Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data

In this study, our focus is on obtaining an estimated solution for the nonlinear fractional time diffusion equation. Specifically, we have utilized the Riemann Liouville fractional derivative. Additionally, we have concerned Gaussian white noise in the input data. As we are aware, this problem is considered ill-posed according to Hadamard’s definition. To tackle this problem, we have proposed a regularized solution and demonstrated the convergence between the mild solution and the regularized solution


Introduction
In this study, we investigate the nonlinear time-fractional diffusion equations subject to Dirichlet boundary conditions and the terminal value condition The domain Ω ⊂ R d (d ≥ 1) is a bounded open set with a sufficiently smooth boundary ∂Ω, and T > 0 is the terminal time.The symbol D α t represents the Riemann Liouville derivative of order α (0 < α < 1), which will be defined later.Recently, many scientists have become increasingly interested in the problems of fractional diffusion equations.Fractional differential equations have practical applications in modeling anomalous diffusion phenomena in various scientific fields, such as physics, chemistry, engineering, and more [1,2].There are several definitions of fractional derivatives, including Caputo, Riemann-Liouville, Caputo-Riesz, and others, which are nonlocal operators in contrast to the local operators of integer-order derivatives.Depending on the specific application and their experience, researchers may use a certain type of derivative.One of the main differences between fractional derivatives and classical derivatives is the non-local property of fractional derivatives.This property reflects the fact that the change at a specific location in the environment is affected by the state of the entire region.
The main objective of this article is to propose a regularized solution that can approximate the solution of ( 1)- (2).It is crucial to mention that our problem of reverse time is considered ill-posed according to the Hadamard definition.Hence, a regularization method is required to recover an accurate approximation.Assuming that the final value h is observed as ξ ϵ , it is well-known that observations are susceptible to random errors that arise due to the limitations of the measuring device (measurement errors).Consequently, it is common to encounter data that is perturbed or noisy.This work focuses on situations where such perturbations manifest in the form of additive stochastic white noise. (3) Let's examine a situation where ϵ indicates the amplitude of the noise, and W(x) represents a process of Gaussian white noise.Additionally, suppose that the observations given in (3) can not be accurately measured but can only be discretized and therefore observed in a sampled format.
The direct problem of time-fractional diffusion equations featuring various types of fractional derivatives has been widely investigated in recent years.Inverse problems for timefractional diffusion equations seek to retrieve initial data, source function, diffusion coefficient, and other parameters through additional data.However, such problems have received little attention recently [3][4][5].As far as we know, no previous studies have focused on (1)-(2) concerning random noise as depicted in (4).
The structure of this paper is structured as follows.Section 2 presents some preliminary materials.An example of Ill-Posed is provided in Section 3.1.The primary results of this paper are presented in Section 3.3 which is followed by an analysis of the convergent estimates between a mild solution and a regularized solution under some prior assumptions on the exact solution.
2 Preliminaries Definition 2.1 (Podlubny [6]).Let α > 0 and β ∈ R, the Mittag-Leffler function is Definition 2.2 (Podlubny [6]).Let α be a real number such that α ∈ (0, 1).The Riemann-Liouville derivative of fractional order α with power-law of function u(t) is defined as where Γ(•) is the gamma function.The Riemann-Liouville definition conforms to all the mathematical principles within the domain of fractional calculus, especially when utilizing Laplace transform.
Property 2.1.The Laplace transform of the Riemann-Liouville fractional integral operator of order α ∈ (0, 1) can be obtained in the form of: where u is the Laplace transform of u, Property 2.2 (Podlubny [6]).Given reals number α ∈ (0, 1) and λ > 0. There always exists positive constants B 1 (α) and B 2 (α) such that Moreover, the following identity is applicable for t > 0

Some necessary spaces
We have Ω ⊂ R d be an open bounded domain and let ⟨•, •⟩ be the inner product of L 2 (Ω).
Then, there exists an orthonormal basis which is a Hilbert space and endowed with the norm ∥u∥ 2 Definition 2.3 (Zou and Wang [7]).Given a measure probability space (D, F , µ) and Banach space B. The function u : (0, T) → B measurable such that ess sup t∈(0,T) The space of B-valued predictable processes u such that

Integral form of the solution
We denote the inner product The first equation of system ( 1) is transformed into Applying the properties of Laplace transformations Use simple transformations we have and Laplace transform of the convolution integral Taking inverse Laplace transforms the two sides of (10) we get and the terminal condition u p (T) = ξ p give us Thus Definition 2.4 (Mild solution).Suppose that Problem (1)-( 2) has a solution u ∈ L 2 (0, T; Ω), then the mild solution is presented as Lemma 2.6.Let ξϵ ∈ L 2 (Ω) and suppose that ξ ∈ H µ (Ω), µ > 0. The error is estimated as following Where n(ϵ) := n depends on ϵ and satisfies that lim ϵ→0 + n(ϵ) = +∞.
Proof.We begin our argument by recognizing that Since ∼ N(0, 1) implies that EW 2 p = 1.We, therefore, acquire the desired result.
We have statistics estimate for h and f that are obeyed in the models The source Problem (1) associated with the terminal function ξϵ (x) and the source f ϵ (x, t) has the mild solution We consider at the initial time, we have For convenience, we denote Since Lemma 2.2, there exist a constant B 1 such that Where EM 2 1 = ϵ 2 and using H ölder inequality, we have The function f defined as ( 14), one obtain The inequality (16) will become By choosing ϵ = 1/ 3 √ λ n , and according to Lemma 2.6, we have Consequently, we can deduce that the problem is ill-posed in the sense of Hadamard.

Approximate solution
The regularized solution for our problem is constructed in this section using the truncation Fourier series method.To obtain the regularized solution, we first introduce a truncation operator.
N := N(ϵ) and n := n(ϵ) respectively are the regularized parameter and the sample size.Similarly, there is also a mild solution to this problem Definition 3.1 (Regularized solution).Given real numbers µ, ν > 0 and the integrals 0 as the regularized solutin of Problem ( 1)-( 2).

Theorem 3.2 (Existence and uniqueness).
Let B 1 and B 2 be a constant such that Suppose further that G is global Lipschitz, If B 1 T(B 2 + 1)L < 1 and Problem (1)-( 2) has a solution u ∈ L ∞ (0, T; L 2 (Ω)) then the integral equation (19) existence and uniqueness solution.
Proof theorem 3.2.To establish the existence and uniqueness of the solution to the integral equation, we make use of the Banach fixed point theorem.First, we reformulate the operator Φ(u)(x, t) in the form: For any u ϵ 1 , u ϵ 2 ∈ Z(0, T; H), the Parseval identity leads to First, note that there exists a constant B 1 and and using the H ölder inequality, we arrive at One can confirm that Since B 1 T 2 (B 2 + 1)L, using Banach fixed point theorem, we show that Φ(u ϵ ) = u ϵ has a unique solution.

Estimation of errors
The regularization parameter N and the sample size n are chosen such that Proof theorem 3.3.We have First, we realize that from Lemma (2.2), there exists a constant B 2 such that To facilitate the reader, we divided proof into 4 steps: Step 1.We have Lemma 2.6 leads to Step 2. We also have Since Lipschitz property of the source, we have Step 3. Similarly, we have Note that E  .
By rearranging, we get the result of the theorem.