Essential Partial Differential Equations: Analytical and Computational Aspects

370 Pages·2015·6.95 MB·English

by  David F. Griffiths

Springer Undergraduate Mathematics Series
David F. Griffiths
John W. Dold
David J. Silvester
Essential
Partial
Differential
Equations
Analytical and Computational Aspects
Springer Undergraduate Mathematics Series
Advisory Board
M.A.J. Chaplain, University of Dundee, Dundee, Scotland, UK
K. Erdmann, University of Oxford, Oxford, England, UK
A. MacIntyre, Queen Mary University of London, London, England, UK
E. Süli, University of Oxford, Oxford, England, UK
M.R. Tehranchi, University of Cambridge, Cambridge, England, UK
J.F. Toland, University of Cambridge, Cambridge, England, UK
More information about this series at http://www.springer.com/series/3423
fi
David F. Grif ths John W. Dold
(cid:129)
David J. Silvester
Essential Partial Differential
Equations
Analytical and Computational Aspects
123
DavidF. Griffiths DavidJ. Silvester
University of Dundee Schoolof Mathematics
Fife TheUniversity of Manchester
UK Manchester
UK
JohnW.Dold
Schoolof Mathematics
TheUniversity of Manchester
Manchester
UK
ISSN 1615-2085 ISSN 2197-4144 (electronic)
SpringerUndergraduate MathematicsSeries
ISBN978-3-319-22568-5 ISBN978-3-319-22569-2 (eBook)
DOI 10.1007/978-3-319-22569-2
LibraryofCongressControlNumber:2015948777
Mathematics Subject Classification (2010): 35-01, 65L10, 65L12, 65L20, 65M06, 65M12, 65N06,
65N12
SpringerChamHeidelbergNewYorkDordrechtLondon
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In mathematics you don’t understand things.
You just get used to them.
John von Neumann
“Begin at the beginning”, the King said, very
gravely, “and go on till you come to the end:
then stop”.
Lewis Carroll, Alice in Wonderland
Preface
Thistextbookisaself-containedintroductiontothemathematicalaspectsofpartial
differential equations. The book is aimed at undergraduate students studying for a
mathematics degree. Selected chapters could also be of interest to engineers and
scientists looking to develop an understanding of mathematical modelling and
numerical analysis.
The material is organised into 13 chapters, with roughly equal emphasis placed
on analytical and numerical solution techniques. The first four chapters provide a
foundation for the study of partial differential equations. These chapters cover
physical derivation, classification, and well-posedness. Classical solution tech-
niques are discussed in Chaps. 8 and 9. Computational approximation aspects are
developed in Chaps. 6 and 10–12. A clear indication is given in each of these
chapters of where the basic material (suitable perhaps for a first course) ends and
where we begin to probe more challenging areas that are of both a practical and
theoretical interest. The final chapter defines a suite of projects, involving both
theory and computation, that are intended to extend and test understanding of the
material in earlier chapters.
Other than the final chapter, the book does not include programming exercises.
We believe that this strategy is in keeping with the aims and objectives
of the SUMS series. The availability of software environments like MATLAB
(www.mathworks.com), Maple (www.maplesoft.com) and Mathematica
(www.wolfram.com) means that there is little incentive for students to write
low-level computer code. Nevertheless, we would encourage readers who are
ambitioustotrytoreproducethecomputationalresultsinthebookusingwhatever
computational tools that they have available.
Most chapters conclude with an extensive set of exercises (almost 300 in all).
These vary in difficulty so, as a guide, the more straightforward are indicated byI
while those at the more challenging end of the spectrum are indicated byH. Full
solutions to all the exercises as well as the MATLAB functions that were used to
vii
viii Preface
generate the figures will be available to authorised instructors through the book’s
website (www.springer.com). Others will be able to gain access to the solu-
tions to odd-numbered exercises though the same web site.
Distinctive features of the book include the following.
1. The level of rigour is carefully limited—it is appropriate for second-year
mathematicsundergraduatesstudyingintheUK(perhapsthirdorfourthyearin
the USA). The ordering of topics is logical and new concepts are illustrated by
worked examples throughout.
2. Analyticalandnumericalmethodsofsolutionarecloselylinked,settingbothon
an equal footing. We (the authors) take a contemporary view of scientific
computing and believe in mixing rigorous mathematical analysis with informal
computational examples.
3. Thetextiswritteninalivelyandcoherentstyle.Almostallofthecontentofthe
book is motivated by numerical experimentation. Working in the “computa-
tionallaboratory”iswhatultimatelydrivesourresearchandmakesourscientific
lives such fun.
4. Thebookopensthedoortoawiderangeoffurtherareasofstudyinbothapplied
mathematics and numerical analysis.
Thematerialcontainedinthefirstninechaptersreliesonlyonfirst-yearcalculus
andcouldbetaughtasaconventional“introductiontopartialdifferentialequations”
module in the second year of study. Advanced undergraduate level courses in
mathematics, computing or engineering departments could be based on any com-
bination of the early chapters. The material in the final four chapters is more
specialisedand,wouldalmostcertainlybetaughtseparatelyasanadvancedoption
(fourth-yearorMSc)entitled“numericalmethodsforpartialdifferentialequations”.
Our personal view is that numerical approximation aspects are central to the
understandingofpropertiesofpartialdifferentialequations,andourhopeisthatthe
entire contents of the book might be taught in an integrated fashion. This would
mostprobablybeadouble-semester(44hours)second-(orthird-)yearmoduleina
UKuniversity.Havingcompletedsuchanintegrated(core)course,studentswould
beperfectlypreparedforaspecialistappliedmathematicsoption,sayincontinuum
mechanicsorelectromagnetism,orforadvancednumericalanalysisoptions,sayon
finite element approximation techniques.
We should like to extend our thanks to Catherine Powell, Alison Durham, Des
Higham and our colleagues at Manchester and Dundee, not to mention the many
students who have trialled the material over many years, for their careful reading
andfrank opinionsofearlier drafts ofthebook. Itisalso apleasure tothank Joerg
Sixt and his team at Springer UK.
May 2015 David F. Griffiths
John W. Dold
David J. Silvester
Contents
1 Setting the Scene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Some Classical PDEs … . . . . . . . . . . . . . . . . . . . . . . . . . . ..
2
1.2 … and Some Classical Solutions. . . . . . . . . . . . . . . . . . . . . ..
4
2 Boundary and Initial Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1 Operator Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
14
2.2 Classification of Boundary Value Problems . . . . . . . . . . . . . .. 16
2.2.1 Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Nonlinear Problems. . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2.3 Well-Posed Problems. . . . . . . . . . . . . . . . . . . . . . . . .
21
3 The Origin of PDEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.1 Newton’s Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
27
3.1.1 The Wave Equation for a String . . . . . . . . . . . . . . . . .
27
3.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
3.2.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Laplace’s Equation and the Poisson Equation. . . . . . . .
31
3.2.3 The Wave Equation in Water. . . . . . . . . . . . . . . . . . .
32
3.2.4 Burgers’ Equation. . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.1 Characteristics of First-Order PDEs. . . . . . . . . . . . . . . . . . . ..
38
4.2 Characteristics of Second-Order PDES. . . . . . . . . . . . . . . . . .. 44
4.2.1 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 Parabolic Equations. . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2.3 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3 Characteristics of Higher-Order PDEs . . . . . . . . . . . . . . . . . ..
52
4.4 Postscript. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
53
5 Boundary Value Problems in R1 . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.1 Qualitative Behaviour of Solutions . . . . . . . . . . . . . . . . . . . ..
61
5.2 Comparison Principles and Well-Posedness . . . . . . . . . . . . . ..
64
ix
x Contents
5.3 Inner Products and Orthogonality . . . . . . . . . . . . . . . . . . . . ..
68
5.3.1 Self-adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . .
71
5.3.2 A Clever Change of Variables . . . . . . . . . . . . . . . . . .
72
5.4 Some Important Eigenvalue Problems . . . . . . . . . . . . . . . . . ..
74
6 Finite Difference Methods in R1 . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6.1 The Approximation of Derivatives. . . . . . . . . . . . . . . . . . . . ..
86
6.2 Approximation of Boundary Value Problems . . . . . . . . . . . . ..
90
6.3 Convergence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
95
6.4 Advanced Topics and Extensions . . . . . . . . . . . . . . . . . . . . .. 103
6.4.1 Boundary Conditions with Derivatives. . . . . . . . . . . . . 103
6.4.2 A Fourth-Order Finite Difference Method. . . . . . . . . . .
109
7 Maximum Principles and Energy Methods . . . . . . . . . . . . . . . . . .
119
7.1 Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
119
7.2 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
124
8 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.1 The Heat Equation Revisited . . . . . . . . . . . . . . . . . . . . . . . .. 129
8.1.1 Extension to Other Parabolic Problems .... . . . . . . . . . .
136
8.1.2 ... and to Inhomogeneous Data . . . . . . . . . . . . . . . . . .
140
8.2 The Wave Equation Revisited. . . . . . . . . . . . . . . . . . . . . . . ..
143
8.3 Laplace’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
148
9 The Method of Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . .
161
9.1 First-Order Systems of PDEs . . . . . . . . . . . . . . . . . . . . . . . .. 161
9.2 Second-Order Hyperbolic PDEs . . . . . . . . . . . . . . . . . . . . . .. 166
9.3 First-Order Nonlinear PDES. . . . . . . . . . . . . . . . . . . . . . . . ..
171
9.3.1 Characteristics of Burgers’ Equation . . . . . . . . . . . . . .
174
9.3.2 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
9.3.3 Riemann Problems and Expansion Fans. . . . . . . . . . . .
184
10 Finite Difference Methods for Elliptic PDEs . . . . . . . . . . . . . . . . .
195
10.1 A Dirichlet Problem in a Square Domain. . . . . . . . . . . . . . . .. 196
10.1.1 Linear Algebra Aspects . . . . . . . . . . . . . . . . . . . . . . . 198
10.1.2 Convergence Theory . . . . . . . . . . . . . . . . . . . . . . . . .
202
10.1.3 Improving the Solution Accuracy . . . . . . . . . . . . . . . .
204
10.2 Advanced Topics and Extensions . . . . . . . . . . . . . . . . . . . . ..
207
10.2.1 Neumann and Robin Boundary Conditions. . . . . . . . . .
207
10.2.2 Non-rectangular Domains. . . . . . . . . . . . . . . . . . . . . .
211
10.2.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . .
217
10.2.4 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . . . . 221
10.2.5 Anisotropic Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . 223
10.2.6 Advection–Diffusion . . . . . . . . . . . . . . . . . . . . . . . . .
226