5 Partial Differential Equations (PDE)
Waves: a hyperbolic-advective process
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