While partial differential equations (PDEs) are fundamental in mathematics and throughout the sciences, most undergraduate students are only exposed to PDEs through the method of separation of variations. This text is written for undergraduate students from different cohorts with one sole purpose: to facilitate a proficiency in many core concepts in PDEs while enhancing the intuition and appreciation of the subject. For mathematics students, this will in turn provide a solid foundation for graduate study. A recurring theme is the role of concentration as captured by Dirac's delta function. This both guides the student into the structure of the solution to the diffusion equation and PDEs involving the Laplacian and invites them to develop a cognizance for the theory of distributions. Both distributions and the Fourier transform are given full treatment.
The book is rich with physical motivations and interpretations, and it takes special care to clearly explain all the technical mathematical arguments, often with pre-motivations and post-reflections. Through these arguments the reader will develop a deeper proficiency and understanding of advanced calculus. While the text is comprehensive, the material is divided into short sections, allowing particular issues/topics to be addressed in a concise fashion. Sections which are more fundamental to the text are highlighted, allowing the instructor several alternative learning paths. The author's unique pedagogical style also makes the text ideal for self-learning.
Rustum Choksi: McGill University, Montreal, QC, Canada
Preface Chapter 1. Basic Definitions 1.1. ∙ Notation 1.2. ∙ What Are Partial Differential Equations and Why Are They Ubiquitous? 1.3. ∙ What Exactly Do We Mean by a Solution to a PDE? 1.4. ∙ Order, Linear vs. Nonlinear, Scalar vs. Systems 1.5. ∙ General Solutions, Arbitrary Functions, Auxiliary Conditions, and the Notion of a Well-Posed Problem 1.6. ∙ Common Approaches and Themes in Solving PDEs
Chapter 2. First-Order PDEs and the Method of Characteristics 2.1. ∙ Prelude: A Few Simple Examples Illustrating the Notion and Geometry of Characteristics 2.2. ∙ The Method of Characteristics, Part I: Linear Equations 2.3. ∙ An Important Quasilinear Example: The Inviscid Burgers Equation 2.4. ∙ The Method of Characteristics, Part II: Quasilinear Equations 2.5. The Method of Characteristics, Part III: General First-Order Equations 2.6. ∙ Some General Questions 2.7. ∙ A Touch of Numerics, I: Computing the Solution of the Transport Equation 2.8. The Euler Equations: A Derivation 2.9. Chapter Summary Exercises
Chapter 3. The Wave Equation in One Space Dimension 3.1. ∙ Derivation: A Vibrating String 3.2. ∙ The General Solution of the 1D Wave Equation 3.3. ∙ The Initial Value Problem and Its Explicit Solution: D’Alembert’s Formula 3.4. ∙ Consequences of D’Alembert’s Formula: Causality 3.5. ∙ Conservation of the Total Energy 3.6. ∙ Sources 3.7. ∙ Well-Posedness of the Initial Value Problem and Time Reversibility 3.8. ∙ The Wave Equation on the Half-Line with a Fixed Boundary: Reflections 3.9. ∙ Neumann and Robin Boundary Conditions 3.10. ∙ Finite String Boundary Value Problems 3.11. ∙ A Touch of Numerics, II: Numerical Solution to the Wave Equation 3.12. Some Final Remarks 3.13. Chapter Summary Exercises
Chapter 4. The Wave Equation in Three and Two Space Dimensions 4.1. ∙ Two Derivations of the 3D Wave Equation 4.2. ∙ Three Space Dimensions: The Initial Value Problem and Its Explicit Solution 4.3. Two Space Dimensions: The 2D Wave Equation and Its Explicit Solution 4.4. Some Final Remarks and Geometric Optics 4.5. Chapter Summary Exercises
Chapter 5. The Delta “Function” and Distributions in One Space Dimension 5.1. ∙ Real-Valued Functions 5.2. ∙ The Delta “Function” and Why It Is Not a Function. Motivation for Generalizing the Notion of a Function 5.3. ∙ Distributions (Generalized Functions) 5.4. ∙ Derivative of a Distribution 5.5. ∙ Convergence in the Sense of Distributions 5.6. Dirac’s Intuition: Algebraic Manipulations with the Delta Function 5.7. ∙ Distributions Defined on an Open Interval and Larger Classes of Test Functions 5.8. Nonlocally Integrable Functions as Distributions: The Distribution PV 1/𝑥 5.9. Chapter Summary Exercises
Chapter 6. The Fourier Transform 6.1. ∙ Complex Numbers 6.2. ∙ Definition of the Fourier Transform and Its Fundamental Properties 6.3. ∙ Convolution of Functions and the Fourier Transform 6.4. ∙ Other Important Properties of the Fourier Transform 6.5. Duality: Decay at Infinity vs. Smoothness 6.6. Plancherel’s Theorem and the Riemann-Lebesgue Lemma 6.7. The 2𝜋 Issue and Other Possible Definitions of the Fourier Transform 6.8. ∙ Using the Fourier Transform to Solve Linear PDEs, I: The Diffusion Equation 6.9. The Fourier Transform of a Tempered Distribution 6.10. Using the Fourier Transform to Solve PDEs, II: The Wave Equation 6.11. The Fourier Transform in Higher Space Dimensions 6.12. Frequency, Harmonics, and the Physical Meaning of the Fourier Transform 6.13. A Few Words on Other Transforms 6.14. Chapter Summary 6.15. Summary Tables Exercises
Chapter 7. The Diffusion Equation 7.1. ∙ Derivation 1: Fourier’s/Fick’s Law 7.2. ∙ Solution in One Space Dimension and Properties 7.3. ∙ Derivation 2: Limit of Random Walks 7.4. Solution via the Central Limit Theorem 7.5. ∙ Well-Posedness of the IVP and Ill-Posedness of the Backward Diffusion Equation 7.6. ∙ Some Boundary Value Problems in the Context of Heat Flow 7.7. ∙ The Maximum Principle on a Finite Interval 7.8. Source Terms and Duhamel’s Principle Revisited 7.9. The Diffusion Equation in Higher Space Dimensions 7.10. ∙ A Touch of Numerics, III: Numerical Solution to the Diffusion Equation 7.11. Addendum: The Schrödinger Equation 7.12. Chapter Summary Exercises
Chapter 8. The Laplacian, Laplace’s Equation, and Harmonic Functions 8.1. ∙ The Dirichlet and Neumann Boundary Value Problems for Laplace’s and Poisson’s Equations 8.2. ∙ Derivation and Physical Interpretations 1: Concentrations in Equilibrium 8.3. Derivation and Physical Interpretations 2: The Dirichlet Problem and Poisson’s Equation via 2D Random Walks/Brownian Motion 8.4. ∙ Basic Properties of Harmonic Functions 8.5. ∙ Rotational Invariance and the Fundamental Solution 8.6. ∙ The Discrete Form of Laplace’s Equation 8.7. The Eigenfunctions and Eigenvalues of the Laplacian 8.8. The Laplacian and Curvature 8.9. Chapter Summary Exercises
Chapter 9. Distributions in Higher Dimensions and Partial Differentiation in the Sense of Distributions 9.1. ∙ The Test Functions and the Definition of a Distribution 9.2. ∙ Convergence in the Sense of Distributions 9.3. ∙ Partial Differentiation in the Sense of Distributions 9.4. ∙ The Divergence and Curl in the Sense of Distributions: Two Important Examples 9.5. ∙ The Laplacian in the Sense of Distributions and a Fundamental Example 9.6. Distributions Defined on a Domain (with and without Boundary) 9.7. Interpreting Many PDEs in the Sense of Distributions 9.8. A View Towards Sobolev Spaces 9.9. Fourier Transform of an 𝑁-dimensional Tempered Distribution 9.10. Using the Fourier Transform to Solve Linear PDEs, III: Helmholtz and Poisson Equations in Three Space 9.11. Chapter Summary Exercises
Chapter 10. The Fundamental Solution and Green’s Functions for the Laplacian 10.1. ∙ The Proof for the Distributional Laplacian of 1over |𝐱| 10.2. ∙ Unlocking the Power of the Fundamental Solution for the Laplacian 10.3. ∙ Green’s Functions for the Laplacian with Dirichlet Boundary Conditions 10.4. ∙ Green’s Functions for the Half-Space and Ball in 3D 10.5. Green’s Functions for the Laplacian with Neumann Boundary Conditions 10.6. A Physical Illustration in Electrostatics: Coulomb’s Law, Gauss’s Law, the Electric Field, and Electrostatic Potential 10.7. Chapter Summary Exercises
Chapter 11. Fourier Series 11.1. ∙ Prelude: The Classical Fourier Series —the Fourier Sine Series, the Fourier Cosine Series, and the Full Fourier Series 11.2. ∙ Why Cosines and Sines? Eigenfunctions, Eigenvalues, and Orthogonality 11.3. ∙ Fourier Series in Terms of Eigenfunctions of 𝒜 with a Symmetric Boundary Condition 11.4. ∙ Convergence, I: The 𝐿² Theory, Bessel’s Inequality, and Parseval’s Equality 11.5. ∙ Convergence, II: The Dirichlet Kernel and Pointwise Convergence of the Full Fourier Series 11.6. Term-by-Term Differentiation and Integration of Fourier Series 11.7. Convergence, III: Uniform Convergence 11.8. What Is the Relationship Between Fourier Series and the Fourier Transform? 11.9. Chapter Summary Exercises
Chapter 12. The Separation of Variables Algorithm for Boundary Value Problems 12.1. ∙ The Basic Separation of Variables Algorithm 12.2. ∙ The Wave Equation 12.3. ∙ Other Boundary Conditions 12.4. Source Terms and Duhamel’s Principle for the Diffusion and Wave Equations 12.5. ∙ Laplace’s Equations in a Rectangle and a Disk 12.6. ∙ Extensions and Generalizations of the Separation of Variables Algorithm 12.7. ∙ Extensions, I: Multidimensional Classical Fourier Series: Solving the Diffusion Equation on a Rectangle 12.8. ∙ Extensions, II: Polar and Cylindrical Coordinates and Bessel Functions 12.9. Extensions, III: Spherical Coordinates, Legendre Polynomials, Spherical Harmonics, and Spherical Bessel Functions 12.10. Extensions, IV: General Sturm-Liouville Problems 12.11. Separation of Variables for the Schrödinger Equation: Energy Levels of the Hydrogen Atom 12.12. Chapter Summary Exercises
Chapter 13. Uniting the Big Three Second-Order Linear Equations, and What’s Next 13.1. Are There Other Important Linear Second-Order Partial Differential Equations? The Standard Classification 13.2. Reflection on Fundamental Solutions, Green’s Functions, Duhamel’s Principle, and the Role/Position of the Delta Function 13.3. What’s Next? Towards a Future Volume on This Subject Appendix. Objects and Tools of Advanced Calculus A.1. Sets, Domains, and Boundaries in ℝ^{ℕ} A.2. Functions: Smoothness and Localization A.3. Gradient of a Function and Its Interpretations, Directional Derivatives, and the Normal Derivative A.4. Integration A.5. Evaluation and Manipulation of Integrals: Exploiting Radial Symmetry A.6. Fundamental Theorems of Calculus: The Divergence Theorem, Integration by Parts, and Green’s First and Second Identities A.7. Integral vs. Pointwise Results A.8. Convergence of Functions and Convergence of Integrals A.9. Differentiation under the Integral Sign A.10. Change in the Order of Integration A.11. Thinking Dimensionally: Physical Variables Have Dimensions with Physical Units Exercises Bibliography Index