Classifications of PDE and boundary conditions

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==Classification of PDEs==
==Classification of PDEs==
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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, <math>\eta</math> and <math>\zeta</math>, is
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<center><math>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  </math></center>
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This equation is linear if all the coefficients (A, B, C, D, E, F and G) are a function of <math>\eta</math> and <math>\zeta</math>, a constant, or zero.  If they are a function of <math>\Phi</math> 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 <math>\eta</math> and <math>\zeta</math>, a constant, or zero.  If a differential equation is quasilinear, it can be classified as an elliptic (<math>AC - {B^2} > 0</math>), parabolic (<math>AC - {B^2} = 0</math>) or hyperbolic (<math>AC - {B^2} < 0</math>) equation.
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''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]]

Revision as of 09:38, 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

See Main Article Classification of boundary conditions