of B. We have From [28], we see that a k ≥ Ck 2 d for C is a constant and k ∈ N. For all p ≥ 0, let us define the Hilbert scale spaces and fractional powers of B as follows: and   ([29, 30]) For 0 < k < 1, q > 0 and m ∈ N, we obtain (1k) m k q ≤ q q (m + 1) -q < q q m -q . Lemma 2.2 For p, γ , λ 1 be positive constant. Let G be a function defined by in which Z 1 = Z 1 (p, β) and Z 2 = Z 2 (p, β, λ 1 ). Then we have the estimate Proof See [23], Lemma 2.7.
By taking the inner product of the main equation of (1.1) with w k (x), we get the following equality: Thanks to [1], the Laplace transform of the Caputo-Fabrizio derivative has the following expression: . By λ k > 0 and α ∈ (0, 1), we obtain Thanks to [31], we get This implies that the solution of Problem (1.1) is given by the Fourier series By t = T in (2.7) and we get Proof First, we obtain where we noted that exp(-αλ k (T-s) 1+λ k (1-α) ) ≤ 1 for ∀t ≥ 0, and K 1,α (T) = T 1-α . Otherwise, we also get This implies that ).

Ill-posedness and stability estimates
For any v ∈ L 2 ( ), let R : L 2 ( ) → L 2 ( ) be the following operator: where the kernel κ(·, ·) satisfies Since our problem is of finding F which can be transform into It is easy to see that R : L 2 ( ) → L 2 ( ) is compact operator. It leads to the Problem (1.1) to be ill-posed. To show the ill-posedness of this problem, we propose an illustrative example.
then the corresponding source terms are , or lim m→∞ H m (·) L 2 ( ) → 0. Therefore, the problem (1.1) satisfying (1.2) is ill-posed in the sense of Hadamard.

Modified quasi-boundary value method
In this section, we present a regularization method to solve our problem. Now, we study a quasi-boundary value method to solve problem (1.1) and give two convergence estimates under an a priori parameter choice rule, respectively. Applying the modified quasiboundary value method given by [23], we propose the following regularized solution with exact data P: We have the solution with the observation data H ε , Afterwards, we will give an error estimation for F (·) -F ε n(ε) (·) L 2 ( ) and show the convergence rate under a suitable choice for the regularization parameter. If we choose the regularization parameter Proof Using the triangle inequality, we get First, we give an estimate for the first term F ε n(ε) (·) -F n(ε) (·) L 2 ( ) , then we get From (3.5), applying Lemma 2.3 and the inequality (a + b) ≥ 2 √ ab, ∀a, b ≥ 0, then we obtain On the other hand, we estimate the second term F n(ε) (·) -F (·) L 2 ( ) , for brevity, we put in which (3.10) From (3.10), applying the Lemma 2.2, by replacing β = K 2,α (T, λ 1 )Q 1 and γ = n(ε), we can find that (3.11) Combining (3.4) to (3.11), it gives (3.12) By choosing the regularization parameter n(ε) as follows: we conclude that (3.14) The proof is finished.

Fractional Landweber regularization method and convergence rate
In this section, we propose a fractional Landweber regularization method to solve the illposed problem (1.1) satisfying (1.2). The convergence rates for the regularized solution under two parameter choice rules are also considered. From [32], RF = P is equivalent to F = I -cR * R F + cR * P, for any c > 0, where 0 < c < R -2 and R * is the adjoint operator of R.
Applying the fractional Landweber method given by [26], we propose the regularized solution with exact data H as follows: We have the solution with the observation data noised H ε as follows: is the fractional order, and n > 0 is the iterative step and is a regularization parameter. Here, we note that when χ = 1, the fractional Landweber method becomes a standard Landweber regularization.

A priori parameter choice rule and convergence estimate
Let us choose n := n(ε) such that F ε n,χ (·) -F (·) L 2 ( ) → 0 as ε → 0. Using an a priori regularization parameter choice rule, we propose the convergent rate for the fractional Landweber regularized solution F ε n,χ to the exact solution F . In order to give an error estimate, we assume that F D ((-B)

If we choose the regularization parameter
Proof From the triangle inequality, we obtain Firstly, we give an estimate for the first term F ε n,χ (·) -F n,χ (·) L 2 ( ) . Applying Lemma 4.1, then we get Next, we estimate the second term, Because χ ∈ ( 1 2 , 1], we thus get

Lemma 2.3 now shows that
Hence, we see that Choose the regularization parameter This ends the proof.

A posteriori parameter choice rule and convergence estimate
Now, based on Morozov's discrepancy principle [33], we consider the choice of the a posterior regularization. Let us choose the regularization parameter n such that RF ε n,χ -P ε L 2 ( ) ≤ με ≤ RF ε n-1,χ -P ε L 2 ( ) , (4.6) where P ε L 2 ( ) ≥ με > 0. Choosing μ > 1 and the bound for n is given and depends on ε and M.
The absolute error estimation E α between the exact source function F n,θ and the regularized source function F ε n,χ is as follows: The results of this example are shown in Table 1. In Fig. 4, we present the graph of the exact and regularized source functions. From the error table and the figures above, we can see that the smaller the ε, the better the computed approximation. In particular, the regularized source function will gradually approach the exact source function as ε tends to zero.  The exact source function F n,θ and the regularized source function F ε n,χ for x ∈ (0, π )

Conclusions
In this paper, we study the source function for the Caputo-Fabrizio time fractional diffusion equation with the new kernel. We show that the backward problem for this equation is ill-posed (in the sense of Hadamard). Then the fractional Landweber and the modified quasi-boundary value methods are used to deal with this inverse problem and the regularized solution is also obtained. We also give the convergence estimates between the regularized solution and the exact solution by using the a priori regularization parameter choice rule and the a posteriori parameter choice rule. Finally, we give a numerical example to illustrate the proposed method.