Showing papers in "Quarterly of Applied Mathematics in 1977"

Numerical integration methods for the solution of singular integral equations

Abstract: The evaluation of the stress intensity factors at the tips of a crack in a homogeneous isotropic and elastic medium may be achieved with higher accuracy and much less computation if the Lobatto-Chebyshev method of numerical solution of the corresponding system of singular integral equations is used instead of the method of Gauss-Chebyshev commonly applied to such problems. Comparison of results obtained by the two numerical methods when applied to the problem of a cruciform crack in an infinite medium proves the potentialities of the new approach.

Viscous effects on perturbed spherical flows

Abstract: The problem of two viscous, incompressible fluids separated by a nearly spherical free surface is considered in general terms as an initial-value problem to first order in the perturbation of the spherical symmetry. As an example of the applications of the theory, the free oscillations of a viscous liquid drop and of a bubble in a viscous liquid are studied in some detail. It is shown that the oscillations are initially describable in terms of an irrotational approximation, and that the normal-mode results are recovered as t —* <». In between these asymptotic regimes, however, the motion is significantly different from either approximation.

A model for one-dimensional, nonlinear viscoelasticity

Abstract: The problem utt = a(0)a(ux)x + a(t — r)a(ux)x dr + /, 0 < x < 1, t > 0, Jo u(0, t) = m(1, t) = 0, u(x, 0) = u0(x), u,(x, 0) = u,{x) is considered. The essential hypotheses are that a(t) = a„ + A(t), a„ > 0, A £ L'(0, ), (— l)kaa)(t) > 0, k = 0, 1, 2, a(0) = 0, a'(£) > e > 0. It is shown that the problem has a unique classical solution for all t if the data are sufficiently small and, if / is suitably restricted, this solution tends to zero as t tends to infinity. It is shown that the problem provides a special model for elastic materials which exhibit a memory effect.

The notion of approximate eigenvalues applied to an integral equation of laser theory

Abstract: The integral operator with kernel (iri/ir)1'2 exp [—it)(x — yf] on the interval \\x\\, y < 1 serves to model the behavior of a class of lasers. Although the kernel is simple, it is not Hermitian; this presents a major obstacle to a theoretical understanding of the equation—indeed, even the existence of eigenvalues is difficult to prove. We here introduce a definition of approximate eigenvalues and eigenfunctions, and argue that these will model the physical problem equally well. We then show that, for i) sufficiently large, each point X with |X| = 1 is an approximate eigenvalue, and that the number of mutually orthogonal approximate eigenfunctions corresponding to X grows faster than any constant multiple of \\/vThis confirms a conjecture of J. A. Cochran and E. W. Hinds, supported by numerical evidence. In physical terms, it shows that for large Fresnel number the laser cannot be expected to settle to a single mode. Introduction and conclusions. For a laser with rectangular plane-parallel reflectors which are mirror images of each other, the integral operator AJ = 0Vt)'/2 j exp [~iv(x yf]j{y) dy, \\x\\ < 1, with rj a real parameter, describes the way in which the field distribution j(y) is transformed by the laser [1], Interest focuses on those field distributions which reproduce themselves, to within some complex constant X, on a single passage through the laser. These are called \"modes\", and 1 — |X|2 is termed the energy loss of the mode. If a field distribution has the form j(y) = ^2>\" afPi-(y), with each |X3|, the field distribution behaves like a |X/>,(,)/) + '

On a functional equation arising in the stability theory of difference-differential equations

Abstract: : The functional differential equation Q'(t) = AQ(t) + B((Q(tau - t)) to the T-th power), - infinity t infinity, where A,B are n x n constant matrices, tau or = 0, Q(t) is a differentiable n x n matrix and (Q(t)) to the T-th power is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions is given. This equation, of considerable interest in its own right, naturally arises in the construction of Liapunov functionals of difference differential equations of the type dx(t)/dt = Cx(t) + Dx(t-tau), where C,D are constant n x n matrices. The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations. In this paper, we show that, in spite of the functional nature of this equation, the linear vector space of its solutions is n squared; moreover, ,we give a complete algebraic characterization of its solutions and indicate computationally simple methods for obtaining these solutions, which we illustrate through an example. Finally, we briefly indicate how to obtain solutions for the nonhomogeneous problem, through the usual variation of constants method.

Periodic solutions of finite difference equations

Abstract: The eigenvalues of the matrix A are 1 =b hi. Hence all the iterates 2/1,2/2, ■ • • , of (2) spiral away from the origin. On the other hand, every non-trivial solution of (1) is periodic with period 2ir. The essential difficulty with the differential equation (1) is that the equilibrium solution x = x = 0 is a center. Equivalently, the periodic solutions of (I) are not orbitally asymptotically stable, and thus they may be destroyed under an arbitrary small perturbation. The situation is very different, however, when x =

Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

Abstract: The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.

Buckling of cylindrical shells with small curvature

Abstract: Publisher Summary This chapter discusses buckling of cylindrical shells with small curvature. It focuses on the bifurcation buckling of a rectangular plate Ω = (0,l) × (0,1) under a lateral force of magnitude λ applied to x = 0, l. In the absence of forces, the plate has an imperfection of magnitude α and assumes a form z = αw0(x, y) for some known w0: Ω Ω ¯ → R; the buckled state of the plate is given by z = αw0(x, y) + w(x, y) where w is the solution to be determined. The boundary of the plate is simply supported (hinged). The chapter presents this problem, described by the von Karman equations, for parameters (λ, α) varying independently near (λ0, 0) and for w near the unbuckled state w = 0, where λ0 represents the critical buckling load. The chapter also presents the differential equations describing w.

A multi-phase Stefan problem describing the swelling and the dissolution of glassy polymer

Abstract: In the swelling and the dissolution of certain glassy polymers, three distinctive regimes are present. They are (1) the liquid solution wherein the disassociated polymer molecules are carried away by diffusion, (2) the gel layer of rubbery polymer containing large solvent concentration, and (3) the glassy phase of the polymer where there is very little solvent penetration. The gel/liquid interface that separates the diffusion of the disassociated polymer in the liquid solution from that of the solvent in the polymer is characterized by a constant disassociation concentration. The position of this gel/liquid interface is described explicitly either by a relationship between diffusion processes, or by the rate of disassociation at the interface in addition to the diffusion processes, depending on whether the disassociation rate exceeds the diffusion capability in removing the disassociated polymer molecules at the interface. The glassy phase is characterized by a sharp decrease in several orders of magnitude of the diffusion coefficient of solvent in gel-like and glassy polymers. The position of the glass/gel transition, however, has to be determined implicitly from the diffusion problem. The existence of the glass/gel transition presents some unique features in the Stefan problem requiring special numerical considerations.

Polyconics. I. Polyellipses and optimization

Abstract: A. Introduction. 1. Ellipses and hyperbolas may be defined as plane curves consisting of all points the sum, or the difference, of whose distances from two fixed points F, and Ft, the foci, is constant. This generalizes at once to a provisional definition of a polyconic with n foci Fi , ■ ■ , Fn in the plane: a plane locus consisting of all points the sum of whose signed distances to Fu ■ ■■, Fn is constant, the signatures being +1 or —1. If all n signatures are +1 the polyconic is a polyellipse or, more specifically, an n-ellipse. Thus a circle in this terminology is a 1 -ellipse and an ordinary ellipse becomes a 2-ellipse. Why bother with polyconics? There is the possible intrinsic interest, novelty, and educational value. Then, certain properties of conics themselves appear to be easier and more natural in this generalized context; for instance, the radius of curvature of a polyellipse is easier to get from first principles than is that of an ellipse. Also, the numerical and graphical treatment of polyconics introduces early, simply and naturally certain important techniques of numerical analysis, such as e.g. the variable step-length procedures. However, the most important reasons for working with polyconics appear to us to be two: a) polyconics may help to revive interest in the neglected subject of geometry, b) polyconics, and especially polyellipses, arise naturally in the treatment of an important class of optimization problems.

The resolvent of singular integral equations

Abstract: The investigation reported is concerned with the construction of the resolvent for any given kernel function. In problems with ill-behaved inhomogeneous terms as, for instance, in the aerodynamic problem of flow over a flapped airfoil, direct numerical methods become very difficult. A description is presented of a solution method by resolvent which can be employed in such problems.

An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem

Abstract: Development of an iterative solution technique for a certain nonlinear eigenvalue problem supplies an iterative solution technique for the ion case and isolated neutral atom case boundary-value problems for the Thomas-Fermi equation.

Symmetry-axis elastic waves for transversely isotropic media

Abstract: An unbounded, linear transversely isotropic elastic solid is excited by a suddenly applied point body force of arbitrary orientation. Simple closed-form expressions are found for the various displacement components on the symmetry axis as a function of time and distance from the source. Applications are given for 13 hexagonal crystals. Introduction. This paper treats the problem of the transient response of an unbounded, transversely isotropic, linear elastic solid, excited by a suddenly applied point body force. By restricting the analysis to the symmetry axis of the solid, it is possible to obtain explicit closed-form expressions for the various displacement components. These results constitute the first (three-dimensional, time-dependent) extension of Stokes' [1] celebrated solution for an isotropic solid to a (physically realizable) anisotropic solid. Chee-Seng [2], who follows and extends the work of Lighthill [3], has analyzed the axial wave motion in transversely isotropic media with particular emphasis on a magnetohydrodynamic problem. The w3 displacement component of the present paper conforms to the P operator of Chee-Seng, but the Ui component (due to the d2/dx2 M term) does not. Directly related to the present problem is the work of Cameron and Eason [4] who treated the displacement components in the transverse plane of the elastic solid, normal to the symmetry axis. Their results required numerical integration. The steady-state version of the above stated problem has been treated by Buchwald [5], who gave an asymptotic solution valid for the far field, based on the work of Lighthill [3]. The present paper is independent of the above-mentioned references. It follows and extends the two-dimensional wave propagation problem treated by the author [6]. The mathematical technique used for the multidimensional transform inversion in [6] is due to Burridge [7]. In the analysis contained below for radiation along the axis of material symmetry, the two-dimensional inversion technique of Burridge is extended to threedimensions. For a special combination of the elastic parameters, the Q operator (defined in Sec. 2) factors into two wave operators with the result that the problem becomes much more tractable. This situation has been exploited by the author [8] and serves as a useful check on the present work. In Sec. 2 the basic equations are stated and their solution is reduced to a residue calculation by means of integral transforms. The relevant pole locations are treated in Sec. 3 and then combined to give explicit expressions for the displacement components * Received November 7, 1975.

Variational thermodynamics of viscous compressible heat-conducting fluids

Abstract: A new variational principle of virtual dissipation generalizing d'Alembert's principle to nonlinear irreversible thermodynamics is applied to compressible heat- conducting fluids with Newtonian and non-Newtonian viscosity. The principle is applied in the context of Eulerian formalism where the flow is described with reference to a fixed coordinate system. New concepts of entropy displacement and mass displacement are used as well as a new definition of the chemical potential which avoids the usual ambi- guities of the classical thermodynamic approach. The variational principle is used to derive a novel, form of field differential equations for the coupled fluid dynamics and convective heat transfer.

Dual extremum principles for a nonlinear diffusion problem

Abstract: Maximum and minimum principles for a nonlinear boundary value problem in diffusion with concentration-dependent coefficient D(c) are derived in a unified manner from the theory of dual extremum principles. The results are illustrated by a calculation in the case D(c) = exp c. The purpose of this note is to give a new variational formulation of the nonlinear boundary value problem described by the equation c( 0) = 1, c(oo) = o. (2) This problem arises in the study of diffusion into a semi-infinite medium, where the diffusion coefficient D is a function of the concentration c. The associated flux is given by /(z) = D(c) dc/dz, / < 0, (3) and we shall suppose that D(c) £ C'[0, 1] and that D(c) > 0 for c > 0. Shampine [1] has shown that this kind of problem possesses a unique solution and has discussed in [2] some aspects of an approximate solution due to Macey [3], This latter work formulates the solution c implicitly via an expression for z(c). That is, the dependent and independent variables are interchanged. If we adopt this approach, the problem in (1) to (3) may be written as [2] dz/dc = D(c)/f, z(I) = 0, (4) — df/dc = \\z, 0 < c < 1, /(0) = 0. (5) Here z and / are regarded as functions of the independent variable c. To give a variational formulation of the problem we now observe that (4) and (5) are examples of the canonical equations dz/dc — dH/df, (6) -df/dc = dH/dz, (7) with the Hamiltonian H(/, z) given by H(j, z) = D{c) In |/| + i22. (8) We note that d'H/df = -D(e)/f, d2H/dz2 = i (9) Since D(c) > 0, H is concave in / and convex definite in z, which in turn means that we can obtain dual extremum principles [see 4, 5],

Singularity at the apex of pyramidal notches with three equal angles

Abstract: A Green's function approach is employed to study the potential theory problem of determining the strength of singularity at the apex of a pyramid with three equal angles. The problem is reduced to finding the eigenvalue of a singular integral equation. Numerical results are obtained and compared with available literature. Introduction. The potential theory problem of determining the strength of the singularity at the apex of a pyramid with three equal angles has been studied by Bazant [1]. The problem studied there was that of a harmonic function in a region with a pyramidal boundary, with constant potential on the boundary. He developed a solution by use of a general numerical (finite-difference) scheme for potential theory in which several lines of singularity intersect. In particular, he found that when the pyramidal boundary was reentrant to the space considered, the potential behavior with distance was 0(py) (0 < y < 1) as p —> 0, where p is the distance to the pyramidal apex. This means that the gradient of the potential (flux) will have a singularity 0(py~l) asp —» 0. However, if the pyramid is not reentrant, the paramenter y will be greater than unity and will not produce any singularities in the flux. In the present study a Green's function method is employed to examine the title problem. An advantage with the proposed method is that for a prescribed accuracy less numerical effort is required than with the finite-difference scheme [1], although more analysis is required to develop the equations into a form suitable for the numerical analysis. Similar results have been noted in the authors' earlier investigation [2] of the problem of determining the strength of singularity at the corner of a wedge-shaped region. Formulation of the problem. Consider the pyramid OABC with the three edges OA, OB, and OC oriented in such a way that the angle between any two consecutive edges is the same, say, 20 (where< tt/3). The points A, B and C are taken anywhere on the semiinfinite edges of the pyramid with apex O (the lengths of the edges of the pyramid are immaterial for the present study, since an arbitrarily small neighborhood of the apex of the pyramid is all that is of concern here). Introduce the cartesian coordinate system in a way that OA makes the same angle with either of the axes y and z as OB does with z and x. * Received March 1, 1977: revised version received May 4, 1977. Support by the Air Force Office of Scientific Research under Grant No. AFOSR 75-2859 is gratefully acknowledged. The authors are also grateful for helpful discussions with Professor Z. P. Bazant during the course of this research. ** Permanent address: Department of Mathematics, Indian Institute of Technology, Bombay, India.