First-order Nonlinear Equations Research Articles

Relaxation in non-convex optimal control problems described by first-order evolution equations

The problem is considered of minimizing an integral functional with integrand that is not convex in the control, on solutions of a control system described by a first-order non-linear evolution equation with mixed non-convex constraints on the control. A relaxation problem is treated along with the original problem. Under appropriate assumptions it is proved that the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence for the original problem that converges to the optimal solution. Moreover, in the appropriate topologies the convergence is uniform simultaneously for the trajectory, the control, and the functional. The converse also holds. An example of a non-linear parabolic control system is treated in detail.

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.

Postbuckling of non-linear viscoelastic imperfect laminated plates Part I: material considerations

A method of predicting the non-linear relaxation behavior from creep experiments done on non-linear viscoelastic orthotropic materials is presented. Such a method is desirable since creep experiments are those mainly performed for these materials. It is shown that for given non-linear creep properties, and creep compliance represented by the Prony series, the Schapery creep model for non-linear viscoelastic orthotropic material can be transformed into a set of first-order non-linear equations. The solution of these equations makes it possible to obtain the non-linear stress relaxation curves, from which the strain-dependent constitutive equation of any non-linear viscoelastic model can be constructed, as needed for engineering applications.

Differential equations: a systems approach

0. Complex Numbers. Introduction. The Cartesian and Exponential Forms. Roots of Polynomial Equations and Numbers. Matrix Notation and Terminology. The Solution of Simultaneous Equations. The Algebra of Matrices. Matrix Multiplication. The Inverse of a Matrix. The Computation of A-1. Determinants. Linear Independence. 1. First-Order Differential Equations. Preliminaries. Definitions. The First-Order Linear Equation. Applications of First-Order Linear Equations. Nonlinear Equations of First Order. 2. Linear Systems. Introduction. Eigenvalues and Eigenvectors. First-Order Systems. Solution and Fundamental Solution Matrices. Some Fundamental Theorems. Solutions of Nonhomogeneous Systems. Nonhomogeneous Initial-Value Problems. Fundamental Solution Matrices. 3. Second-Order Linear Equations. Introduction. Sectionally Continuous Functions. Linear Differential Operators. Linear Independence and the Wronskian. The Nonhomogeneous Equation. Constant Coefficient Equations. Spring-Mass Systems in Free Motion. The Electric Circuit. Undetermined Coefficients. The Spring-Mass System: Forced Motion. The Cauchy-Euler Equation. Variation of Parameters. 4. Higher Order Equations. Introduction. The Homogeneous Equation. The Nonhomogeneous Equation. Companion Systems. Homogeneous Companion Systems. Variation of Parameters. 5. The Laplace Transform. Introduction. Preliminaries. General Properties of the Laplace Transform. Sectionally Continuous Functions. Laplace Transforms of Periodic Functions. The Inverse Laplace Transform. Partial Fractions. The Convolution Theorem. The Solution of Initial-Value Problems. The Laplace Transform of Systems. Tables of Transforms. 6. Series Methods. Introduction. Analytic Functions. Taylor Series of Analytic Functions. Power Series Solutions. Legendre's Equations. Three Important Examples. Bessel's Equation. The Wronskian Method. The Frobenius Method. 7. Numerical Methods. Introduction. Direction Fields. Notational Conventions. The Euler Method. Heun's Method. Taylor Series Methods. Runge-Kutta Methods. Multivariable Methods. Higher Order Equations. 8. Boundary-Value Problems. Introduction. Separation of Variables. Fourier Series Expansions. The Wave Equation. The One-Dimensional Heat Equation. The Laplace Equation. A Potential about a Spherical Surface.

Optimal coordination strategies for production and marketing decisions

This study develops general optimal coordination strategies for short-term production and marketing decisions, i.e., marketing-dominating coordination approach (MDCA) and production-dominating coordination approach (PDCA). These approaches reflect a general context of coordinating production and marketing decisions through information exchange between the two functions, by solving a first-order nonlinear difference equation iteratively, based on a marginal analysis of the profit function. We show that the MDCA predominant in both practice and academia may not converge to an optimal solution, and suggest an alternative PDCA for this divergent case. Also, we illustrate a coordination strategy by solving a previously unexplored Price-EPQ problem of jointly determining a manufacturer’s profit-maximizing selling price (marketing) and production quantity (production) decisions for a price-dependent demand item with a finite production rate unlike previous Price-EOQ problem studies.

Direct and inverse scattering for transient electromagnetic waves in nonlinear media

Nonlinear propagation of electromagnetic waves is an important problem in optics. Often the properties of the nonlinear media are not fully understood. The solution of an inverse problem can provide an aid to that understanding. An inverse transmission problem is posed; it is one of reconstructing the medium parameters, by measurement of a wave that has been propagated through the nonlinear medium. The nonlinear medium is assumed to be homogeneous and isotropic. The methods have application to nonlinear optics, and the numerical results for both the direct and inverse problems presented are based on the nonlinear Kerr effect, which is observed in the optical wavelength band. However, the mathematical techniques that are developed are applicable to any set of nonlinear first-order equations. The method is therefore model independent.

Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Nonlinear nonplanar dynamics of parametrically excited cantilever beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

Consider the first-order neutral nonlinear difference equation of the form δ (y n−p ny n−r)+ q n ∏ i=l m |y n−σ i| ai sgn y n−σ i = 0, n=0,1,… , where τ > 0, σ i ≥ 0 ( i = 1, 2,…, m) are integers, { p n } and { q n } are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ n=0 ∞ q n = ∞ or Σ n=0 ∞ nq n Σ j= n ∞ q j = ∞ commonly used in the literature.

SH surface waves in layered half-spaces

The analysis of SH surface waves on a layered anisotropic half-space is performed having recourse to a wave-splitting approach. Assuming that the medium be asymptotically homogeneous, the problem is reduced to a first-order nonlinear equation with a boundary condition at the surface. An iterative linearization of the equation is proposed and a sufficient condition for its convergence is given. Accounting for a first-order approximation, we obtain the dispersion equation in terms of the Laplace transform of a function which characterizes the inhomogeneity. The model is applied to several specific dependences of the material parameters on the depth within the half-space.

Three-to-One Internal Resonances in Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.

THE RESPONSE OF A NONLINEAR SYSTEM WITH A NONSEMISIMPLE ONE-TO-ONE RESONANCE TO A COMBINATION PARAMETRIC RESONANCE

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

Nonlinearly Coupled Pitch and Roll Motions in the Presence of Internal Resonance; Part I, Theory

The dynamic stability and complicated motions of a vessel in following or head regular waves are investigated when the frequency in pitch is nearly twice the frequency in roll. The damping in the pitch mode is modeled by a linear visco ins damping term, whereas that of the roll mode is modeled by the sum of a linear viscous part and a quadratic viscous part. The method of multiple scales is used to determine a system of four nonlinear first-order equations governing the modulation of the amplitudes and phases of the pitch and roll. Force-response andfrequency-response curves are generated. Coexistence of multiple solutions is found. The jump phenomenon continues to exist, whereas the saturation phenomenon ceases in the presence of quadratic damping. Hopf bifurcations are found. Near these bifurcations, the modulation equations possess limit-cycle solutions and hence the steady-state motion is a periodically modulated pitch and roll motion. Numerical simulations are used to investigate the bifurcations of these limit cycles and how they lead to chaos and hence chaotically modulated pitch and roll motions.

Prufer transforms and numerical solutions to the Helmholtz equation

The Prufer transformation maps the second-order, linear Helmholtz differential equation to two first-order, nonlinear differential equations. The coupling between the two separates such that the equation for the eigenvalue is a single nonlinear first-order equation. This transformation has long been useful in theoretical studies of Strum–Liouville where many properties of the eigenvalues have been derived using it. It turns out also to have some useful numerical properties especially for problems involving waveguides where the coupling between the boundaries is poorly conditioned. It also identifies explicitly the both stability points in the evanescent sections of the modes and the local oscillation frequency in the ducted section. The numerical aspects of using the Prufer transform and some examples are discussed in the presentation.

A Viscosity Solutions Approach to Shape-From-Shading

The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton–Jacobi–Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions. A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.

Thermal runaway in microwave heated ceramics: A one-dimensional model

The heating of a ceramic slab under TM-illumination is modeled and analyzed in the small Biot number regime. The temperature distribution is almost spatially uniform in this limit and its evolution in time is governed by a first-order nonlinear amplitude equation. This equation admits a time independent solution which is a multivalued function of the microwave power. The graph of this steady temperature as a function of the microwave power gives an S-shaped response curve when the electrical conductivity is modeled either as an exponential function of temperature or an Arranius law. The dynamics of the heating process are deduced from the amplitude equation and the multivalued response, and are dependent upon the microwave power and initial conditions. For certain initial conditions and power levels the system evolves to the upper branch of the response curve which corresponds to thermal runaway. Other initial conditions and power levels force the system to evolve to the lower branch and a safe sintering temperature. This heating process can be controlled in some sense by varying the microwave power in time at a rate commensurate with the thermal changes. Specifically, the power is allowed to change in an exponential fashion from a higher to a lower power level. This relationship is turned into a differential equation, which is appended to the amplitude equation to form an autonomous system of the second order. This system is analyzed using a phase-plane method. The analysis shows the existence of a stable manifold which divides the phase plane into two parts: Trajectories in the region above this curve correspond to runaway heating while those below yield stable sintering. Various heating scenarios are presented and discussed.

Changes in modal parameters of a bridge during fatigue testing: Salane, H.J. and Baldwin, J.W. Exp. Mech. June 1990 30, (2), 109–113

Failure investigations of structural cracking require a crack-growth life prediction method which can be rapidly applied and which directly displays the sensitivity of life to the crack configuration and the random cyclic stress environment. Such predictions are based on first-order nonlinear rate equations which are fitted to material properties measured under constant-amplitude stress. Many digital programs have been developed to simulate cycle-by-cycle crack growth based on these rate equations.This paper outlines a method better suited to rapid assessment. The method is based on the Palmgren-Miner Rule (linear damage summation), which was originally proposed for predicting crack-initiation life from empirical data. The Palmgren-Miner Rule is shown to be exact for crack growth when the stress and crack-length variables in the rate equation are separable. Such applications are limited, but a numerical experiment shows that the Palmgren-Miner Rule also produces acceptably accurate estimates in practical cases when the rate equation embodies commonly observed stress sequence effects. In its present state of development, however, the method cannot be applied to interaction phenomena, i.e. situations where the stress sequence can change the rate equation.

Non-linear axisymmetric response of closed spherical shells to a radial harmonic excitation

A numerical-perturbation approach is used to study the axisymmetric dynamic response of closed spherical shells to an external harmonic excitation having a frequency near one of the natural frequencies of a flexural mode (i.e. primary resonance of a flexural mode) in the presence of a two-to-one internal (autoparametric) resonance between the excited mode and a lower flexural mode. The method of multiple-time scales is used to derive four first-order non-linear equations describing the evolution (modulation) of the amplitudes and phases of the interacting modes. A numerical scheme that combines a shooting technique and a Newton-Raphson procedure is used to calculate limit cycles of the evolution equations. The stability of these limit cycles is determined by using a Floquet analysis. As the excitation amplitude varies, the fixed-point solutions of the evolution equations exhibit the jump and saturation phenomena. They also undergo supercritical and subcritical Hopf bifurcations as the frequency of excitation varies; the supercritical Hopf-bifurcation frequency is lower than the subcritical one. As the excitation frequency increases above the supercritical Hopf-bifurcation value, the fixed-point solutions lose their stability and limit cycles are born. These limit cycles experience a symmetry-breaking (pitchfork) bifurcation followed by a cascade- of period-doubling bifurcations culminating in chaos. On the other hand, as the excitation frequency decreases below the subcritical Hopf-bifurcation value, the resulting limit cycle deforms and eventually loses its stability through a cyclic-fold bifurcation causing a transition to chaos. In the frequency range in between the above two chaotic regions the evolution equations possess limit-cycle solutions.

Non-linear non-planar parametric responses of an inextensional beam

The non-linear integro-differential equations of motion for an inextensional beam are used to investigate the planar and non-planar responses of a fixed-free beam to a principal parametric excitation. The beam is assumed to undergo flexure about two principal axes and torsion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. In both cases, the beam possesses a one-to-one internal resonance with one of the natural flexural frequencies in one plane being approximately equal to one of the natural flexural frequencies in the second plane. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the interaction of the two resonant modes, yielding four first-order non-linear ordinary-differential equations governing the amplitudes and phases of the modes of vibration. The results show that the non-linear inertia terms produce a softening effect and play a significant role in the planar responses of high-frequency modes. On the other hand, the non-linear geometric terms produce a hardening effect and dominate the planar responses of low-frequency modes and non-planar responses for all modes. If the non-linear geometric terms were not included in the governing equations, then non-planar responses would not be predicted. For some range of parameters, Hopf bifurcations exist and the response consists of amplitude- and phase-modulated or chaotic motions.

Estimation of Cauchy data for a first-order nonlinear hyperbolic equation

Let Sz be an open interval in Iw. We study the Cauchy problem 24, + uu, + au = 4(x, t) in Szx (0, t) u(x, 0) = UJX). (1.1) Equation (1.1) may be viewed as a model of a one-dimensional flow in which the constant a > 0 represents a friction term, 4 a forcing term related to a pressure gradient, uO(x) the initial velocity at point x, and u(x, t) the velocity of fluid at the point x and at time t [S]. It is well known (cf. [ll]) that ( 1.1) is associated with an initial value problem for the system of ordinary differential equations dx x = 2, x(0,5)= 5 dz (1.2) z = --az + 4(x, C), m 5) = &l(5) called the characteristic equations for ( 1.1). We study the following problems for (1.1). Suppose that M observations {zk}kM, I at positions {xk}fz, are given at time to. We wish to estimate the initial condition u,, from among an admissible set Qad of initial conditions. Roughly, our approach is to first solve the characteristic equations with the data as initial conditions backward to t = 0. That is, solve dx x = z,