On connections of the li\enard equation with some equations of. K mar 06, 2011 we prove that every positive solution of the maxtype difference equation, authors. The equation 9 can be transformed into an equivalent twodimensional system of ordinary differential equation. An ordinary differential equation or ode is an equation involving derivatives of an unknown quantity with respect to a single variable. The general linear difference equation of order r with constant coef. Pdf a complete classification of lienard equation researchgate.
Ali gelisken, cengiz cinar and ibrahim yalcinkaya citation. The lienards equation describes the oscillating circuits. In simple cases, a di erence equation gives rise to an associated auxiliary equation rst explained in 7. Lienard, equation, equations, exact solutions, solvable. Analysis of lienard iitype oscillator equation by symmetrytransformation methods. Uniqueness of periodic solution for a class of lienard. Existence of positive solutions of the lienard differential equation. Homogeneous difference equations the simplest class of difference equations of the form 1 has f n 0, that is simply. Existence of positive solutions of the lienard differential. Solving the lienard equation by differential transform method mashallah matinfar. In this paper, we investigate a numerical solution of lienards equation. In this paper, the differential transform method dtm is proposed for solving the. In this study, we consider a lienard iitype harmonic.
It is significant that the nonlinear term contains two variables. Jul 29, 2015 we investigate the connection between the linear harmonic oscillator equation and some classes of secondorder nonlinear ordinary differential equations of lienard and generalized lienard type, which physically describe important oscillator systems. Aug 30, 2014 a class of exact solutions is obtained for the lienardtype ordinary nonlinear differential equation. In order to find the first integrals of the form, we can use a practicable procedure and apply it to the lienard iitype harmonic nonlinear oscillator equation. As a first step in our study, the secondorder lienardtype equation is transformed into a first abel kind, first order differential equation. Firstorder constantcoefficient linear homogeneous difference equation. Autonomous equations the general form of linear, autonomous, second order di. We investigate the connection between the linear harmonic oscillator equation and some classes of second order nonlinear ordinary differential equations of. Exact solutions ordinary differential equations secondorder nonlinear ordinary differential equations lienard equation 17. K differential transform method for lienard equation in comparison with the exact solution of u m 4, n. In mathematics, more specifically in the study of dynamical systems and differential equations, a lienard equation is a second order differential equation, named after the french physicist alfredmarie lienard. That is, we have looked mainly at sequences for which we could write the nth term as a n fn for some known function f.
A class of exact solutions of the lienardtype ordinary. An excellent book for real world examples of solving differential equations is that of shampine, gladwell, and thompson 74. Using this method, it is possible to find the exact solution or an approximate solution of the problem. On the criteria for integrability of the li\e nard equation. The authors would like to thank olaf hansen, california state university at san. Ghanbari department of mathematics, university of mazandaran, babolsar 474161468, iran received february 14 2008, accepted june 11 2008. Pdf lienard and riccati differential equations related via. Exact solutions of the lienard and generalized lienardtype. The lienard equation is a nonlinear second order differential equation proposed by lienard 4andis presentedas. For a certain class of differential equations called lienard systems, one can prove the existence of a stable limit cycle. Pdf accurate numerical method for lienard nonlinear.
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given. Accurate numerical method for lienard nonlinear differential. In mathematics, more specifically in the study of dynamical systems and differential equations, a lienard equation is a second order differential equation, named. E is a polynomial of degree r in e and where we may assume that the coef. On explicit exact solutions for the lienard equation and its. With the use of an exact integrability condition for the abel equation chiellini lemma, the exact general solution of the abel equation can be. Solving the lienard equation by differential transform method. Linear di erence equations posted for math 635, spring 2012.
Ordinary differential equations an ordinary differential equation or ode is an equation involving derivatives of an unknown quantity with respect to a single variable. A class of exact solutions of the li\enard type ordinary nonlinear. Hence, our problem is reduced to construct one completely continuous operator in which sends into, such that the fixed points of operator in some openbounded set are the odd antiperiodic solutions of 1. Solving the lienard equation by differential transform. Periodic solutions for a class of conservative lienardtype.
A numerical implementation of the variational iteration method for the lienard equation m. Orhan and ozer advances in difference equations 2016 2016. The proposed method is a combination of the fractional taylor series and the residual functions. A numerical solution of fractional lienards equation by. Accurate numerical method for lienard nonlinear differential equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized lienard type laplacian differential equations are obtained.
Elaydi and others published an introduction to difference equation find, read and cite all the research you need on researchgate. Existence of periodic solutions for a prescribed mean. Then, the given interval is discretized, and the method is formulated by using newtons backward difference interpolation formula. It can be written in a form of a 2dimensional system 163. Analysis of lienard iitype oscillator equation by symmetry. The residual power series rps method is implemented to find an approximate solution to this problem. Lienard and riccati differential equations related via lie algebras article pdf available in discrete and continuous dynamical systems series b 102. Existence of limit cycles system of lienard differential. As for rst order equations we can solve such equations by 1. Linear difference equations with constant coef cients. Existence of periodic solutions for a prescribed mean curvature lienard plaplacian equation with two delays advances in difference equations, nov 2014 zhiyan li, tianqing an, weigao ge. What is more, the lienard equation often appears as a traveling wave re duction of nonlinear partial differential equations. Linear difference and functional equations with one independent variable 1. A numerical implementation of the variational iteration.
Linear difference and functional equations containing unknown function with two different arguments firstorder linear difference equations. It is not to be confused with differential equation. Aug 25, 2010 obviously, the operator is continuous based on the proof of theorem 3. The lienard s equation describes the oscillating circuits.
First, the second order lienard differential equation is transformed into a first order system of equations. Centers of classical lienard equations are related to scalar differential equations x. The suggested scheme is a merger of homotopy analysis technique, classical laplace transform and homotopy polynomials. By means of a method inspired by quantum mechanics, and which consists of the deformation of the phase space coordinates of the harmonic. Pdf exact solutions of the lienard and generalized lienardtype. The numerical results with using differential transform method for lienard equation in comparison with the exact solution of u m 4, n. The lienard type second order nonlinear differential equation of the. The zero on the righthand side signi es that this is a homogeneous di erence equation. Difference equations differential equations to section 1.
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