Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Abstract

The focus of this paper is on utilizing the spectral element method to find the numerical solution of the fractional Klein–Gordon equation. The algorithm employs interpolating scaling functions (ISFs) that meet specific properties and satisfy the multiresolution analysis. Using an orthonormal projection, the equation is mapped to the scaling spaces in this method. A matrix representation of the Caputo fractional derivative of ISFs is presented using matrices representing the fractional integral and derivative operators. Using this matrix, the spectral element method reduces the desired equation to a system of algebraic equations. To find the solution, the generalized minimal residual method (GMRES method) and Newton’s method are used in linear and nonlinear forms of this system, respectively. The method’s convergence is proven, and some illustrative examples confirm it. The method is characterized by its simplicity in implementation, high efficiency, and significant accuracy.