Y. N. Reddy

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.

Asymptotic and numerical analysis of singular perturbation problems: A survey

This paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems. The purpose is to find out what problems are treated and what numerical/asymptotic methods are employed, with an eye toward the goal of developing general methods to solve such problems. A summary of the results of some recent methods is presented, and this leads to conclusions and recommendations about what methods to use on singular perturbation problems. Finally, some areas of current research are indicated. A bibliography of about 130 items is provided.

Numerical treatment of singularly perturbed two point boundary value problems

We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort.

Method of reduction of order for solving singularly perturbed two-point boundary value problems

In this paper, a method of reduction of order is proposed for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. It is distinguished by the following fact: the original 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. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory.

Higher order finite difference method for a class of singular boundary value problems

In this paper, a fourth order finite difference method for a class of singular boundary value problems is presented. The original differential equation is modified at the singular point. The fourth order finite difference method is then employed to solve the boundary value problem. Some model problems are solved, and the numerical results are compared with exact solution.

An initial-value approach for solving singularly perturbed two-point boundary value problems

In this paper an initial-value approach is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. This approach is based on the boundary layer behavior of the solution. The method is distinguished by the following fact: The given singularly perturbed two-point boundary value problem is replaced by three first order initial-value problems. Several linear and non-linear problems are solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution very well.

Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.

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.

An exponentially fitted finite difference method for singular perturbation problems

In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. A fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the system. The stability of the algorithm is investigated. Several linear and nonlinear problems are solved to demonstrate the applicability of the method. It is observed that the present method approximates the exact solution very well.

A numerical method for singular two point boundary value problems via Chebyshev economizition

In this paper we present a numerical method for solving a two point boundary value problem in the interval [0,1] with regular singularity at x=0. By employing the Chebyshev economizition on [0,@d], where @d is near the singularity, we first replace it by a regular problem on some interval [@d,1]. The stable central difference method is then employed to solve the problem over the reduced interval. Some numerical results are presented to demonstrate the applicability of the method.