Regularization method for the problem of determining the source function using integral conditions

In this article, we deal with the inverse problem of identifying the unknown source of the time-fractional diffusion equation in a cylinder equation by A fractional Landweber method. This problem is ill-posed. Therefore, the regularization is required. The main result of this article is the error between the sought solution and its regularized under the selection of a priori parameter choice rule.


Introduction
According to the history of mathematical research, it has been found that the standard diffusion equation has been used to represent the particle motion Gaussian process. To describe anomalous diffusion phenomena, the classical derivative will be replaced with a non-integer derivative. Therefore, it leads to great applications of differential equations with non-integer derivatives. Fractional derivatives and fractional calculus was also considered by many scientists because of applications in potential theory, physics, electrochemistry, viscoelasticity, biomedicine, control theory, and signal processing, see e.g. [1,2] and the references therein. Nigmatullin [3] first applied the fractional diffusion equation to describe diffusion in a medium shaped fractal. Metzler and Klafter [4] gave a proof that a fractional diffusion equation is possible governs a non-Markovian propagation process that has a memory. Among many different interesting topics about the fractional diffusion equation, some types of inverse problems in this genre attract the community interested in research. T. Wei and her group [5,6,7,8] investigated some regularization methods for homogeneous backward problem. Y. Hang and his coauthors [9] used fractional Landweber method for solving backward time-fractional diffusion problem. The diffusion process inverse source problem is intended to detect the source function of the physical field from some indirect measurement (such as last time information or boundary measurement). As we all know, the problem of determining the source function has attracted a lot of mathematicians interested in research because of its applications in practice. Some interesting works on this topic can be found in some previous paper, for example [10]- [38]. In general, the inverse source problem is often ill-posed in the sense of Hadamard. In this work, we focus on the following equation in an axis-symmetric cylinder where the Caputo fractional derivative ∂ β ∂t β is defined as follows: a and Φ and satisfy with the following condition on the final time data The main purpose of this paper is to apply a fractional Landweber method to regularized our inverse source problem. We will demonstrate that the regularized solution will converge on the sought solution. There are two challenges that we need to overcome. The first difficulty is that the problem is considered in the domain of axis-symmetric cylinder making the assessment techniques complicated. The second difficulty is the presence of integral conditions that make estimates of errors cumbersome. It can be said that our result is one of the first results about the source function for the problem (1.1)-(1.4). The outline of the paper is given as follows: In Section 2, we give some preliminary theoretical results. Ill-posed analysis and conditional stability are obtained in Section 3. In Section 4, we propose the Fractional Landweber regularization method and give a convergence estimate under an a-priori regularization parameter choice rule.

Statement of the problem
We introduced the Lesbesgue space associated with the measure rdr, i.e which is a Hilbert space with the scalar product u, ν r = Ω u(r, z)ν(r, z)rdrdz, and norm is given by v L 2 r (Ω) = Ω ν 2 (r, z)rdrdz 1 2 · Throughout this paper, for the convenience of writing, [19]) For any constant γ and r ∈ R, the Mittag-Leffler function is defined as: where γ > 0 and α ∈ R are arbitrary constant.
Proof. This proof can be found at [14].
Lemma 2.4. [13] For λ mn > 0, β > 0, and positive integer j ∈ N, we have: Proof. Please see the proof in [20]. Proof. The solution of problem (1.1) is as follows: where J 0 (z) and J 1 (z) denote the 0th order and 1st order Bessel function, and ζ m are the sequence of solution of the equation J 0 (z) = 0 which satisfy Defining then it is easy to check that the eigenfunctions Ψ m,n (r, z) m,n≥1 from an orthonormal basis in L 2 r (Ω). Using the eigenfunctions Ψ m,n (r, z) as a basic, formula (3.11) can be written for a shorter as follows This implies that (3.17), the latter equality implies that The mild solution is given by Proof. A linear operator P : L 2 r (Ω) → L 2 r (Ω) as follows.
and Ψ(r, z) = +∞ m,n=1 Due to (r, z) = (z, r), we know P is self-adjoint operator. Next, we are going to prove its compactness. Defining the finite rank operators P M,N as follows From P M,N f and Pf , using the inequality (a + b) 2 ≤ 2(a 2 + b 2 ), a, b ≥ 0, we have: (3.23) Therefore, P M,N f − Pf L 2 r (Ω) in the sense of operator norm in L(L 2 r (Ω); L 2 r (Ω)) as M, N → ∞. Also, P is a compact operator. Next, the SVDs for the linear self-adjoint compact operator P are and corresponding eigenvectors is Ψ m,n which is known as an orthonormal basis in L 2 r (Ω). Corresponding eigenvectors is Ψ mn which is known as an orthonormal basis in L 2 r (Ω). From (3.21), the inverse source problem we introduced above can be formulated as an operator equation Pf (r, z) = Ξ(r, z) and by Kirsch [30]. Assume that u 0,m † ,n † = 0 and g m † ,n † is noised data by and g m † ,n † we have estimate By the Lemma 2.3 and the Lemma 2.5, we know that From (3.25) and (3.26), therefore in the computation of (3.25), the small data error can be amplified arbitrarily much by the factor V θ 1 ,θ 2 m,n (β, Φ) −2 which increase without bound, so recovering the source f (r, z) from a measured data g (r, z) is ill-posed. Hence, regularization for this article needs to be considered.

Conditional stability of source term f
Proof. From (3.20), applying the Hölder inequality, we know (3.27) and this inequality leads to Combining (3.27) and (3.28), we get
Proof. This Lemma 4.1 can be found in [9].
, we have the following estimates Proof. The proof can be found in [9].