The dependency of the analytical and numerical solution on the $\varepsilon$ parameter in hyperbolic and pseudo-hyperbolic problems with inverse coefficients

The aim of this study is to analyze the behavior of $\varepsilon$ on the solution of an inverse coefficient nonlinear pseudo-hyperbolic equation $w_{tt}-\varepsilon w_{xxtt}-\varepsilon w_{xx}=\theta (t)f(x,t,w)$ with periodic boundary conditions. We also consider the inverse coefficient problem $w_{tt}-w_{xx}=\theta (t)f(x,t,w).$ The solution function of nonlinear pseudo-hyperbolic equation is found to be convergent to the solution function of nonlinear hyperbolic equation, when $ \varepsilon \rightarrow 0$ is proved. The Fourier method was used to illustrate the theoretically relation between the inverse problems while the Finite Difference Method was used numerically. In order to get more accurate numerical solution higher precision schemes have been applied in implicit finite difference equation. The cases where $\varepsilon =0$ and $\varepsilon \neq 0$ have been solved analytically and numerically, and compared each other.