Aspects of Osillations of Beams and Plates

Boundary Values ◽

Internal Resonances ◽

Specific Parameter ◽

Plate Equations ◽

Initial Boundary Value ◽

Parameter Values

In this paper some initial boundary value problems for beam and plate equations will be studied. These initial boundary values problems can be regarded as simple models describing free oscillations of plates on elastic foundations or describing coupled torsional and vertical oscillations of a beam. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For the plate on an elastic foundation it turns out that complicated internal resonances can occur for specific parameter values.

A class of densely invertible parabolic operator equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270004226x ◽

1969 ◽

Vol 1 (3) ◽

pp. 363-374 ◽

Cited By ~ 3

Author(s):

R.S. Anderssen

Keyword(s):

Variational Methods ◽

Variational Formulation ◽

Lagrange Equation ◽

Boundary Value ◽

Initial Boundary ◽

Parabolic Differential Equations ◽

Operator Equations ◽

Before variational methods can be applied to the solution of an initial boundary value problem for a parabolic differential equation, it is first necessary to derive an appropriate variational formulation for the problem. The required solution is then the function which minimises this variational formulation, and can be constructed using variational methods. Formulations for K-p.d. operators have been given by Petryshyn. Here, we show that a wide class of initial boundary value problems for parabolic differential equations can be related to operators which are densely invertible, and hence, K-p.d.; and develop a method which can be used to prove dense invertibility for an even wider class. In this way, the result of Adler on the non-existence of a functional for which the Euler-Lagrange equation is the simple parabolic is circumvented.

Exact variational representations for the solution of the simple parabolic equation

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700047250 ◽

1971 ◽

Vol 5 (3) ◽

pp. 305-314

Author(s):

R.S. Anderssen

Keyword(s):

Differential Equation ◽

Fourier Series ◽

Parabolic Equation ◽

Boundary Value ◽

Initial Boundary ◽

Parabolic Differential Equation ◽

Series Representations ◽

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

THE DISCRETE BOLTZMANN EQUATION: A REVIEW OF THE MATHEMATICAL ASPECTS OF THE INITIAL AND INITIAL-BOUNDARY VALUE PROBLEMS

Reviews in Mathematical Physics ◽

10.1142/s0129055x91000060 ◽

1991 ◽

Vol 03 (02) ◽

pp. 137-162 ◽

Cited By ~ 29

Author(s):

N. BELLOMO ◽

T. GUSTAFSSON

Keyword(s):

Boltzmann Equation ◽

Boundary Value ◽

Initial Boundary ◽

Future Research ◽

Survey Paper ◽

Initial Boundary Value ◽

Mathematical Literature ◽

Discrete Boltzmann Equation

This paper provides a review of the various results available in the mathematical literature on the solution to the initial and initial-boundary value problem for the discrete Boltzmann equation, a nonlinear mathematical model of the kinetic theory of gases. References include papers published until 1990. The aim of this survey paper is to provide a detailed picture of the state of the art in the field and some necessary information for future research in the field.

Initial-boundary value problems for linear PDEs with variable coefficients

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000084 ◽

2007 ◽

Vol 143 (1) ◽

pp. 221-242 ◽

Cited By ~ 3

Author(s):

P. A. TREHARNE ◽

A. S. FOKAS

Keyword(s):

Boundary Value Problems ◽

Evolution Equations ◽

Boundary Value ◽

Initial Boundary ◽

Variable Coefficients ◽

Linear Evolution ◽

Higher Derivatives ◽

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Bounds for the solution of a non-linear parabolic initial-boundary value problem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100053767 ◽

1977 ◽

Vol 82 (1) ◽

pp. 131-145

Author(s):

M. R. Carter

Keyword(s):

Steady State ◽

Boundary Value ◽

Initial Boundary ◽

The Past ◽

Existence And Stability ◽

Image Position ◽

Initial Boundary Value ◽

Steady State Solutions

A number of papers have appeared over the past decade or so which study questions of the existence and stability of positive steady-state solutions for parabolic initial-boundary value problems of the general form

Abstract initial boundary value problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500022393 ◽

1994 ◽

Vol 124 (5) ◽

pp. 879-908 ◽

Cited By ~ 11

Author(s):

C. Palencia ◽

I. Alonso Mallo

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Time Integration ◽

Initial Boundary ◽

Linear Operators ◽

Hyperbolic Problems ◽

Euler’S Method ◽

We consider abstract initial boundary value problems in a spirit similar to that of the classical theory of linear semigroups. We assume that the solution u at time t is given by u(t) = S(t) ξ + V(t)g, where ξ and g are respectively the initial and boundary data and S(t) and V(t) are linear operators. We take as a departing point the functional equations satisfied by the propagators S and V. We discuss conditions under which a pair (S, V) describes the solution of an abstract differential initial boundary value problem. Several examples are provided of parabolic and hyperbolic problems that can be accommodated within the abstract theory. We study the backward Euler's method for the time integration of the problems considered.

Travelling waves in an initial-boundary value problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001533x ◽

1981 ◽

Vol 90 (1-2) ◽

pp. 41-61 ◽

Cited By ~ 3

Author(s):

E. J. M. Veling

Keyword(s):

Boundary Value Problem ◽

Asymptotic Expression ◽

Travelling Waves ◽

Boundary Value ◽

Initial Boundary ◽

Stationary Problem ◽

Travelling Wave ◽

Boundary Values ◽

SynopsisIn this paper we consider the initial-boundary value problem for the semihnear diffusion equation ul=uxx+f(u) on the half-line x>0, when for 0<a<1 f(0)=f(a)=f(1)=0 and f(u)<0 on (0, a), f(u)>0 on (a, 1). For a wide class of initial and boundary values a uniformly valid asymptotic expression is given to which the solution converges exponentially. This expression is composed of a travelling wave and a solution of the stationary problem.

On a nonclassical problem for the heat equation and the Feller semigroup generated by it

Carpathian Mathematical Publications ◽

10.15330/cmp.12.2.297-310 ◽

2020 ◽

Vol 12 (2) ◽

pp. 297-310

Author(s):

B.I. Kopytko ◽

A.F. Novosyadlo

Keyword(s):

Boundary Integral Equations ◽

Boundary Value ◽

Initial Boundary ◽

Continuous Functions ◽

Boundary Integral ◽

Feller Semigroup ◽

Initial Boundary Value ◽

Bounded Continuous Functions

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.

The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions

Letters in Mathematical Physics ◽

10.1007/s11005-021-01356-7 ◽

2021 ◽

Vol 111 (1) ◽

Author(s):

B. Deconinck ◽

A. S. Fokas ◽

J. Lenells

Keyword(s):

Evolution Equations ◽

Initial Conditions ◽

Boundary Value ◽

Initial Boundary ◽

Boundary Values ◽

Initial Value ◽

Integrable Evolution ◽

Initial Boundary Value ◽

The Given

AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0018 ◽

2018 ◽

Vol 21 (2) ◽

pp. 276-311 ◽

Cited By ~ 18

Author(s):

Adam Kubica ◽

Masahiro Yamamoto

Keyword(s):

Dirichlet Boundary ◽

Boundary Value ◽

Initial Boundary ◽

Fractional Diffusion ◽

Unique Existence ◽

Fractional Diffusion Equations ◽

Time Variables ◽

Abstract We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.