共查询到20条相似文献,搜索用时 31 毫秒
1.
The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented. 相似文献
2.
A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors. 相似文献
3.
This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form. 相似文献
4.
For a weakly nonlinear differential equation in a Banach space, we establish necessary and sufficient conditions for the existence
of solutions bounded on the entire real axis under the assumption that the generating equation has bounded solutions and the
corresponding homogeneous equation admits an exponential dichotomy on the semiaxes.
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Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 151–159, April–June, 2008. 相似文献
5.
An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed. 相似文献
6.
For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that
are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on
the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider
examples of countable systems of ordinary differential equations that have bounded solutions.
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Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006. 相似文献
7.
We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the
asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular,
if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In
this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges
to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation
around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable
comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence
of solutions. 相似文献
8.
The convective cooling of a slab by an ambient fluid under the most general linear and homogeneous boundary conditions is considered. For the roots of the corresponding transcendental eigenvalue equation an explicit formula is written down in a parametric form. The practical consequences of this representation, among them certain “singular solutions” which cannot be obtained by a direct numerical treatment of the original eigenvalue equation, are discussed in detail. 相似文献
9.
T. E. Churkina 《Mechanics of Solids》2014,49(2):127-135
We consider the satellite plane motion about the center of mass in a central Newtonian gravitational field in an elliptic orbit. This motion is described by a second-order differential equation known as the Beletskii equation. In the framework of the plane problem (under the assumption that the body vibrates in the unperturbed orbit plane), there exists a family of periodic solutions of the Beletskii equation near the 3: 2 resonance between the orbital revolution and axial rotation periods. A nonlinear stability analysis of these periodic solutions is carried out both in the presence of third- and fourth-order resonances and in their absence as well as on the boundaries of the stability regions in the first approximation. The problem is solved numerically. For fixed parameter values (the eccentricity of the center-of-mass orbit and the inertial parameter), the construction of a symplectic mapping of the equilibrium into itself is used to calculate the coefficients of the mapping generating function, which are further used to conclude whether the equilibrium is stable or not. 相似文献
10.
E Momoniat 《International Journal of Non》2004,39(3):447-455
The non-classical symmetry method is used to determine particular forms of the arbitrary velocity and forcing terms in a linear wave equation used to model the propogation of waves in a linear elastic fluid. The behaviour of solutions derived using the non-classical symmetry method are discussed. Solutions satisfy a given initial profile and wave velocity. For some solutions the arbitrary forcing terms and wave velocity can be written in terms of the initial wave profile. Relationships between the arbitrary forcing, arbitrary velocity and the solution are derived. 相似文献
11.
For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability. 相似文献
12.
董明德 《应用数学和力学(英文版)》1985,6(3):213-218
For Poiseuille flow the Orr-Sommerfeld equation is solved exactly. Regular solutions are obtained, thereby eigenvalue equation can be analyzed analytically and explicitly. The bifurcation solutions will be discussed subsequently. 相似文献
13.
Yuriy Povstenko 《Nonlinear dynamics》2010,59(4):593-605
The paper is concerned with analysis of time-fractional diffusion-wave equation with Caputo fractional derivative in a half-space.
Several examples of problems with Dirichlet and Neumann conditions at the boundary of a half-space are solved using integral
transforms technique. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the
ordinary diffusion and wave equations. Numerical results are presented graphically for various values of order of fractional
derivative. 相似文献
14.
Manfred Braun Merle Randrüüt 《Archive of Applied Mechanics (Ingenieur Archiv)》2014,84(9-11):1263-1274
The Camassa–Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa–Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa–Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa–Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense. 相似文献
15.
The structure of a normal (direct) shock in a gas for the parameters corresponding to nitrogen is investigated with allowance for the rotational degrees of freedom on the basis of a model kinetic equation. For various Mach numbers the structure is compared with both the known experimental results and the solutions of the Navier-Stokes approximation within the framework of two-temperature hydrodynamics. The possibility of assuming the constancy of the fraction of excited rotational degrees of freedom is studied. 相似文献
16.
Yuriy Povstenko 《Nonlinear dynamics》2008,53(1-2):55-65
The radial diffusion in a sphere of radius R is described using time-fractional diffusion equation. The Caputo fractional derivative of the order 0<α<2 is used. The Laplace and finite sin-Fourier transforms are employed. The solution is written in terms of the Mittag–Leffler functions. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Several examples of signaling, source and Cauchy problems are presented. Numerical results are illustrated graphically. 相似文献
17.
Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partial-differential equations
and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model
of a transversely vibrating string, Kirchhoff’s equation, is derived from another nonlinear model, a partial-differential
equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled
planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to
compare the transverse responses calculated from the two models with the transverse component calculated from the coupled
equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models
of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that
the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision
for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results. 相似文献
18.
We consider symmetry-breaking bifurcations at non-positive, radially symmetric solutions of semilinear elliptic equations on a ball with Dirichlet boundary conditions. For nonlinearities which are asymptotically affine linear, we find solutions at which the symmetry breaks. The kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group O(n). For this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough. Thus we obtain, by the equivariant branching lemma, a large variety of isotropy subgroups of O(n) which occur as symmetries of the bifurcating solution branches. 相似文献
19.
O. Yu. Tsvelodub V. Yu. Shushenachev 《Journal of Applied Mechanics and Technical Physics》2005,46(3):365-374
The flow of a thin film of a nonlinearly viscous fluid whose stress tensor is modeled by a power law, flowing down a vertical
plane in the field of gravity, is considered. For the case of low flow rates, an equation that describes the evolution of
surface disturbances is derived in the long-wave approximation. The domain of linear stability of the trivial solution is
found, and weakly nonlinear, steady-state travelling solutions of this equation are obtained. The mechanism of branching of
solution families at the singular point of the neutral curve is described.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 73–84, May–June, 2005. 相似文献
20.
For the nonlinear kinetic equation describing the one-dimensional motion of a quasineutral collisionless plasma, perturbation
velocities are determined and conditions of generalized hyperbolicity are formulated. Exact (in particular, periodical) solutions
of the model are constructed and interpreted physically for the class of traveling waves. Differential conservation laws approximating
the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation,
which show the possibility of turnover of the kinetic distribution function. 相似文献