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## Partial Differential Equations Notes This is an extract of our Partial Differential Equations document, which we sell as part of our Mathematics for Natural Sciences Notes collection written by the top tier of Cambridge University students.

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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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