On Mesh free Method for Singularly Perturbed Problems

Abstract

In the present thesis, an attempt has been made to demonstrate some robust and efficient mesh-free techniques for approximating the solutions of singularly perturbed problems. It is observed that the proposed mesh-free method are more efficient than the conventional methods and are, at the same time, conceptually simple. We consider one and two-dimensional singularly perturbed problems commonly arise in the field of science and engineering. Besides it, a well-developed theory for their convergence and error estimates has also been proved. Numerical experiments have been carried out extensively to support the theoretical results. The research work started with the following three objectives: To propose and implement a mesh-free method for solving linear/non-linear singularly perturbed problems in one dimension. To implement the proposed mesh-free method for solving the higher dimensional singularly perturbed problem. To implement the proposed mesh-free method for solving a real-life problem. To accomplish these objectives, the research work has been carried out and organized in form of Chapters. The thesis is categorized into seven Chapters. A brief overview of the Chapters is given below: In Chapter 1 of the proposed thesis work focuses on singularly perturbed differential equations and their importance in various fields. It provides an introduction to basic definitions and notations related to singularly perturbed problems (SPP), highlighting their significance in real-world applications. The Chapter emphasizes on the need for efficient and accurate methods for solving SPP, and reviews various analytic and numerical techniques proposed in this regard. The discussion also includes a summary of meshfree schemes and their developments, highlighting their advantages and limitations. The presented literature survey on this topic has led to the identification of gaps in the existing approaches which motivates for the objectives of the Ph.D. thesis. Overall, Chapter 1 lays the foundation for the subsequent