Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method

Le Dinh Long,Nguyen Hoang Luc,Yong Zhou,And Can Nguyen

doi:10.3390/math7100934

Le Dinh Long, Nguyen Hoang Luc + Show 2 more

Open Access

https://doi.org/10.3390/math7100934

Copy DOI

Abstract

In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them.

Highlights

Let Ω be a bounded domain in Rd with sufficiently smooth boundary ∂Ω, β ∈ (1, 2)
We propose a Tikhonov regularization method and give two convergence estimates under an a priori assumption for the exact solution and two regularization parameter choice rules: Section 4 and Section 5
In the following Theorem, we provide the uniqueness property of the inverse source problem

Summary

Tikhonov Method

Le Dinh Long 1,† , Nguyen Hoang Luc 2,† , Yong Zhou 3,4,† , and Can Nguyen 5, *,†. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam.

Introduction

Preliminary Results

A Posteriori Parameter Choice

Simulation Example

Conclusions