Diffusive limits of nonlinear hyperbolic systems with variable coefficients

We consider the initial-boundary value problem for a 2-speed system of first-order nonhomogeneous semilinear hyperbolic equations whose leading terms have a small positive parameter. Using energy estimates and a compactness lemma, we show that the diffusion limit of the sum of the solutions of the hyperbolic system, as the parameter tends to zero, verifies the nonlinear parabolic equation of the p-Laplacian type.     

ASYMPTOTIC SOLUTIONS OF THE NONLINEAR STABILITY OF A TRUNCATED SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

In this paper,the nonlinear stability problem of a clamped truncated shallow sphericalshell with a nondeformable rigid body at the center under a concentrated load is studied bymeans of the singular perturbation method.When the geometrical parameter k is large,theuniformly valid asymptotic solutions are obtained.     

The Method of Multiple Scales Applied to the Nonlinear Stability Problem of a Truncated Shallow Spherical Shell of Variable Thickness with the Large Geometrical Parameter

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Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Existence Results

In this paper,we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.     

The singular perturbation method applied to the nonlinear stability problem of a shallow spherical shell (II)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author’s previous paper[11].     

Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in \(\mathbb {R}^2\) with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Pointwise Growth and Uniqueness of Positive Solutions for a Class of Sublinear Elliptic Problems where Bifurcation from Infinity Occurs

In this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary‐value problem of general sublinear type with a weight function multiplying the nonlinearity. When this function vanishes on some subdomain, the problem exhibits a bifurcation from infinity. In this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs. In this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite Dirichlet boundary conditions. This behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs. Also, we solve the problem using spectral collocation methods coupled with path‐following techniques to show how the main uniqueness result is optimal. Throughout the paper the mathematical analysis aids the numerical study, and the numerical study confirms and illuminates the analysis. (Accepted February 17, 1998)     

The singular perturbation for the buckling of a truncated shallow spherical shell with the large geometrical parameter

A problem of practical interest for nonlinear axisymmetrical stability of a truncated shallow spherical shell of the large geometrical parameter with an articulated external edge and a nondeformable rigid body at the center under compound loads is investigated in this paper. By using modified method of multiple scales, the uniformly valid asymptotic solutions of this boundary value problem are obtained when the geometrical parameter k is large. Project supported by the National Natural Science Foundation of China     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

Plane strain problem for an incompressible nonlinearly elastic solid

The plane strain of an incompressible body is studied with geometrical and physical nonlinearity and potential forces taken into account. A nonlinear system of equations for strains is obtained in actual variables, and conditions of its ellipticity are derived in terms of the elastic potential. Boundary conditions for strains are found from specified loads. Analytical solutions of the boundary problem in strains and their corresponding stress fields are found for the case of identical elongations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 217–225, March–April, 2009     

Asymptotic behavior of the spectrum of an elliptic problem in a domain with aperiodically distributed concentrated masses

In this paper, we consider a spectral problem with singular perturbation of density located near the boundary of the domain, depending on a small parameter. We prove the compactness theorem and study the behavior of eigenelements to the given problem, as the small parameter tends to zero.     

Geometrical nonlinear analysis of thin-walled composite beams using finite element method based on first order shear deformation theory

Based on a seven-degree-of-freedom shear deformable beam model, a geometrical nonlinear analysis of thin-walled composite beams with arbitrary lay-ups under various types of loads is presented. This model accounts for all the structural coupling coming from both material anisotropy and geometric nonlinearity. The general nonlinear governing equations are derived and solved by means of an incremental Newton–Raphson method. A displacement-based one-dimensional finite element model that accounts for the geometric nonlinearity in the von Kármán sense is developed to solve the problem. Numerical results are obtained for thin-walled composite beam under vertical load to investigate the effects of fiber orientation, geometric nonlinearity, and shear deformation on the axial–flexural–torsional response.     

Rigorous Results in Steady Finger Selection in Viscous Fingering

This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension (ɛ²). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results that address the selection problem. We rigorously conclude that for relative finger width λ in the range , with small, analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a. The methodology consists of proving the existence and uniqueness of analytic solutions for a weak half-strip problem for any λ in a compact subset of (0, 1). The weak problem is shown to be equivalent to the original finger problem in the function space considered, provided we invoke a symmetry condition. Next, we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case , for some . This turning point accounts for exponentially small terms in ɛ, as ɛ→0⁺ that generally violate the symmetry condition. We prove that the symmetry condition is satisfied for small ɛ when the parameter a is constrained appropriately. (Accepted July 4, 2002 Published online January 15, 2003) Communicated by F. OTTO     

Singular perturbation solutions of the nonlinear stability of a truncated shallow shperical shell under linear distributed loads along the interior edge

Using a singular perturbation method,the nonlinear stability of a truncated shallow,spherical shell without a nondeformable rigid body at the center under linear distributed loads along the interior edge is investigated in this paper.When the geometrical parameter k is large,the uniformly valid asymptotic solutions are obtained.     

Energy Scaling of Compressed Elastic Films –Three-Dimensional Elasticity¶and Reduced Theories

We derive an optimal scaling law for the energy of thin elastic films under isotropic compression, starting from three-dimensional nonlinear elasticity. As a consequence we show that any deformation with optimal energy scaling must exhibit fine-scale oscillations along the boundary, which coarsen in the interior. This agrees with experimental observations of folds which refine as they approach the boundary. We show that both for three-dimensional elasticity and for the geometrically nonlinear Föppl-von Kármán plate theory the energy of a compressed film scales quadratically in the film thickness. This is intermediate between the linear scaling of membrane theories which describe film stretching, and the cubic scaling of bending theories which describe unstretched plates, and indicates that the regime we are probing is characterized by the interplay of stretching and bending energies. Blistering of compressed thin films has previously been analyzed using the Föppl-von Kármán theory of plates linearized in the in-plane displacements, or with the scalar eikonal functional where in-plane displacements are completely neglected. The predictions of the linearized plate theory agree with our result, but the scalar approximation yields a different scaling.     

Analysis of bifurcation bending modes of arches and panels

This paper reports the results of numerical analysis of the bifurcation solutions of nonlinear boundary-value problems of plane bending of elastic arches and panels. Problems are formulated for a system of six nonlinear ordinary differential equations of the first order with independent fields of finite displacements and rotations. Two loading versions (by follower and conservative pressures) and two versions of boundary conditions (rigid clamping and pinning) are considered. In the case of clamped arches and panels, the set of solutions consists of symmetric and asymmetric bending modes which exist only for positive values of the load parameter. In the case of pinning, the set of solutions includes symmetric and asymmetrical modes which correspond to positive, negative, and zero values of the parameter. In both problems, the phase relations between the state parameter and the load parameter are bifurcated, ambiguous, have isolated branches, and admit a catastrophe — a finite jump from the fundamental equilibrium mode to a buckled mode.     

On the Barenblatt Model for Non-Equilibrium Two Phase Flow in Porous Media

We prove global existence and uniqueness of solutions to a quasilinear Goursat problem, which was proposed by G. I. Barenblatt to describe non-equilibrium two phase fluid flow in permeable porous media. When the equilibrium relaxation time tends to zero, the solution is shown to converge to the entropy solution of the corresponding initial-boundary value problem for the classical Buckley-Leverett equation.     

Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.     

Analytical method for the construction of solutions to the Föppl–von Kármán equations governing deflections of a thin flat plate

We discuss the method of linearization and construction of perturbation solutions for the Föppl–von Kármán equations, a set of non-linear partial differential equations describing the large deflections of thin flat plates. In particular, we present a linearization method for the Föppl–von Kármán equations which preserves much of the structure of the original equations, which in turn enables us to construct qualitatively meaningful perturbation solutions in relatively few terms. Interestingly, the perturbation solutions do not rely on any small parameters, as an auxiliary parameter is introduced and later taken to unity. The obtained solutions are given recursively, and a method of error analysis is provided to ensure convergence of the solutions. Hence, with appropriate general boundary data, we show that one may construct solutions to a desired accuracy over the finite bounded domain. We show that our solutions agree with the exact solutions in the limit as the thickness of the plate is made arbitrarily small.