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

2017 ◽  
Vol 18 (1) ◽  
pp. 41-55 ◽  
Author(s):  
Sunil Kumar ◽  
B. V. Rathish Kumar
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Singularly Perturbed ◽  
Test Problems ◽  
Galerkin Finite Element ◽  
Monotone Iterative ◽  
Singularly Perturbed Differential Equation ◽  
The Stability ◽  
Perturbed Differential Equation ◽  
Semilinear Parabolic

AbstractIn 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.

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

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 213 ◽  
Author(s):  
Assiya Zhumanazarova ◽  
Young Im Cho
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Singularly Perturbed Problems ◽  
Cauchy Function ◽  
Higher Derivatives ◽  
Singularly Perturbed Differential Equation ◽  
Perturbed Differential Equation

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.

THREE-STEP TAYLOR GALERKIN METHOD FOR SINGULARLY PERTURBED GENERALIZED HODGKIN–HUXLEY EQUATION

2010 ◽  
Vol 01 (02) ◽  
pp. 257-276
Author(s):  
VIVEK SANGWAN ◽  
B. V. RATHISH KUMAR ◽  
S. V. S. S. N. V. G. K. MURTHY ◽  
MOHIT NIGAM
Keyword(s):  
Boundary Layer ◽  
Finite Element Method ◽  
Finite Element ◽  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Galerkin Finite Element Method ◽  
Third Order ◽  
Galerkin Finite Element ◽  
Huxley Equation

A numerical study is carried out for the singularly perturbed generalized Hodgkin–Huxley equation. The equation is nonlinear which mimics the ionic processes at a real nerve membrane. A small parameter called singular perturbation parameter is introduced in the highest order derivative term. Keeping other parameters fixed, as this singular perturbation parameter approaches to zero, a boundary layer occurs in the solution. Three-step Taylor Galerkin finite element method is employed on a piecewise uniform Shishkin mesh to solve the equation. To procure more accurate temporal differencing, the method employs forward-time Taylor series expansion including time derivatives of third order which are evaluated from the governing singularly perturbed generalized Hodgkin–Huxley equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The method is third-order accurate in time. The code based on the purposed scheme has been validated against the cases for which the exact solution is available. It is also observed that for the Singularly Perturbed Generalized Hodgkin–Huxley equation, the boundary layer in the solution manifests not only by varying the singular perturbation parameter but also by varying the other parameters appearing in the model.

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