Applied Partial Differential Equations

294 Pages·2015·4.398 MB·English

by  J. David Logan (auth.)

Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
Series editors
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San Francisco State University, San Francisco, CA, USA
Kenneth Ribet
University of California, Berkeley, CA, USA
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Fadil Santosa, University of Minnesota, Minneapolis, MN, USA
Amie Wilkinson, University of Chicago, Chicago, IL, USA
Undergraduate Texts in Mathematics are generally aimed at third- and
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featureexamplesthatillustratekeyconceptsaswellasexercisesthatstrengthen
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For further volumes:
http://www.springer.com/series/666
J. David Logan
Applied Partial Differential
Equations
J. David Logan
Department of Mathematics
University of Nebraska-Lincoln
Lincoln, NE, USA
ISSN 0172-6056 ISSN 2197-5604 (electronic)
UndergraduateTextsinMathematics
ISBN 978-3-319-12492-6 ISBN 978-3-319-12493-3 (eBook)
DOI 10.1007/978-3-319-12493-3
Springer Cham Heidelberg New York Dordrecht London
LibraryofCongressControlNumber:2014955188
MathematicsSubjectClassification:34-01,00-01,00A69,97M50,97M60
(cid:2)c SpringerInternationalPublishingSwitzerland2015
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To Aaron, Rachel, and David
Contents
Preface to the Third Edition.................................... ix
To Students .................................................... xi
1. The Physical Origins of Partial Differential Equations ...... 1
1.1 PDE Models............................................. 2
1.2 Conservation Laws ....................................... 12
1.3 Diffusion ................................................ 29
1.4 Diffusion and Randomness................................. 38
1.5 Vibrations and Acoustics .................................. 49
1.6 Quantum Mechanics* ..................................... 57
1.7 Heat Conduction in Higher Dimensions...................... 60
1.8 Laplace’s Equation ....................................... 66
1.9 Classification of PDEs .................................... 72
2. Partial Differential Equations on Unbounded Domains...... 79
2.1 Cauchy Problem for the Heat Equation ..................... 79
2.2 Cauchy Problem for the Wave Equation ..................... 87
2.3 Well-Posed Problems...................................... 92
2.4 Semi-Infinite Domains..................................... 96
2.5 Sources and Duhamel’s Principle ...........................101
2.6 Laplace Transforms.......................................106
2.7 Fourier Transforms .......................................117
3. Orthogonal Expansions .....................................127
3.1 The Fourier Method ......................................127
3.2 OrthogonalExpansions....................................131
viii Contents
3.3 Classical Fourier Series....................................145
4. Partial Differential Equations on Bounded Domains ........155
4.1 Overview of Separation of Variables.........................156
4.2 Sturm–Liouville Problems .................................167
4.3 Generalization and Singular Problems.......................180
4.4 Laplace’s Equation .......................................186
4.5 Cooling of a Sphere.......................................198
4.6 Diffusion in a Disk........................................202
4.7 Sources on Bounded Domains ..............................207
4.8 Poisson’s Equation* ......................................216
5. Applications in the Life Sciences............................229
5.1 Age-Structured Models....................................229
5.2 Traveling Waves Fronts ...................................238
5.3 Equilibria and Stability ...................................245
6. Numerical Computation of Solutions .......................257
6.1 Finite Difference Approximations...........................258
6.2 Explicit Scheme for the Heat Equation ......................260
6.3 Laplace’s Equation .......................................268
6.4 Implicit Scheme for the Heat Equation ......................273
Appendix A. Differential Equations.............................279
References......................................................285
Index...........................................................287
Preface to the Third Edition
The goal of this new edition is the same as that for the original, namely, to
presentaone-semestertreatmentofthebasicideasencounteredinpartialdiffer-
entialequations(PDEs).Thetextisdesignedfora3-creditsemestercoursefor
undergraduate students in mathematics, science, and engineering. The prereq-
uisites are calculus and ordinary differential equations. The text is intimately
tied to applications in heat conduction, wave motion, biological systems, and
a variety other topics in pure and applied science. Therefore, students should
have some interest, or experience, in basic science or engineering.
Themainpartofthetextisthefirstfourchapters,whichcovertheessential
concepts. Specifically, they treat first- and second-orderequations on bounded
andunboundeddomainsandincludetransformmethods(LaplaceandFourier),
characteristicmethods,andeigenfunctionexpansions(separationofvariables);
there is considerable material on the origin of PDEs in the natural sciences
and engineering.Two additional chapters,Chapter 5 and Chapter 6, are short
introductionstoapplicationsofPDEsinbiologyandtonumericalcomputation
ofsolutions.Thetextoffersflexibilitytoinstructorswho,forexample,maywant
toinserttopicsfrombiologyornumericalmethodsatanytimeinthecourse.A
briefappendixreviewstechniquesfromordinarydifferentialequations.Sections
marked with an asterisk (*) may safely be omitted. The mathematical ideas
arestronglymotivatedbyphysicalproblems,andtheexpositionispresentedin
a concise style accessible to students in science and engineering. The emphasis
is on motivation, methods, concepts, and interpretation rather than formal
theory.
The level of exposition is slightly higher than students encounter in the
post-calculus differential equations course. The philosophy is that a student
should progress in the ability to read mathematics. Elementary texts contain
x Preface to theThird Edition
many examples and detailed calculations, but advanced mathematics and sci-
ence books leave a lot to the reader. This text leaves some of the easy details
to the reader. Often, the arguments are derivations in lieu of carefully con-
structed proofs. The exercises are at varying levels and encourage students to
thinkabouttheconceptsandderivationsratherthanjustgrindoutlotsofrou-
tine solutions. A student who reads this book carefully and who solves many
of the exercises will have a sound knowledge base to continue with a second-
year partial differential equations course where careful proofs are constructed
or with upper-division courses in science and engineering where detailed, and
often difficult, applications of partial differential equations are introduced.
This third edition, a substantial revision, contains many new and revised
exercises, and some sections have been greatly expanded with more worked
examplesandadditionalexplanatorymaterial.Anew,lessdense,formatmakes
key results more apparentand the text easier to read for undergraduates.The
result is a text one-third longer. But the size and brevity of text, contrary to
voluminous other texts, struck a chord with many users and that has been
maintained. Many users provided suggestions that have become part of this
revision, and I greatly appreciate their interest and comments.
Elizabeth Loew, my editor at Springer, deserves special recognitionfor her
continuous and expert support. I have found Springer to be an extraordinary
partner in this project.
Finally,thisbookisveryaffectionatelydedicatedtomytwosonsanddaugh-
ter, Aaron, David, and Rachel, who have often been my teachers with their
challenginganduniqueperspectivesonlife.Forthesegifts Igreatlythankyou.
I welcome suggestions, comments, and corrections. Contact information is
onmywebsite:http://www.math.unl.edu/~jlogan1,whereadditionalitems
can be found. Solutions to some of the exercises can be found on the Springer
web site.
J. David Logan
Willa Cather Professor
Lincoln, Nebraska