A general second order PDE can be written as (Özişik 1993, Tannehill 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:
\[ \dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} = 0 \]
Steady Heat Diffusion Equation:
\[ \dfrac{\partial^2 T}{\partial x^2} + \dfrac{\partial^2 T}{\partial y^2} = 0 \]
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:
Heat or Unsteady Diffusion Equation:
\[ \dfrac{\partial \phi}{\partial t} = \alpha \nabla^2 \phi = 0 \]
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:
\[ \dfrac{\partial u}{\partial t} + c \dfrac{\partial u}{\partial x} = 0 \]
\[ \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} = 0 \]
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
No comments:
Post a Comment