• Bulletin of the Korean Mathematical Society
• /
• v.45 no.2
• /
• pp.241-252
• /
• 2008

By means of a Riccati transform and averaging technique some oscillation criteria are established for perturbed nonlinear differential equations of second order $(P_1)\;(p(t)x'(t))'+q(t)|x({\phi}(t)|^{{\alpha}+1}sgnx({\phi}(t))+g(t,\;x(t))=0$ $(P_2)$ and $(P_3)$ satisfying the condition (H). A comparison theorem and examples are given.

• 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.

• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.13-18
• /
• 2004

We defined Cantor dimensions of a perturbed Cantor set, and investigated a relation between these dimensions and Hausdorff and packing dimensions of a perturbed Cantor set. In this paper, we introduce another expressions of the Cantor dimensions. Using these, we study some informations which can be derived from power equations induced from contraction ratios of a perturbed Cantor set to give its Hausdorff or packing dimension. This application to a deranged Cantor set gives us an estimation of its Hausdorff and packing dimensions, which is a generalization of the Cantor dimension theorem.

• Journal of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.407-426
• /
• 1997

We consider a system of perturbed linear differential equation x' =A(t)x + f(t,x) and obtain conditions to ensure the strong-equiasymptotic stability of the zero solution.

• The Transactions of the Korean Institute of Electrical Engineers
• /
• v.43 no.4
• /
• pp.648-657
• /
• 1994

This paper presents a solution of the H$\infty$ control problem for a linear singularly perturbed system. A sufficient condition for a linear singularly perturbed system to achieve the prescribed disturbance attenuation level is obtained. Based upon this sufficient condition, an H$\infty$ controller design method which involves the solutions of two generalized algebraic Riccati equations(GRE) is developed.

• Journal of applied mathematics & informatics
• /
• v.33 no.5_6
• /
• pp.485-502
• /
• 2015

A computational method for solving singularly perturbed boundary value problem of differential equation with shift arguments of mixed type is presented. When shift arguments are sufficiently small (o(ε)), most of the existing method in the literature used Taylor's expansion to approximate the shift term. This procedure may lead to a bad approximation when the delay argument is of O(ε). The main idea for this work is to deal with constant shift arguments, which are independent of ε. In the present method, we construct the formally asymptotic solution of the problem using the method of composite expansion. The reduced problem is solved numerically by using operator compact implicit method, and the second problem is solved analytically. Error estimate is derived by using the maximum norm. Numerical examples are provided to support the theoretical results and to show the efficiency of the proposed method.

• 제어로봇시스템학회:학술대회논문집
• /
• 1995.10a
• /
• pp.432-435
• /
• 1995

In this paper a new approach to obtain the solution of the linear-quadratic Gaussian control problem for singularly perturbed discrete-time stochastic systems is proposed. The alogorithm proposed is based on exploring the previous results that the exact solution of the global discrete algebraic Riccati equations is found in terms of the reduced-order pure-slow and pure-fast nonsymmetric continuous-time algebraic Riccati equations and, in addition, the optimal global Kalman filter is decomposed into pure-slow and pure-fast local optimal filters both driven by the system measurements and the system optimal control input. It is shown that the optimal linear-quadratic Gaussian control problem for singularly perturbed linear discrete systems takes the complete decomposition and parallelism between pure-slow and pure-fast filters and controllers.

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.139-159
• /
• 2000

In this manuscript we study perturbed Newton-like methods for the solution of nonlinear operator equations in a Banach space and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlierpapers we extend these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is mot true in general. That in why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable all previous results.

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