A Review on Singularly Perturbed Differential Equations with Turning Points and Interior Layers Kapil K. Sharma a1 , Pratima Rai b , Kailash C. Patidar c a Department of Mathematics, South Asian University (SAU), Akbar Bhawan, Chanakyapuri, New Delhi, India, 110021, b Department of Mathematics, Amity Institue of Applied Sciences, Amity University, sector 125, Noida, Uttar Pradesh, India, 201303, c Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa E-mail adresses: a kapil.sharma@sau.ac.in; b prai2@amity.edu; c kpatidar@uwc.ac.za Abstract Singular perturbation problems with turning points arise as mathematical models for various physical phe- nomena. The problem with interior turning point represent one-dimensional version of stationary convection- diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. Boundary turning point problems, on the other hand, arise in geophysics and in modeling thermal bound- ary layers in laminar flow. In this paper, we review some existing literature on asymptotic and numerical analysis of singularly perturbed turning point and interior layer problems. The purpose is to find out what problems are treated and what numerical/asymptotic methods are employed, with an eye towards the goal of developing general methods to solve such problems. Since major work in this area started after 1970 so this paper limits its coverage to the work done by numerous researchers between 1970 and 2011. AMS 2000 classification: 34(K25, K26, K28), 35(A40, B25, B40). Keywords: Singular perturbation; turning points; discontinuous data; ordinary differential equa- tions; partial differential equations; asymptotic analysis; numerical analysis. 1 Introduction Many phenomena in biology, chemistry, engineering, physics, etc., can be described by boundary value problems associated with various type of differential equations or systems. Whenever a mathematical model is associated with a phenomenon, the researchers generally try to capture what is essential, retaining the important quantities and omitting the negligible ones which involve small parameters. The model that would be obtained by maintaining the small parameters is called the perturbed model, whereas the simplified model (the one that does not include the small parameters) is called the unperturbed (or reduced) model. For study purpose the perturbed model can be replaced by its unperturbed counterpart but what matters is that its solution must be “close enough” to the solution of the corresponding perturbed model. This fact holds good in case of regular perturbation but, in the case of singular perturbation it is very unlikely to hold. These singular perturbation problems with or without turning point(s) commonly occur in many branches of applied mathematics, e.g., as boundary layers in fluid mechanics, edge layers in solid mechanics, skin layers in electrical applications, shock layers in fluid and solid mechanics, transition points in quantum mechanics and Stokes lines and surfaces in mathematics. In these kind of problems perturbations are operative over a very narrow region across which the dependent variable undergoes very rapid change. These narrow regions frequently adjoin the boundaries or some interior point of the domain of interest, owing to the fact that the small parameter multiplies the highest derivative. Therefore, these kind of problems exhibit boundary and/or interior layers, i.e., there are thin regions where the solution changes rapidly. Kadalbajoo and Reddy [35] gave survey of various asymptotic and numerical methods developed from 1908 − 1986 for the determination of approximate solution of singular perturbation problems of various kinds. Kadalbajoo and Patidar [36] extended the work done by Kadalbajoo and Reddy and surveyed the work done by various researchers in the area of singular perturbation from 1984 − 2000. In this work they considered one dimensional problems only and discussed the work done on linear, non-linear, semilinear and quasilinear problems. In [37] Kadalbajoo and Patidar covered the survey of singularly perturbed partial differential equations and surveyed the work done in this area from 1980 − 2000. Kadalbajoo and Vikas [38] in continuation with the work done by the first author [35, 36, 37] gave brief survey on the computational techniques for different classes of singular perturbation problems considered by various researchers from 2000 − 2009. In this way one can see that this area has developed so much in the past century that it is not possible to give whole of the survey in a single paper. In particular, singularly perturbed differential equations with turning point form an important class of problems which are very challenging and even today there is a lot to be explored in this area. Also, problems where discontinuity in the data results into interior 1 Corresponding author: Kapil. K. Sharma; email address: kapil.sharma@sau.ac.in; phone: +91 01124195271; 1