Fourier Spectral Methods for Solving Some Nonlinear Partial Differential Equations

The spectral collocation or pseudospectral (PS) methods (Fourier transform methods) combined with temporal discretization techniques to numerically compute solutions of some partial differential equations (PDEs). In this paper, we solve the Korteweg-de Vries (KdV) equation using a Fourier spectral collocation method to discretize the space variable, leap frog and classical fourth-order Runge-Kutta scheme (RK4) for time dependence. Also, Boussinesq equation is solving by a Fourier spectral collocation method to discretize the space variable, finite difference and classical fourth-order RungeKutta scheme (RK4) for time dependence. Our implementation employs the Fast Fourier Transform (FFT) algorithm.