Numerical Study of Partial Differential Equations by an Adaptive Diffusion Wavelet Method

Abstract

In the paper, an adaptive diffusion wavelet method (ADWM) is developed to solve partial differential equations (PDEs) with different boundary conditions. We construct diffusion wavelets using an approximation of second-order differential operators. We test the squeezing error behavior corresponding to different parameters involved in the constructed diffusion wavelet, using two test functions. The diffusion wavelet is used to construct adaptive grid arrangements and the fast computation of the diffusion operator’s dual power. The constructed adaptive grid arrangements and dual power of diffusion operators are more effective for finding the numerical solutions of PDEs. Test problems verify the accuracy and efficiency of the proposed method. For each test problem, the processing time taken by ADWM is compared with the existing methods like the fast diffusion wavelet method (FDWM) and finite difference method (FDM). The proposed method (ADWM) takes lesser processing time than FDWM and FDM. We verify the convergence of the given method.