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.

Placement Inverse Source Problem under Partition Hyperbolic Equation

“…̅̅̅̅̅ then where is a Gaussian normal distribution. The mean of zero was applied, while the standard deviation of was given by [ ] [1][2][3][4]11], and represents the percentage of noise. Zeroth-order Tikhonov regularization was applied after the oscillation being in .…”

“…Because of having the same values of vx(0,t) (i.e. (11)) that appeared in [3], the figure is not represented in this research. Furthermore, from we can obtain the numerical solution .…”

“…In [1,2], the boundary element method (BEM) was combined with a separation variable, where the left end Dirichlet or Neumann conditions, as additional measurements, were considered. Sequentially, in [3], the BEM was switched to the finite element method (FEM). Meanwhile, the papers [4,12] were based on the right end Dirichlet or Neumann boundary conditions for extra conditions, whereas [5] focused on unusual boundary conditions with some theorems of existence and uniqueness, which were also provided in [5][6][7][8].…”

“…represent the initial displacement and velocity, respectively. Furthermore, for solving the inverse problem above, we need to have the following extra measurements in the role of the flux tension of the string at the left end [1][2][3][4]11]:…”

“…Numerically, we next deal with the solutions ( 5)- (10). Firstly, the FDM is applied on the first part of equation ( 5) with the initial condition (6) and boundary condition (8), which is similar to the equations in [3] and thus not rewritten here. Then, in the form of , we have…”