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 Part II: Ch. 5

# 5 Partial Differential Equations (PDE)

Most important in physics: quasilinear PDEs of second order:

$\begin{eqnarray} a_{11} \frac{\textstyle \partial^{2} u}{\textstyle \partial x^{2}}+ 2 a_{12} \frac{\textstyle \partial^{2} u}{\textstyle \partial x \partial y}+ a_{22} \frac{\textstyle \partial^{2} u}{\textstyle \partial y^{2}}+ f(x,y,u, \frac{\textstyle \partial u}{\textstyle \partial x}, \frac{\textstyle \partial u}{\textstyle \partial y})&=&0 \end{eqnarray}$

 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 $c^{2} \frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} - \frac{\textstyle \partial^{2}u}{\textstyle \partial t^{2}} -a\frac{\textstyle \partial u}{\textstyle \partial t} =f(x,t)$ 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 $\frac{\textstyle \hbar^{2}}{\textstyle 2m} \frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} +i\hbar \frac{\textstyle \partial u}{\textstyle \partial t}-U(x)u =0$ 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 $\frac{\textstyle \partial^{2}u}{\textstyle \partial x^{2}} +\frac{\textstyle \partial^{2}u}{\textstyle \partial y^{2}} -\frac{\textstyle 2m}{\textstyle \hbar^{2}}U(x)u =0$     (or $\;\dots =\epsilon u$) Schroedinger equation, stationary case

Generally the physical applications may be categorized as follows:

$\begin{eqnarray} \left. \begin{array}{l} {\rm hyperbolic} \\ {\rm parabolic} \\ \end{array} \right\} & \Longleftrightarrow & \; {\rm initial \; value \; problems} \\ \left. \begin{array}{l} {\rm elliptic \;\;\;\;\;\;\;\;} \end{array} \right. & \Longleftrightarrow & \; {\rm \; boundary \; value \; problems} \\ \end{eqnarray}$

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

$\begin{eqnarray} \frac{\textstyle \partial u}{\textstyle \partial t} = - \nabla \cdot j \end{eqnarray}$

where $u(r,t)$ (scalar or vector) is the density of a conserved quantity, and $j(r,t)$ 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: Derivation of the conservative PDE

Spatial distribution: "density" $u(r,t)$.

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

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

"Flux density", or "current density" $j(r,t)$: local contribution to the total influx (see Figure):

$\begin{eqnarray} J & \equiv & - \int\limits_{O} j (r,t) \cdot dS \;\;\; {\rm (per \; def.)} \\ & = & - \int\limits_{V} (\nabla \cdot j) dr \;\;\;\; {\rm (Gauss \; law)} \end{eqnarray}$

Continuity equation:
$\frac{\textstyle dM_{V}}{\textstyle dt} = J \;\;\;\; {\rm or} \;\;\;\; \int\limits_{V} \left[ \frac{\textstyle \partial u}{\textstyle \partial t} + \nabla \cdot j \right] d r = 0$

Thus

$\frac{\textstyle \partial u}{\textstyle \partial t} = - \nabla \cdot j$

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

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

• $j= j(u)$: conservative-hyperbolic equation

$\frac{\textstyle \partial u}{\textstyle \partial t} = - \nabla \cdot j(u)$

• $j = j(\nabla u)$: conservative-parabolic equation

$\frac{\textstyle \partial u}{\textstyle \partial t} = \frac{\textstyle \partial}{\textstyle \partial x} (\lambda \frac{\textstyle \partial u}{\textstyle \partial x}) \;\;\;\; {\rm or} \;\;\; \frac{\textstyle \partial u}{\textstyle \partial t} = \lambda \frac{\textstyle \partial^{2} u}{\textstyle \partial x^{2}}$

Examples:

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

$\begin{eqnarray} \frac{\textstyle \partial^{2} E_{y}}{\textstyle \partial t^{2}} &=& c^{2} \frac{\textstyle \partial^{2} E_{y}}{\textstyle \partial x^{2}} \end{eqnarray}$

which is equivalent to
$\begin{eqnarray} \frac{\textstyle \partial E_{y}}{\textstyle \partial t} = c \frac{\textstyle \partial B_{z}}{\textstyle \partial x} \;\;\;&& \frac{\textstyle \partial B_{z}}{\textstyle \partial t} = c \frac{\textstyle \partial E_{y}}{\textstyle \partial x} \end{eqnarray}$

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

(2) Consider the diffusion equation in 1D:
$\begin{eqnarray} \frac{\textstyle \partial u}{\textstyle \partial t} = D \frac{\textstyle \partial^{2} u}{\textstyle \partial x^{2}} & \equiv & \frac{\textstyle \partial}{\textstyle \partial x} (D \frac{\textstyle \partial u}{\textstyle \partial x}) \end{eqnarray}$

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

### Sections

vesely 2005-10-10

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