
1342 LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH MATHEMATICA
Example 23.14. First-order linear equation. Method of characteristics. Classical Cauchy
problem. Let us solve the first-order linear equation with the Cauchy data
u
t
−xu
x
= u, u(x, 0) = f (x)
by th e m ethod of characteristics. This equation can be obtaine d from the Fokker–Planck equatio n
[see Blu man e t al. (2010)]
3
u
t
= u
xx
+ (xu)
x
,
by neglecting the term u
xx
.
We can obtain the solution of this Cauc hy problem
u
1
(x,t) = f (xe
t
)e
t
and plot the characteristic curves as follows:
SetOptions[ParametricPlot,ImageSize->300,PlotStyle->{Hue[0.7],Thickness[0.001]}];
{tF=Pi/2, xF=2*Pi, tr1=x->x[t], tr2=x[t]->x, ...