The Müntz spectral element method for the numerical solutions of regularized long wave equation

This paper introduces new bases derived from Müntz polynomials for application in the spectral element method. Within this framework, we demonstrate that the mass and stiffness matrices exhibit matrix with three non-zero diagonals and diagonal structures, respectively. The Müntz spectral element method is employed to numerically solve the regularized long wave equations. Initially, the temporal derivative is discretized using the forward difference operator, leading to the development of a semi-discrete scheme in the time domain. Subsequently, a fully discrete formulation is obtained by discretizing the spatial variable using the Müntz spectral element method. The stability of the semi-discrete time scheme is analyzed and confirmed through Fourier analysis. Additionally, the convergence of the fully discrete scheme is rigorously established. Finally, by solving several numerical examples and comparing the results with outcomes from existing methods, the accuracy and efficiency of the Müntz spectral element method are effectively demonstrated.