Classifications of PDE and boundary conditions

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===Elliptic, Parabolic and Hyperbolic PDEs===
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==Classification of PDEs==
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''See Main Article'' [[Classifications 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]]
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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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===Classification of Boundary Conditions===
 
''See Main Article'' [[Classification of boundary conditions]]
''See Main Article'' [[Classification of boundary conditions]]
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==References==
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Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA
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Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO.
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==Further Reading==
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==External Links==

Current revision as of 13:41, 5 August 2010

Contents

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

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Further Reading

External Links