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. The material covers all the elements that are encountered in any standard university study: first-order equations, including those that take very general forms, as well as the classification of second-order equations and the development of special solutions e.g. travelling-wave and similarity solutions.
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Inhalt
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- 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
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