**1. Introduction**

Most practical problems which model systems in nature lead to nonlinear partial differential equations (PDEs). This is evident in the fields of chemistry, physics, biology, mathematics and engineering. Many assumptions have been made to make some nonlinear PDEs solvable. It has been reported that a vast number of nonlinear PDEs that are encountered in these fields are difficult to solve analytically [1]. The investigation of solutions of such nonlinear PDEs has then been of key interest to many researchers due to their potential applications and more effort has been devoted to search for better and more efficient solution methods for these nonlinear models [2, 3].

The nonlinear PDEs that are solved in this study include the generalized Burger's-Fisher equation, the generalized Burger's-Huxley equation and the Fitzhugh-Nagumo equation. The generalized Burger's-Fisher equation appears in many applications such as shock wave formation, fluid mechanics, turbulence, traffic flows, gas dynamics, heat conduction and sound waves via viscous medium among other fields of applied science [4–6]. The generalized Burger's-Huxley equation models the interaction between reaction mechanisms, diffusion transports and convection effects [7–11]. The Fitzhugh-Nagumo equation arises in genetics, biology, and heat and mass transfer [12, 13].

A number of methods have been applied to solve the nonlinear PDEs such as spectral collocation method [7, 8], Adomian decomposition method [9], homotopy perturbation method [14] and the variational iteration method [4]. The spectral methods have been reported to be strikingly successful if the problem has a smooth solution and falls into various categories, namely Galerkin, Tau and collocation-based methods [15], and therefore, recent advances in the development of numerical methods for solving nonlinear PDEs has focused spectral-based approaches as they require a few grid points to give very accurate results and take less computation time. The spectral collocation-based methods are used often, chiefly because they offer the simplest treatment of boundary conditions. A newly developed spectral collocation method for solving nonlinear PDEs is the bivariate spectral quasi-linearization method (QLM) [16]. This method approximates the solution of the PDE using a bivariate Lagrange interpolation polynomial [17]. It applies quasi-linearization method of Bellman and Kalaba [18] to simplify the nonlinear PDE which is then discretized using spectral collocation on both time and space variables. The method has successfully been used to solve problems defined over shorter time intervals [16]. However, it has been observed that when this method is applied to solve problems defined over large-time intervals, there is no guarantee that the resulting approximate solution will be accurate [16].

In this study, we describe the multidomain bivariate spectral collocation method (MDBSCM) to solutions of nonlinear parabolic PDEs defined over large-time intervals. The MDBSCM is based on decomposing the given domain of approximation in the time variable into smaller subintervals and then solving the PDE independently in each subinterval using the bivariate spectral collocation method. The multidomain approach has been applied to solve nonlinear ordinary differential equations that model chaotic systems described as 1st order systems of equations [19–21]. In this study the same idea is extended to solutions of nonlinear parabolic PDEs. In the description of the method, the algorithm is kept as simple as possible, while retaining the heart of generality to cover many applications. The extent of the discussion of multidomain approach in this study is limited to nonoverlapping subintervals only.
