Numerical Solution for Multi-parameter Singularly Perturbed Initial and Boundary Value Problems

Abstract

The main objective of this thesis is to provide some efficient numerical techniques for solving various class of singularly perturbed ordinary and partial differential equations. Generally, in singular perturbation problems(SPPs), a small parameter " known as “singular perturbation parameter” is involved as the coefficient of the highest order derivatives. When the parameter " goes to zero, the perturbations are operative over a thin region, where the dependent variable undergoes very rapid change on the domain of interest. These thin regions are frequently referred as boundary layers. Due to the layer behavior, classical numerical methods are unfit for solving such problems on uniform meshes unless the mesh size is too small in comparison with the ". Also, these approaches on a uniform mesh fails to decrease the maximum pointwise error until the size of mesh and the parameter " have the same order of magnitude. In this context, sufficient mesh points are placed inside the layers to produce a satisfactory numerical approximation. These drawbacks motivate to develop parameter uniform numerical methods, where the maximum pointwise error on discrete maximum norm are independent of the parameter. The theme of this thesis is, therefore, to analyze, improve and optimize some parameter-uniform numerical methods for singularly perturbed initial and boundary value problems. This is accomplished by constructing special type of layer adapted meshes resolving the boundary layers. At first, a hybrid numerical scheme is proposed for singularly perturbed initial value problems on layer adapted meshes like standard Shishkin mesh (S-mesh), Bakhvalov-Shishkin mesh (B-S mesh) and Vulanovic mesh (V-mesh). The finite difference scheme combines the second order central difference scheme on the fine mesh with a modified midpoint upwind scheme on the coarse mesh which provides a second order optimal accuracy for both numerical solution and scaled numerical derivatives. Then, a new spline based hybrid finite difference technique is introduced which combines the cubic spline difference scheme on the fine mesh with a modified midpoint upwind scheme on the coarse mesh. It is observed, that the newly proposed hybrid scheme on the B-S mesh is optimal and more accurate than the one obtained on the S-mesh. Thereafter, two parameter SPP is considered which contains a delay term. Till date no result exists so far for two parameter SPP containing a delay term. To obtain "-uniform convergence for such model, an upwind scheme is used followed by a hybrid scheme on S-mesh. Then, such idea is extended for a singularly perturbed parabolic reaction-diffusion problem with time delay. In this context, an upwind scheme is used for the space direction on S-mesh and B-S mesh and the implicit Euler scheme for the time variable is used on an uniform mesh to approximate the solution. It is shown that the proposed scheme is of first order rate of convergence which is optimal on B-S mesh. To increase the rate of convergence, a hybrid scheme which consists the upwind scheme, midpoint upwind scheme and the second order central difference scheme for the spatial derivatives and the backward Euler scheme on a uniform mesh in the time derivative is developed, which provides a second second order accuracy. Finally, a singularly perturbed parabolic PDE containing both positive and negative shift arguments (small) in the space variable as well as delay in the time variable is approximated by using the upwind finite difference scheme for space on Shishkin type meshes and the backward Euler scheme for time derivative on uniform mesh. It is observed that the proposed method is " uniform convergent and first order with respect to both space and time. Extensive numerical results are shown in shape of tables and figures which confirm the theoretical findings.