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.     

Recurrence Method for Solving Singularly Perturbed Boundary Value Problems

We base on Taylor series expansions to construct the numerical method for solving singularly perturbed boundary value problems. We use the trapezoid method to approximate the integrals and obtain three‐term recurrence relationship. The efficiency of the proposed method is demonstrated by test problems. The numerical result is found in a good agreement with exact solution.     

Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems

We develop a numerical algorithm for solving singularly perturbed one-dimensional parabolic convection-diffusion problems. The method comprises a standard finite difference to discretize in temporal direction and Sinc-Galerkin method in spatial direction. The convergence analysis and stability of proposed method are discussed in details, it is justifying that the approximate solution converges to the exact solution at an exponential rate. we know that the conventional methods for these problems suffer due to decreasing of perturbation parameter, but the Sinc method handel such difficulty as singularity. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence.     

B-spline solution of a singularly perturbed boundary value problem arising in biology

We use B-spline functions to develop a numerical method for solving a singularly perturbed boundary value problem associated with biology science. We use B-spline collocation method, which leads to a tridiagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical result is found in good agreement with exact solution.     

A novel method for a class of parameterized singularly perturbed boundary value problems

In this paper a novel approach is presented for solving parameterized singularly perturbed two-point boundary value problems with a boundary layer. By the boundary layer correction technique, the original problem is converted into two non-singularly perturbed problems which can be solved using traditional numerical methods, such as Runge–Kutta methods. Several non-linear problems are solved to demonstrate the applicability of the method. Numerical experiments indicate the high accuracy and the efficiency of the new method.     

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].     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.     

On the solution a of second order singularly-perturbed boundary value problem by the Sinc-Galerkin method

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the Sinc-Galerkin method for solving second order singularly perturbed boundary value problems. The method is then tested on linear and nonlinear examples and a comparison with spline method and finite element scheme is made. It is shown that the Sinc-Galerkin method yields better results.Received: January 3, 2003; revised: July 14, 2003     

High-Accuracy Numerical Approximations to Several Singularly Perturbed Problems and Singular Integral Equations by Enriched Spectral Galerkin Methods

Usual spectral methods are not effective for singularly perturbed problems and singular integral equations due to the boundary layer functions or weakly singular solutions. To overcome this difficulty, the enriched spectral-Galerkin methods (ESG) are applied to deal with a class of singularly perturbed problems and singular integral equations for which the form of leading singular solutions can be determined. In particular, for easily understanding the technique of ESG, the detail of the process are provided in solving singularly perturbed problems. Ample numerical examples verify the efficiency and accuracy of the enriched spectral Galerkin methods.     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations

For linear singularly perturbed boundary value problems, we come up with a method that reduces solving a differential problem to a discrete (difference) problem. Difference equations, which are an exact analog of differential equations, are constructed by the factorization method. Coefficients of difference equations are calculated by solving Cauchy problems for first-order differential equations. In this case nonlinear Ricatti equations with a small parameter are solved by asymptotic methods, and solving linear equations reduces to computing quadratures. A solution for quasilinear singularly perturbed equations is obtained by means of an implicit relaxation method. A solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested against solutions to the known Lagerstrom-Cole problem.     

An efficient hybrid numerical method based on an additive scheme for solving coupled systems of singularly perturbed linear parabolic problems

We construct an efficient hybrid numerical method for solving coupled systems of singularly perturbed linear parabolic problems of reaction-diffusion type. The discretization of the coupled system is based on the use of an additive or splitting scheme on a uniform mesh in time and a hybrid scheme on a layer-adapted mesh in space. It is proven that the developed numerical method is uniformly convergent of first order in time and third order in space. The purpose of the additive scheme is to decouple the components of the vector approximate solution at each time step and thus make the computation more efficient. The numerical results confirm the theoretical convergence result and illustrate the efficiency of the proposed strategy.     

Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions

We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ?-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ?-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270].     

Uniform Global Convergence of a Hybrid Scheme for Singularly Perturbed Reaction–Diffusion Systems

We consider a system of coupled singularly perturbed reaction–diffusion two-point boundary-value problems. A hybrid difference scheme on a piecewise-uniform Shishkin mesh is constructed for solving this system, which generates better approximations to the exact solution than the classical central difference scheme. Moreover, we prove that the method is third order uniformly convergent in the maximum norm when the singular perturbation parameter is small. Numerical experiments are conducted to validate the theoretical results.     

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 modified Bakhvalov mesh

We consider a few numerical methods for solving a one-dimensional convection–diffusion singularly perturbed problem. More precisely, we introduce a modified Bakvalov mesh generated using some implicitly defined functions. Properties of this mesh and convergence results for several methods on it are given. Numerical results are presented in support of the theoretical considerations.     

An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.     

Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations

We consider stationary solutions with internal transition layers (contrast structures) for a singularly perturbed elliptic equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an efficient algorithm for constructing an asymptotic approximation to the localization curve of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions.     

Numerical solution of singular perturbation problems via deviating arguments

We present a new approach to numerically solving linear, singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in the implicit form is derived. Then, the outer region problem is solved as a two point boundary value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, a new inner region problem is obtained and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Some numerical examples have been solved to demonstrate the applicability of the method.