Inverse Source Problem for Sobolev Equation with Fractional Laplacian

Abstract

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 β ⟶ 1 + .

On the Convergence Result of the Fractional Pseudoparabolic Equation

“…(M 1 ): from [15], we get that if j ≥ 1, then we fnd that j 2s − j 2 ≤ C θ j s+θ (1 − s) θ , for any θ > 0 and C θ is the constant which depends on θ. For any μ > 0, in view of the inequality |e…”

“…Te main techniques and methods frequently used are the modifed Lavrentiev regularization method and the Fourier truncated regularization method. To the fractional pseudoparabolic equation, sometimes, the inverse source problem is also discussed [15][16][17].…”

“…Recently, in [15], the authors focused on the source problem for the pseudoparabolic equation with fractional Laplacian. In this article, they also investigated the convergence of the source function when the fractional order tends to 1 − .…”

“…Tere are not many results devoted to the convergence of mild solution v(x, t) when the fractional order tends to 1 − . Motivated by the results in [15], we decided to study the fractional pseudoparabolic equation (1) and investigate the convergence of the state function v(x, t) when the fractional order tends to 1 − .…”