Various Numerical Methods for Singularly Perturbed Boundary Value Problems

Abstract

The numerical treatment of singular perturbation problems is currently a field in which active research is going on these days. Singular perturbation problems in which the term containing the highest order derivative is multiplied by a small parameter ε , occur in a number of areas of applied mathematics, science and engineering among them fluid mechanics (boundary layer problems) elasticity (edge effort in shells) and quantum mechanics. In this paper, we consider few numerical methods for singularly perturbed boundary value problems developed by numerous researchers between 2006 to 2013. A Summary of the result of some recent methods is presented and this leads to conclusion and recommendations regarding methods to use on singular perturbation problem.

A new quartic B-spline method for third–order self-adjoint singularly perturbed boundary value problems

“…Bender [2] and the reference in. Various techniques are known to solve these types of problems [3,7,8,13,14]. One of the most important techniques is spline function.…”

Numerical Solution of a Singularly Perturbed Fredholm Integro Differential Equation with Robin Boundary Condition

“…Singularly perturbed problems (SPPs) are mostly characterized by a small parameter ε that multiplies some or all of the higher order terms in the equation, because boundary layers are generally found in their solutions. To cite a few, one can find the exact solutions of SPPs and their applications in [16,19]. SPPs possesses vast number of applications in the fields of fluid dynamics, population dynamics, heat transport problem, nanofluid, neurobiology, mathematical biology, viscoelasticity and simultaneous control systems etc.…”

On the Volterra Delay-Integro-Differential equation with layer behavior and its numerical solution

“…It is wellknown that standard discretization methods do not work well for these problems as they often produce oscillatory solutions which are inaccurate if the perturbed parameter " is small. To obtain robust numerical methods it is necessary to fit the coefficients (fitted operator methods) or the mesh (fitted mesh methods) to the behavior of the exact solution [1,2,5,10,15,19,22] (see also references cited in them). For a survey of early results in the theoretical analysis of singularly perturbed Volterra integro-differential equations (VIDEs) and in the numerical analysis and implementation of various techniques for these problems we refer to the book [11].…”