• Title/Summary/Keyword: Perturbed Equations

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EXPONENTIALLY FITTED NUMERICAL SCHEME FOR SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS INVOLVING SMALL DELAYS

  • ANGASU, MERGA AMARA;DURESSA, GEMECHIS FILE;WOLDAREGAY, MESFIN MEKURIA
    • Journal of applied mathematics & informatics
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    • v.39 no.3_4
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    • pp.419-435
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    • 2021
  • This paper deals with numerical treatment of singularly perturbed differential equations involving small delays. The highest order derivative in the equation is multiplied by a perturbation parameter 𝜀 taking arbitrary values in the interval (0, 1]. For small 𝜀, the problem involves a boundary layer of width O(𝜀), where the solution changes by a finite value, while its derivative grows unboundedly as 𝜀 tends to zero. The considered problem contains delay on the convection and reaction terms. The terms with the delays are approximated using Taylor series approximations resulting to asymptotically equivalent singularly perturbed BVPs. Inducing exponential fitting factor for the term containing the singular perturbation parameter and using central finite difference for the derivative terms, numerical scheme is developed. The stability and uniform convergence of difference schemes are studied. Using a priori estimates we show the convergence of the scheme in maximum norm. The scheme converges with second order of convergence for the case 𝜀 = O(N-1) and for the case 𝜀 ≪ N-1, the scheme converge uniformly with first order of convergence, where N is number of mesh intervals in the domain discretization. We compare the accuracy of the developed scheme with the results in the literature. It is found that the proposed scheme gives accurate result than the one in the literatures.

AN ASYMPTOTIC FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE WITH DISCONTINUOUS SOURCE TERM

  • Babu, A. Ramesh;Ramanujam, N.
    • 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.

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Stability Analysis of a Fluid Dynamic Journal Bearing Considering the Tilting Motion (틸팅 운동을 고려한 유체 동압 베어링의 안정성 해석)

  • Kim, Myung-Gyu;Jang, Gun-Hee
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 2008.11a
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    • pp.394-400
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    • 2008
  • This paper presents an analytical method to investigate the stability of FDBs (fluid dynamic bearings) considering the tilting motion. The perturbed equations of motion are derived with respect to translational and tilting motion for the general rotor-bearing system with five degrees of freedom. The Reynolds equations and their perturbed equations are solved by using the FEM in order to calculate the pressure, load capacity, and the stiffness and damping coefficients. This research introduces the radius of gyration to the equations of notion in order to express the mass moment of interia with respect to the critical mass. Then the critical mass of FDBs is determined by solving the eigenvalue problem of the linear equations of motion. This research is numerically validated by comparing the stability chart of FDBs with the time response of the whirl radius obtained from the direct integration of the equations of motion. This research shows that the tilting motion is one of the major design considerations to determine the stability of rotating system. It also shows that the stability of FDBs considering only translation is overestimated in comparison with the stability of FDBs considering both translational and tilting motion.

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Dynamic Analysis of an Optical Disk Drive with an Automatic Ball Balancer (자동볼평형장치가 부착된 광디스크 드라이브의 동특성해석)

  • Kim, Kang-Sung;Chung, Jin-Tai
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.26 no.12
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    • pp.2511-2518
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    • 2002
  • Dynamic behaviors and stability of an optical disk drive coupled with an automatic ball balancer (ABB) are analyzed by a theoretical approach. The feeding system is modeled a rigid body with six degree-of-freedom. Using Lagrange's equation, we derive the nonlinear equations of motion for a non -autonomous system with respect to the rectangular coordinate. To investigate the dynamic stability of the system in the neighborhood of the equilibrium positions, the monodromy matrix technique is applied to the perturbed equations. On the other hand, time responses are computed by the Runge -Kutta method. We also investigate the effects of the damping coefficient and the position of ABB on the dynamic behaviors of the system.

Dynamic Analysis of an Optical Disk Drive with an Automatic Ball Balancer (자동볼평형장치가 부착된 광디스크 드라이브의 동특성해석)

  • 김강성;정진태
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • 2001.11b
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    • pp.983-988
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    • 2001
  • Dynamic behaviors and stability of an optical disk drive coupled with an automatic ball balancer(ABB) are analyzed by a theoretical approach. The feeding system is modeled a rigid body with six degree-of-freedom. Using Lagrange's equation, we derive the nonlinear equations of motion for a non-autonomous system with respect to the rectangular coordinate. To investigate the dynamic stability of the system in the neighborhood of the equilibrium positions, the monodromy matrix technique is applied to the perturbed equations. On the other hand, time responses are computed by the Runge-Kutta method. We also investigate the effects of the damping coefficient and the position of ABB on the dynamic behaviors of the system.

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HYBRID FIXED POINT THEORY AND EXISTENCE OF EXTREMAL SOLUTIONS FOR PERTURBED NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

  • Dhage, Bapurao C.
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.315-330
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    • 2007
  • In this paper, some hybrid fixed point theorems are proved which are further applied to first and second order neutral functional differential equations for proving the existence results for the extremal solutions under the mixed Lipschitz, compactness and monotonic conditions.

NUMERICAL METHOD FOR SINGULARLY PERTURBED THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

  • ROJA, J. CHRISTY;TAMILSELVAN, A.
    • 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.

A NEW APPROACH FOR STABILIZATION OF NONSTENDAD SINGULARLY PERTYRBED SYSTEMS

  • Xu, Hua;Mukaidani, Hiroaki;Mizukami, Koichi
    • 제어로봇시스템학회:학술대회논문집
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    • 1995.10a
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    • pp.99-102
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    • 1995
  • In this paper, we consider the stabilization problem of nonstandard singularly perturbed systems by using state feedback. Different fro the existing sequenetial designn procedures, we propose a parallel design method to construct the stabilizing controller. The method involves solving two completely independent algebraic Riccati equations.

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