Showing papers in "Quarterly of Applied Mathematics in 1975"

On the use of the interpolation polynomial for solutions of singular integral equations

Abstract: On the basis of integration of singular integral equations by means of Gaussian quadrature, it is demonstrated how to obtain the corresponding approximate polynomial solution. For some special cases compact formulas are given for the strength of the singularities at the endpoints of the integration interval.

A Neumann series representation for solutions to boundary-value problems in dynamic elasticity

Abstract: A regularized integral equation formulation for two exterior fundamental boundary-value problems in elastodynamics is presented. In either case, the displacement vector is assumed to be harmonic in time with a small frequency. It is shown that the solution can be expressed as a Neumann series in terms of the prescribed function; moreover, a sufficient condition for the convergence of the series is established.

Mixed finite-element approximations of linear boundary-value problems

Abstract: A theory of mixed finite-element/Galerkin approximations of a class of linear boundary-value problems of the type T*Tu + ku + / = 0 is presented, in which appropriate notions of consistency, stability, and convergence are derived. Some error estimates are given and the results of a number of numerical experiments are discussed.

Some remarks on path independence in the small in plasticity

Abstract: Introduction. Drucker's [1] definition of work-hardening has had a significant influence on the development of stress-strain relations in the mathematical theory of plasticity. Also, recognition that this definition provides a condition which is sufficient to ensure uniqueness of solution, in problems involving small deformations, has lead to its consideration as a stability postulate with extensions to time-dependent materials [2, 3] and as a basis for idealized models of soil behavior. The stability postulate assists in defining the class of materials covered by the theory. Some materials are excluded and for these a different starting point must be used. Frictional materials provide a number of examples of exceptions to Drucker's postulate. The postulate also excludes materials that soften. If these are to be brought within the scope of the theory of plasticity, a less restrictive postulate is required which allows softening but still provides the accepted forms of flow rule for hardening plasticity. With this in mind, Drucker has suggested an alternative postulate based on the concept of path independence in the small [4], The object of this paper is to examine some of the implications of this idea. Inviscid plasticity is considered first. Following this, an example of a frictional material is taken to show that the new postulate is restrictive and that some forms of material are excluded. Finally, softening is considered in an application to an ideal material which fractures in a progressive manner.

The transverse vibration of a rotating beam with tip mass: The method of integral equations

Abstract: An integral equation method is used to obtain improvable lower bounds for the second eigenvalue of the second-order \"reduced\" problem obtained from the problem described in the title by singular perturbation methods. These lower bounds are compared with results obtained directly by invariant embedding. The computational aspects of the integral equation method are stressed. The method is shown to be quite general and can be applied to a variety of boundary-value problems including those in which the eigenvalue parameter appears in the boundary conditions as well as in the differential operator.

On the breaking of water waves on a sloping beach of arbitrary shape

Abstract: Greenspan [1] considered water waves of finite amplitude on a beach of constant slope. He proved that: (Gi) A wave of elevation with nonzero slope at the front propagating shoreward into quiescent water always breaks before the shore.1 (G2) Under the same conditions a wave of depression never breaks. In this note we do not assume that the beach has constant slope, but rather we allow the depth to be an arbitrary smooth function of position. We show that the appropriate generalizations of Greenspan's results are: (GV) In the above circumstances a wave of elevation always breaks; in particular, it breaks at the shore when the amplitude is sufficiently small, otherwise it breaks before the shore. (G2') A wave of depression breaks if and only if a certain integral (involving only the depth function) is finite, and it never breaks away from shore.

Some functional differential equations

Abstract: Sufficient conditions are given for the solution of the functional differential equations with associated boundary conditions CO dy/dx = X) any(iinx), y{0) = 1, n = 0 dy/dx = / a(u)y([iux) du, y(0) = 1. ^0 A discussion is also given of some possible solutions to the differential equations which do not satisfy the boundary conditions. 1. Functional differential equations involving a parameter n of the forms oo dy/dx = X) any(n\"x), 2/(0) = 1 (1) 71 = 0 and dy/dx = [ a(u)y(iiux) du, y(0) = 1 (2) *0 do not seem to have been considered so far. It is the purpose of this paper to obtain some solutions for these equations together with sufficient conditions for their existence. A short discussion will also be given of sufficient conditions for the nonuniqueness of the solutions. It will be seen that a number of different types of solution to the system (1) exist, and that analogous solutions exist to the system (2). Unless otherwise mentioned, all quantities are real, and if the series aJ\" converges with nonzero radius of convergence r, its sum will be written A(t). A(t) may be termed the generating function. 2. Before proceeding to solutions of the system (1) when the set \\an} is arbitrary, and of the system (2) when the function a(u) is arbitrary, it is worth noting the following obvious solutions: if = 0, then a solution to the system (1) is y = 1. Similarly, if /„\" a(u) du = 0, then a solution to the system (2) is y = 1. 3. The obvious first form of solution to look for is a power series. When such a series converges, it will provide a unique solution. * Received October 15, 1971; revised versions received October 20, 1972 and June 13, 1973.

Asymptotic solutions of partial differential equations with damping and delay

Abstract: Certain hyperbolic partial differential equations with small nonlinearities involving significant damping as well as time delay are investigated. The KrylovBogoliubov-Mitropolskii method is extended and applied. An application is given to longitudinal vibrations of a nonlinear elastic rod.

Transient disturbances in a half-space during the first stage of frictionless indentation of a smooth rigid die of arbitrary shape

Abstract: A solution is obtained by the method of self-similar potentials for the time-spatial distribution of the contact stress between a homogeneous, isotropic, linearly elastic half-space and a smooth rigid die having an arbitrary indenting velocity and shape. The solution holds as long as the outward speed of the contact zone does not fall below the speed of the dilatational wave in the elastic medium. A proof is given that the instantaneous value of the force required to indent the die during this stage of contact is directly proportional to the product of the area of contact and the velocity of indentation at that instant. Introduction. In this paper the method of self-similar solutions [1-7] is used to solve problems in which rigid dies of arbitrary shape are pressed into a linearly elastic, homogeneous isotropic half-space at a rate sufficient to cause the contact to be superseismic. For these problems, no disturbance propagates along the surface of the halfspace more rapidly than the boundary of the region of contact. Consequently, there will be no deformation of the surface at points beyond the region of contact. Moreover, the deformation of the surface within the contact zone will be completely defined by the portion of the rigid die which has crossed the original position of the surface of the halfspace. In [5-7] it was shown that the rate of penetration must exceed a certain value for the contact between a half-space and a die with a cusp at the point of initial contact to be superseismic. However, the situation is quite different if the die is smooth in the region of contact. For problems of this type, the contact must always be superseismic for a finite interval of time. Moreover, the length of this interval of time can easily be computed from the shape and indentation velocity of the die and the velocity of the dilatational wave in the half-space. The problem in the title belongs to that class of elastodynamic problems the boundary conditions of which may be expressed in terms of functions which are homogeneous functions of space and time. For such problems, solutions may readily be obtained by either the self-similar potential approach or by the more familiar transform methods [8]. The virtual equivalence at these methods when applied to self-similar problems has most recently been demonstrated by Norwood [9]. * Received October 14, 1973; revised version received February 21, 1974. 216 A. R. ROBINSON AND J. C. THOMPSON Two-dimensional problems. For plane strain contact problems, it is convenient to denote the vertical displacement of points on the surface of the y = 0 half space by U(x, t). (We use the notation of [5-7] in this paper.) If the contact is frictionless, the boundary conditions defined below are sufficient completely to determine the stress and displacement fields at every point in the half-space: uv(xf 0 ft) — U(Xj f) j

Application of hill functions to circular plate problems

Abstract: A type of finite element—hill functions—is applied to solve circular plate problems in conjunction with the method of Lagrange multipliers which is used to treat various constraint conditions. Results obtained compare very nicely wTith the exact solutions. Introduction. Finite-element methods have been widely applied to obtain numerical solutions in both engineering [1, 2] and applied mathematics [3, 4, 5], The popularity of the methods may be partially due to their ability to deal with more complex problems and to provide accurate numerical results. However, it should be noted that there are different kinds of finite-element methods and the use of hill functions as finite elements which are developed in applied mathematics [3, 4, 5, 6] is one of them. In [6], hill functions are utilized to solve one-dimensional string-beam problems in which the so-called \"method of artificial parameters\" is used for handling various boundary conditions. In the present paper, these same functions are employed to solve axisymmetric (polar coordinate) circular plate problems; the Lagrange multiplier method is used here for the treatment of constraint situations. The striking difference between these two types of applications is in the evaluation of matrix elements of the system equations; for string-beam problems, all the terms in these elements turn out to be simply multiples of hill function coefficients [6], but for circular plate problems, they are not so straight-forward and require, for evaluation, methods of numerical integration. Rayleigh Ritz method. For the sake of completeness and use in the following sections, key equations involved in the Rayleigh-Ritz method are presented in this section. Suppose there exists a sequence 2/1,2/2, • • ■ of admissible functions in the variational problem such that lim F(yn) = d, (1) n—>00 where d is the lower bound of the functional F(y)-. The Rayleigh-Ritz method is a recipe for the construction of such a sequence by choosing an arbitrary system of coordinate functions, «i , u2 , ■ ■ ■ , with the property that any linear combination yn = Cjo>1 + c2w2 + • • • + c„co„ (2) is admissible in the variational problem, and that the solution function y and its relevant * Received October 26, 1973. The research reported here was supported by the Office of Naval Research (N00014-67-A-0377-0011).

Laminar boundary layer at a discontinuity in wall curvature

Abstract: The discontinuity in pressure gradient predicted for two-dimensional inviscid subsonic or supersonic flow at a jump discontinuity in wall curvature is smoothed by means of local solutions which take into account the interaction of a laminar boundary layer with the external flow.

Unsaturated seepage flow from a horizontal boundary

Abstract: Introduction. A knowledge of the time-depth history of infiltration from rainfall or irrigation water into a permeable soil is useful for soil-moisture and ground-water studies and in the analysis of the movement and dispersal of dissolved or suspended materials such as nutrients or pollutants. If these materials are subject to adsorption by soil, the concentration profiles which develop are influenced by the amount and timedistribution of water application. Since the physical system leads to a rather special form of nonlinear transport equation, its properties are reviewed briefly. (a) Since the soil is generally unsaturated in the infiltration zone, flow is induced by capillarity as well as gravity, and the system exhibits strongly nonlinear properties. Comprehensive reviews [1,2] discuss the literature on this problem. (b) Capillarity is subject to hysteresis effects, so that at each point in the porous medium, the relationship between capillary pressure and soil moisture is dependent upon previous history. A convenient mathematical description of the hysteresis phenomenon in porous materials does not exist at present [3, 4], (c) Usually the principal boundary condition—the availability of water at the soil surface expressed as a flux—exhibits time dependence, and may be a variable with 'stochastic' properties. Only in the case of either ponding or zero input can a steady surface condition be expected to prevail. (d) In unsaturated flow, some water may be 'bound', e.g. by adsorption or by entrapment in dead-end pores [5]. If the amount of bound water depends nonlinearly upon the total water present in unit volume, there will be some influence upon the flow characteristics. (e) Variation of permeability with depth may be important, and can become extreme in certain hydrophobic ('water-repellent') surface conditions. It is then possible for the wetting front to become unstable [6] and complicated three-dimensional flows can develop. (f) Infiltration of water into an air-filled porous medium leads to a two-phase flow problem [7], The buildup of pressure due to entrapped air reduces the rate of infiltration, and instability such as that discussed in (e) can again occur. Some indications of 'fingering', suggestive of instability, have been observed [8], Several authors [1, 9, 10] have examined the problem of infiltration into a uniform porous medium when the moisture content at the surface is suddenly increased, e.g. by flooding, so that the boundary moisture condition has a step-function time dependence. Effects of capillary hysteresis were not considered by these authors, since the problem

On the stability of solutions of a second-order differential equation

Abstract: This paper deals with the Hill differential equation d2y/dx2 + — — :—2 y = 0. i\" 1 — 2a cos x + a y Although this equation looks more difficult than Mathieu's, it can be dealt with somewhat more simply than the latter. Stability criteria are obtained in terms of r and a (at least in principle).

A singular perturbation solution to a problem of extreme temperatures imposed at the surface of a variable-conductivity halfspace: small surface conductivity

Abstract: The transient temperature field resulting from a constant and uniform temperature T, (or time-dependent heat flux H = ht~1/2) imposed at the surface of a halfspace initially at uniform temperature T0 is considered. A temperature-dependent thermal conductivity variation, k(T) = k0 exp [X(T — T0)/To], and a constant product of density and specific heat, pC, are assumed to be accurate models for the halfspace for some useful temperature range. The problem is initially formulated in terms of the dimensionless conductivity 4> = k(T)/k0 . Attention is then focused on the singular problem resulting from the limits <£s = 4>(T,) J, 0 and , In *.). ■MO

Lagrangian formulation of bubble dynamics

Abstract: The dynamical problem of two ideal fluids separated by an interface is formulated in terms of general coordinates in Lagrangian variables. The same problem is shown to be equivalent to a Hamiltonian variational principle which takes into account explicitly the surface energy of the interface between the two fluids. The formulation is applied to the motion of slightly nonspherical bubbles. It is shown that, although we start from an entirely different set of differential equations, the results of the linear stability analysis in Eulerian formulation are recovered.

Stochastic models of the scattering of sound by bubbles in the upper ocean

Abstract: Stochastic models are developed to relate the statistics of sound speed fluctuations and bubble density variations as a function of sound frequency in the upper ocean. These predictions from the stochastic model have been compared with ocean experimental data of sound speed modulation in the frequency range 15 to 150 kHz, and show satisfactory agreement. Future experiments and further modification of this model are discussed.