First-order Nonlinear Ordinary Differential Equations Research Articles

THERMAL POST-BUCKLING OF SLENDER ELASTIC RODS WITH DIFFERENT BOUNDARY CONDITIONS

This paper presents mathematical formulation, critical buckling temperature and analytical and numerical solutions for the thermal post-buckling behavior of slender rods subjected to uniform thermal load. The material is assumed to be linear elastic, homogeneous and isotropic. Furthermore, large displacements are considered hence the formulation is geometrically non-linear. Three different boundary conditions are assumed: (i) double-hinged non-movable, (ii) hinged non-movable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring, and (iii) double-fixed non-movable. The governing equations are derived from geometrical compatibility, equilibrium of forces and moments, constitutive equations and strain-displacement relation, yielding a set of six first-order non-linear ordinary differential equations with boundary conditions specified at both ends, which constitutes a complex boundary value problem. The buckling and post-buckling solutions are respectively accomplished assuming infinitesimal and finite rotations. The results are presented in non-dimensional graphs for a range of temperature gradients and different values of slenderness ratios, and it is shown that this parameter governs the rod post-buckling response. The influence of the boundary conditions is evaluated through graphic results for deformed configuration, maximum deflection, maximum inclination angle and maximum curvature in the rod.

Stochastic Control of Sensitive Nonlinear Motions of an Ocean Mooring System

Complex sensitive motions have been observed in ocean mooring systems consisting of nonlinear mooring geometries. These physical systems can be modeled as a system of first-order nonlinear ordinary differential equations, taking into account geometric nonlinearities in the restoring force, quadratic viscous drag, and harmonic excitation. This study examines the controllability of these systems utilizing an embedded approach to noise filtering and online controllers. The system is controlled using small perturbations about a selected unstable cycle and control is instigated for periodic cycles of varying periodicities. The controller, when applied to the system with additive random noise in the excitation, has marginal success. However, the addition of an iterated Kalman filter applied to the system increases the regime under which the controller behaves under the influence of noise. Because the Kalman filter is applied about locally linear trajectories, the feedback of the nonlinearities through the filter has little effect on the overall filtering system.

Resonances in nonlinear structure vibrations under multifrequency excitations

The response of a single-degree-of-freedom system with quadratic, cubic and quartic nonlinearities subjected to a sinusoidal excitation that involves multiple frequencies is considered. The method of multiple scales is used to construct a first order uniform expansion yielding two first-order nonlinear ordinary differential equations that are derived for the evolution of the amplitude and phase. These oscillations involve a subharmonic oscillation of order one-fourth and superharmonic oscillation of order two. Steady state responses and their stability are computed for selected values of the system parameters. The effects of these (quadratic, cubic, and quartic) nonlinearities on these oscillations are specifically investigated. With this study, it has been verified that the qualitative effects of these nonlinearities are different. Regions of hardening (softening) behaviour of the system exist for the case of subharmonic resonance. The response curve is not affected by decreasing the damping factor for the case of superharmonic resonance. It is shown that the response curve contracts or expands as the parameters vary. The multivalued region increases or decreases when some parameters vary.

Vibration control of a cantilever beam of varying orientation

We investigate the problem of suppressing the vibrations of a non-linear system with a cantilever beam of varying orientation subject to parametric and direct excitation. It is known that the growth of the response is limited by non-linearity. Therefore, vibration control and high-amplitude response suppressions of the first mode of a cantilever beam can be performed using a simple non-linear feedback law. This control law is based on cubic velocity feedback. The method of multiples scales is used to construct first-order non-linear ordinary differential equations governing the modulation of the amplitudes and phases. The stability and effects of different system parameters are studied numerically.

New exact travelling waves solutions to the combined KDV-MKDV and generalized Zakharov equations

In this paper, the solutions of a first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term and Fan's direct algebraic method of obtaining exact solution to nonlinear PDEs are applied to two nonlinear evolution equations, the combined KdV-MKdV equation and generalized Zakharov equations. As a result, some special exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, bright and dark solitary wave solutions, triangular periodic wave solutions and singular solutions.

A new generalized algebra method and its application in the (2+1) dimensional Boiti–Leon–Pempinelli equation

Abstract In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.

MATHEMATICAL ANALYSIS OF THE EVOLUTION OF A MODEL OF REGIONAL POPULATION DISTRIBUTION

This paper deals with the mathematical analysis of a model of urban dynamics, which was proposed by Miyata and Yamaguchi in the context of a region of Japan. The model is a complex system of first-order nonlinear ordinary differential equations. The study undertaken by Miyata and Yamaguchi is essentially computational, while an extensive study of the asymptotic behavior of the solutions is performed in this paper related to a detailed analysis of the qualitative properties of the system.

Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers

Difference schemes for two-point boundary value problems for systems of first-order nonlinear ordinary differential equations are considered. 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 0(|h| m ) with respect to the maximal step size |h|. This m-TDS represents a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. In the present paper, new efficient methods for the implementation of an m-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.

4-Point FDF of Muskingum method based on the complete St Venant equations

Using the Froude number, a nonlinear convection–diffusion equation was derived from the St Venant equations of continuity and momentum. It was made applicable to discrete space using a mixing-cell method, resulting in a first-order nonlinear ordinary differential equation. A 4-point finite difference scheme was used to solve the first-order nonlinear ordinary differential equation to achieve a nonlinear algebraic form that is very similar in structure to the traditional Muskingum form thus its naming as the ‘4-point FDF of Muskingum Method (4-point FDF M-M)’. In this method, the space interval Δ x is a characteristic channel length and a variable that is dependent on flow discharge, water depth, and flow velocity. An iterative search method was applied to simultaneously obtain the flow discharge and the optimal space interval Δ x. The method developed was tested with both synthetic numerical examples and observed events and the results were compared with those of the Lambda scheme and the method of characteristics. The outflow hydrographs produced by this new 4-point FDF M-M were of comparable accuracy. The parameters used in the new method are based on the physical attributes of the channel and thus do not need calibration as required for the Muskingum method.

Non-Isochoric Bending and Shearing

Two families of non-homogeneous finite deformations are treated. For each family, the governing system of second-order nonlinear partial differential equations reduces to a system of first-order non-linear ordinary differential equations for homogeneous compressible elastic solids with arbitrary isotropic or anisotropic response.

New Exact Travelling Wave Solutions to Kundu Equation**The projectpartially supported by the State Key Basic Research Program of China under GrantNo. 2004CB318000

Based on a first-order nonlinear ordinary differential equation withsix-degree nonlinear term,we first present a new auxiliary equation expansion method and itsalgorithm. Being concise and straightforward, the method isapplied to the Kundu equation. As a result, some new exacttravelling wave solutions are obtained, which include bright anddark solitary wave solutions, triangular periodic wave solutions,and singular solutions. This algorithm can also be applied toother nonlinear evolution equations in mathematical physics.

Coexistence of Stable Solutions in the Nonlinear Ship Motion

The dynamic stability and complicated motions of a vessel in regular sea are investigated when the frequency in the pitch is nearly twice the frequency in the roll. We use the Method of Multiple Scales to determine four first-order nonlinear ordinary differential equations that govern the modulation of the amplitudes and phases when the forcing frequency is near either the pitch or the roll natural frequency. These equations are used to determine the steady-state solutions and their stability. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found in the presence of quadratic nonlinearities. The linear solution is symmetric. The pitch mode has multi-valued solutions and the roll mode has a single-valued solution for the case of primary resonance of the pitch mode. Both pitch and roll modes have multivalued solutions in small intervals for the case of primary resonance of the roll mode.

Three-to-one Internal Resonance in The Non Linear Oscillation of Shallow Arch

We determine the response of bowed structural elements to a two-frequency, transverse load. The effects of both initial curvature (i.e., quadratic nonlinearities) and mid-surface stretching (i.e., cubic nonlinearities) are included. We investigate possible large-amplitude-low-frequency responses to excitations of such structural elements (drumming) and determine the conditions under which the one-half subharmonics of the transverse load can be excited individually in the presence of three-to-one internal resonance. The method of multiple time scales is used to determine four first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. Numerical solutions are carried out and graphical representations of the results are presented. The region of multivalued disappear for increasing the parameters µ22, d6, Γ2 and decreasing the parameters d2, d4, d5 and ω1. The nontrivial solution has unstable solution for increasing the damping factors. The trivial solution has unstable and stable solutions for decreasing the parameters Γ1 and Γ2 respectively.

Combined axial shearing, extension, and straightening of elastic annular cylindrical sectors

The combined axial shearing, extension, and straightening of an annular cylindrical sector is a deformation that, following Truesdell & Noll (1965, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics III/3) and Hill (1973, Z. Angew. Math. Phys., 24, 609-618), we describe in terms of two prescribed constants and two unknown functions that depend only on the radial material coordinate. Under the assumption that the material is elastic, compressible, and isotropic we show that for equilibrium in the absence of body forces the unknown functions must satisfy a system of first-order nonlinear ordinary differential equations. The system of differential equations can be decoupled for certain material classes, one of which is the class of Hadamard-Green materials. Thus, several new exact solutions are obtained and, under the assumption that the annular cylindrical sector is composed of a Hadamard-Green material that is strongly elliptic, the existence and uniqueness of solutions for two types of boundary conditions is established.

A mathematical study of a two-regional population growth model

The paper provides a mathematical study of a model of urban dynamics, adjusting to an ecological model proposed by Lotka and Volterra. The model is a system of two first-order non-linear ordinary differential equations. The study proposed here completes the original proof by using the main tools such as a Lyapunov function. To cite this article: J. El Ghordaf et al., C. R. Biologies 327 (2004).

THERMAL POSTBUCKLING OF SLENDER ELASTIC RODS WITH HINGED ENDS CONSTRAINED BY A LINEAR SPRING

This paper presents the critical buckling temperature and a closed-form analytical solution for the thermal postbuckling behavior of uniformly heated slender elastic rods. The boundary conditions are assumed hinged and nonmovable at one end, whereas at the other end longitudinal displacement is constrained by a linear spring. The thermal strain–temperature relationship is considered nonlinear and the material is assumed to be linearly elastic, homogeneous, and isotropic. Large displacements are included; hence, the formulation is geometrically nonlinear. A set of six first-order nonlinear ordinary differential equations with boundary conditions defined at both ends yields a complex boundary-value problem. The buckling and postbuckling solutions are respectively accomplished by assuming infinitesimal rotations and employing complete elliptical integrals. The results are presented in nondimensional graphs obtained for different values of spring stiffness, slenderness ratios, and nonlinear thermal strain coefficients. It is shown that these three parameters govern the buckling and postbuckling behavior.

An improved algebra method and its applications in nonlinear wave equations

In this paper, other types of exact solution of a first-order nonlinear ordinary differential equation, which is including the algebraic method, is further investigated. By using the solutions of this equation, we give some types of solutions of the coupled KdV equation, nonlinear dispersion mK( m, n, k) equation, the variant shallow water wave equation, nonlinear dissipative equation and the higher-order nonlinear Schrodinger equation. These solutions includes compacton solutions, solitary pattern solutions, solitary wave solutions, Weierstrass elliptic function solutions, which may be useful to explain some phenomena.

Stability of Chebyshev collocation methods

In [1], a class of global collocation methods for the numerical solution of systems of nonlinear first-order ordinary differential equations was derived. The favorable comparison with other existing methods stimulated us to study them in depth. So in this paper the equivalent implicit Runge-Kutta methods are derived and their stability is studied. Methods derived in [1] are A-stable at least at up to order 20.

On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics

Using series of admissible functional transformations we reduce the one-dimensional axisymmetric nonlinear Schrödinger (NLS) equation, as well as the forced damped nonlinear Duffing (NLD) equation to equivalent nonlinear first-order integrodifferential equations. The forced undamped (NLD) equation results as a special case. The reduced integrodifferential equations are exact. In the limits of small or large values of the parameters characterizing these nonlinear problems, we prove that further reductions lead to first-order nonlinear ordinary differential equations which, except in case of the (NLS) equation, are of the Abel classes. The approximate reduced (NLS) equation admits exact analytic solutions. On the other hand, taking into account the known exact analytic solutions of the equivalent Abel classes of equations we show that there do not exist analytic solutions of the above two nonlinear Duffing oscillators. However, if further asymptotic approximations take place, new approximate analytic solutions concerning the (NLD) equations are constructed.

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.