# Classifications of PDE and boundary conditions

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''See Main Article'' [[Classification of PDEs]] | ''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 [<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]] | ''See Main Article'' [[Classification of boundary conditions]] |

## 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

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 [], the Neumann [, or ], and the mixed [, or ] type.

*See Main Article* Classification of boundary conditions