The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations

Abstract

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

An accurate numerical method for solving the linear fractional Klein–Gordon equation

“…A combination of Eqs. (14), (15) and (13) leads to the desired result and completes the proof of the theorem.…”

“…Most FDEs do not have exact solutions, so approximate and numerical techniques [2][3][4][5][6][7][8][9][10] must be used. Recently, several numerical methods to solve FDEs have been given, such as variational iteration method [8], homotopy perturbation method [11,12], Adomian decomposition method [13], homotopy analysis method [7], collocation method [14][15][16][17][18][19][20][21][22][23][24] and finite difference method (FDM) [25,26]. We describe some necessary definitions and mathematical preliminaries of the fractional calculus theory required for our subsequent development.…”

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

“…In , the mixed Laguerre‐Legendre spectral method and pseudospectral method are proposed for solving the fluid folw problems in an infinite channel. Khader proposed an efficient numerical method for the fractional delay differential equation using the generalized Laguerre polynomials. Rahmoune presented a spectral collocation method using the scaled Laguerre functions for solving the Fredholm integral equations of the second kind on the half‐line.…”

Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

“…Theorem 1 [23] Let ( ) u x be approximated by the generalized Laguerre polynomials as (11) and also suppose 0 ν > then, its Caputo fractional derivative can be written in the following form…”