Self-similarity and beyond: exact solutions of nonlinear problems

315 Pages·2000·1.935 MB·English

by  P.L. Sachdev

Preface
Ihaveenjoyedfindingexactsolutionsofnonlinearproblemsforseveraldecades.
Ihavealsohadthepleasureofassociationofalargenumberofstudents,postdoc-
toral fellows, and other colleagues, from both India and abroad, in this pursuit.
The present monograph is an attempt to put down some of this experience.
Nonlinear problems pose a challenge that is often difficult to resist; each new
exact solution is a thing of joy.
InthewritingofthisbookIhavebeenmuchhelpedbymycolleagueProfessor
V. Philip, and former students Dr. B. Mayil Vaganan and Dr. Ch. Srinivasa
Rao. I am particularly indebted to Dr. Rao for his unstinting help in the
preparation of the manuscript. Mr. Renugopal, with considerable patience and
care, put it in LaTex form.
My wife, Rita, provided invaluable support, care, and comfort as she has
doneinmyearlierendeavours. Oursons,DeepakandAnurag,eachcontributed
in their own ways.
I am grateful to the Council of Scientific and Industrial Research, India
for financial support. I also wish to thank Dr. Sunil Nair, Commissioning
Editor,Chapman&Hall,CRCPress,forhispromptactioninseeingthisproject
through.
P.L. Sachdev
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Contents
1Introduction
2First-OrderPartialDifferentialEquations
2.1LinearPartialDifferentialEquationsofFirstOrder
2.2QuasilinearPartialDifferentialEquationsofFirstOrder
2.3Reductionof u +unu +H(x,t,u )=0
t x
2.4InitialValueProblemfor u +g(u)u +λh(u)=0
t x
2.5InitialValueProblemfor u +uαu +λuβ =0
t x
3ExactSimilaritySolutionsofNonlinearPDEs
3.1ReductionofPDEsbyInfinitesimalTransformations
3.2SystemsofPartialDifferentialEquations
3.3Self-SimilarSolutionsoftheSecondKind
3.4Introduction
3.5ANonlinearHeatEquationinThreeDimensions
3.6SimilaritySolutionofBurgersEquationbytheDirectMethod
3.7ExactFreeSurfaceFlowsforShallow-WaterEquations
3.8AnExamplefromGasdynamics
4ExactTravellingWaveSolutions
4.1TravellingWaveSolutions
4.2SimpleWavesin1-DGasdynamics
4.3ElementaryNonlinearDiffusiveTravellingWaves
4.4TravellingWavesforHigher-OrderDiffusiveSystems
4.5MultidimensionalHomogeneousPartialDifferentialEquations
4.6SystemsofNonhomogeneousPartialDifferentialEquations
4.7ExactHydromagneticTravellingWaves
4.8ExactSimpleWavesonShearFlows
5ExactLinearisationofNonlinearPDEs
5.1Introduction
5.2CommentsontheSolutionofLinearPDEs
5.3BurgersEquationinOneandHigherDimensions
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5.4 Nonlinear Degenerate Diffusion Equation u =[f(u)u−1]
t x x
5.5 MotionofCompressibleIsentropicGasintheHodographPlane
5.6 The Born-Infeld Equation
5.7 Water Waves up a Uniformly Sloping Beach
5.8 Simple Waves on Shear Flows
5.9 C-Integrable Nonlinear PDEs
6 Nonlinearisation and Embedding of Special Solutions
6.1 Introduction
6.2 Generalised Burgers Equations
6.3 Burgers Equation in Cylindrical Coordinates with Axisym-
metry
6.4 Nonplanar Burgers Equation – A Composite Solution
6.5 Modified Burgers Equation
6.6 Embedding of Similarity Solution in a Larger Class
7 Asymptotic Solutions by Balancing Arguments
7.1 Asymptotic Solution by Balancing Arguments
7.2 Nonplanar Burgers Equation
7.3 One-Dimensional Contaminant Transport through Porous
Media
8 Series Solutions of Nonlinear PDEs
8.1 Introduction
8.2 AnalysisofExpansionofaGasSphere(Cylinder)intoVacuum
8.3 Collapse of a Spherical or Cylindrical Cavity
8.4 ConvergingShockWavefromaSphericalorCylindricalPiston
References
©2000 CRC Press LLC
Chapter 1
Introduction
Nonlinear problems have always tantalized scientists and engineers: they
fascinate, but oftentimes elude exact treatment. A great majority of non-
linear problems are described by systems of nonlinear partial differential
equations (PDEs) together with appropriate initial/boundary conditions;
these model some physical phenomena. In the early days of nonlinear sci-
ence, since computers were not available, attempts were made to reduce
thesystemofPDEstoODEsbytheso-called“similaritytransformations.”
The ODEs could be solved by hand calculators. The scenario has since
changed dramatically. The nonlinear PDE systems with appropriate ini-
tial/boundary conditions can now be solved effectively by means of so-
phisticated numerical methods and computers, with due attention to the
accuracy of the solutions. The search for exact solutions is now motivated
by the desire to understand the mathematical structure of the solutions
and, hence, a deeper understanding of the physical phenomena described
by them. Analysis, computation, and, not insignificantly, intuition all pave
the way to their discovery.
The similarity solutions in earlier years were found by direct physical
and dimensional arguments. The two most famous examples are the point
explosion and implosion problems (Taylor (1950), Sedov (1959), Guderley
(1942)).Simplescalingargumentstoobtainsimilaritysolutions,illustrating
also the self-similar or invariant nature of the scaled solutions, were lucidly
given by Zel’dovich and Raizer (1967). Their work was greatly amplified
by Barenblatt (1996), who clearly explained the nature of self-similar solu-
tionsofthefirstandsecondkind.Moreimportantly,Barenblattbroughtout
manifestlytheroleofthesesolutionsasintermediateasymptotics;theseso-
lutionsdonotdescribemerelythebehaviourofphysicalsystemsundercer-
tain conditions, they also describe the intermediate asymptotic behaviour
ofsolutionsofwiderclassesofproblemsintherangeswheretheynolonger
depend on the details of the initial/boundary conditions, yet the system is
still far from being in a limiting state.
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The early investigators relied greatly upon the physics of the problem
to arrive at the similarity form of the solution and, hence, the solution
itself. This methodology underwent a severe change due to the work of
Ovsyannikov (1962), who, using both finite and infinitesimal groups of
transformations, gave an algorithmic approach to the finding of similar-
ity solutions. This approach is now readily available in a practical form
(Bluman and Kumei (1989)). A recent direct approach, not involving the
use of the groups of finite and infinitesimal transformations, may be found
even more convenient in the determination of similarity solutions; the fi-
nalresultsviaeitherapproachare,however,essentiallythesame(Clarkson
and Kruskal (1989); Hood (1995)). So the reduction to ODEs (if the PDEs
originallyinvolvedtwoindependentvariables)isaroutinematter,butthen
the ODEs have to seek their own initial/boundary conditions to be solved
and used to explain some physical phenomenon. On the other hand, given
a mathematical model, one must use both algorithmic and dimensional
approaches suitably to discover if the problem is self-similar, solve the re-
sulting ODEs subject to appropriate boundary conditions, and prove the
asymptotic character of the solution. Since, in the process of reduction to
self-similarform,thenonlinearityisfullypreserved,theself-similarsolution
provides important clues to a wider class of solutions of the original PDE.
Asamathematicalmodelismademorecomprehensivetoincludeother
effects and extend its applicability, it may lose some of its symmetries, and
the groups of infinitesimal or finite transformations to which the model is
invariant may shrink. As a result, the self-similar form may either cease to
exist or may become restricted. A simple example is the system of gasdy-
namic equations in plane geometry. As soon as the spherical or cylindrical
geometry term is included in the equation of continuity, there is a diminu-
tion in the scale invariance (Zel’dovich and Raizer (1967)). Therefore, one
mustrelinquishtheself-similarhypothesisandassumeamoregeneralform
of the solution; that is, one must go beyond self-similarity. In the gasdy-
namic context, several problems in nonplanar geometry, such as flow of a
gasintovacuumorapistonmotionleadingtostrongconvergingshock, are
solved by assuming an infinite series in one of the independent variables,
time, say, with coefficients depending on a similarity variable (Nageswara
Yogi (1995); Van Dyke and Guttman (1982)). This results in an infinite
(instead of finite) system of ODEs with appropriate boundary conditions;
the zeroth order term in the series is the (known) solution in planar ge-
ometry. The series, of course, must be shown to converge in the physically
relevant domain. The infinite system of ODEs, in a sense, reflects loss of
some symmetry and, hence, greater complexity of the solution.
Anotherwaytoovercomethelimitationsimposedbyinvariancerequire-
mentistoexactlylinearisethePDEsystemwhenpossible,orchoosea“nat-
ural” coordinate system such that the boundaries of the domain are level
lines. The linearisation process immediately gives access to the principle
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of linear superposition and, hence, the ease of solution associated with it.
Hodograph transformations for steady two-dimensional gasdynamic equa-
tionsandHopf-ColetransformationfortheBurgersequationarewell-known
examples of exact linearisation. Linearisation, of course, imposes its own
constraints,particularlywithregardtoinitialand/orboundaryconditions.
An example of natural coordinated is again from gas dynamics where the
shocktrajectoryandparticlepathsmaybechosenaspreferredcoordinates.
The transformedsystem isnonlinear, but has its owninvariance properties
leading to new classes of exact solutions of the original system of PDEs
(Sachdev and Reddy (1982)).
There is yet another way of extending the class of similarity solutions.
This is to embed the similarity solutions, suitably expanded, in a larger
family; this family is obtained by varying the constants and introducing
an infinite number of unknown functions into the expanded form of the
similarity solution. These functions are then determined by substituting
the assumed form of the solution into the PDEs and, hence, solving the
resulting (infinite) system of ODEs appropriately. Thus, the similarity
solution becomes a special (embedded) case of the larger family. What is
the role and significance of the extended family of solutions must of course
becarefullyexamined(Sachdev,Gupta,andAhluwalia(1992);Sachdevand
Mayil Vaganan (1993)). This embedding is analogous to that for nonlinear
ODEs(see,forexample,Hille(1970)andBenderandOrszag(1978)forthe
solution of Thomas-Fermi equation).
Exactasymptoticsolutionscanalsobebuiltupfromthe(known)linear
solutions(Whitham(1974)). Theschemeorformofthenonlinearsolutions
is chosen such that they extend far back (in time, say) the validity of the
linearasymptoticsolution. Forexample,forgeneralisedBurgersequations,
the exact solution of the planar Burgers equation for N wave neatly moti-
vates the form of the solution for the former (Sachdev and Joseph (1994)).
In exceptional circumstances, a “composite” solution may be written out
whichspanstheinfinitelylongevolutionoftheNwave, barringafiniteini-
tial interval during which the initial (usually discontinuous) profile loosens
its gradients (Sachdev, Joseph, and Nair (1994)).
The activist approach to nonlinear ODEs (Bender and Orszag (1978);
Sachdev (1991)) suggests how one may build up large time approximate
solutions of nonlinear PDEs by a balancing argument. For this purpose,
oneintroducessomepreferredvariables,thesimilarityvariableandtimefor
instance, into the PDE and looks for possible solutions of truncated PDE
made up of terms which balance in one of the independence variables. The
simpler PDE thus obtained is usually more amenable to analysis than the
originalequation. Theapproximatesolutionsodeterminedcanbeimproved
bytakingintoaccounttheneglectedlowerorderterms. Usually,afewterms
inthisanalysisgiveagooddescriptionoftheasymptoticsolution(Grundy,
Sachdev, and Dawson (1994); Dawson, Van Duijn, and Grundy (1996)).
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Wemayrevertandsaythatwheneversimilaritysolutionsexist,their
existencetheorygreatlyassistsintheunderstandingoftheoriginalPDE
system.Thesesolutionsalsohelpinthequantitativeestimationofhow
thesolutionsofcertainclassesofinitial/boundaryvalueproblemsevolvein
time(Sachdev(1987)).
Theroleofnumericalsolutionofnonlinearproblemsindiscoveringthe
analyticstructureofthesolutionneedhardlybeemphasised;veryoften
thenumericalsolutionthrowsmuchlightonwhatkindofanalyticform
onemustexplore.Besides,understandingthevalidityandplaceofex-
act/approximateanalyticsolutioninthegeneralcontextcanbegreatly
enhancedbythenumericalsolution.Inshort,theremustbe acontinu-
ousinterplayofanalysisandcomputationifanonlinearproblemistobe
successfullytackled.
Theapproachesoutlinedintheabovegobeyondself-similarity,but
theexactsolutionstheyyieldarestillgenerallyasymptoticinnature;these
solutions,perse,satisfysomespecial(singular)initialconditionsbutevolve
tobecomeintermediateasymptoticstowhichsolutionsofacertainlarger
butrestrictedclassofinitial/boundaryvalueproblemstendastimegoes
toinfinity(Sachdev(1987)).
Chapter2dealswithfirst-orderPDEs,illustratingwiththehelpof
manyexamplestheplaceofsimilaritysolutionsinthegeneralsolution.
Exactsimilaritysolutionsviagrouptheoreticmethodsandthedirectsim-
ilarityapproachofClarksonandKruskal(1989)arediscussedinChapter
3,whiletravellingwavesolutionsaretreatedinChapter4.Exactlineari-
sationofnonlinearPDEs,includingviahodographmethods,isdealtwith
inChapter5.InChapter6,constructionofmoregeneralsolutionsfrom
specialsolutionsofagivenorarelatedproblemisaccomplishedvianonlin-
earisationorembeddingmethods.Chapter7usesthebalancingargument
for nonlinear PDEs to find approximate solutions of nonlinear problems.
The concluding chapter expounds series solutions for nonlinear PDEs with
the help of several examples; the series are constructed in one of the inde-
pendent variables, often the time, with the coefficients depending on the
other independent variable.
The approach in the present monograph is entirely constructive in na-
ture; there is very little by way of abstract analysis. The analytic and
numerical solutions are often treated alongside. Most examples are drawn
fromrealphysicalsituations,mainlyfromfluidmechanicsandnonlineardif-
fusion. Theideaistoillustrateandbringoutthemainpoints. Tohighlight
the goals of the present book we could do no better than quote from the
lastchapteronexactsolutionsinthebookbyWhitham(1974),“Doubtless
much more of value will be discovered, and the different approaches have
added enormously to the arsenal of ‘mathematical methods.’ Not least is
the lesson that exact solutions are still around and one should not always
turn too quickly to a search for the (cid:15).”
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Chapter 2
First-Order Partial
Differential Equations
2.1 Linear Partial Differential Equations of
First Order
Themostgeneralfirst-orderlinearPDEintwoindependentvariablesxand
t has the form
au +bu =cu+d (2.1.1)
x t
where a,b,c,d are functions of x and t only. We single out the variable t
(often “time” in physical problems) and write the first-order general PDE
in the “normal” form
u +F(x,t,u,u )=0.
t x
Thegeneralsolutionofafirst-orderPDEinvolvesanarbitraryfunction. In
applications one is usually interested not in obtaining the general solution
of a PDE, but a solution subject to some additional condition such as an
initial condition (IC) or a boundary condition (BC) or both.
A basic problem for first-order PDEs is to solve
u +F(x,t,u,u )=0,x∈R,t>0 (2.1.2)
t x
subject to the IC
u(x,0)=u (x),x∈R (2.1.3)
0
where u (x) is a given function. (The interval of interest for x may be
0
finite.) This is called a Cauchy problem; it is a pure initial value problem.
It may be viewed as a signal or wave at time t = 0. The initial signal
or wave is a space distribution of u, and a “picture” of the wave may be
obtainedbydrawingthegraphofu=u (x)inthexu-space. ThenthePDE
0
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