A fifth order numerical method for singularperturbation problems

P. Pramod Chakravarthy,K. Phaneendra,Y.N. Reddy 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.3-4

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well. In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

A FIFTH ORDER NUMERICAL METHOD FOR SINGULAR PERTURBATION PROBLEMS

Chakravarthy, P. Pramod,Phaneendra, K.,Reddy, Y.N. Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.3-4

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

A FIFTH ORDER NUMERICAL METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH NEGATIVE SHIFT

Chakravarthy, P. Pramod,Phaneendra, K.,Reddy, Y.N. The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.1

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

A fifth order numerical method for singularly Perturbed differential-difference equations with negative shift

P. Pramod Chakravarthy,K. Phaneendra,Y.N. Reddy 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.1

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well. In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

An initial value technique for singularly perturbed differential-difference equations with a small negative shift

R. Nageshwar Rao,P. Pramod Chakravarthy 한국전산응용수학회 2013 Journal of applied mathematics & informatics Vol.31 No.1

In this paper, we present an initial value technique for solvingsingularly perturbed differential difference equations with a boundary layerat one end point. Taylor’s series is used to tackle the terms containing shiftprovided the shift is of small order of singular perturbation parameter andobtained a singularly perturbed boundary value problem. This singularlyperturbed boundary value problem is replaced by a pair of initial valueproblems. Classical fourth order Runge–Kutta method is used to solvethese initial value problems. The effect of small shift on the boundarylayer solution in both the cases, i.e., the boundary layer on the left sideas well as the right side is discussed by considering numerical experiments. Several numerical examples are solved to demonstate the applicability ofthe method.

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH A SMALL NEGATIVE SHIFT

Rao, R. Nageshwar,Chakravarthy, P. Pramod The Korean Society for Computational and Applied M 2013 Journal of applied mathematics & informatics Vol.31 No.1

In this paper, we present an initial value technique for solving singularly perturbed differential difference equations with a boundary layer at one end point. Taylor's series is used to tackle the terms containing shift provided the shift is of small order of singular perturbation parameter and obtained a singularly perturbed boundary value problem. This singularly perturbed boundary value problem is replaced by a pair of initial value problems. Classical fourth order Runge-Kutta method is used to solve these initial value problems. The effect of small shift on the boundary layer solution in both the cases, i.e., the boundary layer on the left side as well as the right side is discussed by considering numerical experiments. Several numerical examples are solved to demonstate the applicability of the method.