# Classification of PDEs

A general transport equation, whether it is mass, momentum, energy, or species, can be written as:

$\frac{\partial }{{\partial t}}\left( {\rho \Phi } \right) + \nabla \cdot \left( {\rho {\mathbf{V}}\Phi } \right) = \nabla \cdot \left( {\Gamma \nabla \Phi } \right) + F\left( {{\mathbf{x}},t,\Phi , \ldots } \right) \qquad \qquad(1)$

where the dependent variable, Φ, is one for mass, any component of velocity (u,v,w), enthalpy h, or the species mass fraction ωi. The gradient operator is the partial derivative of the dependent variable with respect to all the spatial directions x for a given coordinate system.

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

$A\frac{{{\partial ^2}\Phi }}{{\partial {\eta ^2}}} + 2B\frac{{{\partial ^2}\Phi }}{{\partial \eta \partial \zeta }} + C\frac{{{\partial ^2}\Phi }}{{\partial {\zeta ^2}}} + D\frac{{\partial \Phi }}{{\partial \eta }} + E\frac{{\partial \Phi }}{{\partial \zeta }} + F\Phi + G = 0 \qquad \qquad(2)$
Propagation of disturbances for different types of PDEs displayed as the (a) Zone of Dependence, the (b) Zone of influence, and (c) a physical example.

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.

The classification of a PDE describes how disturbances or changes propagate through a domain. If you are interested in a single point on the domain, you need to know two things: what region in the domain affects that point, and, if you do something at that point, what region in the domain it will affect. What region affects a point and what region that point affects are called the zone of dependence and the zone of influence, respectively. A representation of the zone of dependence and the zone of influence of elliptic, parabolic, and hyperbolic equations are displayed in the figure on the right. A multidimensional PDE that varies with time may be classified by looking at only the relation of two independent variables at a time. For instance look at how the PDE would be classified only in the x and y directions. Then examine the PDE in x and t. Finally examine the PDE in y and t. The zone of dependence is where the zone of dependence for the three cases intersects. The zone of influence is where the zone of influence for the three cases intersects.

An example of an elliptic equation is the Laplace equation. This equation governs physical problems, such as two-dimensional steady-state heat conduction, electrical potential, the free stream characteristics in boundary layer equations, and the pressure field in a porous medium with Darcy’s assumption. The independent variables, η and ζ, are spatial coordinates in these problems, and Φ is the potential field, representing temperature, voltage, the stream function, or pressure, respectively, for the given cases.

Some examples of parabolic problems are unsteady heat conduction, boundary layer equations for momentum, energy and species as well as small vibrations in an elastic beam. For the unsteady heat equation, η is a spatial coordinate, and ζ is time. For the boundary layer equations, η is a spatial coordinate normal to the free stream velocity, and ζ is a spatial coordinate parallel to the free stream.

A hyperbolic function is often called the wave equation. The most common examples of the wave equation are the propagation of shock waves and sound waves. The interface between phases is also hyperbolic. Another less intuitive example of the use of a hyperbolic function is heat conduction in micro- and nanoscale. It becomes relevant in these situations because heat actually conducts at a finite rate, and the limits of these rates are relevant at such small scales.

## 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.