Showing papers in "Quarterly of Applied Mathematics in 1980"

Generalized thermoelasticity for anisotropic media

Abstract: The equations of generalized thermoelasticity for an anisotropic medium are derived. Also, a uniqueness theorem for these equations is proved. A variational principle for the equations of motion is obtained.

Translational addition theorems for spheroidal scalar and vector wave functions

Abstract: The translational addition theorems for spheroidal scalar wave functions R„n(h, £)Sm„(/i, rj)exp(jm(j)); i = 1, 3, 4 and spheroidal vector wave functions M£/'2(')(/i; r/, 4>), N„ny'zU)(h; £, t], ); i = I, 3, 4, with reference to the spheroidal coordinate system at the origin 0, have been obtained in terms of spheroidal scalar and vector wave functions with reference to the translated spheroidal coordinate system at the origin 0', where 0' has the spherical coordinates (r0, 60, <£0) with respect to 0. These addition theorems are useful in acoustics and electromagnetics in those cases involving spheroidal radiators and scatterers.

A note on the method of multiple scales

Abstract: A modification of the method of multiple scales makes use of the expansion of parameters of the system in order to remove undesirable characteristics of the solution obtained by the usual multiple-scale method. When applied to the damped harmonic oscillator, the modification leads to the exact solution. For the Duffing equation it leads to an approximation which can be reduced to the solution reported by Nayfeh [1], However, the solution derived here appears to be more accurate and the frequency takes on a form without nonuniformities. The method of multiple scales produces a uniformly valid expansion for systems in which a troublesome term is multiplied by a small parameter and for which an ordinary perturbation expansion leads to a nonuniformly valid series solution. For example, the weakly damped harmonic oscillator 5c + 2ex -Ia>2x = 0, e <| 1, (1) has the general solution x = [a0ei^T~^i' + (2) where a* is the complex conjugate of a0. An ordinary perturbation in e yields results which correspond to expanding e~a and exp[ + i(co2 — e2)1/2f in power series which are nonuniformly valid since t can always be large enough to offset the smallness of e or s2. The derivative expansion method of multiple scales [1, pp. 236-240] makes use of the time expansion tn = e\"', x = x(r0, tu t2 •• ), dx/dt = D0x + sDtx + e2D2x + ■■■, (3) where Dn = d/dtn. The time t0 is the unstretched time coordinate and the remaining tn correspond to longer time scales. To 0(s2) the solution

The nonuniform motion of a supersonic dislocation

Abstract: Introduction. The nonuniform motion of a subsonic dislocation has been analyzed previously [1, 2], Here we treat the transient motion of a screw supersonic dislocation with particular attention to the wave-front analysis and the detailed treatment of the singularities involved. Although dislocations may not actually move at supersonic speeds, the solution for a dislocation may be used in other problems involving cracks, slip or separation, as in [3, 4, etc.]. The analysis presented here may also bear on the treatment of problems involving moving singularities in general [5, 6], Consider a screw dislocation parallel to the y-axis with the discontinuity in the displacement component uy across z = 0 being denoted by Au. The dislocation is at rest in an infinite space until time t = 0 when it begins to move according to x = l(t)(or t = rj(x), equivalently). It may be easily seen that the problem is equivalent to the one in the halfspace occupying the region z > 0 (Fig. 1) with the nonzero displacement component satisfying the equation

Beam bending problems on a Pasternak foundation using reciprocal variational inequalities

Abstract: The present study is concerned with a class of two-body contact problems in linear elasticity. The model problem is a bending problem of the beam resting unilaterally upon a Pasternak foundation. A variational formulation is given by the minimax principle of the functional, and proof of the existence of saddle points is given by the compatibility condition for applied forces and moments on the beam. It has been found that the compatibility condition for equilibrium of the beam and foundation can be achieved by arguments of coerciveness of the functional on the admissible set. An approximation and example of the problem by finite element methods and a numerical method for its solution are also introduced.

The swallowtail and butterfly cuspoids and their application in the initial post-buckling of single-mode structural systems

Abstract: This paper investigates the swallowtail and butterfly catastrophes from the point of view which is applicable in the theory of elastic stability. Thus, the results are concerned with the various forms of these instabilities as well as the determination of the critical load surfaces which are of engineering significance. It is demonstrated that the results are applicable to an axially loaded beam resting on a nonlinear elastic foundation. Introduction. Catastrophe theory [1] and the theory of elastic stability [2, 3, 4] are known to be similar in nature. For example, Thompson and Hunt [5] showed that imperfection sensitivity in the anticlinal point of bifurcation results in an elliptic umbilic catastrophe. Further, they showed that the stability analysis of a two-mode axially loaded stiffened flat plate could be classified as a hyperbolic umbilic catastrophe. In addition, the parabolic umbilic and its application to a two-mode pressurized spherical shell and to a two-mode uniformly compressed plate on a nonlinear elastic foundation were examined by Hui and Hansen [6, 7]. For single-mode systems, the catastrophes which have been considered are the fold, cusp, swallowtail and butterfly cuspoids. The fold model occurs when the cubic term of the potential energy is non-zero. It arises, for example, in the single-mode overall buckling of a wide integrally stiffened flat plate (where the stringers are closely spaced so that overall buckling occurs first) under axial compression [8] and in the Cox buckling problem [2]. The cusp model is found in both the single local mode and the single overall mode buckling of axially stiffened cylindrical shells under compressive loads [9, 10, 11], oval cylindrical shells under axial compression [12], compressed sandwich cylindrical panels [13] as well as in the initial post-buckling analysis of a spherical cap under a concentrated or an axisymmetric distributed load [14, 15]. The reason for the present investigation is that all of the above-mentioned elementary catastrophes have been investigated and applied to structural instability problems except the swallowtail and the butterfly cuspoids. Thus, it is of interest to investigate the implications of these remaining catastrophes and to determine the role they play in the theory of elastic stability. In this paper, the swallowtail and butterfly cuspoids are unfolded and analyzed in terms of the theory of elastic stability. They are solved in the most general form so that * Received June 21, 1979. 18 DAVID HUI AND JORN S. HANSEN the results are applicable to any system where the potential energy expression falls into their standard forms. Previous work [16, 17, 18] has been done on these models; however, it was not specifically toward the theory of elastic stability. In particular, the equilibrium paths of the present work illustrate important physical insight and as a result emphasize the distinction between primary and secondary critical surfaces. The swallowtail and butterfly cuspoids are in general applicable to any cusp catastrophe (including all the aforementioned examples) when the quartic term of the potential energy is either zero or sufficiently small. Therefore, in order to demonstrate the general results, a stability analysis of a beam resting on a nonlinear elastic foundation is presented in the final section. This analysis is based on a Koiter-style approach [2]. Classification of single-mode systems. The most general form of the potential energy of a single-mode system expanded about the classical critical load of the perfect system is PE = A'2(A — Acl) + — A2(A Acl)2 + £2 + [^3 + -4 3^ ~~ Kl) + ' ' ']?3 + [^4 + A'4(A — Acl) + • • ]£4 + [A5 + A'5(A — Acl) + • • ]£5 + [^6 + A'6(A — Acl) + ' ' ']£6 + ' \" + [£1 #1 + £1 — Kl) + ' \"]£ + [£2 ^2 + S2 ̂ '2(A — Acl) + ■ • ]q2 + [£3 ®3 + e3^3(^ — Kl) + \"]^3 + [£4^4 + £4^4^ ~* Kl) + ' ' ']£4 + ' \" (1) where Au A\\, A'[, ... and A2 vanish because of the condition that the first and second variations of the potential energy are zero in the critical state of equilibrium. In the above A is the applied load parameter, Acl is the classical critical load, £ is the amplitude of the buckling mode, A'2, A2, A3, A'3, ..., Bu B\\, B2, ... are constants and e1; e2, e3 and £4 are the amplitudes of various imperfection quantities. That is, these imperfections represent deviations from the idealized model of the structure. Thus, they may appear as initial geometric deformation changes in material specification or any other similar quantity. It should be noted that, for an asymptotic analysis, it is usual to retain only the first non-vanishing term in each of the above square brackets. In catastrophe theory, the form of a single mode instability is classified according to the first non-vanishing coefficient A„, where n = 3, 4 The standard form after division by A„ and a suitable scaling is P£={\" + an-2Z\"-2 + an-3f~3 + ■■■ + a3{3 + oc2{2 + a,? with An ± 0, An_1= A2 = 0 (2) where al5 a2, a„_2 are control parameters and are associated with quantities such as applied load(s) and deviations of the real system from the idealized model. Furthermore, all these control parameters are assumed to be zero in the reference state and it is the stability of the perturbations of the reference state which is under consideration. The fold cuspoid occurs whenever the cubic term of the potential energy is non-zero. The standard form is PE = e + oc1H, A3± 0. (3) THE SWALLOWTAIL AND BUTTERFLY CUSPOIDS 19 Also, the standard form of the cusp model is given by PE= ±{4 +

On optimal temperature paths for thermorheologically simple viscoelastic materials

Abstract: Introduction. When a fiber-reinforced viscoelastic material is cooled from cure temperature to room temperature the different coefficients of thermal expansion of the matrix and fibers lead to residual stresses. To study this phenomenon in detail, we consider a thin, isotropic, thermorheologically simple, linearly viscoelastic plate reinforced by a random system of fibers lying in the plane of the plate. We ask the following question: of all temperature paths 9(t), 0 < t < T, which take on prescribed values at t = 0 and t = T, is there a path which renders the residual stress at t = T a minimum?1 The Euler equation associated with the above problem is a nonlinear integral equation, which we use to show that smooth minimizers are generally not possible (cf. [1]). Indeed, solutions of the Euler equation exhibit a jump discontinuity at t = 0, are smooth during (0, T), and generally suffer a second jump discontinuity at t = T.2 For a Maxwell material with an exponential shift function we are able to solve the Euler equation in closed form. We use this solution to compute the optimal temperature path for polymethyl methacrylate with initial and final temperatures 90°C and 80°C, respectively, and with T = 5 hours. The optimal path produces a residual stress of 32 psi as compared to 220 psi for a linear path.

Theoretical analysis of steady, nonadiabatic premixed laminar flames

Abstract: A one-dimensional, steady, non-adiabatic, premixed laminar flame is assumed semi-infinite with the burner at the origin, and the investigation centers on the asymptotic behavior of the temperature and species mass fractions in the burned region near infinity. After a consideration of the general N-species problem, specific results are obtained for the global two-step reaction v/sub r/R..--> mu../sub i/I, v/sub i/I..--> mu../sub p/P, where R, I, P denote reactant, intermediate, and product species, respectively, and v/sub r/, v/sub i/, ..mu../sub i/, ..mu../sub p/ are stoichiometric coefficients. Assuming Arrhenius kinetics, it is shown that the classical linearized asymptotic theory is not applicable unless v/sub r/=v/sub i/=1, in which case the approach to burned equilibrium is an exponential decay. Consequently, a nonlinear theory applicable to arbitrary v/sub r/ and v/sub i/ is presented which shows that in general the asymptotic decay is algebraic. It is further shown that boundedness of the solution at infinity permits the arbitrary specification of only three boundary conditions on the original sixth-order differential system. This result is illustrated by a comprehensive analytical example and the computational implications for the general N-species problem are discussed.

Further study on one-dimensional shock waves in nonlinear elastic media

Abstract: One may define the growth of a shock wave by the growth in the amplitude of discontinuity in the velocity (denoted by [v]) across the shock wave as the shock wave propagates. One may also define the growth of a shock wave by the growth in the amplitude of discontinuity in the stress [a], strain [e], or entropy [17]. It is shown that one definition predicts the growth of the shock wave while others may predict its decay. In this paper we derive the transport equations for one-dimensional shock waves in nonlinear elastic media in which the shock wave can be defined as the amplitude of either [c], [a], [e] or [tj]. Moreover, the dependent quantity can be any one of, or a linear combination of, the seven quantities behind the shock wave. It is shown that when the region ahead of the shock wave is under a homogeneous deformation, the amplitudes of [v], [a] and [c] grow or decay simultaneously if (a) [c]2 is a strictly increasing function of [tj], or (b) the purely mechanical theory of shock waves is employed in which the effect of the entropy is ignored. Regardless of whether the effect of the entropy is ignored or not, there is no assurance that the amplitudes of [v], [a] and [e] grow or decay simultaneously if the region ahead of the shock wave is not under a homogeneous deformation.

On three-dimensional generalizations of the Boussinesq and Korteweg-de Vries equations

Abstract: Three-dimensional generalizations of two different forms of the Boussinesq equation are derived. They are investigated for stability of slowly varying nonlinear wavetrains. The results obtained are then compared with the stability properties following from the full water wave equations. Agreement is found to be good for h0 k0 (depth times wavenumber) of order one. This is very satisfactory, as the Boussinesq equations are only supposed to be valid for small h0 k0. In particular, one version of the Boussinesq equation is found to yield instability with respect to one-dimensional perturbations for h0 k0 > 1.5 (as against 1.36 for the full equations). Finally, a similar comparison is performed for the three-dimensional Korteweg-de Vries equation.

On a uniform asymptotic expansion of a fourier-type integral

Abstract: An alternative derivation is given for a uniform asymptotic expansion of the integral Z i(h) = | sKvV + <»), which has been obtained recently by Schmidt. Here q varies in the interval [0, q0], q0 is some fixed number less than z, and z is finite. A similar result is obtained for integrals having an infinite range of integration. Realistic bounds are also provided for the error terms associated with the expansions. Our approach is based on a summability method introduced by Olver.

Elastic realizability of force and displacement systems in structures

Abstract: This paper discusses the constraints imposed upon equilibrium and compatibility solutions in structures through the use of constitutive equations. It is shown, for example, that \"realizable\" force systems are bounded by statically determinate force systems for the case of trusses. The analysis used depends heavily upon the concept of \"basic solutions\" (statically determinate substructures) for linear systems which appears most commonly in the theory of linear programming. Introduction. As used in this paper, the term \"physically realizable\" or simply \"realizable\" applies to force and displacement solutions which satisfy all the equations of structures. It is the intention here to study the properties of these solutions with respect to solutions of the equilibrium or compatibility equations which may or may not also satisfy the constitutive equations. The idea, of course, is that if equilibrium or compatibility systems are to be \"realized\" or built they must also satisfy constitutive equations. While questions such as realizability have a certain academic interest, the motivation here is somewhat deeper and lies in structural optimization. Having tried unsuccessfully to take structural optimization problems head on, the next step is to attempt to simplify matters. In some cases [1] a convenient simplification is just to neglect the constitutive equations and work with, for example, objective functions and constraints written in terms of forces. When the constitutive equations are dropped from the set of constraints it can happen—notably in cases of multiple loading conditions—that the resulting force system cannot be realized (built). Rather than a practical solution, these results must be regarded as bounds. The question then to be addressed here is the effect of constitutive equations on force and compatibility solutions. It will be shown that the answer lies within the segment of the theory of linear systems which deals with the question of non-negative solutions of linear equations and some concepts of convex analysis. In spite of the fact that this type of analysis is basic to the study of linear programming, much of the supporting material is not easily available. There is, however, an excellent summary of this material in the first chapter of Gale's book on economic models [2] which is highly recommended to the reader. In an earlier effort [3] the authors examined some aspects of the realizability problem for a simple example while here it is hoped to develop a general theory of realizability. In this earlier case it was in fact true that realizable forces were bounded by statically determinate force systems, although this fact was not noted. It is proposed to show here that * Received November 16, 1978. This work has been supported by the National Science Foundation. 412 SPILLERS and LEFCOCHILOS in general realizable solutions are bounded by statically determinate force systems. Other properties of realizable solutions will also be developed. The work presented here proceeds in the following manner. First of all a notation is introduced. Then the case of a single redundant is considered. Finally the general case of an arbitrary number of redundants is developed from the case of the single redundant. The entire paper relies heavily on the concept of a \"basic\" solution of a linear system which of course corresponds to a statically determinate substructure of a given structure. When specific examples are considered they will be trusses, but it may be added that the notation used applies to any type of structure including continuous systems. (The case of the truss, which has a diagonal primitive stiffness matrix, must, however, be extended to the general case in which the primitive stiffness matrix is partitioned-diagonal.) Notation. It is proposed to present the node and mesh methods of structural analysis in the following form (see [5] or any good book on matrix structural analysis): The Node Method: NF = P—node equilibrium, F = KA—constitutive equation, (1) A = N8—member/joint displacement equation, The Mesh Method: CA = 0—compatibility equations, A = K 'F—constitutive equation, (2) F = F°+CF„—member/mesh force equation. In these equations F, A—member force and displacement, P, 8—node force and displacement, K—primitive stiffness matrix, F°—any equilibrium force system (N F° = P), Fm—mesh force matrix, N—generalized incidence matrix, C—generalized branch-mesh matrix. Ordinarily the node and mesh methods are solved as (.NKN)8 = P or (CK'C)F„= -CK'F° (3) in which 8 and Fm are to be computed given the other matrices. In this paper the interest lies then in determining the ranges of 5 and Fm as K varies in some arbitrary manner while remaining positive definite (as required by the particular class of structure under study). For the case of trusses the primitive stiffness matrix is particularly simple: it is just a diagonal matrix with non-negative diagonal terms. For this case the terms in the system matrices in Eq. (3) are linear in either the elements of AT or AT\"1 and these equations can be rewritten in the form ELASTIC REALIZABILITY 413 Dk= p and = 0 (4) where k and k~l are simply column matrices whose elements are the diagonal terms of K and K~' are simply column matrices whose elements are the diagonal terms of K and K ' respectively. The elements of the matrices D and .^are the linear in the displacements S and forces Fm . From Eq. (4) the readability problem is reduced to finding all values of S and Fm for which these equations have semi-positive solutions k and k'\\ (In Gale's terminology, x is semi-positive if x > 0 but x ¥= 0.) There are basic differences in the forces and displacement formulations as they appear in Eqs. (3-4). From one point of view the displacement formulation deals with the solutions of a non-homogeneous system while the force formulation deals with a homogeneous system; from another point of view the displacements S vary inversely along a ray in K-space while the forces are constant along such a ray. Finally it should be noted that the form of Eq. (4) is reminiscent of work on linear inequalities. But the fact that the coefficients of D and <^are linear in the displacements S and the forces Fm adds a degree of difficulty not common in this area. Structures with a single redundant. In this section it will be shown that realizable force systems are bounded by statically determinant force systems for structures which are statically indeterminant to the first degree. Displacement realizability will also be discussed. In this case it is convenient to start with the node equilibrium equation, NF=P. (5) If n is the number of nodal degrees of freedom and b is the number of branch forces (b is the number of bars in the case of the truss), a single redundant implies that b = n + 1. It is furthermore assumed that the structure geometrically stable, which implies that the rank of the matrix N is n. Fig. 1 shows a plane truss which will be useful in discussing this case of a single redundant. The loading itself is of a certain interest since it corresponds to a situation in which the sign of a bar force can be changed by changing the values of the member stiffnesses. For example, as k} -* 0 (bar 3 is removed from the structure) bar 2 goes into tension, while as k, —» 0 it goes into compression. Some of the appropriate matrices are also indicated in this figure. At this point it is convenient to invoke the following theorem. ([5, theorem 2.9]): Theorem 1. Exactly one of the following alternatives holds. Either the equation Ax = 0 has a semi-positive solution or the inequality Ay > 0 has a solution. When Theorem 1 is applied to the system J*k~l = 0 for the case of a single redundant, it simply states that either (a) the system is realizable or (b) all the terms in the row matrix & must have the same sign. The regions of realizability are therefore defined by points at which the terms in ^change sign (pass through zero). Since a term in ^becoming zero corresponds to the formation of a statically determinate substructure, the region of real414 SPILLERS and LEFCOCHILOS

On stability and periodicity in phosphorus nutrient dynamics

Abstract: The stability of an equilibrium and the existence of limit cycles in a three-dimensional dynamical system arising in predator-prey-nutrient dynamics are demonstrated, using center manifold theory. Some implications of this result for limnological applications are discussed.

On a conjecture concerning the means of the eigenvalues of random Sturm-Liouville boundary value problems

Abstract: It has been known for several years that the expected value (Aj> of the smallest eigenvalue of a self-adjoint positive definite random Sturm-Liouville boundary value problem satisfies the relation < Hi, where /c, is the smallest eigenvalue of the corresponding deterministic problem obtained by replacing each random coefficient by its mean. It has been an open question whether similar inequalities are valid for the higher eigenvalues. The answer is negative, as shown by the counterexample given in this note. Consider the random Sturm-Liouville eigenvalue problem consisting of the differential equation , d Lu = —— ax du p^<°)Tx + q(x, w)u = Aw, 0 < x < 1 (1) and the boundary conditions u(0) = ii(l) = 0. (2) The coefficients p(x, to) and q(x, to) are given stochastic processes, with to e Q, where (Q, 3F, P) is the underlying probability space. We assume that, as functions of x and with probability one, p(x, co) e C1, q(x, co) e C, p(x, co) > 0, and q(x, co) > 0 on 0 < x < 1. In other words, these conditions hold except possibly on an co-set N such that P(N) = 0. For each co in Q\\N the problem (1), (2) has a sequence of eigenvalues 0 < Xj(co) < A2(c«) < ■ ■ • < A„(co) < ■ ■ • with the corresponding eigenfunctions uj(x, co), u2(x, co), ..., un(x, co), We assume that the eigenfunctions have been normalized so that (ut, Uj) = djj, where 3^ is the Kronecker delta and (u, t;) = u(x)v(x) dx. The explicit dependence of A, and Uj on co emphasizes that each eigenvalue is a random variable and each eigenfunction a stochastic process. Denoting the mathematical expectation by < • >, we can write = p0(x), (q(x, co)> = q0(x), (3) * Received October 9, 1979. 242 WILLIAM E. BOYCE

Wave propagation in an elastic half-space due to surface pressure over a non-uniformly changing circular zone

Abstract: Two problems of pressure distributions applied to an elastic half-space over a circular pressure zone whose center is fixed but whose radius changes nonuniformly with time are considered. In one case, the pressure depends only on time; in the other case, the pressure varies with the radial distance from the pressure zone center. Complete transform solutions are obtained and several wave propagation aspects are briefly studied, with emphasis on the Rayleigh pole contributions and the associated propagating singularities. The effects of some specific zone time-histories on the Rayleigh pole disturbances at the half-space surface are considered. Some characteristics of a given time-history appear to be manifested in the corresponding disturbance.

Diffusion systems reacting at the boundary

Abstract: The problem considered is that of two species or chemical concentrations which independently diffuse within the same or adjacent regions. The coupling interaction takes place only along a common boundary. This boundary reaction is allowed to be either totally dissipative wherein both species are removed by the interaction, or semi-dissipative wherein one species is stimulated at the expense of the other. This physical situation is modeled by two independent, linear heat equations, each defined over a one-dimensional, semi-infinite domain. Associated with each heat equation is a boundary flux condition containing a nonlinear interactive term which couples the solutions of the two heat equations. With only boundary interaction, the problem can be reduced to the study of two coupled Volterra integral equations. By using monotone operator methods these integral equations are shown to have positive solutions. Uniqueness is also established. The large-time asymptotic behavior of the solutions is examined for the cases of both fast and slow decay of data.

Power series solutions for the th-order-matrix ordinary differential equation

Abstract: A description of the fundamental solution of the mth-order linear ordinary differential equation with matrix coefficients is given in terms of power series and the Green function. The second-order equation is discussed.

On continuous dependence in finite elasticity

Abstract: We investigate the relation between stability and continuous dependence for a nonlinearly elastic body at equilibrium. We show that solutions of the governing equations that lie in a convex, stable set of deformations depend continuously on the body forces and the surface tractions. The definition of stability used is essentially due to Hadamard.

A temperature equation for a rigid heat conductor with memory

Abstract: Here u(t, x) is the temperature, e(t, x) the internal energy, q(t, x) the heat flux, and r(t, x) represents heat supplied to the rod from external sources. Our aim is to give conditions for boundedness and asymptotic behavior of u(t, x), given u(t, x) for t < 0, 0 < .x < n, and satisfying the boundary conditions u(t, 0) = u(t, n) = 0, t > 0. Miller in [4] discusses a more general case of (1) where .x = (.x1;.x„) e R\" and d/dx is replaced by the gradient operator V. He obtains existence theorems for u(t, x), t > 0, .x e B, an open subset of R\", where boundary conditions like u(t, .x) = 0 for all f > 0 and x e SB, the boundary of J3, are imposed. He also obtains conditions for the stability and asymptotic stability of the trivial solution in case r(t, x) = 0. His results are in terms of three types of solutions: distribution solutions, generalized distribution solutions, and classical solutions. His methods use semigroup theory, and standard results for Volterra integrodifTerential equations, using Laplace transform criteria for stability and asymptotic stability. We propose to study an integrodifferential equation for u(t, x) equivalent to (1) with «(0) > 0 and k(0) > 0, by studying, as Miller does, the equations satisfied by the coefficients of the Fourier series for u(t, x), in our case, a sine series, but instead of using Laplace transform techniques exclusively we use some results obtained in [5]. For simplicity, we also confine our study to classical solutions, in the sense of Miller [4], but similar results for distribution solutions can easily be obtained. In addition to conditions for stability and asymptotic stability for the trivial solution in case r(t, x) = 0, we also obtain conditions for asymptotic periodicity and asymptotic almost-periodicity in case r(t, x) is asymptotically periodic or asymptotically almost-periodic, again using results in [5],

Generic bifurcation in the obstacle problem

Abstract: We consider a class of singularities, locally of the form y2 = p(x) near the origin in R2, describing the shape of a free boundary curve arising from an elliptic free boundary value problem. The point of view taken is that of generic bifurcation, in particular with more than one parameter present. Of prime interest is a description of the unfoldings of such singularities, their normal forms, and generic conditions for oneand two-parameter unfoldings. The two simplest cases corresponding to perturbations of singularities y2 — x\" + 0(x\"+l), n = 4, 5 are treated in greater detail and the bifurcation diagram for a generic two-parameter unfolding is given. Our results do not rigorously concern the free boundary problem itself, but rather set down a formal framework, or model, for studying this problem in terms of bifurcation theory. We prove theorems describing this model. Nevertheless, our results have a bearing on any rigorous analysis of this problem since they form the necessary first step to such an analysis. The theory for computing the normal forms of solutions up to first order, for example, is given here.