Wavelet-Galerkin Method For Differential Equation

The wavelet analysis theory and reproducing kernel theory are important branch 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 very important to find the numerical solution of the equation. In this thesis, it is the main problem to investigate that the numerical solution and solution space of the different differential equations by making use of the theory of the Wavelet-Galerkin method and the reproducing kernel Hilbert space, respectly. The main works are as following: On the one hand, using the wavelet analysis theory and the Galerkin methods to solve a kind of second-order variable coefficient differential equation is discussed. Firstly, the Littlewood-paley wavelet basis, that is an orthonormal basis of the space L2 (R), is selected as a basis in the Galerkin methods. Through the "folded"transform, the "folded"orthonormal wavelet basis for the space L2 [0,1] is got. Secondly, it is proved that the "folded"orthonormal wavelet basis satisfies the boundary condition of the equation. In the end, it is given that the Galerkin numerical solution of the equation in the wavelet subspace, and the numerical solution of the equation is also got. Wavelet-Galerkin method is applied to solve the differential equations, and this provides not only the theoretic bases for solving the differential equations, but also a new method to extand applications for the wavelets analysis theory . On the other hand,the reproducing kernel space of the wave equation is investigated in this thesis. Firstly, the reproducing kernel of the solution space of the second-order wave equation is gotten. Secondly, it is proved that the solution space of second-order wave equations can form the reproducing kernel Hilbert space H 1 [0,+∞). Then we prove the isometrical identity and inversion formula of the solution of the wave equation. Besides, the solution of another wave equation is discussed by the same way. The solution space is another RKHS H (C). It is very beautiful that the difference of two kernels is constant. It not only explores a novel application of the reproducing kernel Hilbert space theory, but also provides a new view on solving the differential equation.