Dr. Ilhame Amirali, Associate Professor, Mathematics Department, Düzce University, Turkiye

Biography: Dr. Ilhame Amirali was born in 19th November 1972 in Baku, Azerbaijan. After graduating from high school with high degree in Baku she started her university education at the Baku State University Department of Mathematics. She graduated with MSc degree from Baku State University in 1994. She finished her second MSc and then PhD with a title “Numerical Solutions of Singularly Perturbed Boussinesque System” in 2006 in Turkey. She is working at the Duzce University, Faculty of Sciences, Department of Mathematics, Turkey. Currently She is a Visiting Professor at the Oklahoma University, College of Arts & Sciences, Department of Mathematics, Oklahoma, USA. Statement of her research interests Numerical Analysis, Numerical Methods for Differential and Integral Equations, Design and Analysis of Numerical Methods for Singularly Perturbed Differential Equations.

Speech Title: Numerical Methods for the Singularly Perturbed Problems Containing Integral Boundary Condition

Abstract: Singularly perturbed differential equations are typically characterized by a small parameter ε multiplying some or all of the highest order terms in the differential equation as normally boundary layers occur in their solutions. These equations play an important role in today’s advanced scientific computations. Many mathematical models starting from fluid dynamics to the problems in mathematical biology are modelled by singularly perturbed problems. Typical examples include high Reynold’s number flow in the fluid dynamics, heat transport problem etc. The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problem undergo rapid changes within very thin layers near the boundary or inside the problem domain. It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter is small. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value, i.e. methods that are convergence ε−uniformly.

In this work, parameter-uniform numerical method for a singularly perturbed ordinary differential equation containing integral boundary condition is studied. Asymptotic estimates on the solution and its derivatives are derived. A numerical algorithm based on upwind finite difference operator and appropriate piecewise uniform mesh is constructed. Parameter-uniform error estimate for the numerical solution is established. Numerical results are presented, which illustrate the theoretical results.