Order Nonlinear ODE Research Articles

On an isomonodromy deformation equation without the Painlevé property

We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.

Some Reduction and Exact Solutions of a Higher-Dimensional Equation

The conservation laws of the(3+1)-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitutionu=vxto the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.

Double Reduction and Exact Solutions of Zakharov—Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the (1+2)-dimensional Zakharov—Kuznetsov modified equal width (ZK-MEW) equation with power law nonlinearity are constructed by using Noether's approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.

$${\mathcal{N} = 2}$$ N = 2 Supersymmetric AdS4 Solutions of M-theory

We analyse the most general N=2 supersymmetric solutions of D=11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y_7. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structure on Y_7. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS_4 x S^7 solution with non-trivial four-form flux.

A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-Bénard Convection

The simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type is considered where components of are known at one of the boundaries and components of are specified at the other boundary. The map is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of Ha (2001) to the BVPs arising in the higher order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-Bénard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role.

Closed form analytical approach for a second order non-linear ODE interpreting EPDM vulcanization with peroxides

In this paper, the recently presented kinetic model proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) to interpret EPDM peroxide vulcanization is extensively revised and the resultant second order ODE is solved by means of an approximate but effective closed form analytical approach. The model has kinetic base and it is aimed at predicting, by means of a very refined approach, the vulcanization degree of rubber vulcanized with peroxides. Such a procedure takes contemporarily into consideration, albeit within a simplified scheme, the actual reactions occurring during peroxidic curing, namely initiation, H-abstraction, combination and addition, and supersedes the simplified approach used in practice, which assumes for peroxidic curing a single first order reaction. The main drawback of the overall procedure proposed in Milani and Milani (J Math Chem 51(3):1116–1133, 2013) is that the single second order non-linear differential equation obtained mathematically and representing the crosslink evolution with respect to time, was solved numerically by means of a Runge–Kutta approach. Such a limitation is here superseded and a major improvement is proposed allowing the utilization of an approximate but still effective closed form solution. After some simplifications applied on some parts of the solving function not allowing direct closed form integration, an analytical function is proposed. Kinetic parameters within the analytical model are evaluated through least squares where target data are represented by few experimental normalized rheometer curve values. In order to have an insight into the reliability of the numerical approach proposed, a case of technical interest of an EPDM with low unsaturation and crosslinked with three different peroxides at three increasing temperatures is critically discussed.

Pairs of Nodal Solutions for a Class of Nonlinear Problems with One-sided Growth Conditions

Abstract Boundary value problems of Sturm-Liouville and periodic type for the second order nonlinear ODE uʺ + λf(t, u) = 0 are considered. Multiplicity results are obtained, for λ positive and large, under suitable growth restrictions on f(t, u) of superlinear type at u = 0 and of sublinear type at u = ∞. Only one-sided growth conditions are required. Applications are given to the equation uʺ + λq(t)f(u) = 0, allowing also a weight function q(t) of nonconstant sign.

Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions

We study the problem of existence and multiplicity of subharmonicsolutions for a second order nonlinear ODE in presence oflower and upper solutions. We show how suchadditional informationcan be used to obtain more precise multiplicity results.Applications are given to pendulum type equations and toAmbrosetti-Prodi results for parameter dependent equations.

Fold points and singularity induced bifurcation in inviscid transonic flow

Transonic inviscid flow equation of elliptic–hyperbolic type when written in terms of the velocity components and similarity variable results in a second order nonlinear ODE having several features typical of differential–algebraic equations rather than ODEs. These features include the fold singularities (e.g. folded nodes and saddles, forward and backward impasse points), singularity induced bifurcation behavior and singularity crossing phenomenon. We investigate the above properties and conclude that the quasilinear DAEs of transonic flow have interesting properties that do not occur in other known quasilinear DAEs, for example, in MHD. Several numerical examples are included.

Analytical solution for the unsteady MHD flow of a viscous fluid between moving parallel plates

Two-dimensional unsteady MHD flow of a viscous fluid between two moving parallel plates is considered. We allow the plates to move together as well as apart: When the plates move together it corresponds to squeezing flow problem. The governing Navier–Stokes equations for the flow are reduced to a fourth order nonlinear ODE and analytical solutions are obtained for the ODE via the homotopy analysis method. We show that the flow is strongly influenced by the strength of the magnetic field and the density of the fluid. Furthermore, an error analysis for the obtained solutions is provided.

Two-point difference schemes of an arbitrary given order of accuracy for nonlinear BVPs

In this paper we consider difference schemes for two-point BVPs for systems of first order nonlinear ODEs. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which a priori possesses an arbitrary given order of accuracy m . Here, we demonstrate that the TDS can be reduced to the numerical solution of some IVPs defined on each segment [ x j − 1 , x j ] of the grid by an arbitrary IVP-solver of the order m . Using the difference schemes of the orders of accuracy m and m + 1 we develop an a posteriori error estimator for the numerical solution of the order m . An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. It is based on embedded Runge–Kutta methods. Some numerical results confirming the efficiency of the algorithm are given.

Connected Branches of Initial Points for Asymptotic BVPs, With Application to Heteroclinic and Homoclinic Solutions

Abstract We consider the second order nonlinear ODE uʺ – f(t, u) = 0 and assume that f(·, υ0) ≡ 0; for some υ0 ∈ ℝ. We prove the existence of closed connected sets Γ ⊆ ℝ2 of initial points such that for each (α, β) ∈ Γ there exists a solution u(·) of the given differential equation, with (u(t0), uʹ(t0)) = (α, β) and (u(t), uʹ(t)) → (υ0, 0) as t → –∞ (or as t → +∞). These results are then applied to the search of heteroclinic and homoclinic solutions.

Bound Sets and Two-Point Boundary Value Problems for Second Order Differential Equations

Abstract Using Mawhin's continuation principle we obtain a general result on the existence of solutions to a boundary value problem for second order nonlinear vector ODEs. Applications are given to the existence of solutions which are contained in suitable bound sets with possibly non-smooth boundary.

A Two Point Difference Scheme of an Arbitrary Order of Accuracy for BVPS for Systems of First Order Nonlinear Odes

AbstractWe consider two-point boundary value problems for systems of first-order nonlinear ordinary differential equations. Under natural conditions we show that on an arbitrary grid there exists a unique two-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection onto the grid of the exact solution of the corresponding system of differential equations. A constructive algorithm is proposed in order to derive from the EDS a so-called truncated difference scheme of an arbitrary rank. The m-TDS represents a system of nonlinear algebraic equations with respect to the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Analytical and numerical examples are given which illustrate the theorems proved. Keywords: systems of nonlinear ordinary differential equations, difference scheme, exact difference scheme, truncated difference scheme of an arbitrary order of accuracy, fixed point iteration.

Identifying attractors in a ferroresonant circuit: a simulated perturbation approach

Abstract In a previous paper (S.K. Chakravarthy, Nonlinear oscillations due to spurious energisation of transformers, Proc IEE, EPA, Vol. 145, No. 6, Nov. 1998, 585–592) we have demonstrated the nature of oscillations that can occur in a spuriously energised transformer. The system, in that case, was a third order nonlinear (non-autonomous) ode from which bifurcation equations were determined by the method of multiple scales. The computational difficulty became awesome when analysing oscillations due to spurious energisation of power transformers. The order of the system increased from a third order to a fourth order system in this case. We further demonstrated that nonlinear interaction in power system involves dynamic phenomena of widely different speeds. Using the ‘brute force’ of simulation to search the phase space for the attractors of the system is not feasible. In this paper, we hence study higher order nonlinear systems by an approach similar to the method of perturbations. We propose to use numerical simulation as the basic tool for assessing the effect of perturbations and have coined the term ‘simulated perturbation’. In this approach, the starting point is the conservative autonomous system for which all possible oscillatory modes can be determined. We then introduce other variables in the equations one parameter at a time and then, study the changes in the nature of the oscillations. As we shall show, the method allows one to study a complex nonlinear ode with relative ease and yet obtain a feel for the factors behind the behaviour observed.

Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis

The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete. The 1-dimensional vibrating string satisfying wtt − wxx = 0 on the unit interval x ∈ (0, 1) is an infinite dimensional harmonic oscillator. Consider the boundary conditions: at the left end x = 0, the string is fixed, while at the right end x = 1, a nonlinear boundary condition wx = αwt − βw3 t , α, β > 0, takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type un+1 = F (un), where F is the nonlinear reflection relation. We say that the PDE system is chaotic if the mapping F is chaotic as an interval map. Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities. We then define a rotation number, following J.P. Keener [11], and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin [17] to determine the chaotic regime of α for the nonlinear reflection relation F , thereby rigorously proving chaos. Nonchaotic cases for other values of α are also classified. Such cases correspond to limit cycles in nonlinear second order ODEs. Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics. Received by the editors July 20, 1995 and, in revised form, October 16, 1996. 1991 Mathematics Subject Classification. Primary 35L05, 35L70, 58F39, 70L05. The first and third authors’ work was supported in part by NSF Grant DMS 9404380, Texas ARP Grant 010366-046, and Texas A&M University Interdisciplinary Research Initiative IRI 96-39. Work completed while the first author was on sabbatical leave at the Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C. The second author’s work was supported in part by Grant NSC 83-0208-M-007-003 from the National Council of Science of the Republic of China. c ©1998 American Mathematical Society

Oscillations of Second Order Nonlinear ODE with Impulses

Oscillations of Second Order Nonlinear ODE with Impulses

Exact nonlinear travelling hydromagnetic wave solutions

Exact travelling wave solutions for hydromagnetic waves in an exponentially stratified incompressible medium are obtained. With the help of two integrals it becomes possible to reduce the system of seven nonlinear PDE's to a second order nonlinear ODE which describes an one dimensional harmonic oscillator with a nonlinear friction term. This equation is studied in detail in the phase plane. The travelling waves are periodic only when they propagate either horizontally or vertically. The reduced second order nonlinear differential equation describing the travelling waves in inhomogeneous conducting media has rather ubiquitous nature in that it also appears in other geophysical systems such as internal waves, Rossby waves and topographic Rossby waves in the ocean.

Optimal oscillatory time for a class of second order nonlinear dissipative ODE

Optimal oscillatory time for a class of second order nonlinear dissipative ODE