Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic

Abstract

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply two models to the COVID-19 epidemic in China and one for the Predator-Prey model in mathematical ecology. Keywords Nonlinear Volterra delay integro-differential equations • Collocation method • Taylor polynomials • Epidemic mathematical model • Corona virus Mathematics Subject Classification (2010) 45G15 • 34K28 • 45D05 • 45L05 • 65L60

Taylor collocation method for solving two‐dimensional partial Volterra integro‐differential equations

“…The main aim of this paper is to extend the Taylor collocation method of one-dimensional Volterra integral and integro-differential in earlier studies [8,9,24,25] to solve two-dimensional PVIDEs of type (1.1).…”

“…Methods based on Taylor analysis found their way into approximation theory and have been used to find numerical solutions for different types of equations, such as integral equations [35], delay integral equations [8], integro-differential equations [20,24,25], delay integro-differential equations [9], and partial integro-differential equations [7,12,14,18,36]. For example, Gurbuz and Sezer [18] used a matrix collocation method based on the Laguerre and Taylor polynomials to solve certain nonlinear PVIDEs.…”

“…Partial integro‐differential equations arise in problems of applied sciences and engineering to model dynamical systems; they can be found in financial mathematics [1–29], biological models [17], fluid dynamics [5], and many other areas. Partial Volterra integro‐differential equations (PVIDEs) have many different types, such as elliptic [30], hyperbolic [12], and parabolic [21] in one‐ or multidimensional, that is why numerous numerical schemes have been well‐evaluated by researchers to approximate the solution of these equations.…”

Taylor collocation method for high-order neutral delay Volterra integro-differential equations

“…Equation (1.1) includes many important kinds of equations (see for example [4,6,12,13,15,18,25,29]) and this method can be used to obtain numerical solutions of high order differential equations (for {k 1,v } k−1 v=0 = {k 2,v } k−1 v=0 = {M v } k−1 v=0 = 0), high order integro-differential equations (for {k 2,v } k−1 v=0 = {M v } k−1 v=0 = 0), high order delay differential equation (for {k 1,v } k−1 v=0 = {k 2,v } k−1 v=0 = 0). There are many existing numerical methods for solving Volterra integrodifferential equations, such as Legendre spectral collocation method [26], Runge-Kutta method [7], spectral method [27,28], Polynomial collocation method [8][9][10]23], Tau method [14], operational matrices [2], Homotopy perturbation method [16,24], Haar wavelet method [21], Taylor polynomial [20,22]. The aim of the present paper is to apply a direct collocation method based on the use of Taylor polynomials so that we generalize the Taylor collocation method for delay VIEs [3], first and second order delay IDEs [4,5], and high order VIDEs [19].…”

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

“…Jaroudi et al [ 35 ] studied a brain tumor growth model with reaction–diffusion equations and two three-dimensional different numerical simulations. Laib et al [ 36 ] developed a numerical model for general form of a system of nonlinear Volterra delay integro-differential equations and applied this model to novel coronavirus (COVID-19) epidemic in China, Spain, and Italy. He and Meng [ 37 ] investigated the lump and interaction solutions of generalized (3 + 1)-dimensional nonlinear wave propagation for fluid dynamics.…”