Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-25T12:17:39.636Z Has data issue: false hasContentIssue false

Part II - General theory

Published online by Cambridge University Press:  05 June 2012

Marcus Pivato
Affiliation:
Trent University, Peterborough, Ontario
Get access

Summary

Differential equations encode the underlying dynamical laws which govern a physical system. But a physical system is more than an abstract collection of laws; it is a specific configuration of matter and energy, which begins in a specific initial state (mathematically encoded as initial conditions) and which is embedded in a specific environment (encoded by certain boundary conditions). Thus, to model this physical system, we must find functions that satisfy the underlying differential equations, while simultaneously satisfying these initial conditions and boundary conditions; this is called an initial/boundary value problem (I/BVP).

Before we can develop solution methods for PDEs in general, and I/BVPs in particular, we must answer some qualitative questions. Under what conditions does a solution even exist? If it exists, is it unique? If the solution is not unique, then how can we parameterize the set of all possible solutions? Does this set have some kind of order or structure?

The beauty of linear differential equations is that linearity makes these questions much easier to answer. A linear PDE can be seen as a linear equation in an infinite-dimensional vector space, where the ‘vectors’ are functions. Most of the methods and concepts of linear algebra can be translated almost verbatim into this context. In particular, the set of solutions to a homogeneous linear PDE (or homogeneous linear I/BVP) forms a linear subspace.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • General theory
  • Marcus Pivato, Trent University, Peterborough, Ontario
  • Book: Linear Partial Differential Equations and Fourier Theory
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810183.008
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • General theory
  • Marcus Pivato, Trent University, Peterborough, Ontario
  • Book: Linear Partial Differential Equations and Fourier Theory
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810183.008
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • General theory
  • Marcus Pivato, Trent University, Peterborough, Ontario
  • Book: Linear Partial Differential Equations and Fourier Theory
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511810183.008
Available formats
×