Shannon Wavelet Collocation Method Applied To Partial Differential Equation

The wavelet analysis theory is one of the important branches of mathematics. In nature, many physical phenomena can be described by differential equations, but the general differential equations have no exact solutions, therefore it is significant to find the numerical solutions for the equations. In this thesis, how to solve one dimensional and two dimensional differential equations was studied by the wavelet collocation method.On the one hand, how to solve a kind of heat conduction equation by using the wavelet analysis theory and the collocation methods was discussed. Firstly, the Shannon scale basis function was selected as the basis of the differential equations. Secondly, the scale function was proved satisfy orthonormal property interpolate property and reproducing property. According to one dimensional multi-resolution analysis theory, the approximate formulas of solutions for differential equations were given. The spatial domain was separated by the wavelet collocation method, and the system of ordinary differential equations with respect to time was built. Thirdly, the system of differential equations was solved by the Runge-kutta method. The computation was simplified and with higher precision.On the other hand, the two dimensional differential equations were solved by using two dimensional multi-resolution analysis theory. Firstly, a two dimensional scale function was chosen and was applied to the wavelet collocation method. Secondly, the two dimensional scaling basis function was proved satisfy the interpolate property. And according to two dimensional multi-resolution analysis theory, the approximate formulas of solutions for the two dimensional differential equations were given. The two dimensional partial...