Wavelet solution of a class of two-dimensional nonlinear boundary value problems

Abstract

By combining techniques of boundary extension and Coiflet-type wavelet expansion, an approximation scheme for a function defined on a two-dimensional bounded space is proposed. In this Wavelet approximation, each expansion coefficient can be directly obtained by a single-point sampling of the function. And the boundary values and derivatives of the bounded function can be embedded in the modified wavelet basis. Based on this approximation scheme, a modified wavelet Galerkin method is developed for solving two-dimensional nonlinear boundary value problems, in which the interpolating property makes the solution of such strong nonlinear problems very effective and accurate. As an example, we have applied the proposed method to the solution of two-dimensional Bratu-like equations. Results demonstrate an excellent numerical accuracy, a 2.5-order convergence rate for the present wavelet method, and a very good capability in dealing with the nonlinear boundary value problems with multiple solution branches. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the two-dimensional equations.