  A high precision, and space time fully decoupled, wavelet formulation
numerical method is developed for a class of nonlinear initial boundary value
problems. This method is established based on a proposed Coiflet based
approximation scheme with an adjustable high order for a square integrable
function over a bounded interval, which allows expansion coefficients to be
explicitly expressed by function values at a series of single points. In
applying the solution method, the nonlinear initial boundary value problems are
first spatially discretized into a nonlinear initial value problem by combining
the proposed wavelet approximation scheme and the conventional Galerkin method.
A novel high order step by step time integrating approach is then developed for
the resulting nonlinear initial value problem using the same function
approximation scheme based on wavelet theory. The solution method is shown to
have Nth-order accuracy, as long as the Coiflet with [0, 3N-1] compact support
is adopted, where N can be any positive even number. In addition, the stability
property of the method is analyzed, and the stable domain is determined.
Numerical examples are considered to justify both the accuracy and efficiency
of the method. Results show that the proposed solution method has better
accuracy and efficiency than most other methods.
