A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations

Abstract

This paper reports two successive spectral collocation methods, that enable easy and highly accuracy discretization, for 1 + 1 and 2 + 1 fractional percolation equations (FPEs). The first step depends mainly on the shifted Legendre Gauss–Lobatto collocation method for spatial discretization. An expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the FPE is investigated. In addition, the Legendre-Gauss–Lobatto quadrature rule is established to treat the boundary conditions. Thereby, the expansion coefficients are then determined by reducing the FPE with its boundary conditions to a system of ordinary differential equations for these coefficients. The second step is to propose the shifted Chebyshev Gauss–Radau collocation scheme, for temporal discretization, to reduce such a system to a system of algebraic equations, which is far easier to solve. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to the numerical solution of two-dimensional FPEs. An upper bound of the absolute error is obtained of the approximate solution for the two-dimensional case. Convergence properties of the method are discussed through numerical examples. Several numerical examples with comparisons are reported to highlight the high accuracy of the present method over other numerical techniques.