A class of densely invertible parabolic operator equations

1969 ◽  
Vol 1 (3) ◽  
pp. 363-374 ◽  
Author(s):  
R.S. Anderssen
Keyword(s):  
Variational Methods ◽  
Variational Formulation ◽  
Lagrange Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Parabolic Differential Equations ◽  
Operator Equations ◽  
Initial Boundary Value

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.

Aspects of Osillations of Beams and Plates

2005 ◽  
Author(s):  
Mariya A. Zarubinska ◽  
W. T. van Horssen
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
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.

scholarly journals Exact variational representations for the solution of the simple parabolic equation

1971 ◽  
Vol 5 (3) ◽  
pp. 305-314
Author(s):  
R.S. Anderssen
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Differential Equation ◽  
Initial Boundary Value Problems ◽  
Series Representations ◽  
Initial Boundary Value

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

1991 ◽  
Vol 03 (02) ◽  
pp. 137-162 ◽  
Author(s):  
N. BELLOMO ◽  
T. GUSTAFSSON
Keyword(s):  
Boltzmann Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Future Research ◽  
Initial Boundary Value Problems ◽  
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

2007 ◽  
Vol 143 (1) ◽  
pp. 221-242 ◽  
Author(s):  
P. A. TREHARNE ◽  
A. S. FOKAS
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Initial Boundary Value Problems ◽  
Linear Evolution ◽  
Higher Derivatives ◽  
Initial Boundary Value

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.

An investigation of the Green–Lindsay three-dimensional model

2017 ◽  
Vol 23 (7) ◽  
pp. 987-1003 ◽  
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
Wolfgang H Müller
Keyword(s):  
Boundary Value Problem ◽  
Variational Formulation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Relaxation Times ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Three Dimensional Model ◽  
Initial Boundary Value

In this paper we consider the Green and Lindsay nonclassical model for inhomogeneous anisotropic thermoelastic bodies with two relaxation times, which depend on space variables. We obtain a variational formulation for the initial-boundary value problem corresponding to the Green–Lindsay model. On the basis of the variational formulation we define the spaces of vector-valued distributions corresponding to the initial-boundary value problem and by applying suitable a priori estimates we prove the existence and uniqueness of the solution, an energy equality, and the continuous dependence of the solution on given data.

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

1977 ◽  
Vol 82 (1) ◽  
pp. 131-145
Author(s):  
M. R. Carter
Keyword(s):  
Steady State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
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

1994 ◽  
Vol 124 (5) ◽  
pp. 879-908 ◽  
Author(s):  
C. Palencia ◽  
I. Alonso Mallo
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Initial Boundary Value Problems ◽  
Hyperbolic Problems ◽  
Euler’S Method ◽  
Initial Boundary Value

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.

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

2020 ◽  
Vol 12 (2) ◽  
pp. 297-310
Author(s):  
B.I. Kopytko ◽  
A.F. Novosyadlo
Keyword(s):  
Boundary Integral Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Boundary Integral ◽  
Initial Boundary Value Problems ◽  
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.

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

2018 ◽  
Vol 21 (2) ◽  
pp. 276-311 ◽  
Author(s):  
Adam Kubica ◽  
Masahiro Yamamoto
Keyword(s):  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Initial Boundary Value Problems ◽  
Unique Existence ◽  
Fractional Diffusion Equations ◽  
Time Variables ◽  
Initial Boundary Value

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.

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