A Numerical Method Of Several Nonlocal Boundary Value Problems For Differential Equation

Nonlocal boundary value problems are important problem in di?erential equations,which is applied extensively to scientific research and engineering fields. Considering nu-merical calculation, the appropriate and reliable accuracy numerical solution is dependenton numerical methods. Thus it is important to obtain algorithm which possesses excellentproperties such as high accuracy, strong stability, good convergence and little labor of cal-culation. Good local nature of reproducing kernel in calculation such that it has becomean important function approximation tool. Many researchers made the valuable work bythis tool in numerical calculation of di?erential equations, integral equations, operatorequations and so on.This dissertation aims at finding reproducing kernel numerical approximate method-s for solving nonlocal boundary value problems of di?erential equations. Its advantageis that complex nonlocal boundary conditions of di?erential equations can be containedeasily in reproducing kernel spaces, which removes the in?uence of complex nonlocalboundary conditions to solving equations, furthermore, solve di?erential equations satis-fying the boundary conditions by combining the good properties of reproducing kernelfunction and computational techniques in reproducing kernel space. The main resultsobtained in this thesis are summarized as follows:One-dimensional and two-dimensional reproducing kernel space with complex non-local boundary value conditions are constructed, the calculative method and expression ofreproducing kernel function are presented. This makes that it is possible to solve complexnonlocal boundary value problems in reproducing kernel space, and extends the range ofthe reproducing kernel theory.Reproducing kernel numerical approximate method is provided for solving nonlocalboundary value problems of nonlinear Du?ng equation. Using constructive technique,existence of solution is proved, iterative sequence of approximation is given, moreover,the approximation and its derivatives converge uniformly to the exact solution and itsderivatives. By constructing the corresponding reproducing kernel space, this approachcan be extended to weakly singular two point boundary value problems, two point bound-ary value problems of di?erential equation and systems. Numerical experiments showthat the numerical techniques is a very e?ective and can be applied widely. Reproducing kernel numerical approximate method for solving nonlocal boundaryvalue problems of partial di?erential equation is studied. For linear parabolic nonlocalboundary value problems, the exact solution is given in the form of series and an approx-imate solution is obtained by truncating the series. The partial derivatives of approxima-tion converge uniformly to the partial derivatives of exact solution. For nonlocal boundaryvalue problems of nonlinear reaction di?usion equations, a new reproducing kernel nu-merical technique is presented, its main idea is to seek the minimum of nonlinear termfor di?erential equation in reproducing kernel space, then take the minimum into approx-imation to calculate. The method is implemented easily and good numerical results areobtained.By modifying the traditional reproducing kernel method, a reproducing kernel nu-merical approximate method is presented for solving singular integro-di?erential equationwith Cauchy kernel. Using a transformation, the singular term is removed, requirementfor image space of operator is weakened comparing with traditional reproducing kernelmethod, which illustrate the new method has broader applicability.In short, reproducing kernel space is an ideal space to carry on the numerical cal-culus. This thesis constructs the reproducing kernel space with nonlocal boundary valueconditions, applies the reproducing kernel space theory to solve numerical solution fornonlocal boundary value problems of di?erential equations, numerical simulation illus-trate the presented method possesses broader validity and applicability.