Numerical Solution of Time-Dependent Partial Differential Equations Via Haar Wavelet

Abstract

Some numerical techniques are proposed to approximate time-dependent linear and nonlinear partial differential equations including vibration equation, nonlinear parabolic equations (including semi-linear parabolic, quasi-linear parabolic etc.) and nonlinear Kortweg-de Vries equations of higher order. The proposed methods are based on sim ple algorithms. The finite difference formula is used for discretization of time deriva tives and Haar wavelet integration formula is used to discretize the space derivatives. The presented techniques have converted partial differential equations into algebraic systems of equations (linear or nonlinear), linear systems are solved by Gauss elim ination technique and nonlinear systems are solved by Broyden’s technique that are implemented using computer software MATLAB. The accuracy and efficiency of the given techniques are tested upon several test problems and maximum absolute errors and point wise errors are calculated for different number of grid points. The experimental rate of convergence is also calculated for different collocation points that approaches to two and validate the theoretical results. The computed results are compared with the analytic solutions, that indicate the robustness of the pro posed methods by providing accurate results in a few number of collocation points. The accuracy of the numerical results may be increased by increasing the number of grid points or decreasing the size of time step. Furthermore, MAPLE package and MATHEMATICA are also used for the computational purposes.