• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.109-130
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• 2010

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.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1001-1015
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• 2009

In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a small parameter multiplying the highest derivative. We prove that the schemes are almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and its derivative are established. Numerical results are provided to illustrate the theoretical results.

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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• pp.25-45
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• 2022

In this study, the non-standard finite difference method for the numerical solution of singularly perturbed parabolic reaction-diffusion subject to Robin boundary conditions has presented. To discretize temporal and spatial variables, we use the implicit Euler and non-standard finite difference method on a uniform mesh, respectively. We proved that the proposed scheme shows uniform convergence in time with first-order and in space with second-order irrespective of the perturbation parameter. We compute three numerical examples to confirm the theoretical findings.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.491-505
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• 2022

This paper is concerned with singularly perturbed convection-diffusion parabolic partial differential equations which have time-delayed. We used the Crank-Nicolson(CN) scheme to build a fitted operator to solve the problem. The underling method's stability is investigated, and it is found to be unconditionally stable. We have shown graphically the unstableness of CN-scheme without fitting factor. The order of convergence of the present method is shown to be second order both in space and time in relation to the perturbation parameter. The efficiency of the scheme is demonstrated using model examples and the proposed technique is more accurate than the standard CN-method and some methods available in the literature, according to the findings.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.4
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• pp.511-517
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• 2008

In this paper, we investigate h-stability of nonlinear integro-differential equations.

• Kyungpook Mathematical Journal
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• v.33 no.2
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• pp.163-171
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• 1993

• Kyungpook Mathematical Journal
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• v.32 no.3_spc
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• pp.311-320
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• 1992

• 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 applied mathematics & informatics
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• v.36 no.1_2
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• pp.135-154
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• 2018

In this paper, an almost second order overlapping Schwarz method for singularly perturbed third order convection-diffusion type problem is constructed. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the combination of classical finite difference scheme and central finite difference scheme on a uniform mesh while on the non-layer region we use the midpoint difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. We proved that, when appropriate subdomains are used, the method produces convergence of second order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme are it reduce iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

• Journal of applied mathematics & informatics
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• v.38 no.1_2
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• pp.113-132
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• 2020

In this paper, a class of linear second order singularly perturbed delay differential turning point problems containing a small delay (or negative shift) on the reaction term and when the solution of the problem exhibits twin boundary layers are examined. A hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the problem. We proved that the method is almost second order ε-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results.