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.     

Existence of solutions for nonlinear singular boundary value problems

Existence of solutions to the two-point boundary value problem (p(t)y')' = q(t)f(t, y,p(t)y'), y(l) = 0, lim_t→0+ p(t)y'(t) = 0 is established under a variety of conditions. Here p(0) = 0 is allowed, and q is not assumed to be continuous at 0, so the problem may be doubly singular. In addition, the Dirichlet problem for this differential equation is investigated     

Constructive approximation of solution for fourth‐order nonlinear boundary value problems

In this paper, we obtain a sequence of approximate solution converging uniformly to the exact solution of a class of fourth‐order nonlinear boundary value problems. Its exact solution is represented in the form of series in the reproducing kernel space. The n‐term approximation u_n(x) is proved to converge to the exact solution u(x). Moreover, the derivatives of u_n(x) are also convergent to the derivatives of u(x). Copyright © 2008 John Wiley & Sons, Ltd.     

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method.     

Jacobi-Picard iteration method for the numerical solution of nonlinear initial value problems

In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.     

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.     

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.     

Fixed-sign solutions for a system of singular focal boundary value problems

We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution.     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

一类具非线性边值条件的非线性方程的奇摄动问题

研究了一类具非线性边值条件的非线性方程的奇摄动问题,运用合成展开法构造了问题的形式渐近解,并用微分不等式理论证明了所得渐近解的一致有效性.     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

On the numerical solution of singular two‐point boundary value problems: A domain decomposition homotopy perturbation approach

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two‐point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.     

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.     

High order generalized upwind schemes and numerical solution of singular perturbation problems

High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods. AMS subject classification (2000)  65L10, 65L12, 65L50     

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.