Analytical and Numerical Solution for Inverse Coefficient Non-Linear Hyperbolic Equation with Periodic Boundary Condition

Abstract

This study investigates an inverse problem of unknown time-dependent coefficients in the one-dimensional nonlinear hyperbolic equation with periodic boundary conditions. The generalized Fourier method is employed to construct the Fourier coefficient for the solutions, and using iteration method convergence, the uniqueness and stability of the solution to the nonlinear problem are proved. Additionally, in order to solve the inverse problem numerically Finite Difference Method (FDM) with Gauss Seidel Iteration process is proposed. Two different implicit finite difference schemes are applied, namely, implicit and Crank-Nicolson. A numerical example is presented to illustrate the method's behavior. Both numerical predictions are close to experimental results, however, estimation of implicit scheme has lower true error and relative true error than Crank-Nicolson scheme.