On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.


Introduction
In recent decades, the study of noninteger diffusion equations has received great attention from mathematicians around the world. These models have many applications in various types of research fields, for example, thermal diffusion in fractal domains [1] and protein dynamics [2], finance [3], systems biology [4], physics [5] and medicine [6], and besides, there are also some references as follows [7][8][9][10][11], and [12]. In this work, we consider the following problem: (1.1) Here Caputo fractional derivative D β t is defined as follows: t 0 u s (r, s) (ts) β ds, 0<β < 1, (1.2) and the source function G(r, t) ∈ L ∞ ([0, R]; r 2 ), the final data f (r) ∈ L 2 ([0, R], r 2 ) are given. Note that when the fractional order β is equal to 1, the fractional derivative D β t u(r, t) is equal to the first-order derivative du dt (see in [13]), and thus problem (1.1) reproduces the classical diffusion problem. In practice, the input data (f , G) is noisy by the observed data f ε , G ε which satisfy ff L 2 ([0.R];r 2 ) + G -G L ∞ (0,T;L 2 ([0,R];r 2 )) ≤ . (1.3) Our problem is called inverse problem and its solution is not stable. This property is called ill-posed in the sense of Hadamard. In other words, easier to understand, if is small, it will lead to large errors for the corresponding solution if using an unapproximate model for observed data f ε , G ε . The question mentioned in this paper is: Find an approximation method for the solution of the problem with noisy input data f ε , G ε . Before discussing the main results, we would like to outline a few previous papers that mentioned problem (1.1).
Our novel point in this paper is to replace the final condition (1.4) with the nonlocal condition ξ 1 u(r, T) + ξ 2 T 0 u(r, t) dt = f (r) as introduced in the last condition of our problem. This condition is proposed in the paper by Dokuchaev [29]. Very recently, Tuan and coauthors used this condition to solve some nonlocal problem, for example [30][31][32], and [33]. Motivated by this above reason, in this paper, we apply the fractional Tikhonov method to solve problem (1.1). To the best of authors' knowledge, there are not any results concerning problem (1.1). Our paper investigates problem (1.1), and the main results of this work are as follows: • We give the stability and the regularity of the mild solution.
• We show the ill-posedness and the conditional stability of solution in L 2 ([0, R]; r 2 ).
• We propose a regularized method and prove the convergence rate under a priori parameter choice rule and a posteriori parameter choice rule. Let us say that in an analytical sense, our problem seems to be more complicated than the models studied before.
This paper is organized as follows. Section 2 gives some preliminaries that are needed throughout the paper. In Sect. 3, we show the sought solution of problem (1.1), and an example describes the ill-posedness of the problem. In Sect. 4, we study the fractional Tikhonov method to solve problem (1.1) and show the convergence rate under a priori parameter choice rule and a posteriori parameter choice rule. Finally, we add the conclusion for this paper.
Proof This lemma provision can be found in [20].

Lemma 2.4
For any j ≥ 1, we have the following estimate: Proof From Lemma 2.2, we need to show that This leads to Next, due to the fact, we also get which implies that In this section, we need the solution of the direct problem of (1.1) (2.14) From [20], by using the Fourier expansion, we know that , and j 0 (y) denotes the 0th order spherical Bessel functions of the first kind. Besides, we know that {ψ j (r)} ∞ j=1 from an orthonormal basis in L 2 ([0, T], r 2 ).

The mild solution of problem (1.1)
Theorem 3.1 Let f ∈ L 2 ([0, R]; r 2 ) and G ∈ L ∞ (0, T; L 2 ([0, R]; r)). Let us further assume that Then problem (1.1) has a unique solution u given as follows: Then we get the following regularity: ) and the regularity result holds Proof From (2.15) and using the nonlocal condition in problem (1.1), we obtain that By integrating both sides from 0 to T for equation (2.15), we immediately have the following equality: From two observations (3.5) and (3.6), we get the following equality: Our next aim is to express the formula of the function in terms of two input data f and G.
In view of the nonlocal condition as in the last condition of problem (1.1) we find the following identity for the Fourier coefficient of the function : , Therefore, by taking Fourier series for the term u j (t), the formula of the mild solution to problem (1.1) can be given by . (3.9) Using the inequality (a Now, we give the regularity result of a mild solution. First of all, from Lemma 2.4, it gives (3.11) Next, let us evaluate for A i (·, t) 2 L 2 ([0,T];r 2 ) , i = 1, 4, one by one.

The fractional Tikhonov method
In this section, we apply the fractional Tikhonov method given by Morigi [35]. From now on, we denote we propose the following regularized solution with exact data (f , G): However, if the measured data {f , G} are noised by {f , G }, then we get where δ ∈ ( 1 2 ; 1] and α > 0. Noting that when δ = 1, the fractional Tikhonov method becomes a standard Tikhonov regularization. β,j (π, R, T) with Z depending on δ.
Proof The proof of lemma can be found in [36].

An a priori parameter choice rule
Let us consider the operator for ν ∈ L 2 ([0, R]; r 2 ) and 0 ≤ t ≤ T. By applying the fractional Tikhonov method, we can see that (Ts) β G(·, s), ψ j ds ψ j (r). (5.7) By choosing the regularization parameter α, the following theorem gives that the choice α is valid by using suitable assumptions. In order to give error estimate, let us assume that H s ([0,R];r 2 ) ≤ C for any s > 0, where C is a positive constant. Before going to the main theorem, we have auxiliary lemmas as follows.

An a posteriori parameter choice rule
In this subsection, considering the choice of the a posteriori regularization parameter in Morozov's discrepancy principle, [37] we choose the regularization parameter α such that
Lemma 5.5 From (5.23), if we can find that ζ is satisfied, then we have the estimate of α as follows: (5.26) We have the estimate ζ through two steps as follows, one by one.
Step 1: Estimate of X 1 , to do this, we recall , from expressions (5.2) and (5.4), and we have Step 2: Estimate of X 2 , using again the a priori bound condition of , we obtain α, i fs ≥ 1.

(5.29)
From the analytics assessment on the side, we get This yields     This ends the proof of this theorem.

Conclusion
In this paper, we focus on the spherically symmetric backward time-fractional diffusion equation with the nonlocal integral condition. By using some properties of the Mittag-Leffler function, we show two results as follows. First of all, we show the properties of the well-posedness and regularity of the mild solution to this problem. Next, we present that our problem is ill-posed. In addition, we construct a regularized solution and present the convergence rate between the regularized and exact solutions by the fractional Tikhonov method under a priori parameter choice rule and under a posteriori parameter choice rule.