Study On Wavelet Precise Integral Method For Partial Differential Equation

The object of the thesis is to develop a new method, wavelet precise integral method, to solve a problem of 2-dimention diffusion partial differential equation (PDE) with fractal boundary condition. The new method is developed by combining the wavelet collocation method with high precise integral. Space problem and time problem in the 2-dimention defuse PDE could be treated separately by using the new method, namely, Shannon interval wavelet collocation for the space problem, fictitious domain method for the irregularity boundary, and time domain precise integral method for the time problem.The thesis consists of three parts as follows:1, Quasi Shannon interval wavelet and quasi Shannon interval wavelet collocation method were constructed based on quasi Shannon wavelet and the property of wavelet. The difference between quasi Shannon interval wavelet method and quasi Shannon wavelet method in the numerical approximation was compared by a concrete example. It can be send that the quasi Shannon wavelet method could erase border effect and increase computational precision more efficiently than the quasi Shannon wavelet method.2, A new method to solve nonlinear dynamic system, self adapting precise integral method, was proposed based on the 'precise integral method (PIM)' proposed by Prof. Wanxie Zhong . Extrapolation was quoted in the proposed new method so that it could self adapt the time step. Due to the compatibility of PIM and extrapolation method in the collocating of time domain results, the amount of computation would not increases significantly.3,The properties of fractal geometry, such as fractional dimension, self-similar and the effect on the PDE, were studied. In order to solve the diffusion problem inside the object with fractal boundary condition, a 2-dimension diffusion PDE was constructed. A Von Koch snow-flake space was constructed as the boundary. Under the first initial value-boundary condition, the constructed PDE was solved by using the wavelet collocation method combined with fictitious domain in the space domain and PIM in the time domain. The numerical solution indicates that the new method is effective for the parabolic PDE with irregularity boundary condition. The proposed method is worthy being further explored in the further.