**Pure and Applied Undergraduate Texts**

Volume: 54;
2022;
612 pp;
Softcover

MSC: Primary 35;

**Print ISBN: 978-1-4704-6491-2
Product Code: AMSTEXT/54**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-6867-5
Product Code: AMSTEXT/54.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Partial Differential Equations: A First Course

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*Rustum Choksi*

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.

#### Readership

Undergraduate and graduate students interested in partial differential equations.

#### Table of Contents

# Table of Contents

## Partial Differential Equations: A First Course

Table of Contents pages: 1 2 3

- Preface 2424
- Chapter 1. Basic Definitions 3434
- 1.1. ∙ Notation 3434
- 1.2. ∙ What Are Partial Differential Equations and Why Are They Ubiquitous? 3535
- 1.3. ∙ What Exactly Do We Mean by a Solution to a PDE? 3737
- 1.4. ∙ Order, Linear vs. Nonlinear, Scalar vs. Systems 3838
- 1.5. ∙ General Solutions, Arbitrary Functions, Auxiliary Conditions, and the Notion of a Well-Posed Problem 4040
- 1.6. ∙ Common Approaches and Themes in Solving PDEs 4343
- Exercises 4646

- Chapter 2. First-Order PDEs and the Method of Characteristics 5050
- 2.1. ∙ Prelude: A Few Simple Examples Illustrating the Notion and Geometry of Characteristics 5252
- 2.2. ∙ The Method of Characteristics, Part I: Linear Equations 5757
- 2.2.1. ∙ A Few Examples 6060
- 2.2.2. ∙ Temporal Equations: Using Time to Parametrize the Characteristics 6262
- 2.2.3. ∙ More Than Two Independent Variables 6464
- 2.2.4. ∙ Transport Equations in Three Space Dimensions with Constant Velocity 6464
- 2.2.5. Transport Equations in Three Space Dimensions with Space Varying Velocity 6666
- 2.2.6. The Continuity Equation in Three Space Dimensions: A Derivation 6969
- 2.2.7. Semilinear Equations 7070
- 2.2.8. Noncharacteristic Data and the Transversality Condition 7171

- 2.3. ∙ An Important Quasilinear Example: The Inviscid Burgers Equation 7272
- 2.4. ∙ The Method of Characteristics, Part II: Quasilinear Equations 7777
- 2.5. The Method of Characteristics, Part III: General First-Order Equations 8181
- 2.5.1. The Notation 8181
- 2.5.2. The Characteristic Equations 8282
- 2.5.3. Linear and Quasilinear Equations in 𝑁 independent variables 8585
- 2.5.4. Two Fully Nonlinear Examples 8686
- 2.5.5. The Eikonal Equation 8787
- 2.5.6. Hamilton-Jacobi Equations 8989
- 2.5.7. The Level Set Equation and Interface Motion 9090

- 2.6. ∙ Some General Questions 9292
- 2.7. ∙ A Touch of Numerics, I: Computing the Solution of the Transport Equation 9393
- 2.8. The Euler Equations: A Derivation 9999
- 2.8.1. Conservation of Mass and the Continuity Equation 100100
- 2.8.2. Conservation of Linear Momentum and Pressure 101101
- 2.8.3. Gas Dynamics: The Compressible Euler Equations 105105
- 2.8.4. An Ideal Liquid: The Incompressible Euler Equations 106106
- 2.8.5. A Viscous Liquid: The Navier-Stokes Equations 106106
- 2.8.6. Spatial vs. Material Coordinates and the Material Time Derivative 107107

- 2.9. Chapter Summary 109109
- Exercises 110110

- Chapter 3. The Wave Equation in One Space Dimension 116116
- 3.1. ∙ Derivation: A Vibrating String 117117
- 3.2. ∙ The General Solution of the 1D Wave Equation 121121
- 3.3. ∙ The Initial Value Problem and Its Explicit Solution: D’Alembert’s Formula 122122
- 3.4. ∙ Consequences of D’Alembert’s Formula: Causality 124124
- 3.5. ∙ Conservation of the Total Energy 129129
- 3.6. ∙ Sources 131131
- 3.7. ∙ Well-Posedness of the Initial Value Problem and Time Reversibility 135135
- 3.8. ∙ The Wave Equation on the Half-Line with a Fixed Boundary: Reflections 136136
- 3.9. ∙ Neumann and Robin Boundary Conditions 142142
- 3.10. ∙ Finite String Boundary Value Problems 143143
- 3.11. ∙ A Touch of Numerics, II: Numerical Solution to the Wave Equation 145145
- 3.12. Some Final Remarks 147147
- 3.13. Chapter Summary 151151
- Exercises 152152

- Chapter 4. The Wave Equation in Three and Two Space Dimensions 158158
- 4.1. ∙ Two Derivations of the 3D Wave Equation 158158
- 4.2. ∙ Three Space Dimensions: The Initial Value Problem and Its Explicit Solution 163163
- 4.3. Two Space Dimensions: The 2D Wave Equation and Its Explicit Solution 173173
- 4.4. Some Final Remarks and Geometric Optics 175175
- 4.5. Chapter Summary 178178
- Exercises 179179

- Chapter 5. The Delta “Function” and Distributions in One Space Dimension 184184
- 5.1. ∙ Real-Valued Functions 185185
- 5.2. ∙ The Delta “Function” and Why It Is Not a Function. Motivation for Generalizing the Notion of a Function 189189
- 5.3. ∙ Distributions (Generalized Functions) 194194
- 5.4. ∙ Derivative of a Distribution 200200
- 5.5. ∙ Convergence in the Sense of Distributions 206206
- 5.5.1. ∙ The Definition 206206
- 5.5.2. ∙ Comparisons of Distributional versus Pointwise Convergence of Functions 207207
- 5.5.3. ∙ The Distributional Convergence of a Sequence of Functions to the Delta Function: Four Examples 207207
- 5.5.4. 𝜀 vs. 𝑁 Proofs for the Sequences (5.23) and (5.24) 210210
- 5.5.5. ∙ The Distributional Convergence of sin𝑛𝑥 213213
- 5.5.6. ∙ The Distributional Convergence of the Sinc Functions and the Dirichlet Kernel: Two Sequences Directly Related to Fourier Analysis 215215

- 5.6. Dirac’s Intuition: Algebraic Manipulations with the Delta Function 218218
- 5.7. ∙ Distributions Defined on an Open Interval and Larger Classes of Test Functions 222222
- 5.8. Nonlocally Integrable Functions as Distributions: The Distribution PV 1/𝑥 223223
- 5.9. Chapter Summary 230230
- Exercises 232232

- Chapter 6. The Fourier Transform 236236
- 6.1. ∙ Complex Numbers 237237
- 6.2. ∙ Definition of the Fourier Transform and Its Fundamental Properties 239239
- 6.3. ∙ Convolution of Functions and the Fourier Transform 246246
- 6.4. ∙ Other Important Properties of the Fourier Transform 251251
- 6.5. Duality: Decay at Infinity vs. Smoothness 253253
- 6.6. Plancherel’s Theorem and the Riemann-Lebesgue Lemma 255255
- 6.7. The 2𝜋 Issue and Other Possible Definitions of the Fourier Transform 258258
- 6.8. ∙ Using the Fourier Transform to Solve Linear PDEs, I: The Diffusion Equation 259259
- 6.9. The Fourier Transform of a Tempered Distribution 262262
- 6.9.1. Can We Extend the Fourier Transform to Distributions? 262262
- 6.9.2. The Schwartz Class of Test Functions and Tempered Distributions 263263
- 6.9.3. The Fourier Transform of 𝑓(𝑥)≡1 and the Delta Function 266266
- 6.9.4. The Fourier Transform of 𝑓(𝑥)=𝑒^{𝑖𝑎𝑥} and the Delta Function 𝛿ₐ 268268
- 6.9.5. The Fourier Transform of Sine, Cosine, and Sums of Delta Functions 269269
- 6.9.6. The Fourier Transform of 𝑥 269269
- 6.9.7. The Fourier Transform of the Heaviside and Sgn Functions and PV 1/𝑥 270270
- 6.9.8. Convolution of a Tempered Distribution and a Function 271271

- 6.10. Using the Fourier Transform to Solve PDEs, II: The Wave Equation 274274
- 6.11. The Fourier Transform in Higher Space Dimensions 276276
- 6.12. Frequency, Harmonics, and the Physical Meaning of the Fourier Transform 282282
- 6.13. A Few Words on Other Transforms 288288
- 6.14. Chapter Summary 291291
- 6.15. Summary Tables 293293
- Exercises 296296

- Chapter 7. The Diffusion Equation 302302