JГјrgen Jost - Partial Differential Equations

Partial Differential Equations — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Partial Differential Equations" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...155

Springer Science+Business Media New York 2013

Jrgen Jost Partial Differential Equations Graduate Texts in Mathematics 10.1007/978-1-4614-4809-9_1

1. Introduction: What Are Partial Differential Equations?

Jrgen Jost 1

(1)

Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

As a first answer to the question, What are PDEs, we would like to give a definition:

Definition 1.

A PDE is an equation involving derivatives of an unknown function , where is an open subset of , d 2 (or, more generally, of a differentiable manifold of dimension d 2).

Often, one also considers systems of PDEs for vector-valued functions , or for mappings with values in a differentiable manifold.

The preceding definition, however, is misleading, since in the theory of PDEs one does not study arbitrary equations but concentrates instead on those equations that naturally occur in various applications (physics and other sciences, engineering, economics) or in other mathematical contexts.

Thus, as a second answer to the question posed in the title, we would like to describe some typical examples of PDEs. We shall need a little bit of notation: A partial derivative will be denoted by a subscript,

In case d =2, we write x , y in place of . Otherwise, x is the vector .

Examples.

1. (1)
   The Laplace equation

   

   or, more generally, the Poisson equation

   

   For example, the real and imaginary parts u and v of a holomorphic function ( open) satisfy the Laplace equation. This easily follows from the CauchyRiemann equations:

   

   implies

   

   The CauchyRiemann equations themselves represent a system of PDEs. The Laplace equation also models many equilibrium states in physics, and the Poisson equation is important in electrostatics.
2. (2)
   The heat equation: Here, one coordinate t is distinguished as the time coordinate, while the remaining coordinates represent spatial variables. We consider

   

   and pose the equation

   

   The heat equation models heat and other diffusion processes.
3. (3)
   The wave equation: With the same notation as in (2), here we have the equation

   

   It models wave and oscillation phenomena.
4. (4)
   The Kortewegde Vries equation

   

   (notation as in (2), but with only one spatial coordinate x ) models the propagation of waves in shallow waters.
5. (5)
   The MongeAmpre equation

   

   or in higher dimensions

   

   with a given function f , is used for finding surfaces (or hypersurfaces) with prescribed curvature.
6. (6)
   The minimal surface equation

   

   describes an important class of surfaces in .
7. (7)
   The Maxwell equations for the electric field strength and the magnetic field strength as functions of :

   

   where div and curl are the standard differential operators from vector analysis with respect to the variables .
8. (8)
   The NavierStokes equations for the velocity v ( x , t ) and the pressure p ( x , t ) of an incompressible fluid of density and viscosity :

   

   (0)

   ( d =3, ).
9. (9)
   The Einstein field equations of the theory of general relativity for the curvature of the metric ( g ij ) of space-time:

   

   Here, is a constant, T ij is the energymomentum tensor (considered as given), while

   

Next page

1234567...155

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «Partial Differential Equations»

Look at similar books to Partial Differential Equations. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Nakhle H. Asmar

Partial differential equations with fourier series and boundary value problems : third edition

Shigeaki Koike

Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017

Hillen Thomas

Partial Differential Equations: Theory and Completely Solved Problems

Michael E. Taylor

Partial differential equations. II, Qualitative studies of linear equations

Michael E. Taylor

Partial Differential Equations I

Jost

Partial Differential Equations

J. David Logan

Applied Partial Differential Equations

David Borthwick

Introduction to Partial Differential Equations

Stanley J. Farlow

Partial Differential Equations for Scientists and Engineers

Stanley J. Farlow

An Introduction to Differential Equations and Their Applications

Dorina Mitrea

Distributions, Partial Differential Equations, and Harmonic Analysis

Carroll

Abstract Methods in Partial Differential Equations

Discussion, reviews of the book Partial Differential Equations and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.