## Info

Mtdt = —[lim m(x, t) — m(x, 0)] = —w(x) (8.63)

and combining this with Eq. (8.62) we conclude that

We are now going to understand certain properties of w(x), the solution of Eq. (8.60) without actually solving it. To do so, we shall find it handy to employ Dawson's integral (Abramowitz and Stegun (1974); it is also kind of fun to do a web search with key words "Dawson's Integral''):

D( y) = exp(-y2) so that we can rewrite w(x) as exp(s2)ds

Now recall that the main contribution to the integral component of Dawson's integral will come from the end point (and to leading order is (1/2y)exp(y2) ) so that D(y) ~ 1/2y when y is large. Using this relationship allows us to rewrite Eq. (8.66) as w(x)~2 A ^—f) d( -L

Using an integrating factor, we can rewrite Eq. (8.67) as d dx hxexp( — 7

We integrate this equation to obtain

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