# 4.1.4. ODEs and PDEs¶

## 4.1.4.1. How to decide if a PDE is elliptic, hyperbolic or parabolic?¶

$a {{\partial^2 u} \over {\partial x^2}} + b {{\partial^2 u} \over {\partial x \partial y}} + c {{\partial^2 u} \over {\partial y^2}} = g(x, y)$

or for a $$n \times n$$ system:

${\partial \vec{U} \over \partial t} + \vec{A} {{\partial \vec{U}} \over {\partial x}}=0$

The type depends on the determinant:

• $$b^2-4ac>0 \longrightarrow$$ all n eigenvalues are real $$\longrightarrow$$ Hyperbolic

• $$b^2-4ac=0 \longrightarrow$$ all n eigenvalues are real and identical $$\longrightarrow$$ Parabolic

• $$b^2-4ac<0 \longrightarrow$$ all n eigenvalues are complex $$\longrightarrow$$ Elliptic

## 4.1.4.2. What common flow conditions are hyperbolic, elliptic and parabolic?¶

• Hyperbolic $$\longrightarrow$$ (Steady $$\lor$$ Unsteady) $$\land$$ Compressible $$\land$$ Inviscid $$\land$$ Supersonic

• Parabolic $$\longrightarrow$$ (Compressible $$\lor$$ Incompressible) $$\land$$ Viscous Boundary Layer

• Elliptic $$\longrightarrow$$ (Compressible $$\lor$$ Incompressible) $$\land$$ Inviscid $$\land$$ Steady $$\land$$ Subsonic

• Mixed (most cases are mixed) $$\longrightarrow$$ There is a parabolic boundary layer, but parts of the domain away from the boundary are hyperbolic or elliptic

## 4.1.4.3. What are some examples of hyperbolic, elliptic and parabolic PDEs?¶

• Wave Equation $$\longrightarrow$$ Hyperbolic $$\longrightarrow$$ $${\partial u \over \partial t} + a {{\partial u} \over {\partial x}}=0$$

• Laplace Equation $$\longrightarrow$$ Elliptic $$\longrightarrow$$ $${\partial^2 u \over \partial x^2} + {{\partial^2 u} \over {\partial y^2}}=0$$

• Poisson Equation $$\longrightarrow$$ Elliptic $$\longrightarrow$$ $${\partial^2 p \over \partial x^2} + {{\partial^2 p} \over {\partial y^2}}=f(u)$$

• Stokes Flow $$\longrightarrow$$ Parabolic $$\longrightarrow$$ $${\partial p \over \partial x} = \mu {\partial^2 u \over \partial x^2}$$

• Heat Conduction $$\longrightarrow$$ Parabolic $$\longrightarrow$$ $${\partial T \over \partial t} = \alpha {\partial^2 T \over \partial x^2}$$

• Boundary Layer $$\longrightarrow$$ Parabolic $$\longrightarrow$$ $$\rho \left( {u {\partial u \over \partial x} + v {\partial u \over \partial y}} \right) = \mu {\partial^2 u \over \partial y^2}$$