Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 First-order equations
- 3 Second-order linear equations in two indenpendent variables
- 4 The one-dimensional wave equation
- 5 The method of separation of variables
- 6 Sturm–Liouville problems and eigenfunction expansions
- 7 Elliptic equations
- 8 Green's functions and integral representations
- 9 Equations in high dimensions
- 10 Variational methods
- 11 Numerical methods
- 12 Solutions of odd-numbered problems
- References
- Index
10 - Variational methods
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 First-order equations
- 3 Second-order linear equations in two indenpendent variables
- 4 The one-dimensional wave equation
- 5 The method of separation of variables
- 6 Sturm–Liouville problems and eigenfunction expansions
- 7 Elliptic equations
- 8 Green's functions and integral representations
- 9 Equations in high dimensions
- 10 Variational methods
- 11 Numerical methods
- 12 Solutions of odd-numbered problems
- References
- Index
Summary
The PDEs we have considered so far were derived by modeling a variety of phenomena in physics, engineering, etc. In this chapter we shall derive PDEs from a new perspective. We shall show that many PDEs are related to optimization problems. The theory that associates optimization with PDEs is called the calculus of variations. It is an extremely useful theory. On the one hand, we shall be able to solve many optimization problems by solving the corresponding PDEs. On the other hand, sometimes it is simpler to study (and solve) certain optimization problems than to study (and solve) the related PDE. In such cases, the calculus of variations is an indispensable theoretical and practical tool in the study of PDEs. The calculus of variations can be used for both static problems and dynamic problems. The dynamical aspects of this theory are based on the Hamilton principle that we shall derive below. In particular, we shall show how to apply this principle for wave propagation in strings, membranes, etc.
We shall see that the connection between optimization problems and the associated PDEs is based on the a priori assumption that the solution to the optimization problem is smooth enough for the PDE to make sense. Can we justify this assumption? In many cases we can. Moreover, even if the solution is not smooth, we would like to define an appropriate concept of weak solutions as we already did earlier in this book in different contexts.
- Type
- Chapter
- Information
- An Introduction to Partial Differential Equations , pp. 282 - 308Publisher: Cambridge University PressPrint publication year: 2005