Asymptotic properties of solutions of the Cauchy problem for a degenerate singularly perturbed system of differential equations in the case of the multiple spectrum of the principal operator

We prove the asymptotic character of a solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of the coefficients of derivatives in the case where the limit matrix pencil is regular and has multiple “finite” and “infinite” elementary divisors. We establish conditions under which the constructed formal solutions are asymptotic expansions of the corresponding exact solutions. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 247–257, April–June, 2007.     

Conditions for the existence of a solution of a Noether boundary-value problem for a second-order system

We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 368–375, July–September, 2006.     

Boundary-value problems for differential equations unresolved with respect to the derivative in the space of bounded number sequences

We study a nonlinear countable-point boundary-value problem for a differential equation unresolved with respect to the derivative. This equation and a nonlinear boundary condition are defined in the Banach space of bounded number sequences. We study the reducibility of the posed problem to a multipoint boundary-value problem in a finite-dimensional space. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 391–415, July–September, 2007.     

On the Nield-Kuznetsov Theory for Convection in Bidispersive Porous Media

We revisit the problem of thermal convection in a bidispersive porous medium, first addressed by Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006). We investigate the possibility of oscillatory convection by using a highly accurate Chebyshev tau numerical method. We also develop a nonlinear energy stability theory for the same problem. This yields a global stability threshold below which instabilities cannot arise. These thresholds together with the linear instability boundaries yield a zone where thermal instability may be found. The results and theory of Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006) are thus proven to be a highly important development in the modern theory of designer porous materials, cf. Nield and Bejan (Convection in Porous Media, Springer, New York, 2006), pp. 94–97. This work was supported in part by a Research Project Grant of the Leverhulme Trust—Grant Number F/00128/AK.     

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

An analog of the Saint-Venant principle and the uniqueness of a solution of the first boundary-value problem for a third-order equation of combined type in unbounded domains

We consider the first boundary-value problem for a third-order equation of combined type. Using the Saint-Venant principle, we study the uniqueness class for solutions of the problem in an unbounded domain. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 117–126, January–March, 2006.     

Effect of yaw on the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases)

Lighthill (Proc. R. Soc. A 198, 454–470, 1949) considered the diffraction of a normal shock wave passing over a small bend. The bend being small Lighthill was able to linearize the flow equations and solved the problem through several mathematical techniques. Following Lighthill (Proc. R. Soc. A 198, 454–470, 1949), Srivastava and Chopra (J. Fluid Mech. 40, 821–831, 1970) extended the work to the diffraction of oblique shock waves. Srivastava (AIAAJ 33, 2230–2231, 1995) considered the problem of starting point of curvature and extended the work to yawed wedges (Srivastava in Proceedings of the 14th International Mach reflection symposium Sun Marina Hotel, Yonezawa, Japan, 1–5 October 2000, pp. 225–249, 2002). Srivastava (Shock waves 13, 323–326, 2003) considered the problem for starting point of curvature when the relative outflow behind reflected shock before diffraction has been subsonic and sonic. The present work is an extension of the work published in Srivastava (Shock waves 13, 323–326, 2003) when the wedge has been yawed through an angle. The results have been obtained for two angles χ = 60° and χ = 40° (χ is the angle of yaw).      

A Complete Solution of the Wave Equations for Transversely Isotropic Media

A transversely isotropic material in the sense of Green is considered. A complete solution in terms of retarded potential functions for the wave equations in transversely isotropic media is presented. In this paper we reduce the number of potential functions to only one, and we discuss the required conditions. As a special case, the torsionless and rotationally symmetric configuration with respect to the axis of symmetry of the material is discussed. The limiting case of elastostatics is cited, where the solution is reduced to the Lekhnitskii–Hu–Nowacki solution. The solution is simplified for the special case of isotropy. In this way, a new series of potential functions (to the best knowledge of the author) for the elastodynamics problem of isotropic materials is presented This solution is reduced to a special case of the Cauchy–Kovalevski–Somigliana solution, if the displacements satisfy specific conditions. Finally, Boggio's Theorem is generalized for transversely isotropic media which may be of interest to the reader beyond the present application. Dedicated to Morton E. Gurtin     

Projection method for the solution of integro-differential equations with restrictions and control

We consider an application of the projection method to a boundary-value problem for integro-differential equations with restrictions and control and propose a calculation scheme for the method. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 208–216, April–June, 2008.     

An inverse elastoplastic problem for plates

We study an inverse elastoplastic problem of determining the residual stresses, the plasticity zone, and the external loads for a plate for known residual deflections which occur after removal of these loads and elastic unloading. Assuming that the deformation theory of plasticity is valid at the active stage of deformation, we prove the theorem of unique solution. An iterative method of solution is proposed and a variational formulation of the problem is given. Some simple examples are considered. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 186–194, July–August, 1999.     

Eigenvalues of the antiplane-shear crack problem for a power-law material

This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 173–180, January–February, 2008.     

Regular integral equations for the second boundary-value problem of the bending of an anisotropic elastic plate

Two systems of Fredholm equations of the second kind are constructed for the solution of the second boundary-value problem of the bending of an anisotropic plate (a normal bending moment and a generalized shear force are specified on the boundary of the simply-connected domain) under the assumption of validity of the Kirchhoff-Love hypotheses. Correct equilibrium conditions are specified for the examined boundary-value problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 108–119, May–June, 2005.     

Quasi-Neutral Limit for a Model of Viscous Plasma

We perform a rigorous analysis of the quasi-neutral limit for a model of viscous plasma represented by the Navier–Stokes–Poisson system of equations. It is shown that the limit problem is the Navier–Stokes system describing a barotropic fluid flow, with the pressure augmented by a component related to the nonlinearity in the original Poisson equation.     

Exact solution of the problem of convective heat exchange for a circular cylinder at small Prandtl numbers

The problem of the heat exchange of a circular cylinder in an incompressible flow at small Prandtl numbers Pr ≪ 1 is solved. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–48, January–February, 1994.     

Domain of Convergence of an Iteration Procedure for a Weakly Nonlinear Boundary-Value Problem

We obtain estimates for the values of a small parameter for which the iteration procedure used for the construction of solutions of the Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations is convergent in both critical and noncritical cases. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 278–288, April–June, 2005.     

Solving the torsion problem for an anisotropic prism by the advanced Kantorovich–Vlasov method

The torsion problem for a rectangular prism with general anisotropy loaded on the lateral surface is solved using the advanced Kantorovich–Vlasov method, which reduces the original three-dimensional problem to three coupled one-dimensional problems, each for one of the variables of the domain. The warping of the cross-section and the deformation of the axis of the prism for different types of anisotropy are analyzed     

A new approach for the ellipsoidal statistical model

In this paper we aim to introduce a systematic way to derive relaxation terms for the Boltzmann equation based on the minimization problem for the entropy under moments constraints (Levermore in J. Stat. Phys. 83:1021–1065, 1996; Schneider in M2AN 38:541–561, 2004). In particular the moment constraints and corresponding coefficients are linked with the eigenfunctions and eigenvalues of the linearized collision operator through the Chapman–Enskog expansion. Then we deduce from this expansion a single relaxation term of BGK type. Here we stop the moments constraints at order two in the velocity v and recover the ellipsoidal statistical model (Holway in Rarefied Gas Dynamics, vol I, pp 193–215, 1966).      

Second-Order Fredholm Equations for the First Boundary-Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

A complete potential theory is constructed for the first boundary-value problem in the two-dimensional anisotropic theory of elasticity (the force vector is specified on the boundary) in a bounded domain on a plane with a Lyapunov boundary. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 85–94, March–April, 2006.     

Existence of an invariant torus for one class of systems of differential equations

We consider the problem of the existence of an asymptotically stable toroidal set for a system of linear differential equations defined on an m-dimensional torus. We establish conditions under which a nonlinear system of differential equations has an invariant toroidal manifold. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 520–529, October–December, 2008.