Numerical solution for a general class of nonlocal nonlinear wave equations
This work provides numerical analysis and simulation tools for researchers studying nonlocal wave phenomena in elastic media.
The paper develops numerical methods for a class of nonlocal nonlinear wave equations, proving convergence of a semidiscrete Fourier spectral scheme and using a fully-discrete scheme with Fourier pseudo-spectral method and 4th order Runge-Kutta to study kernel effects. Petviashvili's iteration is employed to generate solitary wave solutions.
A class of nonlocal nonlinear wave equation arises from the modeling of a one dimensional motion in a nonlinearly, nonlocally elastic medium. The equation involves a kernel function with nonnegative Fourier transform. We discretize the equation by using Fourier spectral method in space and we prove the convergence of the semidiscrete scheme. We then use a fully-discrete scheme, that couples Fourier pseudo-spectral method in space and 4th order Runge-Kutta in time, to observe the effect of the kernel function on solutions. To generate solitary wave solutions numerically, we use the Petviashvili's iteration method.