The Research Of Algorithms For Some Nonlinear Problems

Many natural phenomena can be modeled by linear and nonlinear differential equations. These equations as mathematical models have important applications in physics, biology, control science and so on. In general, the analysis of these phenomena can reduce to solving differential equations. It is difficult to obtain the analytical representation of exact solutions, therefore, the research of an efficient numerical method for differential equations is of theoretical and applicable significance and how to solve these significant equations becomes more and more important.In this thesis, several numerical methods of solving some classes of linear and nonlinear differential equations are presented by using reproducing kernel theory.This thesis introduces application background and history of reproducing kernel spaces and runs back over the development of reproducing kernel spaces. Moreover, we give the concrete representation of the reproducing kernel spaces in which there are the corresponding reproducing kernel functions in the every chapter. Some numerical tests are given in every chapter and numerical results verify the validity of conclusion.Firstly, the representation of reproducing kernel is simplied by improving definition of inner product in the original reproducing kernel spaces. The orthonormal basis can be obtained from the representation of reproducing kernel functions. Through Fourier series expanding on this basis, the solutions of fifth-order linear equations can be obtained in Chapter 2. In the mean time, an approximate sequence that is implemented easily on computers is constructed to solve singular linear problems in Chapter 3. Its construction is simple and it can approximate effectively the function with large number of nodes. The error of approximate sequence is decreasing in the sense of Sobolev norm and the uniform convergence can be guaranteed.Next, a convergent iterative sequence is constructed. The approximate solutions can be obtained by truncating series. If the solutions of equations are not unique, the particular solutions satisfying addictive conditions can be given. This iterative method is suitable for solving general nonlinear equations. Using this method, we solve singular nonlinear systems and nonlinear infinite-delay-differential with proportion delay respectively in Chapter 4 and Chapter 5. Finally, second order nonlinear partial differential equations are solved in Chapter 6. In terms of the reproducing property of reproducing kernel spaces, we convert them to operator equations .After homogenizing the boundary conditions , we put them into two-dimensional reproducing kernel spaces. By using this method in this thesis, we can obtain the presentation of solutions with a unknown. Then by using least squares algorithm, we can give the solutions of operator equations .In summary, the algorithms in this thesis have the following advantages: First, they are very simple; Second, the rate of convergence is fast; Third, the derivatives of approximate solutions can also approximate the derivatives of exact solutions well respectively; Fourth, the algorithms are continuous approximation, that is , the value of arbitrary point can be obtained, which is different from numerical algorithms.