Piecewise reproducing kernel method for singularly perturbed delay initial value problems

A direct application of the reproducing kernel method presented in the previous works cannot yield accurate approximate solutions for singularly perturbed delay differential equations. In this letter, we construct a new numerical method called piecewise reproducing kernel method for singularly perturbed delay initial value problems. Numerical results show that the present method does not share the drawback of standard reproducing kernel method and is an effective method for the considered singularly perturbed delay initial value problems.     

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.     

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.     

The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems

A new computational method for solving the second-order nonlinear singularly perturbed boundary value problems (SPBVPs) is provided in this paper. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an x-domain to a t-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method (LGSM) to search a missing initial condition through the finding of a suitable value of a parameter r ∈ [0, 1]. Moreover, we can derive a closed-form formula to express the initial condition in terms of r, which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.     

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.     

Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems

We present an exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problem. The convergence analysis is given and the method is shown to have second order uniform convergence. Numerical experiments are conducted to demonstrate the efficiency 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.     

Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives

We develop a numerical technique for a class of singularly perturbed two-point singular boundary value problems on an uniform mesh using polynomial cubic spline. The scheme derived in this paper is second-order accurate. The resulting linear system of equations has been solved by using a tri-diagonal solver. Numerical results are provided to illustrate the proposed method and to compared with the methods in [R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput., 172 (2006) 531–544; M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 82 (2005) 67–76].     

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.     

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.     

一类具有无穷边界值的二次奇摄动边值问题

研究一类具有无穷边界值的二次奇摄动Robin边值问题解的存在性与解的渐进行为,重点关注边界值的奇异程度对解的边界层行为的影响;同时将所得的结果与Chang及Howes的结果(带正常边界值)进行比较.研究表明:(1)当边界值大小为O(1/)时,得到的边界层大小为O( ln ),这比Chang及Howes带正常边界值的情形提高了O(ln )量级;(2)增大边界值的奇性至O(1/ r),这里r >1,边界层大小的量级不变,依然为O( ln );(3)若要使得边界层大小为O(1),则边界值的大小需为O(e?1/).最后给出一个算例验证得到的结果.     

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.     

Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems

Extended one-step schemes of exponential type are introduced for solving singularly perturbed Volterra integro-differential problems. These schemes are of order (m + 1), m = 0, 1, 2, …, when the perturbation parameter, ε, is fixed. These schemes have the property that if ε is of order h they reduced to first order of accuracy and optimal when ε → 0. Stability analysis of these schemes are presented. Numerical results and comparisons with other schemes are presented.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Multiresolution exponential B-splines and singularly perturbed boundary problem

The paper considers how cardinal exponential B-splines can be applied in solving singularly perturbed boundary problems. The exponential nature and the multiresolution property of these splines are essential for an accurate simulation of a singular behavior of some differential equation solutions. Based on the knowledge that the most of exponential B-spline properties coincide with those of polynomial splines (smoothness, compact support, positivity, partition of unity, reconstruction of polynomials, recursion for derivatives), one novel algorithm is proposed. It merges two well known approaches for solving such problems, fitted operator and fitted mesh methods. The exponential B-spline basis is adapted for an interval because a considered problem is solved on a bounded domain.      

Third-order variable-mesh cubic spline methods for nonlinear two-point singularly perturbed boundary-value problems

A family of third-order variable-mesh methods for singularly perturbed two-point boundary-value problems of the form y=f(x,y,y),y(a)=A, y(b)=B is derived. The convergence analysis is given, and the method is shown to have third-order convergence properties. Several test examples are solved to demonstrate the efficiency of the method.     

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.     

The hp finite element method for singularly perturbed problems in nonsmooth domains

We consider the numerical approximation of singularly perturbed elliptic boundary value problems over nonsmooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh‐degree combinations are given that yield finite element spaces with robust approximation properties. In particular, we construct an hp finite element space that approximates all components uniformly, at a near exponential rate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 63–89, 1999     

On a rational differential quadrature method in irregular domains for problems with boundary layers

A rational differential quadrature method in irregular domains (RDQMID) is investigated to deal with a kind of singularly perturbed problems with boundary layers. Through a transformation, the boundary layer, which may be not straight, is transformed into a segment of a line parallel to one of the Cartesian axes. The rational differential quadrature method (RDQM) is applied to discretize the governing equation. Finally, a direct expansion method of the boundary conditions (DEMBC) is raised to deal with the boundary conditions. Numerical experiments show that RDQMID is of high accuracy, efficiency and easy to programme.