Fast diffusion wavelet method for partial differential equations

Abstract

A fast diffusion wavelet method for solving partial differential equations (PDEs) is developed. Classes of operators which can be used for the construction of diffusion wavelet include approximation of second order differential operators. The efficiency of the method is that the same diffusion operator is used for the construction of diffusion wavelet as well as for approximation of second order differential operator. As a part of the wavelet method the behavior of compression error with respect to different parameters involved in the construction of diffusion wavelet is tested for two test functions. Furthermore, the diffusion wavelet is used for the compression of operators and hence for the fast and efficient computing of the dyadic powers of the diffusion operator T which are required for solving the PDE. We have considered PDEs with Dirichlet and periodic boundary conditions on one, two, and three dimensional domains. For each test problem, the CPU time taken by the fast diffusion wavelet method is compared with the CPU time taken by the finite difference method. We have also verified the convergence of the proposed method.