Classifications of PDE and boundary conditions

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Revision as of 09:45, 28 June 2010

Classification of PDEs

A partial differential equation (PDE) is an equation of a function and its partial derivatives. In general, a PDE is classified by its linearity or nonlinearity, and by its order. Its order is considered by its highest derivative. A general second order PDE for two independent variables, $η$ and $ζ$ , is

$A\frac{{{\partial ^2}\Phi }}{{\partial {\eta ^2}}} + 2B\frac{{{\partial ^2}\Phi }}{{\partial \eta \partial \zeta }} + C\frac{{{\partial ^2}\Phi }}{{\partial {\zeta ^2}}} + D\frac{{\partial \Phi }}{{\partial \eta }} + E\frac{{\partial \Phi }}{{\partial \zeta }} + F\Phi + G = 0$

This equation is linear if all the coefficients (A, B, C, D, E, F and G) are a function of $η$ and $ζ$ , a constant, or zero. If they are a function of $Φ$ or any of its derivatives, then the PDE is nonlinear. A partial differential equation is called quasi-linear if it is linear in the highest derivatives. In eq. (2), this means that A, B and C are a function of $η$ and $ζ$ , a constant, or zero. If a differential equation is quasilinear, it can be classified as an elliptic ( $A C - B 2 > 0$ ), parabolic ( $A C - B 2 = 0$ ) or hyperbolic ( $A C - B 2 < 0$ ) equation.

See Main Article Classification of PDEs

Classification of Boundary Conditions

For any problem to be well defined, there are boundary/initial conditions that must be applied. There are three basic boundary conditions for second order PDEs. These boundary conditions are the Dirichlet [ $\Phi = f\left( {\eta ,\zeta } \right)$ ], the Neumann [ $\partial \Phi /\partial \eta = f\left( \zeta \right)$ , or $\partial \Phi /\partial \zeta = f\left( \eta \right)$ ], and the mixed [ $a\,(\partial \Phi /\partial \eta ) + b\Phi = f(\zeta )$ , or $a(\partial \Phi /\partial \zeta ) + b\Phi = f(\eta )$ ] type.

See Main Article Classification of boundary conditions

Retrieved from "http://wwww.thermalfluidscentral.org/encyclopedia/index.php/Classifications_of_PDE_and_boundary_conditions"

@@ Line 8: / Line 8: @@
 ''See Main Article'' [[Classification of PDEs]]
-===Classification of Boundary Conditions===
+==Classification of Boundary Conditions==
+For any problem to be well defined, there are boundary/initial conditions that must be applied. There are three basic boundary conditions for second order PDEs.  These boundary conditions are the Dirichlet [<math>\Phi  = f\left( {\eta ,\zeta } \right)</math>], the Neumann [<math>\partial \Phi /\partial \eta  = f\left( \zeta  \right)</math>, or  <math>\partial \Phi /\partial \zeta  = f\left( \eta  \right)</math>],  and the mixed [<math>a\,(\partial \Phi /\partial \eta ) + b\Phi = f(\zeta )
+</math>, or  <math>a(\partial \Phi /\partial \zeta ) + b\Phi  = f(\eta ) </math>] type.
 ''See Main Article'' [[Classification of boundary conditions]]

Classifications of PDE and boundary conditions

From Thermal-FluidsPedia

Revision as of 09:45, 28 June 2010

Classification of PDEs

Classification of Boundary Conditions

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