Difference Matrices for ODE and PDE. A MATLAB Companion

212 Pages·2023·5.252 MB·English

by  John M. Neuberger

John M. Neuberger
Difference Matrices
for ODE and PDE
A MATLAB® Companion
Difference Matrices for ODE and PDE
John M. Neuberger
Difference Matrices
for ODE and PDE
®
A MATLAB Companion
123
JohnM.Neuberger
Department ofMathematics andStatistics
Northern ArizonaUniversity
Flagstaff, AZ,USA
ISBN978-3-031-11999-6 ISBN978-3-031-12000-8 (eBook)
https://doi.org/10.1007/978-3-031-12000-8
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Preface
This text will be useful for the four different audiences listed below. It is expected
that all readers will have knowledge of basic calculus, linear algebra, and ordinary
differential equations, and that the successful student will either already know
elementary partial differential equations, or be concurrently learning that subject.
ThematerialisintendedtobeaccessibletothosewithoutexpertiseinMATLAB®,
although a little prior experience with programming is probably required.
1. This text serves as a supplement for the student in an introductory partial dif-
ferentialequationscourse.Aselectionoftheincludedexercisescanbeassigned
asprojectsthroughoutthesemester.Throughtheuseofthistext,thestudentwill
develop the skills to run simulations corresponding to the primarily theoretical
course material covered by the instructor.
2. These notes work well as a standalone graduate course text in introductory
scientific computing for partial differential equations. With prerequisite
knowledge of ordinary and partial differential equations and elementary
numerical analysis, most of the material can be covered and many of the
exercisesassignedinaone-semestercourse.Someoftheharderexercisesmake
substantial projects, and relate to topics from the other graduate mathematics
coursesgraduatestudentstypicallytake,e.g.,differentialequationsandtopicsin
nonlinear functional analysis.
3. Established researchers in theoretical partial differential equations may find
thesenotesusefulaswell,particularlyasanintroductoryguidefortheirresearch
students. Those unfamiliar with MATLAB can use the included material as a
reference in quickly developing their own applications in that language.
A mathematician who is new to the practical implementation of methods for
scientific computation in general can with relative ease, by working through a
selection of exercises, learn how to implement and execute numerical simula-
tionsofdifferentialequationsinMATLAB.Thesenotescanserveasapractical
guide in independent study, undergraduate or graduate research experiences, or
for help simulating solutions to specific thesis- or dissertation-related
v
vi Preface
experiments. The author hopes that the ease and brevity with which the notes
provide solutions to fairly significant problems will serve as inspiration.
4. The text can serve as a supplement for the instructor teaching any course in
differentialequations.Manyoftheexamplescanbeeasilyimplementedandthe
resulting simulation demonstrated by the instructor. If the course has a
numerical component, a few exercises of suitable difficulty can be assigned as
student projects. Practical assistance in implementing algorithms in MATLAB
can be found in this text.
Scientist and engineers have valid motivations to become proficient at imple-
menting numerical algorithms for solving PDE. The text’s emphasis on enforcing
boundary conditions, eigenfunctions, and general regions will be useful as an
introductiontotheiradvancedapplications.Forthemathematician,accomplishedor
student,amorepowerfulbenefitcanbethetangible,visualrealizationoftheobjects
of calculus, differential equations, and linear algebra. The high-level programs are
developed by the reader from earlier programs and short fragments of relatable
code. The resulting simulations are demonstrations of the properties of the under-
lying mathematical objects, where vectors represent functions, matrix operations
represent differentiation and integration, and calculations such as solving linear
systemsorfindingeigenvaluesareeasilyaccomplishedwithonelineofcode.Even
withoutmuchpriorknowledgeofprogrammingorMATLAB,byworkingthrough
a selection of exercises in this text, the reader will be able to create working
programs that simulate many of the classic problems from PDE, while gaining an
understanding of the underlying fundamental mathematical principles.
The approach of the text is to first review MATLAB and a small selection of
techniques from elementary numerical analysis, and then introduce difference
matricesinthecontextofordinarydifferentialequations.Wethenapplytheseideas
to PDE, including topics from the heat, wave, and Laplace equations, eigenvalue
problems, and semilinear boundary value problems. We enforce boundary condi-
tions on regions including the interval, square, disk, and cube, and more general
domains. We push the general notion that linear problems can be expressed as a
single linear system, while many nonlinear problems can be solved via Newton’s
method.
Flagstaff, USA John M. Neuberger
Acknowledgements
I would first and foremost like to acknowledge my father J. W. Neuberger. He
taught me much of what I know about differential equations and the numerical
solution of them. His belief in my pedagogical approach and the practical useful-
ness of my course notes was my main motivation in publishing them.
IalsowouldliketothankmyPh.D.advisorAlfonsoCastrowhointroducedme
to nonlinear functional analysis, particularly as it pertains to semilinear elliptic
boundaryvalueproblems.Chapter5contentofthistextreflectsasmallsliceofmy
lifetime pursuits to understand and computationally simulate the equations of that
subject.
Ihavebeenextremelyluckyinhavingawonderfullysupportivefamily.Mywife
Dina has been the main source of inspiration over my career as a professor of
mathematics. My son Nick has an interest in mathematics as well, and has been
insistent that I refine the notes so that others could use them.
I would also like to thank my 20-plus years of undergraduate and graduate
partial differential equations and numerical analysis students. Their feedback was
invaluableintheprocessofdevelopingandrefiningmynotes.Themuch-improved
textcontainsnumerousclarificationsandhintsasadirectresultoftheirsuggestions,
and the exercises are more robust and complete, thanks to their countless hours of
coding and grinding out project reports. I learned a few MATLAB® tricks from
them too. A special thanks goes to my former and current students Tyler Diggans,
Ryan Kelly, and Ian Williams, who each took a turn at proofreading, and made
some valuable editorial suggestions from the student perspective.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Summary of the Differential Equations We Will Consider. . . . 1
1.2 The Use of MATLAB® and the Student Exercises . . . . . . . . . . . 2
1.2.1 Using MATLAB®’s Debugger . . . . . . . . . . . . . . . . . . . 3
1.2.2 Line Numbering in MATLAB® Examples. . . . . . . . . . . 4
1.2.3 Reproducing Codes and Exercises. . . . . . . . . . . . . . . . . 4
1.2.4 Guidelines to the Homework Exercises . . . . . . . . . . . . . 5
1.3 The Organization of This Text . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Review of Elementary Numerical Methods and MATLAB®. . . . . . . 11
2.1 Introduction to Basic MATLAB® at the Command Line. . . . . . . 12
2.1.1 MATLAB®:sum,prod,max,min,abs,norm,linspace,
for loop, eigs, and sort . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Runge–Kutta Method for Initial Value Problems . . . . . . . . . . . . 17
2.2.1 The Shooting Method for ODE BVP—an IVP
Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Comparison of Approximate Solutions to Exact
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 MATLAB®: Ones, Zeros, the ‘:’ Iterator, the @
Syntax for Defining Inline Functions, Subfunctions . . . . 23
2.2.4 Rows Versus Columns Part I, Diagnosing Dimension
and Size Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Numerical Differentiation and Integration. . . . . . . . . . . . . . . . . . 27
2.3.1 Higher Order Differences and Symbolic Computation. . . 32
2.3.2 MATLAB®: kron, spdiags, ‘backslash’ (mldivide), tic,
and toc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Newton’s Method for Vector Fields. . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 MATLAB®: if, else, while, fprintf, meshgrid, surf,
reshape, find, single indexing, and rows versus
columns Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ix
x Contents
2.5 Cubic Spline Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Making Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Theory: ODE, Systems, Newton’s Method, IVP Solvers,
and Difference Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6.1 Some ODE Theory and Techniques . . . . . . . . . . . . . . . 48
2.6.2 Convergence and Order of Newton’s Method . . . . . . . . 52
2.6.3 First-Order IVP Numerical Solvers: Euler’s
and Runge–Kutta’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.4 Difference Formulas and Orders of Approximation. . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Second-Order Semilinear Elliptic Boundary Value Problems. . . . 64
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Linear Ordinary Second-Order BVP. . . . . . . . . . . . . . . . . . . . . . 69
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Eigenvalues of (cid:1)D and Fourier Series . . . . . . . . . . . . . . . . . . . 71
2
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 Enforcing Zero Dirichlet, Zero Neumann, and Periodic
Boundary Conditions Using Either Point or Cell Grid.... . . . . . . 77
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5 First-Order Linear Solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6 Systems of First-Order Linear Equations
for Second-Order IVP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.7 First-Order Nonlinear IVP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.8 A Practical Guide to Fourier Series . . . . . . . . . . . . . . . . . . . . . . 89
4 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1 The Laplacian on the Unit Square . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Creating the Sparse Laplacian Matrix D and Eigenvalues . . . . . 95
2
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Semilinear Elliptic BVP on the Square. . . . . . . . . . . . . . . . . . . . 100
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Laplace’s Equation on the Square . . . . . . . . . . . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5.1 Explicit Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5.2 Implicit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5.3 Explicit–Implicit Method . . . . . . . . . . . . . . . . . . . . . . . 111
Contents xi
4.5.4 The Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.5 Fourier Expansion with Numerical Integration . . . . . . . . 113
4.5.6 Block Matrix Systems . . . . . . . . . . . . . . . . . . . . . . . . . 117
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.6.1 The Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.2 A Good Explicit Method . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.3 Block Matrix Systems and D’Alembert Matrices . . . . . . 126
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.7 Tricomi’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.8 General Regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.8.1 The Laplacian on the Cube. . . . . . . . . . . . . . . . . . . . . . 140
4.8.2 The Laplacian on the Disk . . . . . . . . . . . . . . . . . . . . . . 141
4.8.3 Accurate Eigenvalues of the Laplacian on Disk,
Annulus, and Sections . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.8.4 The Laplace–Beltrami Operator on a Spherical
Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.8.5 A General Region Code . . . . . . . . . . . . . . . . . . . . . . . . 151
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.9 First-Order PDE and the Method of Characteristics. . . . . . . . . . . 161
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.10 Theory: Separation of Variables for PDE on Rectangular
and Polar Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.10.1 Eigenfunctions of the Laplacian . . . . . . . . . . . . . . . . . . 164
4.10.2 Laplace’s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.10.3 The Heat Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.10.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5 Advanced Topics in Semilinear Elliptic BVP . . . . . . . . . . . . . . . . . . 181
5.1 Branch Following and Bifurcation Detection . . . . . . . . . . . . . . . 181
5.1.1 The Tangent Newton Method for Branch Following. . . . 182
5.1.2 The Secant Method for Bifurcation Detection . . . . . . . . 183
5.1.3 Secondary Bifurcations and Branch Switching . . . . . . . . 186
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.2 Mountain Pass and Modified Mountain Pass Algorithms
for Semilinear BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2.1 The MPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2.2 The MMPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.3 The p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
References.... .... .... .... ..... .... .... .... .... .... ..... .... 203