Let
![\begin{displaymath}
\tilde L = \tilde D^n + a_{n-1}(t)\tilde D^{n-1} + \ldots + a_1(t)\tilde D + a_0(t)
\end{displaymath}](g_2244.gif) |
(1) |
be a differential Operator in 1-D, with
Continuous for
, 1, ...,
on the interval
, and assume we wish to find the solution
to the equation
![\begin{displaymath}
\tilde L y(t) = h(t),
\end{displaymath}](g_2248.gif) |
(2) |
where
is a given Continuous on
. To solve equation (2), we look for a function
such that
, where
![\begin{displaymath}
y(t)=g(h(t)).
\end{displaymath}](g_2252.gif) |
(3) |
This is a Convolution equation of the form
![\begin{displaymath}
y=g*h,
\end{displaymath}](g_2253.gif) |
(4) |
so the solution is
![\begin{displaymath}
y(t) = \int^t_{t_0} g(t-x)h(x)\,dx,
\end{displaymath}](g_2254.gif) |
(5) |
where the function
is called the Green's function for
on
.
Now, note that if we take
, then
![\begin{displaymath}
y(t)=\int_{t_0}^t g(t-x)\delta(x)\,dx = g(t),
\end{displaymath}](g_2258.gif) |
(6) |
so the Green's function can be defined by
![\begin{displaymath}
\tilde Lg(t)=\delta(t).
\end{displaymath}](g_2259.gif) |
(7) |
However, the Green's function can be uniquely determined only if some initial or boundary conditions are given.
For an arbitrary linear differential operator
in 3-D, the Green's function
is defined by
analogy with the 1-D case by
![\begin{displaymath}
\tilde LG({\bf r},{\bf r}') = \delta({\bf r}-{\bf r}').
\end{displaymath}](g_2261.gif) |
(8) |
The solution to
is then
![\begin{displaymath}
\phi({\bf r}) = \int G({\bf r},{\bf r}')f({\bf r}')\,d^3{\bf r}'.
\end{displaymath}](g_2263.gif) |
(9) |
Explicit expressions for
can often be found in terms of a basis of given eigenfunctions
by expanding
the Green's function
![\begin{displaymath}
G({\bf r}_1,{\bf r}_2)=\sum_{n=0}^\infty a_n({\bf r}_2)\phi_n({\bf r}_1)
\end{displaymath}](g_2265.gif) |
(10) |
and Delta Function,
![\begin{displaymath}
\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty b_n\phi_n({\bf r}_1).
\end{displaymath}](g_2266.gif) |
(11) |
Multiplying both sides by
and integrating over
space,
![\begin{displaymath}
\int\phi_m({\bf r}_2)\delta^3({\bf r}_1-{\bf r}_2)\,d^3{\bf ...
...\infty b_n\int\phi_m({\bf r}_2)\phi_n({\bf r}_1)\,d^3{\bf r}_1
\end{displaymath}](g_2268.gif) |
(12) |
![\begin{displaymath}
\phi_m({\bf r}_2)=\sum_{n=0}^\infty b_n \delta_{nm} = b_m,
\end{displaymath}](g_2269.gif) |
(13) |
so
![\begin{displaymath}
\delta^3({\bf r}_1-{\bf r}_2)=\sum_{n=0}^\infty \phi_n({\bf r}_1)\phi_n({\bf r}_2).
\end{displaymath}](g_2270.gif) |
(14) |
By plugging in the differential operator, solving for the
s, and substituting
into
, the original nonhomogeneous equation then can be solved.
References
Arfken, G. ``Nonhomogeneous Equation--Green's Function,'' ``Green's Functions--One Dimension,'' and
``Green's Functions--Two and Three Dimensions.''
§8.7 and §16.5-16.6 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 480-491 and 897-924, 1985.
© 1996-9 Eric W. Weisstein
1999-05-25