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.
The following is a more accessble plain text extract of the PDF sample above, taken from our Mathematics for Natural Sciences Notes. Due to the challenges of extracting text from PDFs, it will have odd formatting:
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 ,
[?]2 f [?] 2 f [?]2 f , ,... , 2 , 2 , , ... =const.
[?]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
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):
and/or values of its partial derivatives on surfaces in
and/or values of its
ordinary derivatives at points in
(x , y , ...) space is required.
Information in the form of values of function
( 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.
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
LaPlace's equation: 2
2 [?] ph [?] ph
+ 2 =0 2
[?]x [?] y
Buy the full version of these notes or essay plans and more in our Mathematics for Natural Sciences Notes.