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

Sami Ullah Khan,Ishtiaq Ali

doi:10.1177/1687814020922113

Sami Ullah Khan, Ishtiaq Ali

Open Access

https://doi.org/10.1177/1687814020922113

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Abstract

The numerical techniques are regarded as the backbone of modern research. In literature, the exact solution of time delay differential models are hardly achievable or impossible. Therefore, numerical techniques are the only way to find their solution. In this article, a novel numerical technique known as Legendre spectral collocation method is used for the approximate solution of time delay differential system. Legendre spectral collocation method and their properties are applied to determined the general procedure for solving time delay differential system with detail error and convergence analysis. The method first convert the proposed system to a system of ordinary differential equations and then apply the Legendre polynomials to solve the resultant system efficiently. Finally, some numerical test problems are given to confirm the efficiency of the method and were compared with other available numerical schemes in the literature.

Highlights

Delay differential equations (DDEs) have been applied to model some real phenomena in a wide range of chemical, physical, engineering and biological systems and their networks
1.581Â10À8 1.217Â10À8 9.305Â10À8 1.082Â10À8 1.039Â10À8 1.313Â10À8 (a). It has been the aim of this article to apply an efficient and reliable numerical technique for the approximate solution to a system of time DDEs
We apply a numerical technique based on Legendre spectral collocation method

Summary

Introduction

Delay differential equations (DDEs) have been applied to model some real phenomena in a wide range of chemical, physical, engineering and biological systems and their networks. We perform the convergence analysis of the LSCM for the numerical solution to the system of time DDE given in equation (1). To show that the function y should satisfy the global Lipschitz condition, for this, there exists L1; L2 > 0 such that for all v1; v2 2 RNþ1 and t; s 2 R for all X1; X2 2 CðE;RÞ and assume P 1⁄4 bðQL1 þ NL2Þ where Q 1⁄4 kKk and N 1⁄4 kHk. Here we will assume the case P < 1; and the operator v given in equation (10) must satisfies equation (12); from the stead of BFP theorem, a unique fixed point y 2 CðE; RNþ1Þ exists for v, which is a unique solution of equation (1). For the convergence of the proposed method, we obtained the error for different values

Current Method

Conclusion