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.

scholarly journals Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Yu Zhang
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Boundary Value Problems ◽  
Fractional Differential Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Fourier Series Method ◽  
Fractional Differential ◽  
Initial Boundary Value

The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.

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.

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.

scholarly journals ON A 3D INITIAL-BOUNDARY VALUE PROBLEM FOR DETERMINING THE DYNAMICS OF IMPURITIES CONCENTRATION IN A HORIZONTAL LAYERED FINE-PORE MEDIUM

2019 ◽  
Vol 3 ◽  
pp. 60
Author(s):  
Sharif E. Guseynov ◽  
Ruslans Aleksejevs ◽  
Jekaterina V. Aleksejeva
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Analytical Approach ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Unsteady State ◽  
State Boundary ◽  
Initial Boundary Value

In the present paper, we propose an analytical approach for solving the 3D unsteady-state boundary-value problem for the second-order parabolic equation with the second and third types boundary conditions in two-layer rectangular parallelepipedic domain.

Existence Theorem for the Initial-Boundary Value Problem for a Singular Parabolic Partial Differential Equation

1972 ◽  
Vol 15 (2) ◽  
pp. 229-234
Author(s):  
Julius A. Krantzberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Image Position ◽  
Initial Boundary Value

We consider the initial-boundary value problem for the parabolic partial differential equation1.1in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

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