• Kyungpook Mathematical Journal
• /
• v.61 no.3
• /
• pp.591-612
• /
• 2021

In this paper, a uniformly convergent numerical scheme is designed for solving singularly perturbed reaction-diffusion problems. The problem is converted to an equivalent weak form and then a Galerkin finite element method is used on a piecewise uniform Shishkin mesh with linear basis functions. The convergence of the developed scheme is proved and it is shown to be almost fourth order uniformly convergent in the maximum norm. To exhibit the applicability of the scheme, model examples are considered and solved for different values of a singular perturbation parameter ε and mesh elements. The proposed scheme approximates the exact solution very well.

• Journal of applied mathematics & informatics
• /
• v.39 no.5_6
• /
• pp.655-676
• /
• 2021

A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.

• Journal of applied mathematics & informatics
• /
• v.22 no.1_2
• /
• pp.317-329
• /
• 2006

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

• Journal of applied mathematics & informatics
• /
• v.27 no.5_6
• /
• pp.1279-1292
• /
• 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
• /
• v.27 no.3_4
• /
• pp.517-529
• /
• 2009

In this paper, a singularly perturbed reaction-convection-diffusion problem with two parameters is considered. A parameter-uniform error bound for the numerical derivative is derived. The numerical method considered here is a standard finite difference scheme on piecewise-uniform Shishkin mesh, which is fitted to both boundary and initial layers. Numerical results are provided to illustrate the theoretical results.

• Proceedings of the KIEE Conference
• /
• 2002.07d
• /
• pp.2155-2157
• /
• 2002

In this paper, we present a stabilizing controller for the singularly perturbed discrete time bilinear systems. The proposed control method guarantees the robust stability for the resulting closed loop system with multi-input. We verify the proposed algorithm by a numerical example.

• 제어로봇시스템학회:학술대회논문집
• /
• 2001.10a
• /
• pp.34.5-34
• /
• 2001

This paper presents a reliable H$\infty$ controller design for singularly perturbed systems by the delta operator approach that guarantees stability with a known H$\infty$ norm bound in case of failures in some control channels. Prespecified are the control channels that may experience failures. Sensor outage is covered in this paper It is shown that the delta systems have improved finite wort length characteristics in the example.

• Communications of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.1015-1027
• /
• 2019

A class of systems of singularly perturbed convection diffusion type equations with integral boundary conditions is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The suggested method is of almost first order convergence. An error estimate is derived in the discrete maximum norm. Numerical examples are presented to validate the theoretical estimates.

• Journal of applied mathematics & informatics
• /
• v.37 no.3_4
• /
• pp.183-200
• /
• 2019

We have constructed a robust numerical method on Shishkin mesh for a class of convection diffusion type turning point problems with Robin type boundary conditions. Supremum norm is used to derive error estimates which is of order O($N^{-1}$ ln N). Theoretical results are verified by providing numerical examples.

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