Simultaneously reconstructing potentials and internal sources for fractional Schr\"odinger equations

TL;DR

This paper addresses the inverse problem of simultaneously reconstructing potentials and internal sources in fractional Schrödinger equations, providing theoretical uniqueness results and a numerical method with experimental validation.

Contribution

It introduces a novel inverse problem framework for fractional Schrödinger equations, proving uniqueness and developing a conjugate gradient-based numerical approach.

Findings

Proved uniqueness for reconstructing potentials and sources from boundary measurements.

Developed a Tikhonov regularization-based numerical method.

Numerical experiments demonstrate the effectiveness of the proposed approach.

Abstract

The inverse problems about fractional Calder\'on problem and fractional Schr\"odinger equations are of interest in the study of mathematics. In this paper, we propose the inverse problem to simultaneously reconstruct potentials and sources for fractional Schr\"odinger equations with internal source terms. We show the uniqueness for reconstructing the two terms under measurements from two different nonhomogeneous boundary conditions. By introducing the variational Tikhonov regularization functional, numerical method based on conjugate gradient method(CGM) is provided to realize this inverse problem. Numerical experiments are given to gauge the performance of the numerical method.