Reproducing Kernel Methods For Solving A Class Of Fourth-order Nonlinear Differential Equations

Many phenomena in nature can be described with nonlinear differential equations. The study of nonlinear differential equations plays a very important role in penetrating the inner structure, analyzing the relationship of things as well as interpreting various physical phenomena. It is well known that the bending of an elastic beam can be modeled with some fourth-order nonlinear ordinary differential equations. Thus, how to solve the differential equations with theoretical value and actual background becomes more and more important.The advantage of reproducing kernel methods for numerically solving nonlinear dif-ferential equations is that many complex boundary conditions such as period boundary conditions, integral boundary conditions can be imposed easily on the reproducing kernel spaces and the corresponding reproducing kernels can be calculated. Then the problems for determining solutions of differential equations can be transferred into an equivalent operator equation. At last the operator equations can be solved by combining the good properties of reproducing kernel spaces with the computational techniques.The aim of this dissertation is to explore iterative methods for obtaining the existence of solution and representation of approximate solutions to some fourth-order nonlinear ordinary differential equations. The proof of the main results is based on the reproducing kernel theory. The main contents are summarized as follows:At first, reproducing kernel theory is studied further. Some of important conclusions about bounded sets and compact sets have been given in the reproducing kernel spaces which possess reproducing kernels with the form of polynomial.Secondly, a method of solving some fourth-order nonlinear ordinary differential equations is proposed. A new iterative reproducing kernel method is designed success-fully to get the representation of approximate solutions to this kind of fourth-order non-linear ordinary differential equations. Since the iterative sequence is the approximation under the project, it is the best one. Moreover, the derivatives of the sequence are also convergent to the derivatives of the exact solution uniformly. It is worthwhile to point out that this new iterative method avoids the Gram-Schmidt orthogonalization process originally in the process of reproducing kernel iteration. Therefore, this can improve the accuracy of computation by preventing accumulating more errors of calculation.Thirdly, a numerical treatment for a nonlinear system of fourth-order and second-order differential equations is discussed. This new iterative method is generalized to solve the nonlinear system of ordinary differential equations in Hilbert space. A convergent iterative sequence is constructed in terms of an orthogonal projection and an orthogonal system of functions in Hilbert space. Consequently, the approximate solutions for the system of equations is provided. It should also be pointed out that this extended method can solve various nonlinear systems of ordinary differential equations.Finally, the existence proof of solution for a class of fourth-order nonlinear differen-tial equations with linear boundary conditions is obtained. This paper firstly applies the reproducing kernel method in the field of the existence of solution for differential equa-tions. The inner product in reproducing kernel space is redefined and another iterative method is presented by an inverse operator in the reproducing kernel space. The exis-tence of solution for this class of fourth-order nonlinear differential equations with linear boundary conditions is proved by employing contract mappings principle. In the mean-time, an iterative formula for solving equations is developed because the limit of iterative sequence is the exact solution. The advantage of the method is that not only the existence of solution is proved, but also the accuracy of approximate solutions obtained by using this iterative method is much higher.Two iterative reproducing kernel methods are presented in this dissertation, which are all convergent. This breaks the old model of only solving differential equations, not proving the existence of solution by the reproducing kernel method. Therefore, the work enriches the reproducing kernel theory and extends the applications of reproducing kernel theory.