Numerical Analysis of Singularly Perturbed Differential Equations:
Most of the physical, chemical and bio-chemical problems can be designed as mathemat-
ical model, which enhance our understanding process and solution strategies of those problems. These problems may have complex properties, like non-uniform behaviour, boundary layer phenomenon, which leads to unstable/oscillatory and inappropriate solutions. In particular, physical oceanography is an exciting, fruitful and important field of study, relevant to the current discourse on, and the effects of climate change. Life depends on climate. For many years people have been trying to understand how the climate system works and how to forecast its behaviour and variability. The natural way to describe the ocean dynamics is by using Navior- Stokes equation with free surface. The Navior-Stokes equation and the Saint-Venant equation infact composes the properties of Singularly Perturbed Differential Equation(SPDE). A differential equation with a small positive parameter multiplying the highest derivative term subject to boundary conditions belongs to a class of problems known as singular perturbation problems. The solution of singular perturbation problems has non-uniform behaviour, i.e., there are thin layer(s)(Boundary layer region) where the solution varies rapidly while away from the layer(s) (outer region) the solution behaves regularly and varies slowly. Often these mathematical problems are extremely difficult (or even impossible) to solve exactly and in these circumstance,
best approximate solution are necessary.

Mathematical models of semi-conductor devices:
Due to the great importance of semiconductor devices in the modern electronics industry, it
is desirable that the design and behavior of a device can be predicted to a known accuracy in
advance of its actual fabrication. The cost of rectifying a design error a later stage is enormous. One of the ways of guarding against such a disaster is to simulate the device using a suitable numerical model. The numerical modeling of the electromagnetic and thermal behavior of the semiconductor devices is a challenging task which requires the solution of many numerical problems. The size of semiconductor devices have been shrinking down to nanoscale levels for the future of very large scale integration (VLSI) technology. In integrated circuit applications, field effect transistors (FETs) are preferred because of their features involving construction and bias.In nano structures, the quantum confinement effects is considered due to the changes in atomic structures. The confinement ranges up to 25nm in the nanoscale devices where the confinement size limits depend upon the effective mass of the electron/hole and the dielectric constant of the nano layer M oS 2 . In this context, quantum mechanical effects are considered for the short channel device of the proposed structure, where the wave function and the behavior of quantum effects are interrupted by the SchrÃ¶dinger equation. Hence the coupled Poisson – SchrÃ¶dinger
equations for the FET device with two dimensional material has been studied.

##### Department & Campus

Department of Mathematics, Coimbatore

##### Any Specific Skillsets Required by Applicants

Expected to be good in computational skills

### Faculty Profile

###### Dr. A. Ramesh Babu

Assistant Professor (Sr. Gr.), Department of Mathematics, School of Engineering, Coimbatore