Research Of Methods For Nonlinear Equations In Reproducing Kernel Spaces

Many natural phenomena can be modeled by linear and nonlinear differentialequations. These equations as mathematical models have important applications inphysics, biology, control science and so on. In general, the analysis of these phe-nomena can be reduced to solving differential equations. It is difficult to obtain theanalytical representation of exact solutions. Therefore, the research of an efficient nu-merical method for differential equations is of theoretical and applicable significanceand how to solve these significant equations becomes more and more important.In this thesis, several numerical methods of solving some classes of nonlineardifferential equations are presented by using reproducing kernel theory.This thesis introduces the application background and history of reproducingkernel spaces and runs back over the development of reproducing kernel spaces.Moreover, we give the concrete representation of the reproducing kernel spaces inwhich there are corresponding reproducing kernel functions in the every chapter.Some numerical tests are given in every chapter and the numerical results verify thevalidity of conclusion.Firstly, the representation of reproducing kernel is simplied by improving defini-tion of inner product in the original reproducing kernel spaces. The new representationof reproducing kernel with the form of polynomial is obtained, which has a simplerrepresentation. Using it can reduce the size of computation in solving problems. Also,it can improve the accuracy of computation by preventing accumulating error of cal-culation.Next, two large-range convergent iterative sequences are constructed. The ap-proximate solutions can be obtained by truncating series. If the solutions of equationsare not unique, the particular solutions satisfying addictive conditions can be givenwith the second method. Those iterative methods are suitable for solving general non-linear equations. Using the first method, we solve Burgers equation in Chapter 2 andthe singular nonlinear initial value problems in Chapter 3. Using the second method,we solve the generalized Burgers equation in Chapter 4.Finally, first order fully nonlinear differential equations are solved by raising dimension in Chapter 5. In terms of the reproducing property of reproducing kernelspaces, we convert them to partial differential equations. After homogenizing theboundary 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 anunknown solution. Then by using least squares algorithm, we can give the solutionsof operator equations.Reproducing kernel theory has important application in model description, curvefitting, function estimation, probability and statistics. In this dissertation, based on re-producing kernel theory, a few analytical techniques are presented to solve nonlineardifferential equations such as Burgers equation, generalized Burgers equation, fullynonlinear differential equations and singular nonlinear initial value problems. The so-lution obtained using the present methods has its own advantages. The main character-istic features of the reproducing kernel methods are, in contrast with other numericalmethods like Runge-Kutta and linear multistep methods, that global approximationcan be established on the whole interval and the convergence is uniform. Moreover, italso permits the study of the behaviour of derivatives of the approximate solution.