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Inverse Source Problem for Sobolev Equation with Fractional Laplacian
In this paper, we are interested in the problem of determining the source function for the Sobolev equation with fractional Laplacian. This problem is ill-posed in the sense of Hadamard. In order to edit the instability instability of the solution, we applied the fractional Landweber method. In the theoretical analysis results, we show the error estimate between the exact solution and the regularized solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Finally, we investigate the convergence of the source function when fractional order .
Let be a bounded domain in with sufficiently smooth boundary . In this paper, we are interested to study the following pseudo-parabolic equation where is the diffusion coefficient, is the source function, and desribe the distribution of the temperature at position and time . The paramater is the fractional order of Laplacian operator with .
Pseudo-parabolic equations or called Sobolev equation describe describing various important physical phenomena, such as heat conduction involving two temperatures , homogeneous liquid permeability in fractured rock , unidirectional propagation of long waves in a nonlinearly dispersed medium , and its references.
Until now, the results on fractional pseudo-parabolic equations equation are not rich we can mention them in a few some few papers, for example, [1, 4–6]. From the fraction operator appearing in the main equation which is nonlocal, many scientists believe that it describes some physical phenomena more accurately than classical integrals differential equation. Properties of fractional operator have been described in detail in .
For equation (1) we usually divide it into three forms. (i)The first type is an initial value problem, i.e., determining when the initial value and the source function is known. The results in this category are vibrant and plentiful ([7, 8])(ii)The second type is terminal value problem, i.e., recovering the function from the terminal value data and the source function data . To the best of our knowledge, there are limited results for the terminal value problem. We can list some recent papers, for example, [9–13]. In general, the terminal value problem is an ill-posed problem; namely, a solution does do not exist, and if a solution exists, it does not depend continuously on the data. The results of the regularized method for this form were recently investigated by [14, 15](iii)The last type is inverse source problem, i.e., recovering the source function if we know the initial value data and the terminal data
The main purpose of this paper is to determine the source function with the split form when we know that
The question of determining the function when we know and will be studied carefully in this paper. It is surprising that the problem of determining the source function for the pseudo-parabolic equation has not been investigated before. We detail the objective of the problem. In practice, the given datais noisy by the observed data by level such that
Our main task here is to construct a regularized method which looking for the function and claims claim that in the appropriate norm. It can be claimed that our paper was one of the first works on the inverse source problem for the Sobolev equation.
In , Tuan-Long-Thinh used the Tikhonov regularization method to regularize regularized an inverse source problem for time fractional diffusion equation. They also introduced two methods, a priori and a posteriori parameter choice rules, to obtain the convergence estimate of the regularized methods. In , the authors studied the problem of finding the source distribution for the linear biparabolic equation when we have the final observation. Ma et al.  identified the unknown space-dependent source term in a time-fractional diffusion equation by applying the generalized and revised generalized Tikhonov regularization methods. There are many different regularized methods, and in this paper, we choose the fractional Landweber regularization method. The Landweber regularization method was first derived from  where the authors applied the filter regularization technique for solving a linear inverse problem. Up to now, the Landweber regularization method has been applied to solve many inverse problems, for example, [19–21] and references therein. This method is beneficial very useful for investigating for the linear ill-posed equation. Recently, Binh et al.  studied an inverse source problem for the Rayleigh–Stokes problem using the Tikhonov method.
For the reader’s convenience, we would like to outline the main results and novelties of the paper briefly: (i)The first goal of this paper is to provide the fractional Landweber method to solve this inverse space-dependent source problem for pseudo-parabolic equation. We give the ill-posedness of our inverse source problem and introduce the convergence rate of the fractional Landweber regularized solution. In addition, we obtain the convergence rate by using an apriori parameter choice rule and an a posteriori parameter choice rule. Looking back at the articles [19–21], we realize that the source functions in these papers do not depend on the time function. So, the computation is not complicated. Meanwhile, the source function of the current paper depends on the function which makes the calculation more cumbersome. The presence of (3) makes our problem more clearly complex complex than [19–21]. One point to note is that the method in the article  can be applied to our model, but we approach it differently, in a different way.(ii)The second interesting point in the paper is the investigation of the convergence of the source function when the order of derivative approaches 1. Comparing the difference between the source function of equation (1) with and the classical pseudo-parabolic equation will help us understand more information about problem (1).
The paper is organized as follows. Section 2 states some preliminary theoretical knowledge. In Section 3, we give the Fourier formula of the source function and also present the ill-posedness of our problem. The conditional stability for the source function source function is also discussed in the same section. Section 4 provides the fractional Landweber regularization method and states a convergence estimate under a priori assumption on the exact solution. The posteriori parameter choice rule is also shown in section 4. Finally, in Section 5, we prove the convergence of the source function in Hilbert scales space with the appropriate assumption of and .
2. Preliminary Results
Let us consider the operator on , and assume that the operator has the eigenvalues such that which approach as goes to , The corresponding eigenfunctions are denoted by . Now, let us define fractional powers of and its domain. For all , we define by the following operator:
The domain is the Banach space equipped with the norm
We introduce the following two lemmas, which are useful and helpful in the next proofs.
Lemma 1. Let such that where and are positive numbers. Let us assume that . Then, the following estimates are true:
Proof. Since , we infer that
Since , we infer that
Let us consider the following function:
The derivative of it is equal to
This implies that is an increasing function on . Therefore, we get that
It follows from (9) that
The proof of the Lemma 1 is completed.
Lemma 2. Let be positive constants such that . By choosing , and , we obtain
Proof. The proof is completed in , page 4.
3. Inverse Source Problem: Explicit Form and Ill-Posedness
Let us first give the explicit of Fourier form of the mild solution to problems (1) and (2). First, taking the inner product of both sides of (1) with , we find that and from the initial condition , we have that since ; we know that
Hence, the source function is defined as follows:
Let us prove the ill-posedness of inverse source problems (1) and (2). Logically, we will consider the source function problem as the problem of finding satisfying (18). From now on, we only treat the source term (18).
Theorem 3. The problem of determining that satisfies (18) is ill-posed in the sense of Hadamard.
Proof. We defined a linear operator as follows:
Due to , we know is a self-adjoint operator. Next, its compactness is explained as follows. Let us define the finite rank operators as follows:
By some simple calculations and using Lemma 1, we have
From (21), we have
Therefore, is a compact operator. The SVDs for the linear self-adjoint compact operator are and corresponding eigenvectors are which is an orthonormal basis in . Therefore, the inverse source problem we introduced above can be formulated as an operator equation where by is the numerator in formula (18), and by Kirsch, we can conclude that it is ill-posed. The final time data , by (18), the source term corresponding to is whereby is defined in formula (21). The input final data , by (18), the source term corresponding to is . We have error in norm between and
Then, the error in norm between and is estimated as follows:
Next, we consider stability of the inverse source problem.
Theorem 4. Ifsuch thatthen we get where .
4. A Fractional Landweber Method and Convergent Rate
In this section, we apply the fractional Landweber regularization method to solve the inverse source problem (1) and give a convergence estimate. The construction of this method and its iterative implementation are clarified in . We denote the fractional Landweber regularization solution with the observed data by
It is obvious to see that formulas (33) and (34) are more complicated. For simplicity, we put . Expressions (33) and (34) become where is called the fractional parameter and is a regularization parameter and . If , it is the classical Landweber method. Next, we have the following lemmas:
Lemma 5. For , let , we get where
Proof. Please see in .
Lemma 6. For , choosing then , by denoting , we have the following estimates:
Proof. The proof can be found in .
4.1. A Priori Parameter Choice Rule
Proof. By using the triangle inequality, we have
We receive as follows:
From (42), we get
Next, we have the estimation of
Combining (42) to (45), we derive that
Next, we give
The above estimate (49) implies that
From observation above, using Lemma 6, we conclude that
By substituting in the above expression, we deduce that where
The proof of Theorem 7 is completed.
4.2. A Posteriori Parameter Choice Rule
In order to obtain a posteriori convergence error estimate, we apply Morozov’s discrepancy principle, which is introduced in . Furthermore, we learn the analysis techniques from previous papers [19–21].
From here on, in this subsection, we need to assume further to further assume that is a natural number that greater than If , then the equation (56) exists in a unique solution.
Lemma 8. Set where . Then, we declare that (a) is a continuous function(b) as (c) as (d) is a strictly increasing function for
Lemma 9. Let us assume that (56) holds. Then, satisfies
Proof. From the definition of , , , we have Using the inequality , we derive that (Step 1)Due to and , from (59), the estimate of is as follows:(Step 2) can be bounded as follows:where Thanks for the two articles [24, 25], we get the following inequality: for . Combining (58) and (63), we deduce that This implies that
Proof. By the triangle inequality, we receive Firstly, we have Secondly, we find that In view of Hölder inequality, we follow from (69) that where