NUMERICAL METHODS FOR SINGULARLY PERTURBED DIFFERENTIAL AND DELAY DIFFERENTIAL EQUATIONS

Abstract

This thesis contributes to the numerical methods for singularly perturbed differential and delay differential equations. The purpose of this research is to propose numerical tech niques for solving singularly perturbed differential and delay differential equations. We investigate, develop, and analyse numerical methods, as well as their implementations, for such challenging problems. A chapter-by-chapter structure of the thesis is as follows: • Chapter 1 presents introduction to SPPs, a brief survey on numerical analysis of SPDDEs and the need for parameter uniform numerical techniques. Objectives, motivations and a brief summary of the present work is also included in this chapter. • In Chapter 2, we have implemented finite element method for the class of SPP PDDEs with time delay. The solution of this class of problems exhibits parabolic boundary layers. The domain is discretized with a piecewise uniform mesh (Shishkin mesh) for spatial variable to capture the exponential behaviour of the solution in the boundary layer region and backward-Euler method on equidistant mesh in time direction. The error analysis id carried out in maximum norm and the proposed method is shown to be of order [O(N −1 lnN) 2+∆t]. The effect of shifts on the bound ary layer behaviour of the solution is shown by numerical experiments. The results of this chapter have been published in the journal “Numerical Method for Partial Differential Equation”. • Chapter 3 is devoted to develop numerical collocation method based on Bernstein polynomial for nonlinear singularly perturbed parabolic reaction-diffusion problems. The existence uniqueness of the proposed problem is carried out. The strategy behind this mesh is to deal with delay term and capture boundary as well as interior xiii layer behaviour of the solution. The performance of the method is corroborated by numerical examples. The results of this chapter have been accepted for publication in the journal “Journal of Mathematical Chemistry". • The main aim of Chapter 4 is to provide finite element method with Richardson extrapolation techniques for singularly perturbed parabolic time delay reaction dif fusion problem and to improve the order of convergence of the numerical scheme proposed in Chapter 2. The solution of this class of problems is polluted by a small positive parameter due to which the solution of the said problem exhibits parabolic boundary layers. The spatial variable domain is evaluated by implementing finite element method along with piecewise uniform mesh (Shishkin mesh) to capture the exponential behaviour of the solution in the boundary layer region and for time variable author has implemented implicit backward-Euler method with Richardson extrapolation on equidistant mesh in time direction to attain a good accuracy along with the higher order convergence. The proposed method is shown to be accurate of order [O(N −1 lnN) 2 + ∆t 2 ] in maximum norm. The results of this chapter have been communicated. • The main purpose of Chapter 5 is to overcome the well-known difficulties asso ciated with numerical methods and to remove restriction on the choice of mesh generation for singularly perturbed problems. In this chapter a closed-form iterative analytic approximation to a class of nonlinear singularly perturbed parabolic partial differential equation is developed and analysed for convergence. We have con sidered both parabolic reaction diffusion and parabolic convection diffusion type of problems in this chapter. The solution of this class of problem is polluted by a small dissipative parameter, due to which solution often shows boundary and interior lay ers. A sequence of approximate analytic solution for the above class of problems is constructed using Lagrange multiplier approach. Numerical experiments are pro vided to illustrate the performance of the method. The results of this chapter have been communicated. • Finally, the Chapter 6 is devoted to conclusion of the study and discussion on some future directions of the current research work.