Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Structural theory of special functions

A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painlevé classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painlevé-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painlevé-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 3–19, April, 1999.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine originalmodulation equations for laminar finite-amplitude non-linearwaves in an incompressible fluid. Three idealized problems areanalysed. The modulation equations comprise conservation ofwaves, averaged conditions for conservation of mass, momentum,kinetic energy and angular momentum and the averaged projectionof the Navier–Stokes equations onto the vorticity vector.The last of these modulation equations, which is related tovortex stretching, only appears in 3D problems. The techniqueof Reynolds averaging is also employed to obtain equations forthe mean velocities and pressure. The Reynolds-averaged Navier–Stokesequations correspond to the modulation equations for conservationof mass and momentum. However, the Reynolds stress transportequations are shown to be inconsistent with the other necessarymodulation equations. In two further idealized problems, exactsolutions of the Navier–Stokes equations are obtainedby employing the modulation equations.     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).     

Difference Equations and Lettenmeyer's Theorem

Many classical results for ordinary differential equations have counterparts in the theory of difference equations, although, in general, the technical details for the difference versions are more involved than the corresponding ones for differential equations. This note surveys material related to a difference analogue of Lettenmeyer's theorem. The projection method of Harris et al. , developed to treat certain questions in the analytic theory of ordinary differential equations is used to obtain counterparts for linear difference equations as well as extensions to certain nonlinear differential and difference equations.     

First‐order system least squares for the Oseen equations

Following earlier work for Stokes equations, a least squares functional is developed for two‐ and three‐dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady‐state, incompressible Navier–Stokes equations with a nested iteration‐Newton‐FOSLS‐AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Periodic solutions of functional dynamic equations with infinite delay

In this paper, sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales, which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers, respectively. The approach is mainly based on the Krasnosel’ski? fixed point theorem, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales. This study shows that one can unify such existence studies in the sense of dynamic equations on general time scales.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

Coulomb Gas Representation for Rational Solutions of the Painlevé Equations

We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II–VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations.     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

On Error Bounds for Approximation Schemes for Non-Convex Degenerate Elliptic Equations

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations.     

Stability analysis of methods employing reducible rules for volterra integral equations of the first kind

The concept of (A ₀,S)-stability for Volterra integral equations of the second kind will be extended to that of the first kind equations. We will show that stability polynomials for methods employing reducible quadrature rules, as applied to the first kind equations, can be trivially obtained from the results for the second kind equations.     

Indefinite integrals of special functions from hybrid equations

ABSTRACT

Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.