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my Partial differential equations cheat sheet
July 7, 2016 compiled on — Thursday July 07, 2016 at 12:20 AM
Contents
1 introduction
These are part of my study notes on PDE's.
Trying to classify PDE's, here is current diagram. It is very large, but it is meant to include a summary of
many methods in one place. Easier to view in a browser than in the pdf.
Some diagrams I made
To derive the PDE, we start by setting up the state quantities and the flow quantities, and relate these to
each others by the use of the constitutive law. Then substiting this into the local conservation law, lead to the
PDE.



state quantity 
constitutive law 
flow quantity 






density 

tension 
tempreture   velocity 
pressure   momentum 
specific internal energy 

heat flux 
entropy 






2 linear PDE's
2.1 Elliptic
Some properties

Solution to the PDE represents steady state of .

Only boundary conditions are used to solve. No initial conditions.

Relation to complex analytic functions: If is analytic, then
and are solutions to Laplace pde's

Solutions to Laplace PDE are called harmonic functions.

constitutive law: Either consider them as stationary process, or take the time dependent pde, and
set those terms in that which depend on time to zero.
Examples of elliptic PDE's

Laplace or in general

Poisson

Helmholtz in 1D

Helmholtz in 2D
2.2 Parabolic
Some properties

Diffusion. Material spread is one specific example of diffusion. Here the state variable is the
concentration of the diffusing matrial. The flow quantity is its flux. The constitutive law is Fickś
law.

Heat spread. Here the state variable is the temprature, and the flow quantity is the heat flux.
The constitutive law is Fourierś law.

Stiff PDE, hence requires small time step, solved using implicit methods, not explicit for stability.

Numerically, use CrankNicleson, in 2D, can use ADI.

Requires initial and boundary conditions to solve.
Examples of parabolic PDE's

Diffusion. where is the diffusion constant, must be positive quantity. For heat
PDE, is the thermal diffusivity where is thermal conductivity, is specific
heat capacity, is density of medium.

In higher spatial dimension

FollerPlank, BlackSholes PDEs

DiffusionReaction where is the reaction term, which can be stiff or
not. Examples

Fischer equation, nonlinear PDE for modeling population
growth. where is carrying capacity, and is growth
rate.
2.3 Hyperbolic
Some properties

Advection PDE (or Transport or convection?). , Transport or drift of conserved
substance (pollutant) in Fluid or Gas where is speed of fluid. Analytic solution is
where is the initial conditions.

The state variable is the concentration of the contaminant, and the flow quantity is its flux
. The constitutive law is .

Wave
equation . Analytic solution is
where and .
Examples

Advection, Wave (See above)

nonhomogenouse advection and wave: and .

KleinGordon

Telegraphy
3 hints
reference
Characteristics are curves in the space of the independent variables along which the
governing PDE has only total differentials
4 references

Elements of partial differential Equations, Pavel Drabek and Gabriela Holubova, 2007.

Applied partial differental equations. 4th edition, Richard Haberman

http://www.phy.ornl.gov/csep/pde/node3.html

http://www.me.metu.edu.tr/courses/me582/files/PDE_Introduction_by_Hoffman.pdf

http://en.wikibooks.org/wiki/Partial_Differential_Equations/Introduction_and_
Classifications

http://www.scholarpedia.org/article/Partial_differential_equation

http://how.gi.alaska.edu/ao/sim/chapters/chap3.pdf good discussion on classification via
Characteristics lines

http://gwu.geverstine.com/pde.pdf table on classification, diagram for discriminant sign