This paper is devoted to the numerical treatment of singularly perturbed differential-difference equations with small delay whose solutions exhibiting boundary layer. The reproducing kernel method presented in the previous work is not valid for singularly perturbed differential-difference equations with small delay. In this work, we will improve the reproducing kernel method in order to obtain accurate approximation to the solutions of considered singularly perturbed differential-difference equations. Two numerical examples are provided to show the performance of the present scheme.

A new analytical method for the computation of reproducing kernel is proposed and tested on some examples. The expression of reproducing kernel on infinite interval is obtained concisely in polynomial form for the first time. Furthermore, as a particular effective application of this method, we give an explicit representation formula for calculation of reproducing kernel in reproducing kernel space with boundary value conditions.

We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.

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.

In this paper,the weakly regular singular boundary value problem(p(x)y')' = f(x,y),0<x≤1,with p(x) = x~(b_0)g(x),0≤b_0<1,and the boundary conditions y(0) = A,αy(1) +βy'(1) =γ,or y'(0) = 0,αy(1) +βy'(1) =γ(R.K.Pandey and Arvind K.Singh presented the second order finite difference methods is considered.The existence of the solution and a new iterative algorithm which is large-range convergent are established for the problems in reproducing kernel space.Illustrative examples are included to demonstrate the validity and applicability of the technique through comparing the method with the method given by R.K.Pandey and Arvind K.Singh.

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.

The exact solution of two-equation system of second-order nonlinear ordinary differential equations is obtained in the reproducing kernel space W 32 .The exact solution is represented in the form of series.The n-term approximation un(x),vn(x) are proved to converge to exact solution u(x),v(x).More importantly,the method's implementation requires no additional conditions,whether the equations are singular or non-singular and linear or nonlinear.Examples are presented to demonstrate the reliability and efficiency of the algorithm developed.

In this paper, two point boundary value problems of 2mth‐order nonlinear differential equations are considered. The existence of the solution and a new iterative algorithm which is large‐range convergent are proposed for the problems in reproducing kernel space. The advantage of the approach must lie in the fact that, on the one hand, for the arbitrary fixed initial value function, the iterative method is convergent. On the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives, respectively. Some examples are displayed to demonstrate the computation efficiency of the method. Foundation item: Supported by National Natural Science Foundation of China (No. 60572125); Heilongjiang Institute of Science and Technology (No. 07–17); Heilongjiang province education department science and technology (No. 11531324).

In this paper, a computational method is presented for solving a class of nonlinear singularly perturbed two-point boundary value problems with a boundary layer at the left of the underlying interval. First a zeroth order asymptotic expansion for the solution of the given singularly perturbed boundary value problem is constructed. Then the reduced terminal value problem is solved analytically using reproducing kernel Hilbert space method. This method is effective and easy to implement. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other not only in the boundary layer, but also away from the layer.

In this paper, a new numerical algorithm is provided to solve nonlinear three-point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions. Its reproducing kernel function is discussed in detail. We also prove 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 second order three-point boundary value problems.

The singular third-order boundary value problem arises in the study of draining and coating flows.Based on the reproducing kernel space,a new algorithm is suggested in this paper to solve such problems.Representation of the exact solution is given in the form of series in the reproducing kernel space W_2~4[0,1].Some examples are displayed to demonstrate the validity and applicability of the proposed method.

We are concerned with general third-order nonlinear boundary value problems. An existence theorem of solution is given under weaker conditions. In the meantime, an iterative algorithm with global convergence is presented. The higher order derivatives of approximate solution is obtained by using this method can approximate the corresponding derivatives of exact solution well.

This paper obtains a searching least value (SLV) method for a class of fourth-order nonlinear boundary value problems is investigated. The argument is based on the reproducing kernel space W 5 [ 0 , 1 ] . The approximate solutions u n ( x ) and u n ( k ) ( x ) are uniformly convergent to the exact solution u ( x ) and u ( k ) ( x ) ( k = 1 , 2 , 3 , 4 ) respectively. Numerical results are verified that the method is quite accurate and efficient for this kind of problem.

In this paper,we give a new method to solve the semilinear heat equation with integral boundary conditions.Its exact solution is represented in the form of series in the reproducing kernel space.The n-term approximation u_n(x,t) of the exact solution u(x,t) is proved to converge to the exact solution.Some numerical examples are displayed to demonstrate the accuracy of the present method.

Based upon He's homotopy perturbation and variational iteration methods, we present a method for approximate solutions of nonlinear second-order multi-point boundary value problems (BVPs) in bridge design. Two numerical experiments are carried out to demonstrate the efficiency of the present method. The results reveal that the proposed method is very effective for second-order multi-point BVPs in bridge design.

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's 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. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.

A new existence proof of solutions for a class of fourth-order nonlinear boundary value problems is proposed. The proof of the main results is based on the reproducing kernel theorem. It is worthwhile to point out that the method presented in this paper can be applied for the existence proof of diverse kinds of boundary conditions.

Numerical solutions to the initial value problem for the fully nonlinear differential equation f( x,u,u') = g( x) are discussed in the reproducing kernel space. In terms of the reproducing property of reproducing kernel spaces,ordinary differential equations are converted to partial differential equations to solve by adding variables. By homogenizing initial conditions and putting them into two-dimensional reproducing kernel spaces,a presentation of solutions with an unknown parameters is obtained. Then by using the least square technique,the solutions to operator equations are given. A numerical example is given,and its results demonstrate that this method is feasible.

In this paper, we propose a new method to solve the forced Duffing equation with integral boundary conditions. Its exact solution is represented in the form of a series in the reproducing kernel space. The n-term approximation un(x) of the exact solution u(x) is proved to converge to the exact solution. Some numerical examples are displayed to demonstrate the accuracy of the present method.

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.