经典数学物理中的偏微分方程 英文版
作者：巴勒（Lev Rubinstein）著
出版时间：2000年版
内容简介
The unique characteristic of this book is that it considers the theory of partial differential equations in mathematical physics as the language of continuous processes， that is to say， as an interdisciplinary science that treats the hierarchy of mathematical phenomena as reflections of their physical counterparts. Special attention is drawn to tracing the development of these mathematical phenomena in different natural sciences， with examples drawn from continuum mechanics， electrodynamics， transport phenomena， thermodynamics， and chemical kinetics. At the same time， the authors trace the interrelation between the different types of problems elliptic， parabolic， and hyperbolic – as the mathematical counterparts of stationary and evolutionary processes. This interrelation is traced through study of the asymptotics of the solutions of the respective initial boundaryvalue problems both with respect to time and the governing parameters of the problem. This combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs， one that will be appreciated by both graduate students and researchers alike.本书为英文版。
目录
Preface
Chapter1.
Introduction
1.Mathematicalphysics
2.Basicconceptsofcontinuum
mechanics
3.Elementsof
electrostatics
4.Elementsof
electrodynamics
5.Elementsofchemical
kinetics
6.Elementsofequilibrium
thermodynamics
7.Integrallawsofconservation
ofextensiveparameters
8.Elementsofthermodynamics
ofirreversibleprocesses
Problems
Chapter2.Typical
equationsofmathematical
physics.Boundary
conditions
1.Lawsofconservationand
continuity.Three
prototypicsecond-order
equationsofmathematical
physics
2.Equationsofcontinuity.
Convectiveanddiffusion
fluxinnonelectrolyte
solutionsinpresenceof
chemicalreactions.Fick\’s
equationofdiffusionin
binarysolutions.Diffusion
ofelectrolytes.
Nernst-Planck
equation
3.Equationofmotionof
continuousmedium
4.Equationofheat
conductionincontinuous
media.Heatconductionin
movinghomogeneous
compressiblefluid
5.Potentialmotionofinviscid
incompressibleliquid.
Equations\’ofvibrationsof
elasticbodyandofslightly
compressibleinviscid
liquid
6.Chainofspringsoscillating
inmediumwithfriction.
Waveequation
7.Maxwell\’sequationsof
electrodynamics
8.Theoryofpercolationof
multicomponent
liquids
9.Brownianmotion.
Langevin\’sequationand
hyperbolicdiffusion
equation
10.Boundaryandinitial
conditions
11.Examplesoftypicalfree
boundary-value
problems
12.Well-posednessin
Hadamard\’ssense.
Examplesofill-posed
problems
13.Terminology.Concluding
remark.Notation
Problems
Chapter3.Cauchy
problemforfirst-order
partialdifferential
equations
1.LocalCauchyproblemfor
quasilinearequationwith
twoindependent
variables
2.LocalCauchyproblemfor
nonlinearfirst-orderpartial
differentialequation
3.GlobalCauchyproblemfor
quasilinearpartial
differentialfirst-order
equationwithtwo
independentvariables.Need
forbroaderclassof
generalized(discontinuous)
solutions
4.Necessaryconditionsof
discontinuity.Problemof
decayofarbitrary
discontinuity.Gelfand\’s
heuristictheory
Problems
Chapter4.Classificationof
second-orderpartial
differentialequationswith
linearprincipalpart.
Elementsofthetheoryof
characteristics
1.Classificationof
second-orderpartial
differentialequations
2.Reductionofsecond-order
equationtocanonical
form
3.Canonicalformoflinear
partialdifferentialequations
withconstant
coefficients
4.Cauchyproblemforpartial
differentialequationswith
linearprincipalpart.
Classificationof
equations
5.Cauchyproblemforsystem
oftwoquasilinearfirst-order
partialdifferentialequations
withtwoindependent
variables;conceptof
characteristics
6.Characteristicsascurvesof
weakdiscontinuityofsecond
orhigherorder
7.Riemann\’sformula.
Characteristicsascurvesof
weakdiscontinuityoffirst
orderorascurvesofstrong
discontinuity
Problems
Chapter5.Cauchyand
mixedproblemsforthe
waveequationinR1.
Methodoftraveling
waves
1.Smallvibrationsofinfinite
string.Methodoftraveling
waves
2.Smallvibrationsof
semi-infiniteandfinite
stringswithrigidlyfixedor
freeends.Methodof
prolongation
3.Generalizedsolutionof
problemofvibrationof
loadedstringwith
nonhomogeneousboundary
conditions
Problems
Chapter6.Cauchyand
Goursatproblemsfora
second-orderlinear
hyperbolicequationwith
twoindependentvariables.
Riemann\’smethod
1.Riemann\’smethod
2.Goursatproblem.Existence
anduniquenessof
Riemann\’sfunction
3.Dynamicsofsorptionfrom
solutionpercolatingthrough
layerofporousadsorbent.
Riemannfunctionfora
linearhyperbolicequation
withconstant
coefficients
Problems
Chapter7.Cauchy
problemfora2-dimensional
waveequation.The
Volterra-D\’Adhemar
solution
1.Characteristicmanifoldof
second-orderlinear
hyperbolicequationwithn
independentvariables
2.Cauchyproblemforthe
2-dimensionalwave
equation.
Volterra-D\’Adhemar
solution
Problems
Chapter8.Cauchy
problemforthewave
equationinRs.Methodsof
veraginganddescent.
Huygens\’sprinciple
1.Methodofaveraging
2.Methodofdescent
3.Huygens\’sprinciple
Problems
Chapter9.Basicproperties
ofharmonicfunctions
1.Convex,linear,andconcave
functionsinRi
2.Classesoftwicecontinuously
differentiablesuperharmonic,
harmonic,andsubharmonic
functionsin
multidimensional
regions
3.Hopf\’slemmaandstrong
maximumprinciple
4.Green\’sformulas.Fluxof
harmonicfunctionthrough
closedsurface.Uniqueness
theorems
5.Integralidentity.Mean
valuetheorem.Inversemean
valuetheorem
Problems
Chapter10.Green\’s
functions
1.Definitions.Main
properties
2.Sommerfeld\’smethodof
electrostaticimages(method
ofsuperpositionofsources
andsinks)
3.Poissonintegral
Problems
Chapter11.Sequencesof
harmonicfunctions.
Perron\’stheorem.Schwarz
alternatingmethod
1.Harnack\’stheorems
2.Completeclassesof
(continuous)superharmonic
andsubharm0nic
functions
3.BasicPerrontheorem
4.Existencetheoremfor
Dirichletproblem.
Barriers.
5.Schwarzalternating
method
Problems
Chapter12.Outer
boundary-valueproblems.
Elementsofpotential
theory
1.Isolatedsingularpointsof
harmonicfunctions
2.Regularityofharmonic
functionsatinfinity
3.Extensionofthe
fundamentalidentityto
unboundedregions.
Liouville\’stheorem
4.Electrostaticpotentials
5.Integralswithpolar
singularities
6.Propertiesofelectrostatic
volumepotential
7.Propertiesofelectrostatic
potentialsofdoubleand
singlelayers
8.DirichletandNeumann
boundary-valueproblems.
Reductiontointegral
equations.Existence
theorems
Problems
Chapter13.Cauchy
problemfor
heat-conduction
equation
1.Fundamentalsolutionof
Fourierequation.Heaviside
unitfunctionandDirac5
function
2.SolutionofCauchyproblem
for1-dimensionalFourier
equation.Poisson
integral
3.Momentsofsolutionof
Cauchyproblem.
Asymptoticbehaviorofthe
Poissonintegralas
tToo
4.Prigogineprinciple,
Glansdorf-Prigogine
criterion,andsolutionof
Cauchyproblemfor
heat-conduction
equation
5.Fundamentalsolutionof
multidimensional
heat-conduction
equation
Problems
Chapter14.Maximum
principleforparabolic
equations
1.Notation
2.Weakmaximum
principle
3.Nirenberg\’sstrongmaximum
principle
4.Vyborny-Friedmananalog
ofHopf\’slemma
5.Uniquenesstheorems.
Tichonov\’scomparison
theorem
6.Remarksontime
irreversibilityinparabolic
equations
Problems
Chapter15.Applicationof
Green\’sformulas.
Fundamentalidentity.
Green\’sfunctionsfor
Fourierequation
1.Fundamentalidentity
2.ApplicationoffirstGreen\’s
formulaanduniqueness
theorems
3.Green\’sfunctions
4.Relationshipbetween
Green\’sfunctionsof
DirichletprobleminR3,
correspondingtoLaplace
andFourieroperators
(Tichonov\’stheorem)
5.ExamplesofGreen\’s
functions
Problems
Chapter16.Heat
potentials
1.Volumeheatpotential
2.Heatpotentialsofdouble
andsinglelayers
Problems
Chapter17.Volterra
integralequationsandtheir
applicationtosolutionof
boundary-valueproblemsin
heat-conductiontheory
1.Reductionoffirst,second,
andthirdboundary-value
problemsforFourier
equationtoVolterraintegral
equations.Existence
theorems
2.Asymptoticbehaviorof
solutionoffirst
boundary-valueproblemand
respectiveintegral
equations
3.Solutionofquasilinear
Cauchyproblem
4.One-dimensionalone-phase
Stefanproblemwith
ablation
5.Determinationof
temperatureofhalf-space
z>0radiatingheat
accordingto
Stefan-Boltzmannlaw
Problems
Chapter18.Sequencesof
parabolicfunctions
1.Parabolicanalogsof
Harnack\’stheorems
2.Spaceofcontinuoussuper-
andsubparabolic
functions
3.Perron-Petrovsky\’stheorem.
Parabolicbarriers
4.Caseofcylindricalregion.
Tichonov\’stheorem.
Duhameltest
5.ApplicationofSchwarz
alternatingmethodto
solutionofDirichletproblem
forheat-conductionequation
innoncylindricalregion
Problems
Chapter19.Fourier
methodforbounded
regions
1.Vibrationofabounded
string.D\’Alembert\’s
solutionandsuperposition
ofstandingwaves.Formal
schemeofthemethodof
separationofvariables
2.Heattransferthrougha
homogeneousslab
3.Two-dimensionalDirichlet
problemforPoisson
equationinarectangle
4.Vibrationofcircular
membranewithrigidlyfixed
boundaryunderactionof
instantpointimpulse
initiallyappliedatan
interiorpointof
membrane
5.Heattransferthrough
two-layercirculardiskwith
Newtonianirradiationfrom
mediumofprescribed
temperature
6.ApplicationofFourier
methodtosolutionofmixed
problems.Reductionto
denumerablesystemof
algebraicequations.Perfect
systems
Problems
Chapter20.Integral
transformmethodin
unboundedregions
1.Integraltransformsin
solutionofboundary-value
problemsinunbounded
regions
2.Fouriertransform,sineand
cosineFouriertransform.
DoubleFourierintegraland
Fourier-Lebesguetheorem.
Fouriertransformof
derivatives
3.UseofFouriertransformsto
solveCauchyproblemof
heatconduction
4.Fourier-Bessel(Hankel)
transformandsolutionof
boundary-valueproblems
withcylindricalsymmetry.
Fundamentalsolutionof
heat-conductionequation
withforcedconvection,
generatedbycontinuously
actingsourceof
incompressibleliquid
5.Laplace-Carsontransform
anditssimplest
properties
6.Relationshipbetween
LaplaceandFourier
transforms.Bromwich
integralandJordan
lemma
7.Relationshipbetweenlimits
offunctionsandtheir
transforms.Asymptotic
expansion
Problems
Chapter21.Asymptotic
expansions.Asymptotic
solutionofboundary-value
problems
1.SolutionofCauchyproblem
for1-dimensionalFourier
equation.Shortrelaxation
timeasymptoticsfor
solutionofhyperbolic
heat-conduction
equation
2.Asymptoticsequences.
Expansionsinasymptotic
series.Definitionsand
preliminarystatements
3.Regularandsingular
perturbations.Differential
equationsdependingon
parameters.Scaling.Outer
andinnerexpansions.
Matching
4.Electrodiffusionandthe
nonequilibriumspacecharge
inthe1-dimensionalliquid
junction
Problems
Appendix1.Elementsof
vectoranalysis
1.Definitions
2.Gaussdivergencetheorem
andStokes\’stheorem
3.Orthogonalcurvilinear
coordinatesystems.Lame
coefficients.Basicoperators
ofvectoranalysis
Appendix2.Elementsof
theoryofBessel
functions
1.Introduction.Euler\’s
gammafunction
2.Generatingfunctionsand
Besselfunctionsoffirst
kind.Neumann
functions
3.BesselandLipschitz
integrals
4.Neumann\’saddition
theorem
5.Potentialofdoublelayerof
dipolesdistributedwith
unitdensityalongsurface
ofinfinitelylongcircular
cylinder.Discontinuous
Weber-Schafheitlin
integral.Fourier-Bessel
doubleintegral
6.Besselfunctionsof
imaginawargument.
SphericalBessel
functions
7.Asymptoticbehaviorof
Besselfunctions
8.Methodofaveraging.
Weber\’sintegrals
9.RepresentationofBessel
functionsbycontourand
singularintegrals
10.Asymptoticrepresentation
ofBesselfunctionsin
complexplane
11.Hintforsolutionof
cylindricalStefan
problem
Problems
Appendix3.Fourier\’s
methodand
$turm-Liouville
equations
1.Separationofvariablesand
eigenvalueproblem
2.Elementarytheoryof
regularSturm-Liouville
equations
3.Expansionoffunctionsin
m*inseriesof
eigenfunctionsofregular
Sturm-Liouville
operator
4.Remarksoncaseofsingular
operator
5.Expansionsinto
Fourier-BesselandDini
series
Problems
Appendix4.Fourier
integral
1.Riemann-Lebesgue
lemma
2.FundamentalFourier
theorem
3.Fouriertransformof
functionofexponential
growthatinfinity.
Relationshipbetweendouble
FourierintegralandFourier
series
4.Convolutiontheoremand
evaluationofdefinite
integrals
5.Abel-summableintegrals
andsolutionofproblems
withconcentrated
capacity
Problems
Appendix5.Examplesof
solutionofnontrivial
engineeringandphysical
problems
1.Heatlossininjectionof
heatintooilstratum
2.Nonlineareffectsin
electrodiffusionequilibrium.
Saturationofforceof
repulsionbetweentwo
symmetricallycharged
spheresinelectrolyte
solution
3.Linearstabilityof
Neumann\’ssolutionof
two-phaseCauchy-Stefan
problem
References
Index
