Numerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Barycentric Jacobi spectral (BJS) method is employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with a fourth-order exponential time differencing predictor corrector. Comparative numerical results for the values of options are presented. The proposed method is very elegant from the computational point of view. Numerical computations for a wide variety of problems, show that the present method offers better accuracy and efficiency in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.