Adaptive techniques for solving chaotic system of parabolic-type

Time-dependent partial differential equations of parabolic type are known to have a lot of applications in biology, mechanics, epidemiology and control processes. Despite the usefulness of this class of differential equations, the numerical approach to its solution, especially in high dimensions, is still poorly understood. Since the nature of reaction-diffusion problems permit the use of different methods in space and time, two important approximation schemes which are based on the spectral and barycentric interpolation collocation techniques are adopted in conjunction with the third-order exponential time-differencing Runge-Kutta method to advance in time. The accuracy of the method is tested by considering a number of time-dependent reaction-diffusion problems that are still of current and recurring interests in one and high dimensions.