Eighth-order Equation Research Articles

Infinite-dimensional symmetry group, Kac–Moody–Virasoro algebras and integrability of Kac–Wakimoto equation

An eighth-order equation in $$(3+1)$$ dimension is studied for its integrability. Its symmetry group is shown to be infinite-dimensional and is checked for Virasoro-like structure. The equation is shown to have no Painlev $$\acute{\mathrm{e}}$$ property. One- and two-dimensional classifications of infinite-dimensional symmetry algebra are also given.

Uniqueness results for higher order Lane-Emden systems

In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet’s: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier’s boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

Toward Numerical Solution of the Problem on Stressed State of Thermoelastic Physically Orthotropic Cylindrical Shells

To investigate the stressed-strained state of thermoelastic anisotropic cylindrical shells the original system of nineteen differential equations of eighth order in partial derivatives of the general theory of shells is transformed into a system of eight differential equations of first order. It is suggested to carry out numerical realization by the method of orthogonal marching. Numerical results are presented for the case of temperature drop over the shell thickness.

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.

A Family of Three‐Point Methods of Ostrowski′s Type for Solving Nonlinear Equations

A class of three‐point methods for solving nonlinear equations of eighth order is constructed. These methods are developed by combining two‐point Ostrowski′s fourth‐order methods and a modified Newton′s method in the third step, obtained by a suitable approximation of the first derivative using the product of three weight functions. The proposed three‐step methods have order eight costing only four function evaluations, which supports the Kung‐Traub conjecture on the optimal order of convergence. Two numerical examples for various weight functions are given to demonstrate very fast convergence and high computational efficiency of the proposed multipoint methods.

Properties of the heavy-fermion spectrum in the canted phase of antiferromagnetic intermetallides

We study the effect of the magnetic-field-induced canting of magnetic sublattices on the energy structure of heavy fermion quasiparticles in intermetallides with antiferromagnetic-type ordering. We work in the framework of an effective Hamiltonian of the periodic Anderson model in the regime of strong electron correlations. With the virtual transfers into high-energy double states taken into account, this Hamiltonian involves exchange interactions between spin moments of the f ions and the s-f-exchange coupling between the two subsystems. For a noncollinear problem geometry, we introduce a unitary transformation that allows reducing an eighth-order equation for the spectrum of heavy fermions to two fourth-order equations. We show that the quasimomentum dependence of the heavy-fermion energy changes qualitatively under the transition from the antiferromagnetic phase to a phase with a considerable canting angle emerging in an external magnetic field.

Integrability conditions for the coupled electromagnetic- gravitational perturbation equations of the Kerr-Newman spacetime

We show by considering integrability conditions for the perturbation equations of the Kerr-Newman spacetime that it is possible, in principle, to obtain decoupled equations of eighth order for at least one of the perturbed quantities.

STABILITY OF FLOWS IN FLUIDIZED BEDS

ABSTRACT We study the linearlized stability of the state of uniform fiuidization, within the context of the theory of mixtures. The mixture is assumed to be made up of a classical linearly viscous fluid infused with particles. In marked departure from most of the previous studies, we model the solid as a granular material and assume a constitutive relation that stems from classical motions in continuum mechanics. The linearized stability analysis of the state of uniform fiuidization, in general, for a three-dimensional disturbance leads to an eighth order equation for the characteristic whose root implies instability, when positive.

A quartic dispersion equation for internal gravity waves in the thermosphere

A new quartic dispersion equation in the square of the complex vertical wave number is derived by employing the ‘shallow atmosphere’ approximation and an ion drag approximation. These approximations allow the coefficients of the quartic equation to be given in terms of the corresponding cubic equation, which neglects the Coriolis force and the zonal ion drag component, but modified to take into account these neglected effects. Coupling between the extraordinary viscosity wave mode and the other three wave modes is highlighted and numerical solutions are compared for this quartic equation, an exact eighth order equation and the cubic equation. For the first time the validity of using the ‘shallow atmosphere’ approximation to describe internal gravity wave motions is demonstrated.

A comparison of some equations for the flexural vibration of damped sandwich beams

This paper compares the theories of flexural vibration of damped, three-layer sandwich beams as presented by Yan and Dowell, and by DiTaranto and Mead and Markus. Depending on the assumptions made about the internal shear stress distribution, the differential equation of transverse flexural displacement is either of fourth or sixth order. The inclusion of the effects of face-plate shear deformation and longitudinal inertia in the analysis yields a sixth order differential equation if the beam section is symmetric, and an eighth order equation if the section is unsymmetric. Flexural wave speeds and loss factors computed from the theories are presented and compared. The DiTaranto and Mead and Markus equations yield reliable values provided the flexural wavelength is greater than about four face-plate thicknesses. The Yan and Dowell equations yield reliable values only at much greater wavelengths or when the central layer in the sandwich is very thick.

On equations of the state of stress in a plate of variable thickness

Method of asymptotic integration of three-dimensional equations of the theory of elasticity is used to construct the internal state of stress in a plate of variable thickness /1/. It is shown that it can generally be described by a system of differential equations of eighth order in the components of the displacement vector of the points of a plane projected inside the plate, with the equations of flexure and of plane state of stress not separated. In a particular case when the face surfaces are symmetrical with respect to this plane, the flexure and the plane state of stress are described by separate equations. The accuracy of the equations obtained is of the order of square of relative thickness of the plate away from the edge and other distortion lines of the stress state. The boundary layer is not considered and conditions at the plate edge are not formulated.

Electrohydrodynamic Instability in a Viscoelastic Liquid Layer

The problem of the onset of instability in a horizontal layer of viscoelastic dielectric liquid under the simultaneous action of a vertical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, an equation of eighth order is derived. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. It is shown that oscillatory modes of instability exist only when the thickness of the liquid layer is smaller than about 0.5 mm and for such a thin layer the force of electrical origin is much more important than buoyancy force.

Shell Vibration and Buckling Analysis Using a New Eighth-Order Equation

Doubly curved plates and regions of general shells may be easily investigated for both vibratory behavior and instability using a form of Donnell's equation. The new eighth-order equation provides more precision than Donnell's and also offers simple closed form relations that provide insight to the physical behavior of the shell. Examples are presented to depict the increased accuracy and the range of applicability of the equation. One result is a particularly simple relation between instability loading and natural frequency. In another demonstration the cylinder form of the general shell equation is shown to yield Levy's result for ring mode buckling of a cylinder under external pressure, which Donnell's equation does not do.

Boltzmann Equation for Hard Spheres in Differential Form

It is shown that it is possible to recast the Boltzmann integro-differential equation for hard spheres into a pair of linear partial differential equations of first and eighth orders respectively. The eighth order equation is shown to be the nine-dimensional Laplacian four times repeated. The first order equation is the familiar df/dt with a different forcing function on the right hand side.