Splitting the One-Dimensional Wave Equation. Part I: Solving by Finite-Difference Method and Separation Variables

Abstract

In this study, an unknown force function dependent on the space in the wave equation is investigated. Numerically wave equation splitting in two parts, part one using the finite-difference method (FDM). Part two using separating variables method. This is the continuation and changing technique for solving inverse problem part in (1,2). Instead, the boundary element method (BEM) in (1,2), the finite-difference method (FDM) has applied. Boundary data are in the role of overdetermination data. The second part of the problem is inverse and ill-posed, since small errors in the extra boundary data cause errors in the force solution. Zeroth order of Tikhonov regularization, and several parameters of regularization are employed to decrease errors for output force solution. It is obvious from figures how error affects the results and zeroth order stables the solution.

Keywords

• Finite difference method, Inverse force problem, Regularization, Separation variables method, Wave equation.