Classifications of PDE and boundary conditions
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==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, <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
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 (AC − B2 > 0), parabolic (AC − B2 = 0) or hyperbolic (AC − B2 < 0) equation.
See Main Article Classification of PDEs
Classification of Boundary Conditions
See Main Article Classification of boundary conditions