• Title, Summary, Keyword: ordinary differential equations

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ON THE STABILITY AND INSTABILITY OF A CLASS OF NONLINEAR NONAUTONOMOUS ORDINARY DIFFERENTIAI, EQUATIONS

  • Sen, M.DeLa
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.2
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    • pp.243-251
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    • 2003
  • This note Presents sufficient conditions for Lyapunov's stability and instability of a class of nonlinear nonautonomous second-order ordinary differential equations. Such a class includes as particular cases a remarkably large number of differential equations with specific physical applications. Two successive nonlinear transformations are applied to the original differential equation in order to convert it into a more convenient form for stability analysis purposes. The obtained stability / instability conditions depend closely on the parameterization of the original differential equation.

BIFURCATION OF BOUNDED SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

  • Ward, James--Robert
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.707-720
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    • 2000
  • Conley index is used study bifurcation from equilibria of full bounded solutions to parameter dependent families of ordinary differential equations of the form {{{{ {dx} over {dt} }}}} =$\varepsilon$F(x, t, $\mu$). It is assumed that F(x, t,$\mu$) is uniformly almost periodic in t.

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SIMPLIFYING AND FINDING ORDINARY DIFFERENTIAL EQUATIONS IN TERMS OF THE STIRLING NUMBERS

  • Qi, Feng;Wang, Jing-Lin;Guo, Bai-Ni
    • Korean Journal of Mathematics
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    • v.26 no.4
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    • pp.675-681
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    • 2018
  • In the paper, by virtue of techniques in combinatorial analysis, the authors simplify three families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind and establish a new family of nonlinear ordinary differential equations in terms of the Stirling numbers of the second kind.

MULTIPLE PERIODIC SOLUTIONS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS ACROSS RESONANCE

  • Cai, Hua;Chang, Xiaojun;Zhao, Xin
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1433-1451
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    • 2014
  • In this paper we study the existence of multiple periodic solutions of second-order ordinary differential equations. New results of multiplicity of periodic solutions are obtained when the nonlinearity may cross multiple consecutive eigenvalues. The arguments are proceeded by a combination of variational and degree theoretic methods.

SIMPLIFYING COEFFICIENTS IN A FAMILY OF ORDINARY DIFFERENTIAL EQUATIONS RELATED TO THE GENERATING FUNCTION OF THE MITTAG-LEFFLER POLYNOMIALS

  • Qi, Feng
    • Korean Journal of Mathematics
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    • v.27 no.2
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    • pp.417-423
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    • 2019
  • In the paper, by virtue of the $Fa{\grave{a}}$ di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials.

Fluid Flow in a Multi-Layer Porous Medium (多層多孔質媒體內의 流體流動)

  • 이충구;서정윤
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.9 no.5
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    • pp.621-626
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    • 1985
  • Unsteady groundwater flow in a three-layer unconfined aquifer has been studied theoretically and experimentally. Two different methods have been used in solving the governing equations of the flow, the nonlinear partial differential equations; (1) The governing equations are linearized for each layer and approximate solutions are obtained. (2) The governing equations are transformed to nonlinear ordinary differential equations, which are solved numerically by Runge-Kutta procedure. Fine, middle sized and coarse sands are used in the experiments. It is found that the solutions from the method(2) ( the reduction of partial differential equations to ordinary differential equations) give better agreement with the experimental results than the solution from the method(1).

GEGENBAUER WAVELETS OPERATIONAL MATRIX METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS

  • UR REHMAN, MUJEEB;SAEED, UMER
    • Journal of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1069-1096
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    • 2015
  • In this article we introduce a numerical method, named Gegenbauer wavelets method, which is derived from conventional Gegenbauer polynomials, for solving fractional initial and boundary value problems. The operational matrices are derived and utilized to reduce the linear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method. We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasilinearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.

A study on the solutions of the 2nd order linear ordinary differential equations using fourier series (Fourier급수를 응용한 이계 선형 상미분방정식의 해석에 관한 연구)

  • 왕지석;김기준;이영호
    • Journal of the Korean Society of Marine Engineering
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    • v.8 no.1
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    • pp.100-111
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    • 1984
  • The methods solving the 2nd order linear ordinary differential equations of the form y"+H(x)y'+G(x)y=P(x) using Fourier series are presented in this paper. These methods are applied to the differential equations of which the exact solutions are known, and the solutions by Fourier series are compared with the exact solutions. The main results obtained in these studies are summarized as follows; 1) The product and the quotient of two functions expressed in Fourier series can be expressed also in Fourier series and the relations between the Fourier coefficients of the series are obtained by multiplying term by term. 2) If the solution of the 2nd order lindar ordinary differential equation exists in a certain interval, the solution can be obtained using Fourier series and can be expressed in Fourier series. 3) The absolute errors of Fourier series solutions are generally less in the center of the interval than in the end of the interval. 4) The more terms are considered in Fourier series solutions, the less the absolute errors.rors.

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