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Parameter Uniform Numerical Methods for Singularly Perturbed Parabolic Partial Differential Equations

Publisher : Journal Article

Source : Journal Article

Campus : Coimbatore

School : School of Engineering

Center : Amrita Innovation & Research

Department : Mathematics

Verified : Yes

Year : 2019

Abstract : This thesis deals with some efficient and higher order numerical methods for solving singularly perturbed parabolic partial differential equations (SPPDEs) in one and two dimensions. The model problems includes one dimensional SPPDEs, time delay SPPDEs, two dimensional SPPDEs and mixed type of parabolic-elliptic problems. In general, these problems are described by partial differential equations in which the highest order derivative is multiplied by a small parameter ε, known as the “singular perturbation parameter” (0 < ε ≪ 1). If the parameter ε tends to 0, the problem has a limiting solution, which is called the solution of the reduced problem. The regions of nonuniform convergence lie near the boundary of the domain, which are known as boundary/interior layers. Due to this layer phenomena, it is a very difficult and challenging task to provide parameter uniform numerical methods for solving SPPDEs. The term “parameter uniform” is meant to identify those numerical methods, in which the approximate solution converges (measured in the supremum norm) independently with respect to the perturbation parameter. The purpose of the thesis is to analyze, develop and optimize the parameter uniform fitted mesh methods for solving SPPDEs on Shishkin-type meshes like the standard Shishkin mesh (S-mesh), the Bakhvalov-Shishkin mesh (B-S mesh), and the modified Bakhvalov-Shishkin mesh (M-B-S mesh) in the spacial direction. This thesis contains eight chapters. It begins with introduction along with the objective and the motivation for solving SPPDEs. Next, Chapter 2 contains a time delay SPPDE which is solved using a hybrid scheme

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