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.     

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.     

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.     

New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions

This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the 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.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.     

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.     

A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method

In this paper, a novel method is presented for solving a class of singularly perturbed boundary value problems. Firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation; then the reproducing kernel method is employed to solve the boundary value new problem. Numerical results show that the present method can provide very accurate analytical approximate solutions.     

基于再生核三次样条基底的构造及其应用

将三次样条理论与再生核理论相结合,利用再生核函数巧妙地构造了三次样条函数空间的一组基底.基于三次样条插值的高收敛特点,得到了微分方程边值问题近似解的一种新的求解方法.数值算例展现出算法简单、有效.     

A reproducing kernel method for solving nonlocal fractional boundary value problems

In our previous works, we proposed a reproducing kernel method for solving singular and nonsingular boundary value problems of integer order based on the reproducing kernel theory. In this letter, we shall expand the application of reproducing kernel theory to fractional differential equations and present an algorithm for solving nonlocal fractional boundary value problems. The results from numerical examples show that the present method is simple and effective.     

A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM

In this paper, a novel method is proposed for solving nonlinear two-point boundary value problems (BVPs). This method is based on a combination of the Adomian decomposition method (ADM) and the reproducing kernel method (RKM). A major advantage of this method over standard ADM is that it can avoid unnecessary computation in determining the unknown parameters. The proposed method can be applied to singular and nonsingular BVPs. Numerical results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for solving nonlinear two-point boundary value problems.     

三阶微分方程边值问题的高效数值算法

基于再生核空间法提出了一个高效的数值算法来解决三阶微分方程的边值问题.利用再生性以及正交基的构造,得到了模型精确解的级数表示形式,并通过截断级数获得了其近似解.通过数值算例说明了此方法的有效性.     

Simplified reproducing kernel method for fractional differential equations with delay

This paper is devoted to the numerical scheme for the delay initial value problems of a fractional order. The main idea of this method is to establish a novel reproducing kernel space that satisfies the initial conditions. Based on the properties of the new reproducing kernel space, the simplified reproducing kernel method (SRKM for short) is applied to obtain accurate approximation. The Schmidt orthogoralization process which requires a large number of calculation is less likely to be employed. Numerical experiments are provided to illustrate the performance of the method.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

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.     

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.     

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.     

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.     

Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method

In this paper, reproducing kernel theorem is employed to solve anti-periodic solutions for Rayleigh-type equations. A simple algorithm is given to obtain the approximate solutions of the equations. By comparing the approximate solution with the exact analytical solution, we find that the simple algorithm is of good accuracy and it can be also applied to some ordinary or partial differential equations with initial-boundary value conditions and nonlocal boundary value conditions.     

轴对称弹性动力学问题的重构核插值法

重构核插值法是近年来提出的一种新型无网格方法.该方法的形函数具有点插值性和高阶光滑性,不仅能够直接施加本质边界条件,而且能保证较高的计算精度.为了更有效地求解三维轴对称弹性动力学问题,对重构核插值法(reproducing kernel interpolation method, RKIM)应用于此类问题进行了研究,并发展了相应的数值模拟方法.由于几何形状和边界条件的轴对称性,计算时只需要横截面上离散节点的信息,因而前处理变得简单.采用Newmark-β法进行了时域积分.数值算例表明,轴对称弹性动力学分析的重构核插值法既有无网格方法的优势,又有较高的计算精度.