Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

A Finite Element Domain Decomposition Approximation for a Semilinear Parabolic Singularly Perturbed Differential Equation

In this paper, we propose a Monotone Schwarz Iterative Method (MSIM) under the framework of Domain Decomposition Strategy for solving semilinear parabolic singularly perturbed partial differential equations (SPPDEs). A three-step Taylor Galerkin Finite Element (3TGFE) approximation of semilinear parabolic SPPDE is carried out during each of the stages of the MSIM. Appropriate Interface Problems are introduced to update the subdomain boundary conditions in the Monotone Iterative Domain Decomposition (MIDD) method. The convergence of the MIDD method has been established. In addition, the stability and $\epsilon$-uniform convergence of 3TGFE based MIDD has been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed MSIM has been established. Also, the proposed 3TGFE based MIDD has been successfully implemented on a couple of test problems.