A wavelet method for solving a class of nonlinear boundary value problems

We develop an approximation scheme for a function defined on a bounded interval by combining techniques of boundary extension and Coiflet-type wavelet expansion. Such a modified wavelet approximation allows each expansion coefficient being explicitly expressed by a single-point sampling of the function, and allows boundary values and derivatives of the bounded function to be embedded in the modified wavelet basis. By incorporating this approximation scheme into the conventional Galerkin method, the interpolating property makes the solution of boundary value problems with strong nonlinearity to be very effective and accurate. As an example, we have applied the proposed method to the solution of the Bratu-type equations. Results demonstrate a much better accuracy than most methods developed so far. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the equations.