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Partial Differential Equations Notes

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Mathematics for NST Part IA Cambridge University, 2012-2013

Notes for Partial Differential Equations General properties of PDEs: A partial differential equation (PDE) is an equation relating a function

f (x , y , ...) of more

than one variable and its partial derivatives with respect to these variables. It therefore has the form:

(

F x, y ,

[?]f [?]f
[?]2 f [?] 2 f [?]2 f , ,... , 2 , 2 , , ... =const.
[?]x [?]y
[?]x [?] y [?]x[?]y

)

Where, in general, PDEs which represent physical systems never exceed the second order. A PDE is considered to be 'linear' if it is of the form

Ly=const . , where

L is a linear

y .

operator on differentiable functions of

Parallels between ODEs and PDEs: The boundary conditions required to specify unique solutions to a PDE can be compared to those required to specify unique solutions to an ODE: Ordinary differential equation (ODE): Information in the form of values of

Partial differential equation (PDE):

function

and/or values of its partial derivatives on surfaces in

f

and/or values of its

ordinary derivatives at points in

(x , y , ...) space is required.

Information in the form of values of function

f

( x , y , ...) space is required.

The boundary conditions fix the arbitrary constants of integration, thus one is needed for a first-order equation, and two are needed for a second-order equation.

[?]D D

The boundary conditions fix the arbitrary functions of integration. Usually, if a solution is sought for variables region

(x , y , ...) in some

D , then the 'boundary conditions' need be given on all or

part of the region boundary

[?] D . It is generally difficult to work

out how much information is sufficient.

Physically-important PDEs: Wave equation: 2

2 [?]ps 1 [?]ps
= 2 2 2
[?]x c [?]t

Diffusion equation: 2

[?]Ph
[?]Ph
=k 2
[?]t
[?]x

LaPlace's equation: 2

2 [?] ph [?] ph
+ 2 =0 2
[?]x [?] y

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