# Classifications of PDE and boundary conditions

(Difference between revisions)
 Revision as of 09:02, 28 June 2010 (view source) (→Elliptic, Parabolic and Hyperbolic PDEs)← Older edit Revision as of 09:38, 28 June 2010 (view source)Newer edit → Line 1: Line 1: ==Classification of PDEs== ==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, $\eta$ and $\zeta$, 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 $\eta$ and $\zeta$, a constant, or zero.  If they are a function of $\Phi$ 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 $\eta$ and $\zeta$, a constant, or zero.  If a differential equation is quasilinear, it can be classified as an elliptic ($AC - {B^2} > 0$), parabolic ($AC - {B^2} = 0$) or hyperbolic ($AC - {B^2} < 0$) equation. + ''See Main Article'' [[Classification of PDEs]] ''See Main Article'' [[Classification of PDEs]] ===Classification of Boundary Conditions=== ===Classification of Boundary Conditions=== ''See Main Article'' [[Classification of boundary conditions]] ''See Main Article'' [[Classification of boundary conditions]]

## 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 (ACB2 > 0), parabolic (ACB2 = 0) or hyperbolic (ACB2 < 0) equation.

See Main Article Classification of PDEs

### Classification of Boundary Conditions

See Main Article Classification of boundary conditions