Numerical Methods for Singularly Perturbed Integro Differential Equation with Not All Data Necessarily Smooth

Abstract

This thesis explores mainly the issues related to singularly perturbed secondorder newlineFredholm integro-differential equations. The primary focus is on comprehending newlinetheir behavior in the presence of abrupt source terms and systems of these equations. newlineThe study deals into the intricacies of this phenomenon, examining both qualitative and newlinequantitative aspects. newlineThe introduction of this thesis initiates by highlighting key aspects of integrodifferential newlineequations. It includes a thorough examination of existing literature, focusing newlinespecifically on Singularly Perturbed Integro-Differential Equations (SPIDEs) and their newlinecorresponding solutions. Historically, mathematicians have employed various methods, newlineincluding RK, Taylor series expansion and others, to address such equations. However, newlinethis thesis advocates the use of Finite Difference Methods (FDMs) as the most appropriate newlinetechnique for the considered problems. newlineThe focus is on addressing scenarios involving non-smooth source term in the newlinecontext of second-order Singularly Perturbed Fredholm Integro-Differential Equations newline(SPFIDEs) and a specific type of convection-diffusion SPFIDEs, respectively. A numerical newlinemethod, specifically an exponentially fitted approach on a Shishkin mesh, is newlineemployed to tackle these issues. The method is demonstrated to exhibit uniform convergence newlineconcerning the singular perturbation parameter. Theoretical findings are substantiated newlineby presenting numerical results that validate the efficacy of the method. newlineEmploying the exponentially-fitted numerical method on a Shishkin mesh proves newlineinstrumental in navigating the intricacies of system of Singularly Perturbed Fredholm newlineIntegro-Differential Equations (SPFIDEs) with smooth data and non-smooth data problems. newlineThe method consistently demonstrates uniform convergence with respect to the newlinesingular perturbation parameter. Almost first order convergence for the system obtain. newlineThe method s effectiveness remains robust across various perturbation parameters, offering newlinereliable solutions.