APPROXIMATION THEORY OF THREE DIMENSIONAL ELASTIC PLATES AND ITS BOUNDARY CONDITIONS WITHOUT USING KIRCHHOFF－LOVE ASSUMPTIONS

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Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form u _t–u=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form u/n _a=g(u(t), u(t–r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

THE GENERALIZED GALERKIN’S EQUATION OF THE FINITE ELEMENT,THE BOUNDARY VARIATIONAL EQUATIONS AND THE BOUNDARY INTEGRAL EQUATIONS

Based on[1 ],we have further applied the variational princi-ple of the variable boundary to investigate the discretizationanalysis of the solid system and derived the generalized Ga-lerkin’s equations of the finite element.the boundary varia-tional equations and the boundary integral equations.These e-quations indicate that the unknown functions of the solid sys-tem must satisfy the conditions in the element S_a or on the boun-dariesГ_a.These equations are applied to establishing the discreti-zation equations in order to obtain the numerical solution ofthe unknown functions.At a time these equations can be usedas the basis for the simplified calculation in various corres-ponding cases.In this paper,the results of boundary integral equationsshow that the calculation of integration is not accurate alongthe surfaceГ_a of interior element S_a by J-integral suggestedby Rice[2].     

On the Linearization of Semilinear Wave Equations

We consider the class of wave equations u _tt−u _xx=f(u, u _t, u _x). By using the differential invariants, with respect to the equivalence transformation algebra of this class, we characterize subclasses of linearizable equations. Wide classes of general solutions for some nonlinear forms of f(u, u _t, u _x) are found.     

Perturbation for fractional-order evolution equation

Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution equation for ^CD^a-e₀₊u(t)=A ^CD^d₀₊u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t), u(0)=u _o , α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C _o -semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified.     

Effective Boundary Conditions Resulting from Anisotropic and Optimally Aligned Coatings: The Two Dimensional Case

We discuss a thermally conducting body insulated by a thin anisotropically conducting coating. The coating is “optimally aligned” in the sense that the normal vector inside the coating is always an eigenvector of the thermal tensor. We study the effects of the coating by investigating the limiting behavior of solutions u of the heat equation with either Dirichlet or Neumann boundary conditions imposed on the outer boundary of the coating, as the thickness of the coating shrinks to zero. In the two-dimensional case, we find the complete list of “effective boundary conditions” satisfied by the limit of u on the boundary of the uncoated body. This list contains not only the usual Dirichlet, Neumann and Robin boundary conditions, but also some new and even nonlocal ones involving the Dirichlet-to-Neumann mapping and the Hilbert transform on the circle. We also prove that u converges to its limit in various norms that include the L ², the Sobolev and the Hölder ones. During the course of this study, we establish a Schauder theory for the regularity of weak solutions of general second order parabolic equations near an interface where the “transmission condition” is satisfied.     

The asymptotic solving equations of thick ring shell and its solution under moment Mo

In this paper, from the fundamental equations of three dimensional elastic mechanics, we have found a sequence of asymptotic solving equations of thick ring shell (or body) applied arbitrary loads by the perturbation method based upon a geometric small parameter a=r_o/R_o, which may be divided into two independent equation groups which are similar to the equation groups for plane strain and torsional problems. Using these equations, we have also found first order and second order approximate solutions of thick ring shell applied moment M_o.     

Direct displacement method in crack theory (numerical resolution)

Solutions in crack theory can be defined directly by opening displacement v _○=v _○(x) and u _○=u _○(x) for the first and the second mode, respectively. In this case, the boundary conditions are expressed by singular integrals of the second order. Aiming to solve numerically the problems, we apply the finite-part definition for the singular integrals and the discretization procedure. Contributed to the memory of my Mother.     

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L¹₊(R) for solutions of general nonlinear diffusion equations u_t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d₂. In the particular case of ϕ(u) ~ u^m at first order when u ~ 0, we also obtain an optimal rate of convergence in L¹ towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.     

Resonant non-linear vibrations in continuous systems—I. Undamped case

A method is presented for obtaining periodic solutions to forced oscillations of non-linear systems governed by equations of the form u_ss?u_yy?εf(u,u,_y,u_yy…,s) = 0. The method is presented by application to the equation u_ss?u_yy?εu²_yu_yy= 0 which governs the vibrations of a soil layer that is free on the top surface and is forced harmonically at the bedrock. It is shown that unlike the ODE case (Duffing equation), the PDE requires an infinite number of periodicity conditions to correctly characterize the resonant region and these conditions lead to an infinite number of branches in the dispersion spectrum. Calculations indicate that these branches tend to an envelope curve. The uniform approach presented by Millmann and Keller is discussed in order to determine in what sense it can be viewed as an effective approximation for the fundamental mode.     

New numerical method for partial differential equations. 1: Application to the diffusion equation

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one-dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three-point implicit scheme is obtained with a truncation error of order O(k⁴, h⁶), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h² ? S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h⁴). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other-implicit or explicit method.     

Stationary problem of third-grade fluids in two and three dimensions: existence and uniqueness

This paper is devoted to the stationary problem of third-grade fluids in two and three dimensions. In two dimensions, we show existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, by generalizing the method used by J.M. Bernard for the stationary problem of second-grade fluids (we deal with a polynomial of four degrees instead of two degrees). Contrary to the case of two dimensions, the resolution of the problem of third-grade fluids in three dimensions requires the physical condition |α₁+α₂|<(24νβ)^1/2. From this condition, we derive two “pseudo ellipticities” for the operator ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, where A(u) is a 3-order symmetric matrix such that tr(A(u))=0. Thus, with, in addition, a sharp estimate of the scalar product (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), we are able to prove existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, in three dimensions.

Résumé

Cet article est consacré au problème stationnaire des fluides de grade trois en dimension deux et trois. En dimension deux, nous montrons l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en généralisant la méthode utilisée par J.M. Bernard pour le problème stationnaire des fluides de grade deux (nous avons affaire à un polynôme de degré quatre au lieu de deux). Contrairement au cas de la dimension deux, la résolution du problème des fluides de grade trois en dimension trois requière la condition physique |α₁+α₂|<(24νβ)^1/2. De cette condition, nous déduisons deux “pseudo matrice” pour l’opérateur ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, où A(u) est une matice symétrique d’ordre 3 à trace nulle. De là, avec, en plus, une fine estimation du produit scalaire (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), nous sommes capables de prouver l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en dimension trois.     

Probability distribution functions for small scale turbulence

The exact expression for the probability distribution function (pdf),P(Δu_r), of a velocity difference Δu_r, over a distancer, in incompressible fluid turbulence, obtained from the Navier-Stokes equations, is used as a basis for deriving approximate profiles forP(Δu_r). These approximate forms are deduced from an approximate factorisation of the underlying functional probability distribution of the flow field, in which the individual factors capture different physical effects.P(Δu_r) is represented as the integral, with respect to the spatially averaged dissipation rateε _r, of the product of the conditionalpdf of Δu_r givenε _r, and thepdf ofε _r. The approximation yields the latter as a log-Poissonpdf, while the conditionalpdf is found to be a Gaussian for a transverse increment, and the product of a Gaussian and a cubic polynomial for a longitudinal increment. This approximation is equivalent to the refined similarity hypothesis coupled with the log-Poisson distribution, and it possesses the characteristic features ofP(Δu_r), including symmetric profiles for transverse increments, asymmetric profiles for longitudinal increments, and the development of pronounced non-Gaussian features at small separations. The associated scaling exponents for longitudinal and transverse structure functions are shown to be identical, in this approximation, and to assume the log-Poisson form.     

On the First Boundary Value Problem for Quasilinear Parabolic Equations with Two Independent Variables

This paper is concerned with the global solvability of the first initial boundary value problem for the quasilinear parabolic equations with two independent variables: a(t,x,u,u_xINF>)u_xxm u_t=f(t,x,u,u_xINF>). We investigate the case when the growth of [(|f(t,x,u,p)|)/(a(t,x,u,p))]{{|f(t,x,u,p)|}\over {a(t,x,u,p)}} with respect to p is faster than p² when |p|M X. Conditions which guarantee the global classical solvability of the problem are formulated.     

Similarity solutions of the equations of a boundary layer in a nonuniform external flow

Similarity solutions of the equations of a laminar incompressible boundary layer, formed in a rotational external flow, are investigated. Such problems arise in the analysis of the flow in a boundary layer when there is an abrupt change in the boundary conditions (for example, in the case of a discrete inflation of the boundary layer, in hypersonic flow about blunt bodies, etc.). Various approaches to their solution have been proposed earlier in [1–4]. Solved below is the so-called inverse problem of boundary layer theory (see [3], p. 200), where the contour of the body that causes a given flow outside the boundary layer is unknown beforehand and is found during the course of solution of the problem in connection with the coupling of the longitudinal and transverse velocity components. The cases of a parabolic (u_e ~ y²) and a linear (u_e=a(x)+b(x)y) variation in the velocity of the external flow with distance along the transverse direction are considered in detail. The latter includes an investigation of the flow in the neighborhood of the critical point of a blunt body, taking account of the vorticity of the flow in the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 78–83, March–April, 1971.     

Entropy Solutions for Nonlinear Degenerate Problems

We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)_t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.     

Approximate solution to nonself-similar problem of motion of a piston after an impact

An approximate solution is obtained to the problem of the motion of a piston after an impact and under the influence of gas pressure under the assumption that the parameter = u_o/a _o, where u_o is the initial velocity of the piston anda _o is the velocity of sound in the gas at rest, is small. Functions that determine the law of motion of the piston and the shock wave, and also the gas flow in the disturbed region are found explicitly to terms of order ³ Translated from Izvestiya Akadeinii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1982.     

Contact Transformation Group Classification of Nonlinear Wave Equations

Interest in nonlinear wave equations has been stimulated bynumerous physical applications, such as telecommunication (e.g.nonlinear telegrapher equation), gasdynamics, anisotropic plasticity andnonlinear elasticity, etc. Mathematical models of these phenomena canoften be reduced to particular types of the equation u _tt = f(x, u _x ) u _xx + g(x, u _x ). In this paper, the problem ofclassification of the latter equation with respect to admitted contacttransformation groups is reduced to the investigation of pointtransformation groups of the equivalent system of first-orderquasi-linear equations v _t =a(x, v)w _x , w _t = b(x,v)v _x .