Classifications of PDE and boundary conditions

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''See Main Article'' [[Classification of PDEs]]
''See Main Article'' [[Classification of PDEs]]
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===Classification of Boundary Conditions===
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==Classification of Boundary Conditions==
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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 )
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</math>, or  <math>a(\partial \Phi /\partial \zeta ) + b\Phi  = f(\eta ) </math>] type.
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''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

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

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