## Wednesday, September 21, 2011

### Partial Differential Equations - Types

One very important step for solving, working with, and understanding partial differential equations (PDEs) is knowing the classifications of types.

A general second order PDE can be written as (Özişik 1993Tannehill et al 1997, Hoffman 2001, Chung 2002, Arfken & Weber 2005, Chung 2010):

$A \dfrac{\partial^2 \phi}{\partial x^2} + B \dfrac{\partial^2 \phi}{\partial x \partial y} + C \dfrac{\partial^2 \phi}{\partial y^2} + D \dfrac{\partial \phi}{\partial x}+ E \dfrac{\partial \phi}{\partial y} + F \phi + G \left( x, y \right) = 0$

The types of PDEs is based on the coefficients $$A$$,  $$B$$, and  $$C$$ which produces the characteristic equations also known as discriminants:

$\begin{array}{lll} B^2 - 4AC < 0 & \text{Elliptic} & \text{Complex characteristic curves} \\ B^2 - 4AC = 0 & \text{Parabolic} & \text{Real and repeated characteristic curves}\\ B^2 - 4AC > 0 & \text{Hyperbolic} & \text{Real and distinct characteristic curves} \end{array}$

Some examples of each type of equation are:

Elliptic:

Laplace's Differential Equation:

$\nabla^2 \phi = 0$

in 2-d Cartesian:

$\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} = 0$

$\dfrac{\partial^2 T}{\partial x^2} + \dfrac{\partial^2 T}{\partial y^2} = 0$

Poisson's Differential Equation (non-homogeneous Laplace Equation):

$\nabla^2 \phi = f$

in 2-d Cartesian:

$\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} = f \left( x, y \right)$

Parabolic:

$\dfrac{\partial \phi}{\partial t} = \alpha \nabla^2 \phi = 0$

in 1-d Cartesian:

$\dfrac{\partial \phi}{\partial t} = \alpha \dfrac{\partial^2 \phi}{\partial x^2} = 0$

Hyperbolic:

Wave Equation:

$\dfrac{\partial^2 \phi}{\partial t^2} = c \nabla^2 \phi = 0$

in 1-d Cartesian:

$\dfrac{\partial^2 \phi}{\partial t^2} = c \dfrac{\partial^2 \phi}{\partial x^2} = 0$

Physical classification breaks up PDE's for further understanding (Tannehill et al 1997, Hoffman 2001).

Equilibrium Problems:

The first is called an equilibrium problem.  The solution of the PDE must be satisfied in a closed domain where conditions are set on the boundary.  Equilibrium problems are also known as boundary value problems and include examples such as steady-state temperature distributions, incompressible inviscid flows, and equilibrium stress distributions in solids.  Dominated by elliptical equations, equilibrium problems are also called jury problems as the boundary conditions determine the solution of every point in the domain.

Marching Problems:

Other equations of note:

First-Order Linear Wave Equation or the Advection Equation:

$\dfrac{\partial u}{\partial t} + c \dfrac{\partial u}{\partial x} = 0$

Inviscid Burgers Equation or Nonlinear First-Order Wave Equation:

$\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} = 0$

Burgers' Equation or Nonlinear Wave Equation with Diffusion:

$\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} = v \dfrac{\partial^2 u}{\partial x^2}$

Tricomi Equation with Diffusion:

$y \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$

References:

Arfken, G. B. & Weber, H.-J. Mathematical Methods for Physicists (6th ed.). Elsevier Academic Press. Burlington, MA. 2005

Chung, T. J. Computational Fluid Dynamics, Cambridge University Press, Cambridge, UK. 2002

Chung, T. J. Computational Fluid Dynamics, 2nd ed. Cambridge University Press, Cambridge, UK. 2010

Hoffman, J. D. Numerical Methods for Engineers and Scientists. CRC Press, Boca Raton, FL, 2nd edition, 2001.

Özişik, M. N. Heat Conduction. John Wiley & Sons, Inc., New York, NY, 2nd edition, 1993.

Smith, G. D. Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford Applied Mathematics & Computing Science Series. Oxford University Press, New York, NY, 3rd edition, 1985.

Tannehill, J. C., Anderson, D. A., and Pletcher, R. H. Computational Fluid Mechanics and Heat Transfer, 2nd ed. Taylor & Francis  Hemisphere, New York, NY. 1997