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On Problems in the calculus of variations in Increasingly Elongated Domains

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Chinese Annals of Mathematics,Series B 2018年第2期39卷 163-182页

作者： Herve LE DRET Amira MOKRANE Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthda

This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.

关键词： calculus of variations Domains becoming unbounded Asymptotic behaviorExponential rate of convergence

A unified approach to the calculus of variations on time scales

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A unified approach to the calculus of variations on time sca...

Variational Approach to Heat Conduction Modeling

2010 Chinese Control and Decision Conference

作者： Ewa Girejko Agnieszka B.Malinowska Delfim F.M.Torres Department of Mathematics University of Aveiro3810-193 AveiroPortugal Faculty of Computer Science Biatystok University of Technology15-351 BiatystokPoland

ISBN: (纸本)9781424451814;9781424451821

＜正＞In this work we propose a new and more general approach to the calculus of variations on time scales that allows to obtain,as particular cases,both delta and nabla *** precisely,we pose the problem of minimizing or maximizing the composition of delta and nabla integrals with Lagrangians that involve directional *** Euler-Lagrange necessary optimality conditions,as well as sufficient conditions under appropriate convexity assumptions, are *** illustrate presented results with simple examples.

关键词： Euler-Lagrange equations calculus of variations delta and nabla calculi time scales directional derivatives

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Journal of Applied Mathematics and Physics 2024年第1期12卷 234-248页

作者： Slavko Đurić Ivan Aranđelović Milan Milotić Faculty of Traffic Engineering University of East Sarajevo Doboj Bosnia and Herzegovina Faculty of Mechanical Engineering University of Belgrade Belgrade Republic of Serbia

It is known that Fourier’s heat equation, which is parabolic, implies an infinite velocity propagation, or, in other words, that the mechanism of heat conduction is established instantaneously under all conditions. This is unacceptable on physical grounds in spite of the fact that Fourier’s law agrees well with experiment. However, discrepancies are likely to occur when extremely short distances or extremely short time intervals are considered, as they must in some modern problems of aero-thermodynamics. Cattaneo and independently Vernotte proved that such process can be described by Heaviside’s telegraph equation. This paper shows that this fact can be derived using calculus of variations, by application of the Euler-Lagrange equation. So, we proved that the equation of heat conduction with finite velocity propagation of the thermal disturbance can be obtained as a solution to one variational problem.

关键词： Telegraph Equation Heat Equation Heat Conduction calculus of variations

Variational Approach to 2D and 3D Heat Conduction Modeling

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Journal of Applied Mathematics and Physics 2024年第4期12卷 1383-1400页

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 × 5 cm aluminium plate and a 5 × 5 × 5 cm aluminum cube. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. Approximate solutions were obtained by applying calculus of variations and Euler-Lagrange equations. In order to verify the correctness of the proposed approximate solutions, they were compared with the exact solutions of parabolic and hyperbolic equations. The paper also presents the research on the influence of time parameters τ as well as the relaxation times τ ∗ to the variation of the profile of the temperature field for the considered aluminum plate and cube.

关键词： Classical Equation of Heat Conduction Generalized Equation of Heat Conduction calculus of variations Approximate Solution

Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles

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Particuology 2020年第4期18卷 184-192页

作者： Mihal Duris Zorana Arsenijevic Radmila Garic-Grulovic Tatjana Kaluderovic Radoicic Institute of Chemistry Technology and Metallurgy(National Institute)Department for Catalysis and Chemical EngineeringUniversity of BelgradeNjegoseva 12BelgradeSerbia Faculty of Technology and Metallurgy University of BelgradeKarnegijeva 4BelgradeSerbia

In this study,we applied the variational model to fluidization of small spherical *** experiments were carried out for spherical particles with 13 diameters between dp=0.13 and 5.00 *** propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficientβby introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes(different flow regimes).By comparing the predictions with the experimental results,we conclude that n=1 should be used for small particles(dp1 mm).This conclusion is generalized by proposing n=1 for particles with Ret*** average mean absolute error was 5.49%in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles *** calculated values ofβwere compared with values of literature models for particles with dp<1.0 *** average mean absolute error of the modified variational model was 8.02%in calculatingβfor different bed voidages for all of the particles examined.

关键词： calculus of variations Isoperimetric problem Bed expansion Drag coefficient Fluidization Spherical particles

ELECTRICALLY CHARGED SOLITONS IN GAUGE FIELD THEORY

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Acta Mathematica Scientia 2010年第6期30卷 1975-2005页

作者：杨亦松 Department of Mathematics Polytechnic Institute of New York University Brooklyn Institute of Contemporary Mathematics Henan University

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.

关键词： gauge field theory monopoles vortices dyons calculus of variations topological invariants

THE STRESS-ENERGY TENSOR AND POHOZAEV'S IDENTITY FOR SYSTEMS

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Acta Mathematica Scientia 2012年第1期32卷 433-439页

作者： N.D.Alikakos A.C.Faliagas Department of Mathematics University of Athens

Utilizing stress-energy tensors which allow for a divergence-free formulation, we establish Pohozaev＇s identity for certain classes of quasilinear systems with variational structure.

关键词： calculus of variations stress-energy tensor p-Lapacian minimal surface

Existence of Optical Vortex Solitons in Pair Plasmas

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Acta Mathematicae Applicatae Sinica 2023年第3期39卷 571-582页

作者： Rui-feng ZHANG School of Mathematics and Statistics Henan UniversityKaifeng 475004China

Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical *** this study,we employ the calculus of variations method to develop an existence theory for the steady state vortex solutions of a nonlinear Schr?dinger type equation to model light waves that propagate in a medium with a new focusing-defocusing ***,we demonstrate the existence of positive radially symmetric solutions by constrained minimization,where we give some interesting explicit estimates related to vortex winding numbers and the wave propagation ***,we establish the existence of saddle-point solutions through a mountain-pass argument.

关键词： calculus of variations mountain-pass theorem pair plasmas nonlinear Schr?dinger type equation optical vortices

Control and Nash Games with Mean Field Effect

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Chinese Annals of Mathematics,Series B 2013年第2期34卷 161-192页

作者： Alain BENSOUSSAN Jens FREHSE International Center for Decision and Risk Analysis School of Management University of Texas-DallasFrance School of Business the Hong Kong Polytechnic University Graduate Department of Financial Engineering Ajou University Institute for Applied Mathematics University of Bonn

Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007,of Gueant-Lasry-Lions in 2011,of HuangCaines-Malham in 2007 and many others).There are a lot of *** general,the applications concern approximating an infinite number of players with common behavior by a representative *** agent has to solve a control problem perturbed by a field equation,representing in some way the behavior of the average infinite number of *** approach does not lead easily to the problems of Nash equilibrium for a finite number of players,perturbed by field equations,unless one considers averaging within different groups,which has not been done in the literature,and seems quite *** this paper,the authors approach similar problems with a different motivation which makes sense for control and also for differential *** the systems of nonlinear partial differential equations with mean field terms,which have not been addressed in the literature so far,are considered here.

关键词： Mean field Dynamic programming Nash games Equilibrium calculus of variations