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**Partial differential equations. A first course.**
*(English)*
Zbl 1500.35001

Pure and Applied Undergraduate Texts 54. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6491-2/pbk; 978-1-4704-6867-5/ebook). xxxii, 612 p. (2022).

This is really an excellent textbook. It covers a wealth of interesting material, it is written in a very clear and convincing style, and it explains ideas, rather than drowning the reader in technicalities, as many other books do. The author takes his task serious, by not restricting himself to just the presentation of definitions, theorems, and proofs, which makes reading often pretty dry, but also by giving some hints which should prevent unexperienced readers from walking into certain traps. (Such sections are called “Examples where this is illegal” and refer, for example, to differentiating under the integral sign.)

To give an idea of the topics covered in the book, let us enumerate, without going into details, the headings of the 13 chapters: 1. Basic definitions; 2. First-order PDEs and the method of characteristics; 3. The wave equation in one space dimension; 4. The wave equation in three and two space dimensions; 5. The delta “function” and distributions in one space dimension; 6. The Fourier transform; 7. The diffusion equation; 8. The Laplacian, Laplace’s equation, and harmonic functions; 9. Distributions in higher dimensions and partial differentiation in the sense of distributions; 10. The fundamental solution and Green’s functions for the Laplacian; 11. Fourier series; 12. The separation of variables algorithm for boundary value problems; 13. Uniting the big three second-order linear equations, and what’s next. Each chapter closes with a summary and some (alas, very few, exercises).

Just reading the Preface is very rewarding and interesting, and the same applies to the Appendix.

Again, this is an excellent textbook. It addresses, according to the author, senior undergraduate students, but it is certainly of interest also for postgraduate or PhD students. If an undergraduate student is looking for a mathematical field with a beautiful theory, some surprising results, methods from other parts of mathematics, and a large variety of applications, a good “test animal” is PDEs. So go ahead and buy this book, and you will believe me.

To give an idea of the topics covered in the book, let us enumerate, without going into details, the headings of the 13 chapters: 1. Basic definitions; 2. First-order PDEs and the method of characteristics; 3. The wave equation in one space dimension; 4. The wave equation in three and two space dimensions; 5. The delta “function” and distributions in one space dimension; 6. The Fourier transform; 7. The diffusion equation; 8. The Laplacian, Laplace’s equation, and harmonic functions; 9. Distributions in higher dimensions and partial differentiation in the sense of distributions; 10. The fundamental solution and Green’s functions for the Laplacian; 11. Fourier series; 12. The separation of variables algorithm for boundary value problems; 13. Uniting the big three second-order linear equations, and what’s next. Each chapter closes with a summary and some (alas, very few, exercises).

Just reading the Preface is very rewarding and interesting, and the same applies to the Appendix.

Again, this is an excellent textbook. It addresses, according to the author, senior undergraduate students, but it is certainly of interest also for postgraduate or PhD students. If an undergraduate student is looking for a mathematical field with a beautiful theory, some surprising results, methods from other parts of mathematics, and a large variety of applications, a good “test animal” is PDEs. So go ahead and buy this book, and you will believe me.

Reviewer: Jürgen Appell (Würzburg)

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35Axx | General topics in partial differential equations |

35Cxx | Representations of solutions to partial differential equations |

35Dxx | Generalized solutions to partial differential equations |

35Exx | Partial differential equations and systems of partial differential equations with constant coefficients |

35Fxx | General first-order partial differential equations and systems of first-order partial differential equations |

35Jxx | Elliptic equations and elliptic systems |

35Kxx | Parabolic equations and parabolic systems |

35Lxx | Hyperbolic equations and hyperbolic systems |

35Pxx | Spectral theory and eigenvalue problems for partial differential equations |