Reproducing Kernel Methods Of Solving Initial And Boundary Value Problems For Differential Equations

An initial and boundary value problem is a field of study of the physical processor phenomenon described by differential equations. It determines the change law of thesystem variable according to initial and boundary value conditions, and it is a study ofmathematical expression of status.Initial and boundary value problems can be classified by several features. Linearityis one way to classify differential equations, linear and nonlinear ones. By the conditions,problems can be divided into initial, boundary or initial-boundary value problems. By theunknown items, problems can be divided into direct problems and inverse problems.This paper explores numerically solving for initial and boundary value problems fordifferential equations, involving in both direct and inverse problems. Nonlinear equa-tions are our main concern while talking about equations. In terms of boundary valueconditions, we mainly choose nonlocal multi-point and integral boundaries.Numerically solving an initial and boundary value problem for a differential equa-tion by the reproducing kernel method can be described as follows: construct reproducingkernel spaces which can absorb initial or boundary value conditions, transfer the initialand boundary value problem into an operator equation in the reproducing kernel space. Inthe reproducing kernel space, the exact solution to the initial and boundary value problemis expressed by the reproducing kernel, and at last solve the operator equation by approxi-mation. It is obvious that constructing reproducing kernel space which satisfies the initialor boundary conditions and effectively solving for the reproducing kernel become the keyto apply reproducing kernel method for initial and boundary value problems.For recent more than 20 years, the reproducing kernel method is employed in thefield of initial and boundary value problems for differential equations. In 1986, MinggenCui built the foundation of solving initial and boundary value problem for differentialequation by giving the reproducing kernel of a two-point boundary value problem. Afterthat, many scholars considered linear and nonlinear differential equation with variousboundary conditions, which consists of two-point boundary, linear boundary and periodboundary. In 2008, Yingzhen Lin calculated the reproducing kernel for nonlocal integral boundary value problems. This made the reproducing kernel method extended to integralboundary problems. However, the method is never applied in the field of multi-pointboundary problems. The reason is that it has been difficult to solve for the reproducingkernel of the multi-point boundary, which has been bothering us.This paper constructs reproducing kernel spaces for multi-point boundary valueproblems, presents the effective way of calculating the reproducing kernel, and success-fully gives the expression of reproducing kernel for multi-point boundary value. It makespossibility for using reproducing kernel methods in the fields of multi-point boundaryvalue problems, and extends the range of the reproducing kernel theory.This paper focuses on numerical solution to the initial and boundary value problemsfor differential equations under the frame of reproducing kernel space, including multi-point boundary value problems, inverse problems, and nonlinear parabolic equations withnonlocal boundary conditions.On the basis of successfully constructing reproducing kernel spaces for multi-pointboundary value problems and calculating their reproducing kernels, the reproducing ker-nel method is discussed in the field of linear and nonlinear multi-point boundary valueproblems. The analytical solutions to the linear and nonlinear multi-point boundary areexpressed. An iterative method is designed for nonlinear multi-point boundary valueproblems, and the iterative sequence is the best approximation.This paper first time applies the reproducing method in the field of inverse, tryingto solve for space-dependent or time-dependent main coefficient inverse problems and asource parameter coefficient inverse problem. For the main coefficient inverse problem,the technology of space decomposition is used for solution representation and determina-tion of main coefficients. For the source parameter coefficient inverse problem, the prob-lem is transferred into an equivalent operator equation in the reproducing kernel space.In the reproducing kernel space, an iterative method is developed. Since the iterativesequence is the approximation under project, it is the best approximation. And partialderivatives of the sequence also are convergent to the derivatives of the solution.The reproducing kernel method is applied to various nonlinear parabolic equationswith nonlocal boundary conditions. Reproducing kernel spaces are constructed for kindsof nonlocal boundary conditions, absorbing the complex boundaries into reproducing ker-nel spaces. The calculation of reproducing kernel for multi-point boundary in the paper breakthe model of calculating reproducing kernel, enrich the reproducing kernel theory. Thereproducing kernel method is used in the field of multi-point problem, integral bound-ary problem, inverse problem and nonlinear parabolic equations, extending the range ofreproducing kernel theory.