K. Phaneendra

A seventh order numerical method for singular perturbation problems

In this paper, a seventh 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 seventh 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 seventh order compact difference scheme to get a two term recurrence relation and is solved. Several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory.

Numerical Treatment of Singularly Perturbed Delay Differential Equations

In this paper, we propose numerical method to solve singularly perturbed delay differential equations which works smoothly in both the cases, i.e., whether the delay is of

Numerical Integration with Exponential Fitting Factor for Singularly Perturbed Two Point Boundary Value Problems

In this paper, we discuss the numerical integration with e xponential fitting factor for singularly perturbed two-point boundary value problems. It is based on t he fact that: the given SPTPBVP is replaced by an asymptotically equivalent delay differenti al equation. Then, numerical integration with exponential fitting factor is employed to obtain a tridiagonal system which is solved efficiently by Thomas algorithm. We discussed con vergence analysis of the method. Model examples are solved and the numerical results are compa red with exact solution.

Solution of Singularly Perturbed Differential-Difference Equations with Mixed Shifts Using Galerkin Method with Exponential Fitting

Galerkin method is presented to solve singularly perturbed differential-difference equations with delay and advanced shifts using fitting factor. In the numerical treatment of such type of problems, Taylor’s approximation is used to tackle the terms containing small shifts. A fitting factor in the Galerkin scheme is introduced which takes care of the rapid changes that occur in the boundary layer. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method. The method is analysed for convergence. Several numerical examples are solved and compared to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shifts on the boundary layer solution.

A Fitted Galerkin Method for Singularly Perturbed Differential Equations with Layer Behaviour

In this paper, we have presented a fitted Galerkin method for singularly perturbed differential equations with layer behaviour. We have introduced a fitting factor in the Galerkin difference scheme which takes care of the rapid changes occur that in the boundary layer. This fitting factor is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method. The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method. Maximum absolute errors in numerical results are presented to illustrate the proposed method.