scholarly journals From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation

2010 ◽  
Vol 28 (3) ◽  
pp. 1101-1119 ◽  
Author(s):  
Ivar Ekeland ◽  
Keyword(s):  
Differential Equation ◽  
Calculus Of Variations ◽  
Classical Problem ◽  
Implicit Differential Equation ◽  
René Thom

Non-LTE Line Transfer with Diffusion of Excited Atoms

1977 ◽  
Vol 32 (2) ◽  
pp. 156-159
Author(s):  
D. F. Düchs ◽  
J. Oxenius
Keyword(s):  
Differential Equation ◽  
Low Temperature ◽  
Radiative Transfer ◽  
Spectral Line ◽  
Source Function ◽  
Homogeneous Medium ◽  
Classical Problem ◽  
Temperature Limit ◽  
Excited Atoms ◽  
Line Transfer

The classical problem of radiative transfer in a spectral line, due to two-level atoms, in a homogeneous medium is reconsidered. It is pointed out that the source function used up to now in the literature neglects the diffusion of the excited atoms. In many cases this assumption is not justified. In the low-temperature limit kT ≪ hv, the correct source function, allowing for diffusion of excited atoms, obeys an integro-differential equation

XIII.—Partial Differential Equations and the Calculus of Variations

1927 ◽  
Vol 46 ◽  
pp. 126-135 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Calculus Of Variations ◽  
Order Equation ◽  
Second Order ◽  
Physical Problem ◽  
Partial Differential ◽  
Hamiltonian Principle

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer

2012 ◽  
Vol 22 (7) ◽  
pp. 896-906
Author(s):  
Lawrence J. De Chant
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Simple Model ◽  
Differential Equations ◽  
Large Scale ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Implicit Differential Equation ◽  
Content Type ◽  
Integration Schemes

PurposeAlthough most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.Design/methodology/approachHere we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.FindingsA predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.Originality/valueThe paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

scholarly journals Index of an implicit differential equation

2010 ◽  
Vol 54 ◽  
pp. 173-186
Author(s):  
L. S. Challapa ◽  
M. A. S. Ruas
Keyword(s):  
Differential Equation ◽  
Implicit Differential Equation

scholarly journals Fractional calculus of variations: a novel way to look at it

2019 ◽  
Vol 22 (4) ◽  
pp. 1133-1144 ◽  
Author(s):  
Rui A.C. Ferreira
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Classical Problem ◽  
Simple Consequence ◽  
Fractional Calculus Of Variations ◽  
Derivative Order

Abstract In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on the derivative-order α.

One method for investigating the solvability of boundary value problems for an implicit differential equation

2021 ◽  
pp. 404-413
Author(s):  
Wassim Merchela
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Linear Boundary ◽  
Implicit Differential Equation ◽  
Carathéodory Conditions ◽  
Covering Set ◽  
The Right ◽  
Scalar Differential Equation

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

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