In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasi-linearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger's-Fisher equation, the Fitzhugh-Nagumo equation and the Burger's-Huxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.

In this work, we explore the application of a novel multi-domain spectral collocation method for solving general non-linear singular initial value differential equations of the Lane-Emden type. The proposed solution approach is a simple iterative approach that does not employ linearisation of the differential equations. Spectral collocation is used to discretise the iterative scheme to form matrix equations that are solved over a sequence of non-overlapping sub-intervals of the domain. Continuity conditions are used to advance the solution across the non-overlapping sub-intervals. Different LaneEmden equations that have been reported in the literature have been used for numerical experimentation. The results indicate that the method is very effective in solving LaneEmden type equations. Computational error analysis is presented to demonstrate the fast convergence and high accuracy of the method of solution.