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Valanarasu, T.,Ramanujam, N. 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.
RAMANUJAM,T. VALANARASU 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.
A. Ramesh Babu,T. Valanarasu 한국전산응용수학회 2019 Journal of applied mathematics & informatics Vol.37 No.5
In this paper, we consider boundary value problem for a weakly coupled system of two singularly perturbed differential equations of convection diffusion type with discontinuous source term. In general, solution of this type of problems exhibits interior and boundary layers. A numerical method based on streamline diffusiom finite element and Shishkin meshes is presented. We derive an error estimate of order O(N^-2 ln^2 N) in the maximum norm with respect to the perturbation parameters. Numerical experiments are also presented to support our theoritical results.
BABU, A. RAMESH,VALANARASU, T. The Korean Society for Computational and Applied M 2019 Journal of applied mathematics & informatics Vol.37 No.5
In this paper, we consider boundary value problem for a weakly coupled system of two singularly perturbed differential equations of convection diffusion type with discontinuous source term. In general, solution of this type of problems exhibits interior and boundary layers. A numerical method based on streamline diffusiom finite element and Shishkin meshes is presented. We derive an error estimate of order $O(N^{-2}\;{\ln}^2\;N$) in the maximum norm with respect to the perturbation parameters. Numerical experiments are also presented to support our theoritical results.
N. Ramanujam,R. Mythili Priyadharshini,T. Valanarasu 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.5
We consider a mixed type singularly perturbed one dimensional elliptic problem with discontinuous source term. The domain under consideration is partitioned into two subdomains. A convection-diffusion and a reaction-diffusion type equations are posed on the first and second subdomains respectively. Two hybrid difference schemes on Shishkin mesh are constructed and we prove that the schemes are almost second order convergence in the maximum norm independent of the diffusion parameter. Error bounds for the numerical solution and its numerical derivative are established. Numerical results are presented which support the theoretical results.
A. Tamilselvan,N. Ramanujam,R. Mythili Priyadharshini,T. Valanarasu 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.1
In this paper, a numerical method for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with the mixed type boundary conditions is presented. Parameter-uniform error bounds for the numerical solution and also to numerical derivative are established. Numerical results are provided to illustrate the theoretical results.
Tamilselvan, A.,Ramanujam, N.,Priyadharshini, R. Mythili,Valanarasu, T. The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.1
In this paper, a numerical method for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with the mixed type boundary conditions is presented. Parameter-uniform error bounds for the numerical solution and also to numerical derivative are established. Numerical results are provided to illustrate the theoretical results.
Priyadharshini, R. Mythili,Ramanujam, N.,Valanarasu, T. The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.5
We consider a mixed type singularly perturbed one dimensional elliptic problem with discontinuous source term. The domain under consideration is partitioned into two subdomains. A convection-diffusion and a reaction-diffusion type equations are posed on the first and second subdomains respectively. Two hybrid difference schemes on Shishkin mesh are constructed and we prove that the schemes are almost second order convergence in the maximum norm independent of the diffusion parameter. Error bounds for the numerical solution and its numerical derivative are established. Numerical results are presented which support the theoretical results.