A new method for solving a class of mixed boundary value problems with singular coefficient

In [1], [2], [3], [4], [5], [6] and [7], it is very difficult to deal with initial boundary value conditions. In this paper, we give a new method to deal with boundary value conditions, the main contribution of this paper is to put mixed boundary value conditions into reproducing kernel Hilbert space. The numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems

This paper presents a new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between our method and other methods for solving an open fourth-order boundary value problem presented by Scott and Watts. The method is also applied to a nonlinear fourth-order boundary value problem. The numerical results demonstrate that the new method is quite accurate and efficient for fourth-order boundary value problems.     

A numerical algorithm for solving a class of linear nonlocal boundary value problems

In order to solve a class of linear nonlocal boundary value problems, a new reproducing kernel space satisfying nonlocal conditions is constructed carefully. This makes it easy to solve the problems. Furthermore, the exact solutions of the problems can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for solving fourth-order nonlocal boundary value problems.     

An efficient computational method for linear fifth-order two-point boundary value problems

In this paper, we present a new algorithm to solve general linear fifth-order boundary value problems (BVPs) in the reproducing kernel space . Representation of the exact solution is given in the reproducing kernel space. Its approximate solution is obtained by truncating the n-term of the exact solution. Some examples are displayed to demonstrate the computational efficiency of the method.     

A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method

In this work, we present an algorithm for solving fourth-order multi-point boundary value problems (BVPs) based on the reproducing kernel method (RKM). In previous works, the RKM has been used to solve various two-point BVPs. However, it cannot be used directly to solve multi-point BVPs, since it is very difficult to obtain a reproducing kernel satisfying multi-point boundary conditions. The aim of this work is to fill this gap. A numerical example is given to demonstrate the efficiency of the present method. The results obtained show that the present method is quite reliable for linear fourth-order multi-point BVPs.     

Numerical solution of nonlinear three‐point boundary value problem on the positive half‐line

In this paper, we present an efficient numerical algorithm to solve the three‐point boundary value problem on the half‐line based on the reproducing kernel theorem. Considering the boundary conditions including a limit form, a new weighted reproducing kernel space is established to overcome the difficulty. By applying reproducing property and existence of the orthogonal basis in the weighted reproducing kernel space, the approximate solution is constructed by the orthogonal projection of the exact solution. Convergence has also been discussed. We demonstrate the accuracy of the method by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Differential operator related to the generalized superradiance integral equation

In this paper we consider a class of problems which are generalized versions of the three-dimensional superradiance integral equation. A commuting differential operator will be found for this generalized problem. For the three-dimensional superradiance problem an alternative set of complete eigenfunctions will also be provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak (cf. Slepian and Pollak (1961) [1], Landau and Pollak (1961, 1962) [2] and [3], Slepian (1964, 1978) [4] and [5]). The uniqueness of the differential operator commuting with that kernel is indicated.     

Constructive proof for existence of nonlinear two-point boundary value problems

Owing to the importance of differential equations in physics, the existence of solutions for differential equations has been paid much attention. In this paper, the existence of solution are obtained for the nonlinear second order two-point boundary value problem in the reproducing kernel space. Under certain assumptions on right-hand side, we propose constructive proof for the existence result, and a method is presented to obtain the exact solution expressed by the form of series. This paper is a extension of previous paper [Wei Jiang, Minggen Cui, The exact solution and stability analysis for integral equation of third or first kind with singular kernel, Appl. Math. Comput. 202 (2) (2008) 666-674], which extends a method of solving linear problems to present method for solving nonlinear problems.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Solving the reaction–diffusion equations with nonlocal boundary conditions based on reproducing kernel space

The reaction–diffusion equations with initial condition and nonlocal boundary conditions are discussed in this article. A reproducing kernel space is constructed, in which an arbitrary function satisfies the initial condition and nonlocal boundary conditions of the reaction‐diffusion equations. Based on the reproducing kernel space, a new algorithm for solving the reaction–diffusion equations with initial condition and nonlocal boundary conditions is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical method for solving the nonlinear four-point boundary value problems

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.     

关于W_2~1[a,b]能否扩大的一些问题

讨论了W12[a,b]能否扩大为含有有间断点函数的再生核空间的问题.结论是:若再生核空间W W12[a,b]含有有间断点的函数,则间断点必固定、间断点个数必有限且非端点a,b.进一步,我们构造了函数含有n个间断点的再生核空间并给出其再生核表达式.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

A numerical solution to nonlinear multi‐point boundary value problems in the reproducing kernel space

In this paper, a new numerical algorithm is provided to solve nonlinear multi‐point boundary value problems in a very favorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel function is discussed in detail. The theorem proves that the approximate solution and its first‐ and second‐order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear multi‐point boundary value problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.     

Numerical solution of eighth order boundary value problems in reproducing Kernel space

In this paper, the approximate solutions to the eighth-order boundary-value problems are presented using the reproducing kernel space method. The procedure is applied on both linear and nonlinear problems. Searching least value (SLV) method is investigated for nonlinear boundary value problems. The argument is based on the reproducing kernel space $W_{2}^{9}[a,b]$ . The approach provides the solution in the form of a convergent series with easily computable components. Analytical results are given for several examples to illustrate the implementation and efficiency of the method. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is more effective and convenient.     

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ^???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ^??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.     

求解奇异三阶边值问题的一个算法

奇异三阶边界值问题出现在排水和涂料流动等研究领域.该文基于再生核理论给出了一个新的算法来求解此类问题.方程的精确解以级数的形式在再生核空间W_2~4[0,1]中给出.同时给出了一些算例说明了这个方法的有效性.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].