Jizeng Wang

Lei Zhang, Jizeng Wang, Xiaojing Liu et al.

A high order wavelet integral collocation method (WICM) is developed for general nonlinear boundary value problems in physics. This method is established based on Coiflet approximation of multiple integrals of interval bounded functions combined with an accurate and adjustable boundary extension technique. The convergence order of this approximation has been proven to be N as long as the Coiflet with N-1 vanishing moment is adopted, which can be any positive even integers. Before the conventional collocation method is applied to the general problems, the original differential equation is changed into its equivalent form by denoting derivatives of the unknown function as new functions and constructing relations between the low and high order derivatives. For the linear cases, error analysis has proven that the proposed WICM is order N, and condition numbers of relevant matrices are almost independent of the number of collocation points. Numerical examples of a wide range of nonlinear differential equations in physics demonstrate that accuracy of the proposed WICM is even greater than N, and most interestingly, such accuracy is independent of the order of the differential equation to be solved. Comparison to existing numerical methods further justifies the accuracy and efficiency of the proposed method.

Jizeng Wang, Lei Zhang, You-He Zhou

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.