On the radius of convergence of series in powers of time for spin correlation functions of the Heisenberg magnet at high temperature

The convergence of series in powers of time for spin autocorrelation functions of the Heisenberg magnet are investigated at infinite temperatures on lattices of different dimensions d. The calculation data available at the present time for the coefficients of these series are used to estimate the corresponding radii of convergence, whose growth with decreasing d is revealed and explained in a self-consistent approximation. To this end, a simplified nonlinear equation corresponding to this approximation is suggested and solved for the autocorrelation function of a system with an arbitrary number Z of nearest neighbors. The coefficients of the expansion in powers of time for the solution are represented in the form of trees on the Bethe lattice with the coordination number Z. A computer simulation method is applied to calculate the expansion coefficients for trees embedded in square, triangular, and simple cubic lattices under the condition that the intersection of tree branches is forbidden. It is found that the excluded volume effect that manifests itself in a decrease in these coefficients and in an increase in the coordinate and exponent of the singularity of the autocorrelation function on the imaginary time axis is intensified with decreasing lattice dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 112. No. 3, pp. 479–491. September, 1997.     

Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.     

A generalized Cauchy problem for the linear differential equations of coupled physical - mechanical fields

A generalized Cauchy problem for a partial differential equation with constant coefficients, which is encountered in the study of physical processes in continuous media with widened physical - mathematical fields (see /1/) (generalized coupled thermoeleasticity /2/, coupled thermoeleasticity, porous media saturated with a viscous fluid /5/, mass and heat transfer /6/, linearized magnetoelasticity /7/, etc.) is considered. The characteristic properties of the solution of the problem, under certain constraints imposed on an equation by the stability condition, are studied. The presence of waves of higher and lower order is characteristic for the solution; in the course of time the lower-order waves are maintained and take a characteristic form. In the general case, the solution is represented in the form of integrals over the segments which link the singular points of Fourier - Laplace transforms with respect to time of the solution under consideration. The methods proposed enable an exact investigation to be made of the processes described by the equation for any time constants, and they also enable one to isolate the singularities at the fronts of propagating perturbations. As an application, the dynamic processes taking place in a thermoelastic subsapce (2) as a result of applying a mechanical and a thermal input at the boundary is studied. It is shown that in the case of unit perturbation of the boundary, the stress and temperature waves in the course of time assume a bell-shaped form and propagate with adiabatic velocity. A numerical analysis of the process which occurs due to sudden application of the force and of the thermal shock at the boundary is given.     

Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. Application in the mechanics of composite materials

A coupled unsteady problem of thermoelasticity for an inhomogeneous body, described by a system of four second-order partial differential equations with coefficients that vary depending on the coordinates, is considered, and the same problem for a homogeneous body of the same shape (the concomitant problem) is examined together with this original problem. Integral formulae are obtained that allow one to express the displacements and temperature in the original problem in terms of the displacements and temperature in the concomitant problem. Integral formulae are used to represent the solution of the original problem in the form of series over all possible derivatives of the solution of the concomitant problem. A system of recurrence problems is written for the coefficients of these series. Expressions are found for the coefficients of the concomitant problem (effective coefficients) and special boundary value problems are formulated, from the solution of which specific expressions are found for the effective thermoelasticity coefficients. A theorem concerning the fact that the effective coefficients satisfy the physicomechanical constraints imposed on the thermoelastic constants of real bodies is proved. The case of a layer that is inhomogeneous in its thickness is considered and explicit analytical expressions for all the thermoelasticity coefficients are obtained for it. The case when the thermoelasticity coefficients depend periodically on the coordinates is examined in detail.     

Two boundary value problems for a strongly anisotropic inhomogeneous elastic ring

The second boundary value problem (displacements are given on the boundary) and the improper mixed problem for a cylindrically orthotropic ring are studied. It is assumed that the coefficients of elasticity are continuously differentiable functions of the coordinates and depend on a small parameter in a specific manner. The form of the dependence of the coefficients on the small parameter is selected in such a way that in the case of constant coefficients it describes bonding of the ring by two families of very rigid fibers located along the radius vectors and concentric circles, where the stiffness of the fiber families is of identical order. Consequently, the coefficients of elasticity are represented in the form of products of constants which will later be called provisionally the “stiffnesses”, and functions of the coordinates. It is assumed that the stiffnesses in the radial and circumferential directions are equal and exceed and shear stiffness considerably. The asymptotic form of the solution of the boundary value problems under consideration is constructed when the ratio between the shear stiffness and the stiffness in the radial direction is used as the small parameter. In the case of the second boundary value problem the limit boundary value problem is described by a hyperbolic system of equations and is not solvable uniquely, since one of the families of characteristics is parallel to the boundary. When constructing the asymptotic form the necessity arises to average the coefficients of elasticity with respect to the circumferential coordinate. In this respect, there is an analogy with the results obtained in /1/ where the boundary value problem was studied for a second-order elliptic equation.     

Exact Closed Form Solution for Two Coupled Inhomogeneous First Order Difference Equations with Variable Coefficients

We present an exact closed form solution for two coupled, homogeneous as well as inhomogeneous, first order difference equations with variable coefficients. The solution is obtained by using the graph theoretic, discrete path formalism. The central parameters in the solution are the crossing index and the crossing number. The transition from an enumerative graph theoretic solution to a closed form combinatorial solution is made possible by an isomorphism in-between paths on the signal flow graph, and n -tuplets of binary numbers.     

可渗透壁面上Falkner-Skan磁流体动力学流动的近似解

研究了可渗透壁面上Falkner-Skan磁流体动力学（MHD）边界层流动问题．利用结合了微分变换法（DTM）和Padé近似的DTM-Padé方法,得到了边界层问题的近似解和壁摩擦因数值．通过建立一个迭代程序,边界层问题的近似解被表示为幂级数的形式,而且以图和表形式对不同参数下的近似解结果与打靶法得到的数值结果进行了对比,近似解和数值解结果高度吻合,从而验证了所得问题近似解和结论的可靠性和有效性．并且,对求得的边界层问题近似解结果进行了讨论,分析了不同物理参数对边界层流动的影响．     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

ANALYSIS OF THE LIMIT CYCLE OF A GENERALISED VAN DER POL EQUATION BY A TIME TRANSFORMATION METHOD

Abstract

The properties of the limit cycle of a generalised van der Pol equation of the form ü + u = ε (1—u²ⁿ)u, where ε is small and n is any positive integer, are investigated by applying a time transformation perturbation method due to Burton. It is found that as n increases the amplitude of the limit cycle oscillation decreases and its period increases. The time transformation solution is compared with the solution derived using the method of multiple scales and with a numerical solution. It is found that, to first order in ε, the time transformation solution for the limit cycle agrees better with the numerical solution than the multiple scales solution. Both perturbation solutions give the same result for the period of the limit cycle to second order in ε. The accuracy of the time transformation solution decreases as n increases.     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

A Nonlocal Finite-Difference Boundary-Value Problem

We investigate the stability of difference schemes for the equation of heat conduction with nonlocal boundary conditions. An example is given which in a certain sense imitates the problem with variable coefficients and has an exact solution in analytical form. It is shown that the difference operator has a simple spectrum and that multiple eigenvalues appear only in the case with constant coefficients. The simple spectrum ensures that the eigenvectors of the finite-difference problem form a basis. This enables us to apply to the nonlocal problem the theory of stability of symmetrizable difference schemes.     

A film coating on a rough surface of an elastic body

A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young's moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occures on a protrusion of the softer substrate. ©2013     

A Smooth Embedding Domain Method Based On the Penalty Approach

It is well known that convergence of the fictitious domain formulation with boundary Lagrange multipliers is slow due to the lower global regularity of its solution. This article presents a smoothed variant of this approach which is based on a formulation in the form of a state constraint optimal control problem. The convergence rate is increased as seen from a model example.     

Bright soliton solution to a generalized Burgers–KdV equation with time-dependent coefficients

We consider a generalized Burgers–KdV type equation with time-dependent coefficients incorporating a generalized evolution term, the effects of third-order dispersion, dissipation, nonlinearity, nonlinear diffusion and reaction. The exact bright soliton solution for the considered model is obtained by using a solitary wave ansatz in the form of sech^s function. The physical parameters in the soliton solution are obtained as functions of the time varying coefficients and the dependent exponents. The dependent exponents and the temporal variations of the model coefficients satisfy certain parametric conditions as shown by the obtained soliton solution. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Burgers–KdV type models.     

Finite Time Merton Strategy under Drawdown Constraint: A Viscosity Solution Approach

We consider the optimal consumption-investment problem under the drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model and we consider a general class of utility functions. On an infinite time horizon, Elie and Touzi (Preprint, [2006]) provided the value function as well as the optimal consumption and investment strategy in explicit form. In a more realistic setting, we consider here an agent optimizing its consumption-investment strategy on a finite time horizon. The value function interprets as the unique discontinuous viscosity solution of its corresponding Hamilton-Jacobi-Bellman equation. This leads to a numerical approximation of the value function and allows for a comparison with the explicit solution in infinite horizon.     

Estimates for the density of a nonlinear Landau process

The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients.     

On a boundary problem with Neumann’s condition for 3D singular elliptic equations

The present work is devoted to the studying of a boundary-value problem with Neumann’s condition for three-dimensional elliptic equation with singular coefficients. The main result is a proof of the unique solvability of the problem considered. An energy integral method and a Green’s function method were used as the main tools in the proof of the main result. The unique solution is found in an explicit form, which contains Appel’s hypergeometric functions.     

Self-similar solution of the plane problem of the evolution of a hydraulic fracture crack in an elastic medium

The plane problem of the evolution of a hydraulic fracture crack in an elastic medium is considered. It is established that a self-similar solution is only possible at a constant rate of fluid injection. The solution for the value of the crack opening is presented in the form of a series expansion in Chebyshev polynomials of the second kind, and expansion coefficients are obtained as a solution of the algebraic set of equations which arise when projecting the balance equation for injected fluid mass on Chebyshev polynomials. When there is no part of the region unfilled with fluid (a fluid lag), the gradient of the crack opening at the crack tip turns out to be singular when the finiteness of the medium stress intensity factor is taken into account. According to the estimate made, the rate of convergence of the series expansion for the solution in Chebyshev polynomials is fairly rapid for a small injection intensity.     

Time-Variable Extension of the Solution of a Nonlocal Multipoint Problem for Partial Differential Equations with Constant Coefficients

A problem with nonlocal multipoint conditions for the nth-order partial differential equation with constant coefficients is considered. In the case where conditions of strict averaging of time intervals are specified, the existence of a solution of the problem in a cylinder that is the Cartesian product of a time interval and a p-dimensional spatial torus is discussed. It is found that under certain conditions of separability of the roots of the characteristic equation for almost all (in the sense of the Lebesgue measure) coefficients of the equation and parameters of the conditions, the solution of the problem cannot be extended in the time variable beyond the extreme points at which the conditions are given.