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Partial Differential EquationsVictor IvriiDepartment of Mathematics,University of Toronto by Victor Ivrii, 2021,Toronto, Ontario, Canada

ContentsContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Introduction1.1 PDE motivations and context . . . .1.2 Initial and boundary value problems1.3 Classification of equations . . . . . .1.4 Origin of some equations . . . . . . .Problems to Chapter 1 . . . . . .iv.117913182 1-Dimensional Waves2.1 First order PDEs . . . . . . . . . . . . . . . . . . . . .Derivation of a PDE describing traffic flow . . . . .Problems to Section 2.1 . . . . . . . . . . . . . . .2.2 Multidimensional equations . . . . . . . . . . . . . . .Problems to Section 2.2 . . . . . . . . . . . . . . .2.3 Homogeneous 1D wave equation . . . . . . . . . . . . .Problems to Section 2.3 . . . . . . . . . . . . . . .2.4 1D-Wave equation reloaded: characteristic coordinatesProblems to Section 2.4 . . . . . . . . . . . . . . .2.5 Wave equation reloaded (continued) . . . . . . . . . . .2.6 1D Wave equation: IBVP . . . . . . . . . . . . . . . .Problems to Section 2.6 . . . . . . . . . . . . . . .2.7 Energy integral . . . . . . . . . . . . . . . . . . . . . .Problems to Section 2.7 . . . . . . . . . . . . . . .2.8 Hyperbolic first order systems with one spatial variableProblems to Section 2.8 . . . . . . . . . . . . . . .2020262932353638444951587478818588i.

Contents3 Heat equation in 1D3.1 Heat equation . . . . . . . . .3.2 Heat equation (miscellaneous)3.A Project: Walk problem . . . .Problems to Chapter 3 . .ii.90. 90. 97. 105. 1074 Separation of Variables and Fourier Series4.1 Separation of variables (the first blood) . . . . . . . . .4.2 Eigenvalue problems . . . . . . . . . . . . . . . . . . .Problems to Sections 4.1 and 4.2 . . . . . . . . . . . .4.3 Orthogonal systems . . . . . . . . . . . . . . . . . . . .4.4 Orthogonal systems and Fourier series . . . . . . . . .4.5 Other Fourier series . . . . . . . . . . . . . . . . . . . .Problems to Sections 4.3–4.5 . . . . . . . . . . . . . .Appendix 4.A. Negative eigenvalues in Robin problemAppendix 4.B. Multidimensional Fourier series . . . .Appendix 4.C. Harmonic oscillator . . . . . . . . . .1141141181261301371441501541571605 Fourier transform1635.1 Fourier transform, Fourier integral . . . . . . . . . . . . . . . 163Appendix 5.1.A. Justification . . . . . . . . . . . . . . . 167Appendix 5.1.A. Discussion: pointwise convergence ofFourier integrals and series . . . . . . . . . . . . . . . . . . . 1695.2 Properties of Fourier transform . . . . . . . . . . . . . . . . 171Appendix 5.2.A. Multidimensional Fourier transform,Fourier integral . . . . . . . . . . . . . . . . . . . . . . . . . 175Appendix 5.2.B. Fourier transform in the complex domain176Appendix 5.2.C. Discrete Fourier transform . . . . . . . 179Problems to Sections 5.1 and 5.2 . . . . . . . . . . . . . 1805.3 Applications of Fourier transform to PDEs . . . . . . . . . . 183Problems to Section 5.3 . . . . . . . . . . . . . . . . . . 1906 Separation of variables6.1 Separation of variables for heat equation . . . .6.2 Separation of variables: miscellaneous equations6.3 Laplace operator in different coordinates . . . .6.4 Laplace operator in the disk . . . . . . . . . . .6.5 Laplace operator in the disk. II . . . . . . . . .195195199205212216

Contents6.6iiiMultidimensional equations . . . . . . . . . . . . . . . . . . 221Appendix 6.A. Linear second order ODEs . . . . . . . . . . 224Problems to Chapter 6 . . . . . . . . . . . . . . . . . . . . 2277 Laplace equation7.1 General properties of Laplace equation7.2 Potential theory and around . . . . . .7.3 Green’s function . . . . . . . . . . . . .Problems to Chapter 7 . . . . . . . .2312312332402458 Separation of variables2518.1 Separation of variables in spherical coordinates . . . . . . . . 2518.2 Separation of variables in polar and cylindrical coordinates . 256Separation of variable in elliptic and parabolic coordinates258Problems to Chapter 8 . . . . . . . . . . . . . . . . . . . . 2609 Wave equation2639.1 Wave equation in dimensions 3 and 2 . . . . . . . . . . . . . 2639.2 Wave equation: energy method . . . . . . . . . . . . . . . . 271Problems to Chapter 9 . . . . . . . . . . . . . . . . . . . . 27510 Variational methods10.1 Functionals, extremums and variations . . . . . . . . . . . .10.2 Functionals, Eextremums and variations (continued) . . . . .10.3 Functionals, extremums and variations (multidimensional) .10.4 Functionals, extremums and variations (multidimensional,continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 Variational methods in physics . . . . . . . . . . . . . . . . .Appendix 10.A. Nonholonomic mechanics . . . . . . . .Problems to Chapter 10 . . . . . . . . . . . . . . . . . . .27727728228811 Distributions and weak solutions11.1 Distributions . . . . . . . . . .11.2 Distributions: more . . . . . . .11.3 Applications of distributions . .11.4 Weak solutions . . . . . . . . .312312317322327.29229930530612 Nonlinear equations32912.1 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . 329

Contents13 Eigenvalues and eigenfunctions13.1 Variational theory . . . . . . . . . . . .13.2 Asymptotic distribution of eigenvalues13.3 Properties of eigenfunctions . . . . . .13.4 About spectrum . . . . . . . . . . . . .13.5 Continuous spectrum and scattering . .iv.33633634034835836514 Miscellaneous37014.1 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . 37014.2 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . 37414.3 Some quantum mechanical operators . . . . . . . . . . . . . 375A Appendices378A.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 378A.2 Some notations . . . . . . . . . . . . . . . . . . . . . . . . . 382

ContentsvPrefaceThe current version is in the online /This online Textbook based on half-year course APM346 at Department ofMathematics, University of Toronto (for students who are not mathematicsspecialists, which is equivalent to mathematics majors in USA) but containsmany additions.This Textbook is free and open (which means that anyone can use itwithout any permission or fees) and open-source (which means that anyonecan easily modify it for his or her own needs) and it will remain this wayforever. Source (in the form of Markdown) of each page could be downloaded:this page’s URL hapter0/S0.htmland its source’s URL hapter0/S0.mdand for all other pages respectively.The Source of the whole book could be downloaded as well. Also couldbe downloaded Textbook in pdf format and TeX Source (when those areready). While each page and its source are updated as needed those three areupdated only after semester ends.Moreover, it will remain free and freely available. Since it free it doesnot cost anything adding more material, graphics and so on.This textbook is maintained. It means that it could be modified almostinstantly if some of students find some parts either not clear emough orcontain misprints or errors. PDF version is not maintained during semester(but after it it will incorporate all changes of the online version).This textbook is truly digital. It contains what a printed textbook cannotcontain in principle: clickable hyperlinks (both internal and external) anda bit of animation (external). On the other hand, CouseSmart and its ilkprovide only a poor man’s digital copy of the printed textbook.One should remember that you need an internet connection. Even if yousave web pages to parse mathematical expression you need MathJax which

Contentsviis loaded from the cloud. However you can print every page to pdf to keepon you computer (or download pdf copy of the whole textbook).Due to html format it reflows and can accommodate itself to the smallerscreens of the tablets without using too small fonts. One can read it onsmart phones (despite too small screens). On the other hand, pdf does notreflow but has a fidelity: looks exactly the same on any screen. Each versionhas its own advantages and disadvantages.True, it is less polished than available printed textbooks but it is maintained (which means that errors are constantly corrected, material, especiallyproblems, added).At Spring of 2019 I was teaching APM346 together with Richard Derryberry who authored some new problems (and the ideas of some of the newproblems as well) and some animations and the idea of adding animations,produced be Mathematica, belongs to him. These animations (animatedgifs) are hosted on the webserver of Department of Mathematics, Universityof Toronto.This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

ContentsviiWhat one needs to know?SubjectsRequired:1. Multivariable Calculus2. Ordinary Differential EquationsAssets: (useful but not required)3. Complex Variables,4. Elements of (Real) Analysis,5. Any courses in Physics, Chemistry etc using PDEs (taken previouslyor now).1. Multivariable CalculusDifferential Calculus(a) Partial Derivatives (first, higher order), differential, gradient, chainrule;(b) Taylor formula;(c) Extremums, stationary points, classification of stationart points usingsecond derivatives; Asset: Extremums with constrains.(d) Familiarity with some notations Section A.2.Integral cCalculus(e) Multidimensional integral, calculations in Cartesian coordinates;(f) Change of variables, Jacobian, calculation in polar, cylindrical, spherical coordinates;(g) Path, Line, Surface integrals, calculations;

Contentsviii(h) Green, Gauss, Stokes formulae;(i) u, A, · A, u where u is a scalar field and A is a vector field.2. Ordinary Differential EquationsFirst order equations(a) Definition, Cauchy problem, existence and uniqueness;(b) Equations with separating variables, integrable, linear.Higher order equations(c) Definition, Cauchy problem, existence and uniqueness;Linear equations of order 2(d) General theory, Cauchy problem, existence and uniqueness;(e) Linear homogeneous equations, fundamental system of solutions, Wronskian;(f) Method of variations of constant parameters.Linear equations of order 2 with constant coefficients(g) Fundamental system of solutions: simple, multiple, complex roots;(h) Solutions for equations with quasipolynomial right-hand expressions;method of undetermined coefficients;(i) Euler’s equations: reduction to equation with constant coefficients.Solving without reduction.Systems(j) General systems, Cauchy problem, existence and uniqueness;(k) Linear systems, linear homogeneous systems, fundamental system ofsolutions, Wronskian;(l) Method of variations of constant parameters;(m) Linear systems with constant coefficients.

ContentsAssets(a) ODE with singular points.(b) Some special functions.(c) Boundary value problems.ix

Chapter 1Introduction1.1PDE motivations and contextThe aim of this is to introduce and motivate partial differential equations(PDE). The section also places the scope of studies in APM346 within thevast universe of mathematics. A partial differential equation (PDE) is angather involving partial derivatives. This is not so informative so let’s breakit down a bit.1.1.1What is a differential equation?An ordinary differential equation (ODE) is an equation for a function whichdepends on one independent variable which involves the independent variable,the function, and derivatives of the function:F (t, u(t), u0 (t), u(2) (t), u(3) (t), . . . , u(m) (t)) 0.This is an example of an ODE of order m where m is a highest order ofthe derivative in the equation. Solving an equation like this on an intervalt [0, T ] would mean finding a function t 7 u(t) R with the propertythat u and its derivatives satisfy this equation for all values t [0, T ].The problem can be enlarged by replacing the real-valued u by a vectorvalued one u(t) (u1 (t), u2 (t), . . . , uN (t)). In this case we usually talkabout system of ODEs.Even in this situation, the challenge is to find functions dependingupon exactly one variable which, together with their derivatives, satisfy theequation.1

Chapter 1. Introduction2What is a partial derivative?When you have function that depends upon several variables, you candifferentiate with respect to either variable while holding the other variableconstant. This spawns the idea of partial derivatives. As an example,consider a function depending upon two real variables taking values in thereals:u : Rn R.As n 2 we sometimes visualize a function like this by considering itsgraph viewed as a surface in R3 given by the collection of points{(x, y, z) R3 : z u(x, y)}.We can calculate the derivative with respect to x while holding y fixed. This leads to ux , also expressed as x u, u, and xu. Similarly, we can hold xx fixed and differentiate with respect to y.What is PDE?A partial differential equation is an equation for a function which dependson more than one independent variable which involves the independentvariables, the function, and partial derivatives of the function:F (x, y, u(x, y), ux (x, y), uy (x, y), uxx (x, y), uxy (x, y), uyx (x, y), uyy (x, y)) 0.This is an example of a PDE of order 2. Solving an equation like thiswould mean finding a function (x, y) u(x, y) with the property that uand its derivatives satisfy this equation for all admissible arguments.Similarly to ODE case this problem can be enlarged by replacing thereal-valued u by a vector-valued one u(t)

PDF version is not maintained during semester (but after it it will incorporate all changes of the online version). This textbook is truly digital. It contains what a printed textbook cannot contain in principle: clickable hyperlinks (both internal and external) and a bit of animation (external). On the other hand, CouseSmart and its ilk provide only a poor man’s digital copy of the printed .