Numerical Method And Its Application Of Several Operator Equation In Reproducing Kernel Space

Generally, many complex equations are hard to find out analytic solution in practical problems, search of effective numerical method has become the main way and research tool to solve the many complex equations. Operator equation can put the complex equation in specific space, it is expressed as operator form, in which space different problems correspond to different operator equation, the study of form and structure for operator equation is particularly important.In this paper, the integrated use of reproducing kernel space and operator equation theory, the properties of the operator equation in reproducing kernel space system research. Introduction of operator equation require that only if complex equation which with various definite conditions be putted in a certain space, then can construct map in the space. Excellent numerical expression of reproducing kernel space embodied by reproducing kernel function can be arbitrary function and fixed variable through the inner product operation show reproducibility in space, namely the inner product into function value of one point. Then it get a continuous expression to most basic numerical computation, at the same time, discrete numerical problems are continuously expressed, calculate progress simplified, speed improved and also calculation accuracy can be guarantee,which make problems get best solution.This article is divided into five chapters. Mainly is to discuss the operator equation properties in reproducing kernel space and of and some differential equations solutions.This article achieved the method via simulation of calculation software Mathematica, and verified the feasibility of this method with numerical examples after compared with the result from other methods. This research work enriched and developed not only reproducing kernel theory but also operator equation theory. The main research contents of this paper summarized as follows:Firstly, basic concepts and properties of operator equation are discussed in the reproducing kernel space, analyzed approximate solution by the condition number, applied the spectral analysis theory to get some useful conclusions.Secondly, the corresponding reproducing kernel space is constructed based on the reproducing kernel theory several different types of specific problems, then find out the corresponding reproducing kernel function solution, use the unique role of reproducingkernel function in inner product, new numerical method is proposed by operator equation as a medium. Advantage of this method is that it can easily deal with the boundary conditions and ensures continuity and uniform convergence of the numerical solution.Thirdly, three numerical method for equation of elastic beam non-resonant conditionsclass of nonlinear fourth-order differential equations is proposed. namely operator spectrum method and reproducing kernel- Adomian decomposition method.Operator spectrum method advantage through discuss the parameters of the corresponding eigenvalue problem of spectrum structure for problem solution, reproducing kernel- Adomian decomposition method based on reproducing kernel method and advantage of Adomian decomposition method, ensure faster convergence speed.Finally, For a class of differential equations with variable coefficient, good properties of reproducing kernel space are applied, starting from the family of reproducing kernel space structure a new reproducing kernel direct space, introduce inner product and norm,the problem using the projection operator approximation for inner product operation, the representation of the exact solution is given in the form of series, truncated series get approximate solution, and the error of the approximate solution is monotonically decreasing with the increasing of nodes.