A document by Manzoor Hussain. Click on the document to view its contents.

Reaction-diffusion equations play important role in problems related to population dynamics, developmental biology, and phase-transition. For such equations, we propose a strong-form local (multiquadric) RBF method that gives sparse well-conditioned differentiation matrices with reduced computational cost and memory storage; thus, avoids solving dense ill-conditioned system matrices, an inherited drawback of strong-form global RBF methods if compared to the limitations of mesh-based methods. After spatial discretization of the time-dependent PDE problem by sparse differentiation matrices, the resultant system of ODEs can be stably integrated in time via a high-order and high-quality ODE solver. The proposed method is tested on two-dimensional Fisher-type equations for its geometric flexibility, accuracy, and efficiency. Unlike the mesh-based methods, the proposed local method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. Some recommendations are also made for further efficient implementation of the proposed local multiquadric method.

This article proposes a simple yet efficient/powerful iterative scheme to obtain approximate solutions to a class of nonlinear parabolic partial differential equations. The approximation method is based on the residual error function and the multiple power series expansion. Error estimates are derived and convergence of the proposed method is established. Several numerical examples are then considered to validate the theoretical results and highlight the computational ease of the developed residual power series method on multi-dimensional nonlinear Burgers’ equations. It is observed that the proposed approximation method captures shock wave propagation and steep gradient formation very well, and delivers more accurate results than the contemporary methods in higher dimensions.