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.     

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.     

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.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.     

带边界层的线性两点边值问题的打靶——小波配点法

本文将打靶法和小波配点法相结合，提出了打靶－小波配点数值算法，用于求解带边界层的常微分方程边值问题。文中给出了数值算例，并进行了分析，验证了这种方法对处理边界层问题的有效性。     

The weighted Ritz‐Galerkin method for elliptic boundary value problems on unbounded domains

Recently Babus?ka‐Oh introduced the method of auxiliary mapping (MAM) which efficiently handles elliptic boundary value problems containing singularities. In this paper, a special weighted residue method, the Weighted Ritz‐Galerkin Method (WRGM), is investigated by introducing special weight functions. Together with this method, MAM is modified to yield highly accurate finite element solutions to general elliptic boundary value problems on the exterior of bounded domains at low cost. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 301–326, 2003.     

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.     

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 reliable treatment of singular Emden-Fowler initial value problems and boundary value problems

In this paper, the variational iteration method (VIM) is used to study the singular Emden-Fowler initial value problems and boundary value problems arising in physics and astrophysics. The VIM overcomes the singularity at the origin. The Lagrange multipliers for all cases of the equations are determined. The work is supported by analyzing few initial value problems and boundary value problems where the convergence of the results is emphasized.     

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.     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

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.     

Weighted extended b-splines for one-dimensional electromagnetic problems

This paper considers the weighted extended b-splines as basis function for finite element method in electromagnetics and compares with the standard finite element method applied to the two-point boundary value problems with different boundary conditions. This new approach, which provides more accurate results than standard finite element method, is presented to compare other numerical techniques and applied to one-dimensional electromagnetic problems. Computed results are compared with other numerical results in literature.     

The Linear Rational Pseudospectral Method with Preassigned Poles

We present a linear rational pseudospectral (collocation) method with preassigned poles for solving boundary value problems. It consists in attaching poles to the trial polynomial so as to make it a rational interpolant. Its convergence is proved by transforming the problem into an associated boundary value problem. Numerical examples demonstrate that the rational pseudospectral method is often more efficient than the polynomial method.     

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

二阶脉冲微分方程三点边值问题解的存在性

建立了二阶脉冲微分方程三点边值问题的比较定理,利用单调迭代方法讨论了二阶脉冲微分方程三点边值问题解的存在性.     

三维Helmholtz方程边值问题的新型边界积分-微分方程及其边界元解法

用双层位势表示的二维Neumann边值问题的边界归化方法,将原始问题归化为新型边界积分-微分方程,由此导出一种新的既能保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程.本文将此法推广到三维Helmholtz方程Neumann边值问题,并给出最优能量模误差估计和内部最大模超收敛估计.