Solution of one-dimensional problems of elasticity and thermoelasticity in stresses for a cylinder

We propose a method of solving problems of elasticity and thermoelasticity in stresses, making it possible to give a simpler construction of the solutions of such one- dimensional problems for a cylinder, compared with the known method of solving in the displacements. We obtain a relation between the components of the stress tensor and the integral conditions of equilibrium and continuity that has important applications in the solution of inverse problems of thermoelasticity.Translated fromMatematichni Metodi ta Fiziko-Mechanichni Polya, Vol. 40, No. 3, 1997, pp. 103–107.     

On the use of characteristic variables in viscoelastic flow problems

The system of partial differential equations governing the flow of an upper converted Maxwell fluid is known to be of mixed elliptic–hyperbolic type. The hyperbolic nature of the constitutive equation requires that, where appropriate, inflow conditions are prescribed in order to obtain a well-posed problem. Although there are three convective derivatives in the constitutive equation there are only two characteristic quantities whichare transported along the streamlines. These characteristicquantities are identified. A spectral element method is describedin which continuity of the characteristic variables is usedto couple the extra stress components between contiguous elements.The continuity of the characteristic variables is treated asa constraint on the constitutive equation. These conditionsdo not necessarily impose continuity on the extra-stress components.The velocity and pressure follow from the doubly constrainedweak formulation which enforces a divergence-free velocityfield and irrotational polymeric stress forces. This meansthat both the pressure and the extra-stress tensor are discontinuous.Numerical results are presented to demonstrate this procedure.The theory is applied to the upper convected Maxwell modelwith vanishing Reynolds number. No regularization techniques such as streamline upwind Petrov Galerkin (SUPG), elastic viscous split stress (EVSS) or explicitly elliptic momentum equation(EEME) are used.     

Integration of equations of one-dimensional problems of elasticity and thermoelasticity for inhomogeneous cylindrical bodies

We propose a method for direct integration of differential equations of equilibrium and continuity in terms of stresses in the case of one-dimensional quasistatic problems of elasticity and thermoelasticity for inhomogeneous and thermosensitive isotropic cylindrical bodies. The solution of each of the one-dimensional problems is reduced to a Volterra integral equation of the second kind, which makes it possible to propose a rapidly convergent iteration method of computations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 41, No. 2, pp. 124–131, April–June, 1998.     

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

On vibrations of an inhomogeneous anisotropic plate

An equation of motion of a flat inhomogeheous anisotropic plate is considered. Formal asymptotic solutions are constructed by applying the space-time ray method. An equation describing the flow of energy is obtained in the form of a continuity equation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 73–82. Translated by A. S. Golubeva.     

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL ² space. It is assumed that the Markov semigroup corresponding to the state process isC ₀ on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.     

DISCRETE-TIME MARTINGALES WITH SPATIAL PARAMETERS

Our analysis of a certain stochastic difference equation driven by a martingale k?M(x,k) that depends on a spatial parameter x∈R ^d requires some regularity properties of the underlying martingale be satisfied. Because of their independent interest, we present these regularity properties in this article. We study first the continuity and Lipschitz continuity properties under corresponding conditions on the quadratic covariation of the martingale. We follow this with differentiability and integrability properties. Our analysis of the stochastic difference equation requires a discrete-time version of Itô's formula. The discrete-time Itô formula we have derived involves a martingale transform term. The purpose of the final section is to introduce linear and nonlinear martingale transforms and analyze their properties.     

2D numerical simulation of submerged hydraulic jumps

Hydraulic jumps are usually used to dissipate energy in hydraulic engineering. In this paper, the turbulent submerged hydraulic jumps are simulated by solving the unsteady Reynolds averaged Navier–Stokes equations along with the continuity equation and the standard k–? equations for turbulence modeling. The Lagrangian moving grid method is employed for the simulation of the free surface. In the developed model, kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water elevation can be obtained along with velocity and pressure fields as part of the solution. Computational results are presented for Froude numbers ranging from 3.2 to 8.2 and submergence factors ranging from 0.24 to 0.85. Comparisons with experimental measurements show that numerical model can simulate the velocity field, variation of free surface, maximum velocity, Reynolds shear and normal stresses at various stations with reasonable accuracy.     

Supercomputer simulation of viscous noncompressible flow past bodies of complex shape

The article reports the results of numerical analysis of viscous thermally conducting noncompressible flow past bodies of complex shape. The effect of physico-chemical processes is ignored. Flow past bodies with various nose cone designs is examined. The process is described using a system of Navier–Stokes equations augmented with the energy equation. The continuity equation is used to control the numerical accuracy. The simulation results are reported in the form of vector fields and surface components of the velocity vector in various channel sections. The results are analyzed for various body configurations.     

Solutions of one-dimensional problems of elasticity and thermoelasticity for cylindrical piecewise-homogeneous bodies

We propose a method of direct integration of the equilibrium and continuity equations in stresses for one-dimensional problems of elasticity and thermoelasticity for piecewise-homogeneous cylinders and disks with an arbitrary number of layers. The solutions are reduced to finding the constants of systems of algebraic equations with nearly triangular matrices of coefficients, making it possible to find the unknown constants in a closed form that is functionally dependent on the bulk forces and temperature field.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 139–148.     

Continuity in functional differential equations with infinite delay

In this paper, we investigate the continuous dependence of solutions of the functional differential equation with infinite delayx(t)=f(t,x _t ) on initial functions. Endowing the phase space ag-norm as well as a supremum norm, we show that if the equation satisfies a mild fading memory dondition, then the continuity off in respect to the topology induced by the supremum norm can yield the continuity of solutions of the equation in respect to the topology induced by theg-norm which is stronger than the ahead one.This research was supported in part by an NSF grant with number NSF-DMS-8521408.On leave from South China Normal University, Guangzhou, PRC. This research was supported in part by the National Science Foundation of PRC.     

Modeling complex systems macroscopically: Case/agent-based modeling,synergetics, and the continuity equation

Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Haken's synergetic approach to modeling certain types of complex, self-organizing social systems macroscopically. The key to this advancement comes from employing a case-based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case-based employment of the continuity equation has four benefits for agent-based and case-based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent-based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.     

An Integrodifferential Equation Arising in Thermo-Viscoelasticity

Viscoelastic material at high temperature is subjected to a cooling process. The stresses built up in the body are determined from a system of equations containing a strongly temperature-dependent viscosity η(T), where the temperature T is given by the heat conduction equation. It is shown that for simple geometries such as infinite cylinders and spheres, the basic equations can be reduced to a single Volterra-type integrodifferential equation, which is shown to have a unique solution.     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.     

Properties of solutions ofu″+g(t) u 2n−1 =0, II

Summary Part I [7] of this paper appeared in this journal in 1962. In Sections 1, 2 of this paper we consider the equation of the title wheng(t)>0. In Section 1 we examine the strength of the hypothesis of continuity ofg(t) in known theorems. In Section 2 we determine the combinations of boundedness and oscillation that may occur for solutions of this equation. Section 3 is devoted to properties of solutions of the equation wheng(t)<0. In general, differentiability ofg(t) is not required in the paper.     

A remark on uniqueness for some ODE

Abstract We present a uniqueness result for the Cauchy problem associated to a particular type of ordinary differential equation (ODE), under the only assumption of continuity of the right hand side at the initial point. Keywords: Polar coordinates, Tangent vector, Inner product Mathematics Subject Classification (2000): 34A12     

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.     

Global well-posedness for the critical 2<Emphasis Type="Italic">D</Emphasis> dissipative quasi-geostrophic equation

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity. Mathematics Subject Classification (1991) Primary: 35Q35; Secondary: 76U05