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Simultaneously reconstructing potentials and internal sources for fractional Schrödinger equations

Xinyan Li

Abstract

The inverse problems about fractional Calderón problem and fractional Schrödinger 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 Schrödinger 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.

Simultaneously reconstructing potentials and internal sources for fractional Schrödinger equations

Abstract

The inverse problems about fractional Calderón problem and fractional Schrödinger 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 Schrödinger 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.
Paper Structure (5 sections, 1 theorem, 30 equations, 2 figures)

This paper contains 5 sections, 1 theorem, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Uniqueness of simultaneous reconstruction
  3. Variational Tikhonov regularization method
  4. Numerical experiments
  5. Conclusions

Key Result

Theorem 2.1

Let $u$ satisfy fseis, $\Omega\subset\mathbb{R}^n,n\geq1$ be bounded open set, $0<s<1$, and $W_1,W_2\subset\Omega_e$ be open sets satisfying $\overline{\Omega}\cap \overline{W_1},\overline{\Omega}\cap \overline{W_2}=\emptyset$. Assume one of the following conditions holds: and further, Assumption ass2 holds. Then for given $f,\tilde{f}\in\tilde{H}^s(W_1)$ being the boundary condition of respecti

Figures (2)

  • Figure 1: The numerical results for Example \ref{['ex1si']} with different noise levels, (a) the reconstruction of $q$; (b) the reconstruction of $g$.
  • Figure 2: The numerical results for Example \ref{['ex2si']} with different noise levels, (a) the reconstruction of $q$; (b) the reconstruction of $g$.

Theorems & Definitions (6)

  • Remark 1.2
  • Theorem 2.1
  • proof
  • Remark 3.1
  • Example 4.1
  • Example 4.2