An inverse problem of reconstructing the time-dependent coefficient in a one-dimensional hyperbolic equation

M J Huntul,Dumitru Baleanu,Muhammad Abbas

doi:10.1186/s13662-021-03608-1

M J Huntul, Dumitru Baleanu + Show 1 more

Open Access

https://doi.org/10.1186/s13662-021-03608-1

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Abstract

In this paper, for the first time the inverse problem of reconstructing the time-dependent potential (TDP) and displacement distribution in the hyperbolic problem with periodic boundary conditions (BCs) and nonlocal initial supplemented by over-determination measurement is numerically investigated. Though the inverse problem under consideration is ill-posed by being unstable to noise in the input data, it has a unique solution. The Crank–Nicolson-finite difference method (CN-FDM) along with the Tikhonov regularization (TR) is applied for calculating an accurate and stable numerical solution. The programming language MATLAB built-in lsqnonlin is used to solve the obtained nonlinear minimization problem. The simulated noisy input data can be inverted by both analytical and numerically simulated. The obtained results show that they are accurate and stable. The stability analysis is performed by using Fourier series.

Highlights

The reconstruction of the unknown coefficients in the inverse problem of the hyperbolic problem has various applications in science and engineering
In the last few decades, various authors have reconstructed the unknown coefficients in the inverse problem of the hyperbolic wave equations
Bellassoued and Yamamoto [3] studied the inverse problem to determine the unknown term in the hyperbolic model with variable terms

Summary

Introduction

The reconstruction of the unknown coefficients in the inverse problem of the hyperbolic problem has various applications in science and engineering. In the last few decades, various authors have reconstructed the unknown coefficients in the inverse problem of the hyperbolic wave equations. The inverse problems of the wave equations for recovering time-dependent potential from over-determination integral condition have been investigated by Tekin [29] while the time-dependent force function has been studied by Hussein and Lesnic [19].

Results

Conclusion