On an efficient hybrid method for solving singularly perturbed difference-differential equations exhibiting turning layer behavior

International Conference on Computational and Experimental Science and Engineering (4. : 2017 : Antalya, Turkey)Singularly perturbed differential equations that involve positive small perturbation parameter(s) 0<ɛ≪1 as the multiplier to the highest order derivative term are important concepts of mathematical and engineering sciences. As ɛ→0, solution of this kind of problems exhibits rapid changes that we call boundary layer behavior since the order of the equation reduces and it is a well-known fact that classical numerical methods are often insufficient to handle them. One may encounter with singular perturbation problems in almost all science branches. Some application areas may be given as modelling of fluid flow problems at high Reynold numbers, electrical and electronic circuits/systems, nuclear reactors, astrophysics problems, control theory problems, combustion theory, quantum mechanics, signal/image processing, etc. This study concerns with finding approximations to the solution of singularly perturbed two-point boundary value problems that exhibit interior layer (turning point) behavior. To achieve this, an efficient and easy-applicable asymptotic-numerical hybrid method is employed. The asymptotic part of the method is based on Successive Complementary Expansion Method (SCEM) and the numerical part is based on finite difference approximations that applies a Lobatto IIIa formula. As the first stage of present method, an asymptotic approximation to the solution of the singularly perturbed problem is proposed using SCEM with the help of stretching variable transformation and later the resulting two-point boundary value problems that come from the SCEM procedure are solved using the numerical procedure. Numerical experiments show that the present method is well-suited for solving this type of problems.No sponso

A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer

The objective of this paper is to construct and analyzea fitted operator finite difference method (FOFDM) forthe family of time-dependent singularly perturbed parabolicconvection–diffusion problems. The solution to the problemswe consider exhibits an interior layer due to the presence ofa turning point. We first establish sharp bounds on the solu-tion and its derivatives. Then, we discretize the time variableusing the classical Euler method. This results in a system ofsingularly perturbed interior layer two-point boundary valueproblems. We propose a FOFDM to solve the system above

A High-Order Method for Stiff Boundary Value Problems with Turning Points

This paper describes some high-order collocation-like methods for the numerical solution of stiff boundary-value problems with turning points. The presentation concentrates on the implementation of these methods in conjunction with the implementation of the a priori mesh construction algorithm introduced by Kreiss, Nichols and Brown [SIAM J. Numer. Anal., 23 (1986), pp. 325–368] for such problems. Numerical examples are given showing the high accuracy which can be obtained in solving the boundary value problem for singularly perturbed ordinary differential equations with turning points