5. Partial Differential Equations (PDE)

Waves: a hyperbolic-advective process

Most important in physics: quasilinear PDEs of second order:

$$a_{11} \frac{\partial^{2} u}{\partial x^{2}}+ 2 a_{12} \frac{\par... ... y^{2}}+ f(x,y,u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}) = 0$$

hyperbolic:      $a_{11}a_{22}-a_{12}^{2}<0$ (e.g. $a_{12}=0,\;a_{11}a_{22}<0$)
parabolic: $a_{11}a_{22}-a_{12}^{2}=0$ (or $a_{12}=0,\;a_{11}a_{22}=0$)
elliptic: $a_{11}a_{22}-a_{12}^{2}>0$ (or $a_{12}=0,\;a_{11}a_{22}>0$)

Examples:

hyperbolic	$c^{2} \frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} - \frac{\textstyle \partial^{2}u}{\textstyle \partial t^{2}}=f(x,t)$	Wave equation
	$c^{2} \frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} - \frac{\texts... ...\partial t^{2}} -a\,\frac{\textstyle \partial u}{\textstyle \partial t} =f(x,t)$	Wave with damping
parabolic	$D \, \frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}}- \frac{\textstyle \partial u}{\textstyle \partial t} =f(x,t)$	Diffusion equation
	$\frac{\textstyle \hbar^{2}}{\textstyle 2m} \, \frac{\textstyle \partial^{2}u}{... ...^{2}} +i\hbar \, \frac{\textstyle \partial u}{\textstyle \partial t}-U(x)\,u =0$	Schroedinger equation
elliptic	$\frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}}+ \frac{\textstyle \partial^{2} u}{\textstyle \partial y^{2}} =- \rho(x,y)$	Potential equation
	$\frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} +\frac{\textstyle \p... ...xtstyle \partial y^{2}} -\frac{\textstyle 2m}{\textstyle \hbar^{2}}\,U(x)\,u =0$	Schroedinger equation,
	$({\rm or}\;\dots =\epsilon \, u)$	stationary case

$\parbox{8em}{hyperbolic \\ parabolic \\ \mbox{} \\ elliptic \\ }$$\parbox{2.5em}{\huge \} \\ \vspace{2ex} }$ $\parbox{15.5em}{$\Longleftrightarrow \;$\ initial value problems \\ \mbox{} \\ [12pt] $\Longleftrightarrow \;$\ boundary value problems \\ }$

Conservative hyperbolic and parabolic equations, describing the transport of conserved quantities, may be written as

$$\fbox{$ \displaystyle \frac{\partial u}{\partial t} = - \mbox{$\bf \nabla$} \cdot \mbox{$\bf j$} $}$$

where $u(\mbox{$\bf r$},t)$ (scalar or vector) is the density of a conserved quantity, and $\mbox{$\bf j$}(\mbox{$\bf r$},t)$ the respective local ``flux density'', or ``current density''.

Proof:

Let the transported quantity (mass, energy, momentum, charge, etc.) be conserved as a whole.
$\Longrightarrow$Law of continuity leads to conservative (hyperbolic or parabolic) equations.

Figure 5.1: Derivation of the conservative PDE

Spatial distribution: ``density'' $u(\mbox{$\bf r$},t)$.

Total amount in a volume $V$: $M_{V}(t) \equiv \int\limits_{V} u(\mbox{$\bf r$},t) \, d\mbox{$\bf r$}$

``Flux'' $J$ through the surface $S$: net amount entering $V$ per unit time.

``Flux density'', or ``current density'' $\mbox{$\bf j$}(\mbox{$\bf r$},t)$: local contribution to the total influx (see Figure):

$$J \equiv - \int\limits_{O} \mbox{$\bf j$} (\mbox{$\bf r$},t) \cdot d \mbox{$\bf S$} \;\;\; {\rm (per \; def.)}$$

$$= - \int\limits_{V} (\mbox{$\bf\nabla$} \cdot \mbox{$\bf j$})\, d \mbox{$\bf r$} \;\;\;\; {\rm (Gauss \; law)}$$

Continuity equation:

$$\frac{dM_{V}}{dt} = J \;\;\;\; {\rm or} \;\;\;\; \int\limits... ...\bf\nabla$} \cdot \mbox{$\bf j$}\, \right] d\mbox{$\bf r$} = 0$$

Thus

$$\fbox{$ \displaystyle \frac{\partial u}{\partial t} = - \mbox{$\bf \nabla$} \cdot \mbox{$\bf j$} $}$$

Usually $\mbox{$\bf j$}$ does not depend explicitly on $\mbox{$\bf r$}$ and $t$, but only implicitly via $u(\mbox{$\bf r$},t)$ or its spatial derivative, $\mbox{$\bf\nabla$} u(\mbox{$\bf r$},t)$:

$$\mbox{$\bf j$} = \mbox{$\bf j$}(u) \;\;\; {\rm or} \;\;\; \mbox{$\bf j$} = \mbox{$\bf j$}(\mbox{$\bf\nabla$} u)$$

• $\mbox{$\bf j$}=\mbox{$\bf j$}(u)$: conservative-hyperbolic equation
  
  $$\fbox{$ \displaystyle \frac{\partial u}{\partial t} = - \mbox{$\bf \nabla$} \cdot \mbox{$\bf j$}(u) $}$$
  
  

• $\mbox{$\bf j$} = \mbox{$\bf j$}(\mbox{$\bf\nabla$} u)$: conservative-parabolic equation
  
  $$\fbox{$ \displaystyle \frac{\partial u}{\partial t} = \frac{... ...\partial t} = \lambda \frac{\partial^{2} u}{\partial x^{2}} $}$$
  
  

Examples:

(1) Consider the electromagnetic wave equation in 2D:

$$\frac{\partial^{2} E_{y}}{\partial t^{2}} = c^{2} \frac{\partial^{2} E_{y}}{\partial x^{2}}$$

which is equivalent to

$$\frac{\partial E_{y}}{\partial t} = c\,\frac{\partial B_{z}}{\partial x} \;\;\; \frac{\partial B_{z}}{\partial t} = c\,\frac{\partial E_{y}}{\partial x}$$

$\Longrightarrow$conservative-hyperbolic, with $u \equiv \mbox{$\bf u$} = (E_{y}, B_{z})$, and $j \equiv \mbox{$\bf j$}(\mbox{$\bf u$}) = -c\, (B_{z}, E_{y})$.



(2) Consider the diffusion equation in 1D:

$$\frac{\partial u}{\partial t} = D \frac{\partial^{2} u}{\partial x^{2}} \equiv \frac{\partial}{\partial x} (D \frac{\partial u}{\partial x})$$

$\Longrightarrow$conservative-parabolic, with $j \equiv j(\nabla u) = D \partial u / \partial x$.