A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

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.     

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.     

求解非线性高阶多点边值问题的数值方法

关于多点边值问题的正解的存在性,目前存在大量的研究文献.方法包括不动点理论、上下解方法等(见文献[4-6]).由于微分方程多点边值问题在理论和实践上的重要性,其数值算法也长期吸引着众多数学家、物理学家和工程师们的注意.     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.     

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 posteriori analysis: error estimation for the eighth order boundary value problems in reproducing Kernel space

In the previous work (Akram and Rehman Numer. Algor. 62 527–540 2013), Akram and Rehman presented the reproducing kernel method (RKM) for solving various eighth order boundary value problems. However, an effective error estimation for this method has not yet been discussed. This work is devoted to deal with this problem. Some other aspects of the RKM will be considered such as convergence analysis and numerical implementations.     

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 continuous method for nonlocal functional differential equations with delayed or advanced arguments

In the previous works, the authors presented the reproducing kernel method (RKM) for solving various differential equations. However, to the best of our knowledge, there exist no results for functional differential equations. The aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments, and give the error estimation for the present method. Some numerical examples are provided to show the validity of the present method.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

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.     

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.     

Adomian decomposition method for solving BVPs for fourth-order integro-differential equations

The Adomian decomposition method (ADM) is applied to solve both linear and nonlinear boundary value problems (BVPs) for fourth-order integro-differential equations. The numerical results obtained with minimum amount of computation or mathematics compare reasonably well with exact solutions.     

On shifted Jacobi spectral method for high-order multi-point boundary value problems

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.     

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.     

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.     

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 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.     

A new method for solving singular fourth-order boundary value problems with mixed boundary conditions

In [1], [2], [3], [4], [5], [6], [7] and [8], it is very difficult to get reproducing kernel space of problem (1). This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space. The numerical results are compared with both the exact solution and its n-order derived functions in the example. It is demonstrated that the new method is quite accurate and efficient for fourth-order problems.