**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.

#### Reviews & Endorsements

Overall, this is a clearly written and very thorough text, one that would support either classroom use or individual study for well-prepared students.

-- Bill Satzer, University of Minnesota

#### Table of Contents

# Table of Contents

## Partial Differential Equations: A First Course

Table of Contents pages: 1 2 3

- 7.3. ∙ Derivation 2: Limit of Random Walks 314314
- 7.4. Solution via the Central Limit Theorem 321321
- 7.5. ∙ Well-Posedness of the IVP and Ill-Posedness of the Backward Diffusion Equation 326326
- 7.6. ∙ Some Boundary Value Problems in the Context of Heat Flow 331331
- 7.7. ∙ The Maximum Principle on a Finite Interval 333333
- 7.8. Source Terms and Duhamel’s Principle Revisited 335335
- 7.9. The Diffusion Equation in Higher Space Dimensions 339339
- 7.10. ∙ A Touch of Numerics, III: Numerical Solution to the Diffusion Equation 340340
- 7.11. Addendum: The Schrödinger Equation 342342
- 7.12. Chapter Summary 344344
- Exercises 346346

- Chapter 8. The Laplacian, Laplace’s Equation, and Harmonic Functions 356356
- 8.1. ∙ The Dirichlet and Neumann Boundary Value Problems for Laplace’s and Poisson’s Equations 357357
- 8.2. ∙ Derivation and Physical Interpretations 1: Concentrations in Equilibrium 358358
- 8.3. Derivation and Physical Interpretations 2: The Dirichlet Problem and Poisson’s Equation via 2D Random Walks/Brownian Motion 360360
- 8.4. ∙ Basic Properties of Harmonic Functions 362362
- 8.5. ∙ Rotational Invariance and the Fundamental Solution 369369
- 8.6. ∙ The Discrete Form of Laplace’s Equation 371371
- 8.7. The Eigenfunctions and Eigenvalues of the Laplacian 372372
- 8.8. The Laplacian and Curvature 377377
- 8.9. Chapter Summary 381381
- Exercises 382382

- Chapter 9. Distributions in Higher Dimensions and Partial Differentiation in the Sense of Distributions 390390
- 9.1. ∙ The Test Functions and the Definition of a Distribution 390390
- 9.2. ∙ Convergence in the Sense of Distributions 392392
- 9.3. ∙ Partial Differentiation in the Sense of Distributions 394394
- 9.4. ∙ The Divergence and Curl in the Sense of Distributions: Two Important Examples 396396
- 9.5. ∙ The Laplacian in the Sense of Distributions and a Fundamental Example 404404
- 9.6. Distributions Defined on a Domain (with and without Boundary) 406406
- 9.7. Interpreting Many PDEs in the Sense of Distributions 407407
- 9.7.1. Our First Example Revisited! 407407
- 9.7.2. Burgers’s Equation and the Rankine-Hugoniot Jump Conditions 408408
- 9.7.3. The Wave Equation with a Delta Function Source 412412
- 9.7.4. Incorporating Initial Values into a Distributional Solution 414414
- 9.7.5. Not All PDEs Can Be Interpreted in the Sense of Distributions 417417

- 9.8. A View Towards Sobolev Spaces 417417
- 9.9. Fourier Transform of an 𝑁-dimensional Tempered Distribution 419419
- 9.10. Using the Fourier Transform to Solve Linear PDEs, III: Helmholtz and Poisson Equations in Three Space 420420
- 9.11. Chapter Summary 421421
- Exercises 423423

- Chapter 10. The Fundamental Solution and Green’s Functions for the Laplacian 430430
- 10.1. ∙ The Proof for the Distributional Laplacian of 1over |𝐱| 430430
- 10.2. ∙ Unlocking the Power of the Fundamental Solution for the Laplacian 433433
- 10.3. ∙ Green’s Functions for the Laplacian with Dirichlet Boundary Conditions 439439
- 10.3.1. ∙ The Definition of the Green’s Function with Dirichlet Boundary Conditions 439439
- 10.3.2. Using the Green’s Function to Solve the Dirichlet Problem for Laplace’s Equation 440440
- 10.3.3. ∙ Uniqueness and Symmetry of the Green’s Function 441441
- 10.3.4. ∙ The Fundamental Solution and Green’s Functions in One Space Dimension 443443

- 10.4. ∙ Green’s Functions for the Half-Space and Ball in 3D 445445
- 10.5. Green’s Functions for the Laplacian with Neumann Boundary Conditions 455455
- 10.6. A Physical Illustration in Electrostatics: Coulomb’s Law, Gauss’s Law, the Electric Field, and Electrostatic Potential 462462
- 10.6.1. Coulomb’s Law and the Electrostatic Force 462462
- 10.6.2. The Electrostatic Potential: The Fundamental Solution and Poisson’s Equation 464464
- 10.6.3. Green’s Functions: Grounded Conducting Plates, Induced Charge Densities, and the Method of Images 466466
- 10.6.4. Interpreting the Solution Formula for the Dirichlet Problem 468468

- 10.7. Chapter Summary 469469
- Exercises 470470

- Chapter 11. Fourier Series 474474
- 11.1. ∙ Prelude: The Classical Fourier Series —the Fourier Sine Series, the Fourier Cosine Series, and the Full Fourier Series 475475
- 11.1.1. ∙ The Fourier Sine Series 475475
- 11.1.2. ∙ The Fourier Cosine Series 477477
- 11.1.3. ∙ The Full Fourier Series 478478
- 11.1.4. ∙ Three Examples 478478
- 11.1.5. ∙ Viewing the Three Fourier Series as Functions over ℝ 481481
- 11.1.6. ∙ Convergence, Boundary Values, Piecewise Continuity, and Periodic Extensions 482482
- 11.1.7. Complex Version of the Full Fourier Series 484484

- 11.2. ∙ Why Cosines and Sines? Eigenfunctions, Eigenvalues, and Orthogonality 486486
- 11.3. ∙ Fourier Series in Terms of Eigenfunctions of 𝒜 with a Symmetric Boundary Condition 491491
- 11.4. ∙ Convergence, I: The 𝐿² Theory, Bessel’s Inequality, and Parseval’s Equality 498498
- 11.5. ∙ Convergence, II: The Dirichlet Kernel and Pointwise Convergence of the Full Fourier Series 503503
- 11.6. Term-by-Term Differentiation and Integration of Fourier Series 509509
- 11.7. Convergence, III: Uniform Convergence 513513
- 11.8. What Is the Relationship Between Fourier Series and the Fourier Transform? 518518
- 11.9. Chapter Summary 521521
- Exercises 523523

- Chapter 12. The Separation of Variables Algorithm for Boundary Value Problems 530530
- 12.1. ∙ The Basic Separation of Variables Algorithm 530530
- 12.2. ∙ The Wave Equation 534534
- 12.3. ∙ Other Boundary Conditions 537537
- 12.3.1. ∙ Inhomogeneous Dirichlet Boundary Conditions 537537
- 12.3.2. ∙ Mixed Homogeneous Boundary Conditions 537537
- 12.3.3. ∙ Mixed Inhomogeneous Boundary Conditions 538538
- 12.3.4. ∙ Inhomogeneous Neumann Boundary Conditions 538538
- 12.3.5. ∙ The Robin Boundary Condition for the Diffusion Equation 539539

- 12.4. Source Terms and Duhamel’s Principle for the Diffusion and Wave Equations 541541
- 12.5. ∙ Laplace’s Equations in a Rectangle and a Disk 543543
- 12.6. ∙ Extensions and Generalizations of the Separation of Variables Algorithm 547547
- 12.7. ∙ Extensions, I: Multidimensional Classical Fourier Series: Solving the Diffusion Equation on a Rectangle 548548
- 12.8. ∙ Extensions, II: Polar and Cylindrical Coordinates and Bessel Functions 550550
- 12.9. Extensions, III: Spherical Coordinates, Legendre Polynomials, Spherical Harmonics, and Spherical Bessel Functions 554554
- 12.10. Extensions, IV: General Sturm-Liouville Problems 563563
- 12.11. Separation of Variables for the Schrödinger Equation: Energy Levels of the Hydrogen Atom 568568
- 12.12. Chapter Summary 572572
- Exercises 573573

- Chapter 13. Uniting the Big Three Second-Order Linear Equations, and What’s Next 582582
- 13.1. Are There Other Important Linear Second-Order Partial Differential Equations? The Standard Classification 582582
- 13.2. Reflection on Fundamental Solutions, Green’s Functions, Duhamel’s Principle, and the Role/Position of the Delta Function 585585
- 13.2.1. Fundamental Solutions/Green’s Functions for the Laplacian 586586
- 13.2.2. Fundamental Solutions/Green’s Functions of the Diffusion Equation 587587
- 13.2.3. Fundamental Solutions/Green’s Functions of the 1D Wave Equation 590590
- 13.2.4. Fundamental Solutions/Green’s Functions of the 3D Wave Equation 593593

- Exercises 596596
- 13.3. What’s Next? Towards a Future Volume on This Subject 598598

- Appendix. Objects and Tools of Advanced Calculus 600600
- A.1. Sets, Domains, and Boundaries in ℝ^{ℕ} 600600
- A.2. Functions: Smoothness and Localization 603603
- A.3. Gradient of a Function and Its Interpretations, Directional Derivatives, and the Normal Derivative 606606
- A.4. Integration 609609
- A.5. Evaluation and Manipulation of Integrals: Exploiting Radial Symmetry 615615
- A.6. Fundamental Theorems of Calculus: The Divergence Theorem, Integration by Parts, and Green’s First and Second Identities 620620
- A.6.1. The Divergence Theorem 620620
- A.6.2. Two Consequences of the Divergence Theorem: Green’s Theorem and a Componentwise Divergence Theorem 621621
- A.6.3. A Match Made in Heaven: The Divergence + the Gradient = the Laplacian 622622
- A.6.4. Integration by Parts and Green’s First and Second Identities 622622

- A.7. Integral vs. Pointwise Results 624624
- A.8. Convergence of Functions and Convergence of Integrals 627627
- A.9. Differentiation under the Integral Sign 629629