Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

Shilan Othman Hussein,M S Hussein

doi:10.24996/ijs.2021.62.1.22

Shilan Othman Hussein, M S Hussein

Open Access

https://doi.org/10.24996/ijs.2021.62.1.22

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Abstract

In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.

Highlights

1 Introduction An unknown force function in the wave equation can be experienced in many engineering applications dealing with wave, wind, seismic, explosion, or noise excitations [2, 4]
The objective of this research is to provide the numerical solution for an inverse force problem for the nonhomogeneous hyperbolic equation, by considering the initial condition with boundary condition
In order to extend the range of applicability, a different boundary condition has been applied in this study

Summary

Introduction

An unknown force function in the wave equation can be experienced in many engineering applications dealing with wave, wind, seismic, explosion, or noise excitations [2, 4]. It can be found in physical problems as well; for instance, the vibrations of a spring or membrane, acoustic scattering, etc. The objective of this research is to provide the numerical solution for an inverse force problem for the nonhomogeneous hyperbolic equation, by considering the initial condition with boundary condition. In a previous study [3], we used the finite difference method (FDM) to numerically discretize the wave equation with the method of separating the variables. In order to extend the range of applicability, a different boundary condition has been applied in this study

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