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Showing papers in "Quarterly of Applied Mathematics in 1988"


Journal ArticleDOI
TL;DR: In this article, the authors considered a more realistic generalization of a previous model proposed by V. C. and co-workers to describe fecal-orally transmitted diseases (cholera, typhoid fever, infectious hepatitis, etc.).
Abstract: ■^-vi(t.x) + Pv\\{t,x) = [ K(x,x')v2(t,x')dx', (1.1b) °n J n on (0, oo) x T and also subject to suitable initial conditions. Here Q denotes an open bounded domain in R\" (n — 2, 3) with boundary T. As usual, A stands for the Laplace operator and d/dn denotes the outward normal derivative on T. The motivation for studying (1.1), (1.1b) comes from the fact that it serves as a model for a class of man-environment epidemic systems when we adopt the following interpretation. We consider Q as the habitat (environment) in which the human population is exposed to an infectious agent, so that an epidemic phenomenon arises because the infected human population acts as a multiplier of the infectious agent itself. In this case vi (t, x) denotes the concentration of the infectious agent at a point x € D. and time t > 0, while v2(t, x) denotes the spatial density of the human infective population. In this context (1.1), (1.1 b) is a more realistic generalization of a previous model proposed by V. C. and co-workers [6, 8] to describe fecal-orally transmitted diseases (cholera, typhoid fever, infectious hepatitis, etc.) which are typical for the European Mediterranean regions. For this kind of epidemic the infectious agent is multiplied by the infective human population and then sent to the sea through the sewage; because of the peculiar eating habits of the population of these regions the agent may return via some diffusion-transport mechanism to any point of the habitat Q, where the infection process is restarted. Thus the kernel in (1.1b) mathematically

142 citations


Journal ArticleDOI
TL;DR: In this article, the Stroh formalism of anisotropic elasticity leads to a 6 x 6 real matrix N that can be composed from three 3x3 real matrices N; (/ = 1,2,3).
Abstract: The Stroh formalism of anisotropic elasticity leads to a 6 x 6 real matrix N that can be composed from three 3x3 real matrices N; (/ = 1,2,3). The eigenvalues and eigenvectors of N are all complex. New identities are derived that express certain combinations of the eigenvalues and eigenvectors in terms of the real matrices N, and the three real matrices H, S, L introduced by Barnett and Lothe. It is shown that the elements of Nj and N3 have simple expressions in terms of the reduced elastic compliances. We prove that -N3 is positive semidefinite and, with this property, we present a direct proof that L is positive definite.

92 citations



Journal ArticleDOI
TL;DR: In this paper, the authors studied the Riemann initial value problem for (1.1) and showed that all the jump discontinuities are required to satisfy the viscosity-capillarity criterion.
Abstract: is of mixed hyperbolic and elliptic type when a is a monotonically increasing function except in an interval (a,/?) (see Fig. 1). System (1.1) has been used to describe dynamic changes of phase in a van der Waals gas [ 10] and to model elastic deformations in a rod under tension [2], Mathematically, changes of phase for Eq. (1.1) are associated with jump discontinuities (shocks) in weak solutions (u,v) of system (1.1), in which u jumps across the interval (a, /?). There is a continuing problem of how to distinguish the phase jumps that are physically relevant. Mathematically, one would like to impose an entropy admissibility condition on all jump discontinuities that selects the physically relevant shocks, including the correct phase jumps, while giving well-posedness of the Cauchy problem. For systems of mixed type, it is not known in general what the appropriate admissibility condition should be, even if the initial data are restricted to lie entirely in the hyperbolic regions. This is largely due to the presence of noncompressive shocks, which fail to satisfy the classical entropy conditions of the theory of conservation laws [3, 4] . In the context of (1.1), noncompressive shocks are phase jumps that are typically nearly stationary (i.e., with nearly zero shock speed). When the shock speed is exactly zero, the phase jump is referred to as the Maxwell line. A requirement of an admissibility condition is that the Maxwell line should be admissible. This paper is a continuation of the study of the viscosity-capillarity criterion for shocks, introduced by Slemrod [11], In particular, I discuss solutions of the Riemann initial value problem for (1.1). This is the Cauchy problem with piecewise constant initial data having a single jump. The main result is that for initial data near the Maxwell line, the Riemann problem has a solution consisting of two weak shock or rarefaction waves, separated by a slowly moving phase jump. All the jump discontinuities are required to satisfy the viscosity-capillarity criterion. Alternative admissibility criteria for shocks in solutions of the Riemann problem for (1.1) are discussed in [1, 5, 9],

54 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a model of the one-dimensional motion of the polytropic ideal gas with adiabatic ends which is put into a vacuum and constructed the solution (u,v,6) in the Holder class Dt>o^7+\" x H2r+a+a x H^+a.
Abstract: (i-5> 6x{0,t) = dx{l,t) = 0. (1.6) This problem is a model of the one-dimensional motion of the polytropic ideal gas with adiabatic ends which is put into a vacuum. (u,v,6), unknown functions, represent the specific volume, the velocity, the absolute temperature of the gas; (R, n,Cy ,k), given positive constants, stand for the gas constant, the coefficient of viscosity, the heat capacity at constant volume, and the coefficient of heat conduction, respectively. The condition (1.5) is called the stress-free condition. Kazhykhov showed the global existence of a unique solution to this problem in [2], He constructed the solution (u,v,6) in the Holder class Dt>o^7+\" x H2r+a x H^+\"} (0 < a < 1) provided (wo, vq, Go) belongs to Hl+a x H2+a x H2+a. (For the definition of the Holder spaces Hn+a etc., see [3].) We call this solution classical in this paper. More recently Okada [5] and Kawashima [1] showed the asymptotic behavior of the solution. The problem has a trivial solution

51 citations


Journal ArticleDOI
TL;DR: In this paper, a non-monotone multivalued law is introduced in order to describe the interlaminar bonding forces and the existence and the approximation of the solution of this inequality are investigated.
Abstract: In this paper the delamination problem for laminated plates is studied. A nonmonotone multivalued law is introduced in order to describe the interlaminar bonding forces. This law is written as the generalized gradient in the sense of F. H. Clarke of an appropriately defined nonconvex superpotential. Moreover, monotone boundary conditions of the subdifferential type are assumed to hold. The problem is formulated as a variational-hemivariational inequality expressing the principle of virtual work in inequality form. By using compactness and monotonicity arguments, the existence and the approximation of the solution of this inequality are investigated.

39 citations


Journal ArticleDOI
TL;DR: On obtient des conditions suffisantes pour que l'equation logistique a retard x˙(t)=r(t)x (t)t) [1−x(t−τ(t))/K] soit respectivement oscillatoire ou non oscillator as mentioned in this paper.
Abstract: On obtient des conditions suffisantes pour que l'equation logistique a retard x˙(t)=r(t)x(t) [1−x(t−τ(t))/K] soit respectivement oscillatoire ou non oscillatoire

34 citations



Journal ArticleDOI
TL;DR: In this paper, the fonction f(z) = J 0 (z)−iJ 1 (z)) for determiner son comportement dans le plan complexe.
Abstract: On etudie la fonction f(z)=J 0 (z)−iJ 1 (z) pour determiner son comportement dans le plan complexe. On montre que f(z) n'a pas de zeros dans le demi-plan superieur

30 citations


Journal ArticleDOI
TL;DR: In this article, single-integral laws of the form (1.3) were proposed to capture the essence of the interaction between nonlinearity and dissipation in the instantaneous response and the dissipation due to memory.
Abstract: Underlying this class of models is the assumption that contributions to the stress superpose additively in the delay time r, an assumption motivated by Boltzmann's superposition principle for linear viscoelasticity. We shall refer to functionals of the form (1.3) as single-integral laws-, for convenience, we normalize 5 so that s(r, e, 0) = 0 for all t > 0 and e e R. An important feature of viscoelasticity—one that is especially relevant to experimental and theoretical studies in wave propagation (cf. [17])—is the interaction between nonlinearity and dissipation, in particular, between nonlinearity in the instantaneous response and dissipation due to memory. Single-integral laws, even though relatively simple, capture the essence of this interaction. Moreover, because of their simplicity, such laws are conducive to the characterization of real materials, and, in addition, lead to initial/boundary-value problems whose analysis is comparatively clean.

28 citations


Journal ArticleDOI
TL;DR: In this article, a point critique is caracterised par un triple zero de la valeur propre dindice un and le systeme is decrit par 3 parametres independant.
Abstract: On etudie la bifurcation et le comportement d'instabilite d'un systeme autonome non lineaire au voisinage d'un point critique compose. Le point critique est caracterise par un triple zero de la valeur propre d'indice un et le systeme est decrit par 3 parametres independants

Journal ArticleDOI
TL;DR: In this paper, a procedure for describing parametrique de toutes les conditions aux limites naturelles sous lesquelles la racine carree de l'operateur derivee d'ordre 4 est positive.
Abstract: On developpe une procedure pour la description parametrique de toutes les conditions aux limites naturelles sous lesquelles la racine carree de l'operateur derivee d'ordre 4 est positive

Journal ArticleDOI
TL;DR: In this article, the problem of scattering an elastic wave by a three-dimensional bounded and smooth body is considered and a one-to-one correspondence between the scattered fields and the corresponding radiation patterns is established.
Abstract: Consider the problem of scattering of an elastic wave by a three-dimensional bounded and smooth body. In the region exterior to a sphere that includes the scatterer, any solution of Navier's equation that satisfies the Kupradze's radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. Moreover, the coefficients of the expansion can recurrently be evaluated from the knowledge of the leading coefficient, known as radiation pattern. Therefore, a one-to-one correspondence between the scattered fields and the corresponding radiation patterns is established. The acoustic and electromagnetic cases are recovered as special cases.

Journal ArticleDOI
TL;DR: In this paper, the problem of determining the shape of the strongest column having a given length and volume V was revisited, and the problem was shown to be NP-hard.
Abstract: We reconsider the problem of determining the shape of the strongest column having a given length / and volume V. Previous results [13,7] have given optimal shapes for which the cross section vanishes at certain points. Although these results are mathematically correct, Theorem 1 below explains what is wrong with these anomalous shapes.

Journal ArticleDOI
TL;DR: In this article, the authors considered the stability of equilibrium for harmonic materials and proved a restricted form of the converse, which implies that a deformation is locally stable only if the strain energy is rank-one convex at each strain involved in the deformation.
Abstract: where and A 2 are the principal stretches. A material with this form of W is called harmonic. The harmonic form of W has been used by a number of investigators [1-4] to obtain explicit analytical solutions of the equations of equilibrium. In the present paper we discuss the stability of equilibrium for harmonic materials. The problems considered are strictly two-dimensional, and we consider stability versus plane alternatives only. Half of the problem of stability is solved by a theorem of Graves [5] which implies that a deformation is locally stable only if the strain energy is rank-one convex at each strain involved in the deformation. We prove a restricted form of the converse. For harmonic materials, and for displacement boundary value problems with no body force, an equilibrium state is stable if W is rank-one convex at each strain involved. Moreover, every locally stable state is globally stable (Section 7). The basic stability theorem can also be stated in terms of Wq, the quasiconvexification of W. For the problems considered, an equilibrium state is stable if and only if W = Wq at each point in the deformed body. We determine Wq explicitly in Sections 5 and 6. With I = Xi + A2 and J = A1A2, it has the form


Journal ArticleDOI
TL;DR: In this paper, the authors considered the delay differential equation y(0 + ay(t) + Pf{y(t r)) = 0, where a, /?, and r are positive constants and / is a continuous function such that uf(u) > 0 for u e [-A, B], 0, and lim = 1, v ; w − o u where A and B are positive numbers.
Abstract: Consider the delay differential equation y(0 + ay(t) + Pf{y(t r)) = 0, (*) where a, /?, and r are positive constants and / is a continuous function such that uf(u) > 0 for u e [-A, B], 0, and lim = 1, v ; w—»o u where A and B are positive numbers. When /(«) = sinw, (*) is the so-called \"sunflower\" equation, which describes the motion of the tip of the sunflower plant. We obtain necessary and sufficient conditions for the oscillation of all solutions of (*), whose graph lies eventually in the strip R+ x [—A, B], in terms of the characteristic equation of the linearized equation z(t) + az(t) + f}z(t r) = 0.


Journal ArticleDOI
TL;DR: In this paper, the authors discuss the existence and theunicite des resultats du problem of problems of potentiels and densites de charge dans les atomes avec des conditions aux limites correspondant a l'atome neutre neutre and a L n neutre isole.
Abstract: Etude de l'existence et de l'unicite des resultats du probleme des potentiels et densites de charge dans les atomes avec des conditions aux limites correspondant a l'atome neutre et a l'atome neutre isole

Journal ArticleDOI
TL;DR: In this article, the exact nonlinear theory for a rotationally symmetric membrane cap deformed by hydrostatic pressure is statically determinant and a small strain theory is obtained without any assumptions on the relative magnitudes of the displacements.
Abstract: It is shown that the exact nonlinear theory for a rotationally symmetric membrane cap deformed by hydrostatic pressure is statically determinant. A small strain theory is obtained without any assumptions on the relative magnitudes of the displacements. This small strain theory can be reduced to a single second-order ordinary differential equation for the determination of the radial stress. A linear shallow cap theory is obtained and solved explicitly for the case of the shallow spherical cap.

Journal ArticleDOI
TL;DR: Propagation d'ondes sonores faiblement non lineaires and faibilityment dispersives dans un cylindre magnetique. Onde solitaire as mentioned in this paper.
Abstract: Propagation d'ondes sonores faiblement non lineaires et faiblement dispersives dans un cylindre magnetique. Onde solitaire

Journal ArticleDOI
TL;DR: On demontre des theoremes d'existence et d'unicite pour des solutions faibles globales de problemes aux valeurs limites and initiales correspondant a l'equation: u tt =(σ(u x )) x +(α(u X ) u xt ) x +f sous des hypotheses qui ne requierent pas la regularite ou la monotonicite de σ.
Abstract: On demontre des theoremes d'existence et d'unicite pour des solutions faibles globales de problemes aux valeurs limites et initiales correspondant a l'equation: u tt =(σ(u x )) x +(α(u x ) u xt ) x +f sous des hypotheses qui ne requierent pas la regularite ou la monotonicite de σ


Journal ArticleDOI
TL;DR: The region d'hyperbolicite is non convex dans le plan de phase as mentioned in this paper, a condition necessaire et suffisante pour resoudre le probleme de Riemann en utilisant des ondes centrees ou de choc.
Abstract: On etudie un systeme a une dimension de lois de conservation. La region d'hyperbolicite est non convexe dans le plan de phase. On trouve une condition necessaire et suffisante pour resoudre le probleme de Riemann en utilisant des ondes centrees ou de choc. On generalise a des donnees initiales arbitraires

Journal ArticleDOI
TL;DR: In this article, a new inverse method for the aerodynamic design of airfoils is presented for subcritical flows, which is mathematically equivalent to solving only one nonlinear boundary value problem subject to known Dirichlet data on the boundary.
Abstract: A new inverse method for the aerodynamic design of airfoils is presented for subcritical flows. The pressure distribution in this method can be prescribed as a function of the arclength of the still unknown body. It is shown that this inverse problem is mathematically equivalent to solving only one nonlinear boundary value problem subject to known Dirichlet data on the boundary.

Journal ArticleDOI
TL;DR: In this paper, a new procedure is given for applying Liapunov's direct method to autonomous discrete equations, which is based on an idea that is closely related to Razumikhin's principle.
Abstract: In this paper, a new procedure is given for applying Liapunov's direct method to autonomous discrete equations. This procedure is based on an idea that is closely related to Razumikhin's principle and it includes Liapunov's direct method as a special case. Examples are given.

Journal ArticleDOI
TL;DR: In this article, the methode du tube de courant is used to etudier les ecoulements de detente supersoniques sur des parois avec une condensation hors d'equilibre.
Abstract: On utilise la methode du tube de courant pour etudier les ecoulements de detente supersoniques sur des parois avec une condensation hors d'equilibre

Journal ArticleDOI
TL;DR: Etablissement des bornes d'erreur pour un developpement asymptotique uniforme de la fonction de Legendre P n −m (chz).
Abstract: Etablissement des bornes d'erreur pour un developpement asymptotique uniforme de la fonction de Legendre P n −m (chz)

Journal ArticleDOI
TL;DR: In this paper, the variational characterization of Korn's constant and Dafermos' technique to reduce it to a boundary value problem have been evaluated for a spherical shell of arbitrary thickness.
Abstract: Upon invoking the variational characterization of Korn's constant and Dafermos' technique to reduce it to a boundary value problem, the Korn constant of a spherical shell of arbitrary thickness has been evaluated. The classical result of Payne and Weinberger for the sphere is recovered as the special case of vanishing interior radius, while as the thickness of the shell tends to zero, Korn's constant tends to infinity in a nonuniform sense.

Journal ArticleDOI
TL;DR: The problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second order hyperbolic systems typical of those arising in I-D surface seismic problems is discussed in this article.
Abstract: The problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second order hyperbolic systems typical of those arising in I-D surface seismic problems is discussed. In addition, the problem of identifying unknown parameters that appear in boundary conditions for the system is treated. A spline-based approximation theory is presented, together with related convergence findings and representative numerical examples.