• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.69-81
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• 1990

In this paper we investigate the problem of exponential asymptotic stability (EAS) in perturbed nonlinear systems of the differential system x' = f(t, x). Also, a simple method for constructing Liapunov functions is used to prove a kind of Massera type converse theorem.

• The Pure and Applied Mathematics
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• v.22 no.3
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• pp.215-227
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• 2015

In this paper, we investigate bounds for solutions of nonlinear functional differential systems using the notion of t_∞-similarity.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.279-287
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• 2012

In this paper, we investigate bounds for solutions of the nonlinear differential systems using the notion of $t_{\infty}$-similarity.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.277-302
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• 2017

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of reaction-diffusion type of third order Ordinary Differential Equations (ODEs). The SPBVP is reduced into a weakly coupled system of one first order and one second order ODEs, one without the parameter and the other with the parameter ${\varepsilon}$ multiplying the highest derivative subject to suitable initial and boundary conditions, respectively. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. The weakly coupled system is decoupled by replacing one of the unknowns by its zero-order asymptotic expansion. Finally the present numerical method is applied to the decoupled system. In order to get a numerical solution for the derivative of the solution, the domain is divided into three regions namely two inner regions and one outer region. The Shooting method is applied to two inner regions whereas for the outer region, standard finite difference (FD) scheme is applied. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing. The main advantage of this method is that due to decoupling the system, the computation time is very much reduced.

• Journal of Korea Society of Industrial Information Systems
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• v.7 no.1
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• pp.75-80
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• 2002

We investigate sorry: asymptotic stabilities of the zero solution for the perturbed linear differential system dx/dt=A(t)x+e(t, x)+f(t,x), by using Perron's method and integral inequalities, etc. and we also find some sufficient conditions that ensure some asymptotic stabilities of the zero solution of the system And hence we obtain several results of it.

• Journal of Institute of Control, Robotics and Systems
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• v.14 no.12
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• pp.1270-1277
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• 2008

A Haar wavelet based numerical method for solving singularly perturbed linear time invariant system is presented in this paper. The reduced pure slow and pure fast subsystems are obtained by decoupling the singularly perturbed system and differential matrix equations are converted into algebraic Sylvester matrix equations via Haar wavelet technique. The operational matrix of integration and its inverse matrix are utilized to reduce the computational time to the solution of algebraic matrix equations. Finally a numerical example is given to demonstrate the validity and applicability of the proposed method.

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.697-705
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• 2014

In this paper, we investigate bounds for solutions of the nonlinear functional differential systems.

• 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.3_4
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• pp.689-706
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• 2008

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.225-238
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• 2019

We show the boundedness and uniform Lipschitz stability for the solutions to the functional perturbed differential system $$y^{\prime}=f(t,y)+{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{t_0}}^t}g(s,y(s),\;T_1y(s))ds+h(t,y(t),\;T_2y(t))$$, under perturbations. We impose conditions on the perturbed part ${\int_{t_0}^{t}}g(s,y(s)$, $T_1y(s))ds$, $h(t,y(t)$, $T_2y(t))$, and on the fundamental matrix of the unperturbed system y' = f(t, y) using the notion of h-stability.