Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation

Abstract

Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.

Applications of Legendre spectral collocation method for solving system of time delay differential equations

“…These polynomials are either even or odd, depends on even or odd orders of N. The first few Legendre polynomials are given below. 25…”

“…Legendre spectral collocation method is also applied for the numerical solution of delay differential and stochastic DDEs. 25 Also, for solving system of nonlinear Fredholm integral equations of second kind, stochastic Volterra integro differential equations and stochastic SIR (Susceptible Infected Recoverable) models are solved by Legendre spectral method. [26][27][28] In this research work, we propose a new computational technique based on Legendre spectral collocation method (LSCM) for solution of system of time DDEs.…”

Numerical analysis of stochastic SIR model by Legendre spectral collocation method

“…The Legendre polynomials have been used by many researchers for approximate solution of differential and integral equations. 28,29 This article is structured as follows: Legendre polynomials are reviewed in section ''Legendre polynomials.'' In section ''Description of LSCM,'' a brief description of LSCM is given to solve equation (2).…”

“…The Legendre polynomials have been used by many researchers for approximate solution of differential and integral equations. 28,29…”

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

“…The explicit solutions of such a type of differential equations are very rare and one must adopt a numerical technique to solve such problems. Recently, for the simulation of complex or smooth physical phenomena, spectral methods have emerged as a very powerful and efficient numerical technique, which are a well-known class of numerical methods for the solutions of various differential equations due to the spectral rate of convergence [10][11][12]. These methods for the numerical solutions of stochastic fractional differential equations were most recently used by Raffaele D' Ambrosio et al [13][14][15], while Jacobi polynomials for the numerical solution of time fractional diffusion systems was used in [16,17].…”