Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Abstract

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear timedelayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply (− 1) nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters (, > −1), and the resulting equations together with the two-point boundary conditions constitute a system of (− 1) ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of (− 1) ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms.

Approximate Solutions of Fisher's Type Equations with Variable Coefficients

“…In [50], Bhrawy proposed an efficient Jacobi-Gauss-Lobatto collocation method for approximating the solution of the generalized Fitzhugh-Nagumo equation in which the Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. Moreover, the Jacobi spectral collocation methods are used to solve some problems in mathematical physics, (see, for instance, [51][52][53]).…”