An introduction to partial differential equations

Most descriptions of physical systems, as used in physics, engineering and, above all, in applied mathematics, are in terms of partial differential equations. This text, presented in three parts, introduces all the main mathematical ideas that are needed for the construction of solutions. You can download the book for free via the link below. Content 1. Part I First-order partial differential equations 2. List of examples 3. Preface 4. Introduction 4.1. Types of equation 4.2. Exercises 1 5. The quasi-linear equation 5.1. Of surfaces and tangents 5.2. The Cauchy (or initial value) problem 5.3. The semi-linear and linear equations 5.4. The quasi-linear equation in n independent variables 5.5. Exercises 2 6. The general equation 6.1. Geometry again 6.2. The method of solution 6.3. The general PDE with Cauchy data 6.4. The complete integral and the singular solution 6.5. Exercises 3 7. Answers 8. Part II Partial differential equations: classification and canonical forms 9. List of Equations 10. Preface 11. Introduction 11.1. Types of equation 12. First-order equations 12.1. The linear equation 12.2. The Cauchy problem 12.3. The quasi-linear equation 12.4. Exercises 2 13. The wave equation 13.1. Connection with first-order equations 13.2. Initial data 13.3. Exercises 3 14. The general semi-linear partial differential equation in two independent variables 14.1. Transformation of variables 14.2. Characteristic lines and the classification 14.3. Canonical form 14.4. Initial and boundary conditions 14.5. Exercises 4 15. Three examples from fluid mechanics 15.1. The Tricomi equation 15.2. General compressible flow 15.3. The shallow-water equations 15.4. Appendix: The hodograph transformation 15.5. Exercise 5 16. Riemann invariants and simple waves 16.1. Shallow-water equations: Riemann invariants 16.2. Shallow-water equations: simple waves 17. Answers 18. Part III Partial differential equations: method of separation of variables and similarity & travelling-wave solutions 19. List of Equations 20. Preface 21. Introduction 21.1. The Laplacian and coordinate systems 21.2. Overview of the methods 22. The method of separation of variables 22.1. Introducing the method 22.2. Two independent variables: other coordinate systems 22.3. Linear equations in more than two independent variables 22.4. Nonlinear equations 22.5. Exercises 2 23. Travelling-wave solutions 23.1. The classical, elementary partial differential equations 23.2. Equations in higher dimensions 23.3. Nonlinear equations 23.4. Exercises 3 24. Similarity solutions 24.1. Introducing the method 24.2. Continuous (Lie) groups 24.3. Similarity solutions of other equations 24.4. More general solutions from similarity solutions 24.5. Exercises 4 25. Answers 26. Index Sisältö Part I First-order partial differential equations List of examples Preface Introduction Types of equation Exercises 1 The quasi-linear equation Of surfaces and tangents The Cauchy (or initial value) problem The semi-linear and linear equations The quasi-linear equation in n independent variables Exercises 2 The general equation Geometry again The method of solution The general PDE with Cauchy data The complete integral and the singular solution Exercises 3 Answers Part II Partial differential equations: classification and canonical forms List of Equations Preface Introduction Types of equation First-order equations The linear equation The Cauchy problem The quasi-linear equation Exercises 2 The wave equation Connection with first-order equations Initial data Exercises 3 The general semi-linear partial differential equation in two independent variables Transformation of variables Characteristic lines and the classification Canonical form Initial and boundary conditions Exercises 4 Three examples from fluid mechanics The Tricomi equation General compressible flow The shallow-water equations Appendix: The hodograph transformation Exercise 5 Riemann invariants and simple waves Shallow-water equations: Riemann invariants Shallow-water equations: simple waves Answers Part III Partial differential equations: method of separation of variables and similarity & travelling-wave solutions List of Equations Preface Introduction The Laplacian and coordinate systems Overview of the methods The method of separation of variables Introducing the method Two independent variables: other coordinate systems Linear equations in more than two independent variables Nonlinear equations Exercises 2 Travelling-wave solutions The classical, elementary partial differential equations Equations in higher dimensions Nonlinear equations Exercises 3 Similarity solutions Introducing the method Continuous (Lie) groups Similarity solutions of other equations More general solutions from similarity solutions Exercises 4 Answers Index