• Title/Summary/Keyword: Nonlinear differential equations

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ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS

  • Huang, Jinghao;Jung, Soon-Mo;Li, Yongjin
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.685-697
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    • 2015
  • We investigate the stability of nonlinear differential equations of the form $y^{(n)}(x)=F(x,y(x),y^{\prime}(x),{\cdots},y^{(n-1)}(x))$ with a Lipschitz condition by using a fixed point method. Moreover, a Hyers-Ulam constant of this differential equation is obtained.

THE COMBINED MODIFIED LAPLACE WITH ADOMIAN DECOMPOSITION METHOD FOR SOLVING THE NONLINEAR VOLTERRA-FREDHOLM INTEGRO DIFFERENTIAL EQUATIONS

  • HAMOUD, AHMED A.;GHADLE, KIRTIWANT P.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.21 no.1
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    • pp.17-28
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    • 2017
  • A combined form of the modified Laplace Adomian decomposition method (LADM) is developed for the analytic treatment of the nonlinear Volterra-Fredholm integro differential equations. This method is effectively used to handle nonlinear integro differential equations of the first and the second kind. Finally, some examples will be examined to support the proposed analysis.

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.

CONTROL PROBLEMS FOR NONLINEAR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

  • Jeong, Jin-Mun;Kim, Han-Geul
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.445-453
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    • 2007
  • This paper deals with the approximate controllability for the nonlinear functional differential equations with time delay and studies a variation of constant formula for solutions of the given equations.

EXISTENCE AND UNIQUENESS OF A SOLUTION FOR FIRST ORDER NONLINEAR LIOUVILLE-CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

  • Nanware, J.A.;Gadsing, Madhuri N.
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.5
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    • pp.1011-1020
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    • 2021
  • In this paper, first order nonlinear Liouville-Caputo fractional differential equations is studied. The existence and uniqueness of a solution are investigated by using Krasnoselskii and Banach fixed point theorems and the method of lower and upper solutions. Finally, an example is given to illustrate our results.

APPLICATION OF EXP-FUNCTION METHOD FOR A CLASS OF NONLINEAR PDE'S ARISING IN MATHEMATICAL PHYSICS

  • Parand, Kourosh;Amani Rad, Jamal;Rezaei, Alireza
    • Journal of applied mathematics & informatics
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    • v.29 no.3_4
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    • pp.763-779
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    • 2011
  • In this paper we apply the Exp-function method to obtain traveling wave solutions of three nonlinear partial differential equations, namely, generalized sinh-Gordon equation, generalized form of the famous sinh-Gordon equation, and double combined sinh-cosh-Gordon equation. These equations play a very important role in mathematical physics and engineering sciences. The Exp-Function method changes the problem from solving nonlinear partial differential equations to solving a ordinary differential equation. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works.

SOME RESULTS ON MEROMORPHIC SOLUTIONS OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

  • Li, Nan;Yang, Lianzhong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1095-1113
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    • 2020
  • In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^nf^{(k)}+Q_{d_*}(z,f)=R(z)e^{{\alpha}(z)}$ and fnf(k) + Qd(z, f) = p1(z)eα1(z) + p2(z)eα2(z), where $Q_{d_*}(z,f)$ and Qd(z, f) are differential polynomials in f with small functions as coefficients, of degree d* (≤ n - 1) and d (≤ n - 2) respectively, R, p1, p2 are non-vanishing small functions of f, and α, α1, α2 are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.

THREE-POINT BOUNDARY VALUE PROBLEMS FOR A COUPLED SYSTEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Yang, Wengui
    • Journal of applied mathematics & informatics
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    • v.30 no.5_6
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    • pp.773-785
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    • 2012
  • In this paper, we establish sufficient conditions for the existence and uniqueness of solutions to a general class of three-point boundary value problems for a coupled system of nonlinear fractional differential equations. The differential operator is taken in the Caputo fractional derivatives. By using Green's function, we transform the derivative systems into equivalent integral systems. The existence is based on Schauder fixed point theorem and contraction mapping principle. Finally, some examples are given to show the applicability of our results.

Oscillation of Second Order Nonlinear Elliptic Differential Equations

  • Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.65-77
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    • 2006
  • By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$\sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.

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NUMERICAL SOLUTION OF A CLASS OF THE NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

  • Saeedi, L.;Tari, A.;Masuleh, S.H. Momeni
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.65-77
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    • 2013
  • In this paper, we develop the operational Tau method for solving nonlinear Volterra integro-differential equations of the second kind. The existence and uniqueness of the problem is provided. Here, we show that the nonlinear system resulted from the operational Tau method has a semi triangular form, so it can be solved easily by the forward substitution method. Finally, the accuracy of the method is verified by presenting some numerical computations.