A HAM-based wavelet approach for nonlinear partial differential equations: Two dimensional Bratu problem as an application

In this paper, a new analytic approach, namely the wavelet homotopy analysis method (wHAM), is developed for boundary value problems (BVPs) governed by nonlinear partial differential equations (PDEs), which successfully combines the homotopy analysis method (HAM) and the generalized Coiflet-type wavelet. To improve the computational efficiency and accuracy, a section-based wavelet approximation for partial derivatives is proposed. The two-dimensional Bratu equation is used as an example to illustrate its basic ideas of the wHAM. Numerical results verify the validity as well as great advantages of the wHAM. Compared with the normal HAM, the wHAM possesses not only larger freedom to choose the auxiliary linear operator, but also better convergence property and higher computational efficiency. In addition, the iteration approach can greatly accelerate convergence.