Partial Differential Equations: An Unhurried Introduction

by Vladimir A. Tolstykh

Overview

This is a clear, rigorous and self-contained introduction to PDEs for a semester-based course on the topic. For the sake of smooth exposition, the book keeps the amount of applications to a minimum, focusing instead on the theoretical essentials and problem solving. The result is an agile compendium of theorems and methods - the ideal companion for any student tackling PDEs for the first time.

Vladimir Tolstykh is a professor of mathematics at Istanbul Arel University. He works in group theory and model-theoretic algebra. Dr. Tolstykh received his Ph.D. in Mathematics from the Ural Institute of Mathematics and Mechanics (Ekaterinburg (Russia) in 1992 and his Doctor of Science degree in Mathematics from the Sobolev Institute of Mathematics (Novosibirsk, Russia) in 2007.

Product Details

ISBN-13:	9783110677249
Publisher:	De Gruyter
Publication date:	06/08/2020
Series:	De Gruyter Textbook
Pages:	276
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

About the Author

Vladimir A. Tolstykh, Istanbul Arel University, Turkey.

Table of Contents

Preface vii

1 Definition 1

1.1 Partial differential equations 1

1.2 Linear PDEs 4

1.3 Separable ODEs: a brief remainder 10

1.4 PDEs reducible to ODEs 16

2 Change of variables in PDEs 29

2.1 Change of variables in a PDE: "direct" approach 29

2.2 Inverse function theorem 39

2.3 Change of variables in a PDE: "indirect" approach 41

3 First-order linear equations 49

4 First-order semilinear equations 63

4.1 Peano-Pickard-Lindelöf theorem 63

4.2 Method of characteristics for semilinear equations 67

5 First-order quasilinear equations: vector fields 85

5.1 Peano-Pickard-Lindelöf theorem for higher dimensions 86

5.2 Vector fields and their characteristic curves 89

5.3 First integrals of vector fields 100

6 First-order quasilinear equations: solution sets 117

6.1 Solutions determined by first integrals 117

6.2 Main theorem on solution sets 124

6.3 Cauchy problem for a PDE 129

6.4 First integrals: up to the task once again 130

7 Method of characteristics for first-order quasilinear equations 157

7.1 Surfaces made up of characteristic curves 157

7.2 Main theorem 170

8 Second-order semilinear equations 205

8.1 Three main types 205

8.2 Hyperbolic equations 214

8.3 Parabolic equations 223

8.4 Elliptic equations 231

A Appendix 249

A.1 Inverse function theorem 249

A.2 Functional dependence 256

Bibliography 263

Index 265

Editorial Reviews

"The main text is enriched with some examples on the usage of Maple and the online resource Wolfram Alpha for solving problems related to the theory of PDEs. The presentation is clear and the book is written so as to be accessible to a wide audience."
Ahmed Lesfari (El Jadida), zbMath (Zentralblatt Math, https://zbmath.org/?q=an%3A07205681, accessed 02/22/2021)

"For me, a high point of reading Partial Differential Equations An Unhurried Introduction was the geometric approach that the author takes, I also really like that　there is a significant treatment of not-necessarily-linear problems. Overall, Partial Differential Equations An Unhurried Introduction is an interesting　textbook option and I believe it is worthy of adoption for a course. "
Jason Graham, MAA (Mathematical Association of America) website, on 12/20/2020, (https://www.maa.org/press/maa-reviews/partial-differential-equations-an-unhurried-introduction, accessed 02/22/2021)

From the Publisher