• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1279-1292
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• 2009

In this paper, a numerical method that uses standard finite difference scheme defined on Shishkin mesh for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.31-48
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• 2010

In this article, we consider singularly perturbed Boundary Value Problems(BVPs) for second order Ordinary Differential Equations (ODEs) with Neumann boundary condition and discontinuous source term. A parameter-uniform error bound for the solution is established using the Streamline-Diffusion Finite Element Method (SDFEM) on a piecewise uniform meshes. We prove that the method is almost second order of convergence in the maximum norm, independently of the perturbation parameter. Further we derive superconvergence results for scaled derivatives of solution of the same problem. Numerical results are provided to substantiate the theoretical results.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.141-152
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• 2007

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.

• Journal of applied mathematics & informatics
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• v.37 no.5_6
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• pp.357-382
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• 2019

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.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.635-648
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• 2015

We consider a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous source term. The diffusion term of each equation is associated with a small positive parameter of different magnitude. Presence of discontinuity and different parameters creates boundary and weak interior layers that overlap and interact. A numerical method is constructed for this problem which involves an appropriate piecewise uniform Shishkin mesh. The numerical approximations are proved to converge to the continuous solutions uniformly with respect to the singular perturbation parameters. Numerical results are presented which illustrates the theoretical results.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.495-508
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• 2016

A singularly perturbed reaction-diffusion fourth-order ordinary differential equation(ODE) with discontinuous source term is considered. Due to the discontinuity, interior layers also exist. The considered problem is converted into a system of weakly coupled system of two second-order ODEs, one without parameter and another with parameter ε multiplying highest derivatives and suitable boundary conditions. In this paper a computational method for solving this system is presented. A zero-order asymptotic approximation expansion is applied in the second equation. Then, the resulting equation is solved by the numerical method which is constructed. This involves non-overlapping Schwarz method using Shishkin mesh. The computation shows quick convergence and results presented numerically support the theoretical results.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1291-1310
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• 2013

This paper develops least-squares pseudo-spectral collocation methods for elliptic boundary value problems having interface conditions given by discontinuous coefficients and singular source term. From the discontinuities of coefficients and singular source term, we derive the interface conditions and then we impose such interface conditions to solution spaces. We define two types of discrete least-squares functionals summing discontinuous spectral norms of the residual equations over two sub-domains. In this paper, we show that the homogeneous least-squares functionals are equivalent to appropriate product norms and the proposed methods have the spectral convergence. Finally, we present some numerical results to provide evidences for analysis and spectral convergence of the proposed methods.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1035-1054
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• 2010

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.

• Journal of Korea Water Resources Association
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• v.38 no.2
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• pp.155-166
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• 2005

Upwind schemes are very well adapted to the discontinuous flow and have become popular for applications Involving dam break flow, transcritical Slow, etc. However, upwind schemes have been applied mainly to the idealized problems not to the natural channels with irregular geometry so far because of the error due to source terms. In this paper, the new type of upwind discretization of source terms, which uses the normalized Jacobian to discretize the source terms, is proposed. As results of tests to flows with source terms by the upwind models, the method proposed in this paper is proved as efficient and accurate. This generalized method for differencing source terms is simple and might beapplicable to diverse type of flux upwind discretization scheme in finite difference method.