Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.     

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

It is shown that by a series of admissible functional transformations the already derived (third-order) strongly nonlinear ordinary differential equation (ODE), describing the elastica buckling analysis of a straight bar under its own weight [Int.J.Solids Struct.24(12), 1179–1192, 1988, The Theory of Elastic Stability, McGraw-Hill, New York, 1961], is reduced to a first-order nonlinear integrodifferential equation. The absence of exact analytic solutions of the reduced equation leads to the conclusion that there are no exact analytic solutions in terms of known (tabulated) functions of this elastica buckling problem. In the limits of large or small values of the slope of the deflected elastica, we expand asymptotically the above integrodifferential equation to nonlinear ODEs of the Emden–Fowler or Abel nonlinear type. In these cases, using the solution methodology recently developed in Panayotounakos [Appl. Math. Lett. 18:155–162, 2005] and Panayotounakos and Kravvaritis [Nonlin. Anal. Real World Appl., 7(2):634–650, 2006], we construct exact implicit analytic solutions in parametric form of these types of equations and thus approximate implicit analytic solutions of the original elastica buckling nonlinear ODE.     

Asymptotic solutions to the non-linear cantilever elastica

In this work it is shown that by a series of admissible functional transformations the constructed higher-order strongly non-linear differential equation (ODE), describing the elastica of a cantilever due to a terminal generalized concentrated, as well as to a lateral uniformly distributed loading, is reduced to a first-order non-linear integrodifferential equation consisting of the first intermediate integral of the original equation. The absence of exact analytic solutions in terms of known (tabulated) functions of the above reduced equation leads to the conclusion that there are no exact analytic solutions of this complicated elastica problem. In the limits of small values of the slope parameter of the deflected elastica, we expand asymptotically the above integrodifferential equation to non-linear ODEs of the generalized Emden–Fowler types, exact analytic solutions of which are constructed in parametric form.     

Exact analytic solutions for the damped Duffing nonlinear oscillatorLes solutions analytiques exactes pour le amortissable non linéaire oscillateur de Duffing

We prove that the second-order damped nonlinear Duffing oscillator is reduced to an equivalent equation of the normal Abel form of the second kind. Based on a recently developed mathematical methodology for the construction of exact analytic solutions of Abel's equation, exact analytic solutions are obtained for the nonlinear damped Duffing oscillator obeying the initial conditions adapted to the physical problem. To improve the general developed methodology an application concerning the nonlinear Van der Pol free oscillator is briefly discussed. To cite this article: D.E. Panayotounakos et al., C. R. Mecanique 334 (2006).     

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Two kinds of second-order non-linear ordinary differential equations (ODEs) appearing in mathematical physics and non-linear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly non-linear including quadratic first-order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in accordance with each specific problem under considerations.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty

The present research work proposes a new systematic approach to the problem of model reduction for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty. The problem of interest is addressed within the context of functional equation theory, and in particular, through a system of invariance functional equations for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system’s slow invariant manifold attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original discrete-time system dynamics on the slow manifold. The analyticity property of the solution to the invariance functional equations enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the system’s transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, the proposed method is evaluated through an illustrative biological reactor example.     

Homotopy analysis method: A new analytic method for nonlinear problems

I.IntroductionAlthoughtherapiddevelopmentofdigitalcomputersmakesiteasierandeasiertonumericallysolvenolllinearproblems,itisstillratherditliculttogivethed'analyticapproximations.Currently,mostofour11onlinearanalytictechlliquesill'cunsatislllctory.Forinstance,althoughpel.turbatiolltechlliquesarewidelyappliedtoalZalyzcnolllillcarproblcllisillscienceandengineerillg,theyarehoweversostronglydependentonsmall13arall,etersappearedinequatiollsunderconsiderationthattheyarerestrictedonlytoweLlklynolllinea…     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term

Using a series of functional transformations we reduce the unforced,damped Duffing oscillator to equivalent equations of the Abel andEmden–Fowler classes. Taking into account the known exact analyticsolutions of these equivalent equations we prove that there does notexist an exact analytic solution of the damped, unforced Duffingoscillator in terms of known (tabulated) analytic functions. It followsthat a new class of solutions must be defined for solving this problem`exactly'. Finally, a new approximate solution of the intermediateintegral of the damped Duffing oscillator with weak damping isconstructed.     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

Restrictions on nonlinear constitutive equations for elastic shells

Constitutive equations for the resultant forces and moments applied to a shell-like body necessarily couple the influences of the shell geometry and the constitutive nature of the three-dimensional material from which the shell is constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the complicated influence of the shell geometry on the constitutive response of the shell is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic shells which ensure that exact solutions of the shell equations are consistent with exact nonlinear solutions of the three-dimensional equations for homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of shells. Examples of the linear theories of a plate and a spherical shell are considered.     

Restrictions on nonlinear constitutive equations for elastic rods

Constitutive equations for the resultant forces and moments applied to a rod-like body necessarily couple the influences of the rod geometry and the constitutive nature of the three-dimensional material from which the rod was constructed. Consequently, even when the nonlinear constitutive equation of the three-dimensional material is known, the influence of the rod geometry on the constitutive response of the rod is not known. The main objective of this paper is to develop restrictions on the constitutive equations of nonlinear elastic rods which ensure that exact solutions of the rod equations are consistent with exact nonlinear solutions of the three-dimensional equations for all homogeneous deformations. Since these restrictions are nonlinear in nature they provide valuable general theoretical guidance for specific constitutive assumptions about the coupling of material and geometric properties of rods. Also, an example of a straight beam clamped at one end and subjected to a shear force at the other end is used to examine the validity of the proposed value for the transverse shear deformation coefficient.     

Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere

The regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.     

A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations

A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

Study of Explicit Analytic Solutions for the Nonlinear Coupled Scalar Field Equations

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Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.