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my Partial diﬀerential equations cheat sheet
May 28, 2020 Compiled on May 28, 2020 at 1:45am
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  \(\Longleftrightarrow \)  momentum 
speciﬁc internal energy   heat ﬂux 
entropy 






2 linear PDE’s
2.1 Elliptic
Some properties

Solution to the PDE represents steady state of \(u\).

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

Relation to complex analytic functions: If \(f(z)=\phi (x,y)+i\psi (x,y)\) is analytic, then \(\phi (x,y)\) and \(\psi (x,y)\) 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 \(u_{xx}=0\) or in general \(\nabla ^2 u=0\)

Poisson \(u_{xx}=f(x)\)

Helmholtz in 1D \(u_{xx}+\lambda u(x) = f(x)\)

Helmholtz in 2D \(u_{xx}+ u_{yy} + \lambda u(x,y) = f(x,y)\)
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. \(u_{t}Du_{xx}=0\) where \(D\) is the diﬀusion constant, must be positive quantity. For heat PDE, \(D\) is the
thermal diﬀusivity \(D=\kappa /{c_p\rho }\) where \(\kappa \) is thermal conductivity, \(c_p\) is speciﬁc heat capacity, \(\rho \) is density of
medium.

In higher spatial dimension \(u_{t}D\nabla ^2 u=0\)

FollerPlank, BlackSholes PDEs

DiﬀusionReaction \(u_tDu_{xx}=F(u(x,t))\) where \(F(u(x,t))\) is the reaction term, which can be stiﬀ or not. Examples

Fischer equation, nonlinear PDE for modeling population growth. \(u_tDu_{xx}=r u(x,t) (1\frac{u(x,t)}{K})\) where \(K\) is carrying
capacity, and \(r\) is growth rate.
2.3 Hyperbolic
Some properties

Advection PDE (or Transport or convection?). \(u_t+a u_x=0\), Transport or drift of conserved substance
(pollutant) in Fluid or Gas where \(a\) is speed of ﬂuid. Analytic solution is \(u(x,t)=f(xat)\) where \(f(x)=u(x,0)\) is the initial
conditions.

The state variable is the concentration \(u\) of the contaminant, and the ﬂow quantity is its ﬂux
\(\phi \). The constitutive law is \(\phi =cu\).

Wave equation \(u_{tt}=c^2 u_{xx}\). Analytic solution is \(u(x,t)=\frac{1}{2} [f(xct)+f(x+ct)]+\frac{1}{2c} \int _{xct}^{x+ct} \! g(y) \, \mathrm{d}y\) where \(f(x)=u(x,0)\) and \(g(x)=u_t(x,0)\).
Examples

Advection, Wave (See above)

nonhomogenouse advection and wave: \(u_t+a u_x=f(x,t)\) and \(u_{tt}=c^2 u_{xx}+f(x,t)\).

KleinGordon \(u_{tt}=c^2 u_{xx}bu\)

Telegraphy \(u_{tt}+ku_t=c^2 u_{xx}+bu\)
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