On the structure of characteristic surfaces related to partial differential equations of first and higher orders. II

The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 529–543, October–December, 2005.     

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

Multiscale Young Measures in almost Periodic Homogenization and Applications

We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport equations, Hamilton–Jacobi equations and fully nonlinear elliptic equations. Motivated by the application in nonlinear transport equations, we also prove basic results on flows generated by Lipschitz almost periodic vector fields, which are of interest in their own. In our analysis, an important role is played by the so-called Bohr compactification of ; this is a natural parameter space for the Young measures. Our homogenization results provide also the asymptotic behavior for the whole set of -translates of the solutions, which is in the spirit of recent studies on the homogenization of stationary ergodic processes.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.     

Henry’ Law and Gas Phase Disappearance

In a two-phase (liquid–gas) two-component (water–hydrogen) system we discuss the formulation of the possible dissolution of hydrogen in the liquid phase. We show how Henry’s law fits in a phase diagram and the problem is formulated as a set of nonlinear partial differential equations with complementarity constraints.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Problems of modeling and optimal stabilization of the gas-lift process

The problems of motion of fluids, gases and gas–liquid mixtures in pipes related to gas-lift oil recovery are mathematically formulated as systems of nonlinear hyperbolic partial differential equations. Optimal-control problems are posed based on the proposed models and some real assumptions. These problems can be used to design programmed paths and controls, which underlie the controllers that stabilize the pressure or volume of injected gas. That the mathematical models agree with available field and laboratory data is demonstrated by examples     

On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type

We study the structure of the set of solutions, continuously differentiable for t ∈ R ⁺ = [0; + ∞), of one limit problem for systems of nonlinear functional differential equations of neutral type with nonlinear deviations of argument that depend on an unknown function. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 277–289, April–June, 2007.     

On the Hartman–Grobman Theorem with Parameters

The Hartman–Grobman Theorem of linearization is extended to families of dynamical systems in a Banach space \mathbb X{\mathbb X} , depending continuously on parameters. We prove that the conjugacy also changes continuously. The cases of nonlinear maps and flows are considered, and both in global and local versions, but global in the parameters. To use a special version of the Banach–Caccioppoli Theorem we introduce equivalent norms on \mathbb X{\mathbb X} depending on the parameters. The functional setting is suitable for applications to some nonlinear evolution partial differential equations like the nonlinear beam equation.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

On the solution of the Chazy system of equations

We obtain a solution of the Chazy system that consists of nine nonlinear algebraic equations. This system gives a necessary condition for the class of nonlinear third-order differential equations with six singularities to be of P-type. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 92–98, January–March, 2009.     

Effects of Chemical Reaction and Double Dispersion on Non-Darcy Free Convection Heat and Mass Transfer

In this article, the effects of chemical reaction and double dispersion on non-Darcy free convection heat and mass transfer from semi-infinite, impermeable vertical wall in a fluid saturated porous medium are investigated. The Forchheimer extension (non-Darcy term) is considered in the flow equations, while the chemical reaction power–law term is considered in the concentration equation. The first order chemical reaction (n = 1) was used as an example of calculations. The Darcy and non-Darcy flow, temperature and concentration fields in this study are observed to be governed by complex interactions among dispersion and natural convection mechanisms. The governing set of partial differential equations were non-dimensionalized and reduced to a set of ordinary differential equations for which Runge–Kutta-based numerical technique were implemented. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) are presented in graphs.     

Nonlinear equations of flexural-flexural-torsional oscillations of rotating beamswith arbitrary cross-section

A system of three nonlinear partial differential equations describing the flexural-flexural-torsional vibrations of a rotating slender cantilever beam of arbitrary cross-section is derived using Hamilton’s principle. It is assumed that the center of gravity and the shear center are at different points. The interaction between flexural and torsional vibrations is accounted for in the linear and nonlinear parts of model Published in Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 123–132, May 2008.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

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.     

Divergence‐Measure Fields and Hyperbolic Conservation Laws

. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to . (Accepted July 16, 1998)     

Stress analysis of a flexible ring plate with circumferentially varying stiffness in a magnetic field

A problem of magnetoelasticity for a ring plate of variable stiffness is solved. A governing system of nonlinear differential equations is derived. It describes the stress–strain state of flexible conductive plates in nonstationary mechanical and electromagnetic fields. A numerical example is given. The stress state of a flexible plate of variable stiffness in a magnetic field is analyzed     

Implicit Difference Methods for First-Order Partial Differential Functional Equations

We present a new class of numerical methods for quasilinear first-order partial differential functional equations. The numerical methods are difference schemes implicit with respect to time variable. We give a complete convergence analysis for the methods and show by an example that the new methods are considerably better than explicit schemes. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 201–215, April–June, 2005.