## difficulties in solving following PDE

Dear Friends,
I encountered with some difficulties in solving following PDE (off course, analytically not numerically), so I would really appreciate it if you help me in this matter.
The PDE is: Uzz+f(t)*Uz=g(t)*Ut
where U(z,t), f(t), and g(t)

B.Cs and I.C are:
U(0,t)=b;
U(infinity,t)=a;
U(z,0)=a;
where a & b are constant.
I am looking forward to hearing from you,
Many Thanks,
Best Regards,
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 It would help if you specify f(t) and g(t)
 U zz+ f(y) ux = g(y)Uy Uzz+ f(y)Ux-g(y) Uy=0 Since f(y) and g(y) are constants we can use b^2-4ac to determine the characteristics don know whether am on right path

## difficulties in solving following PDE

It isn't difficult to find a formal expression for the general solution of the PDE. (see attachment). As usual, the main difficulty is to find the solution fitting with the boundary conditions, among the infinity of solutions provided by the general formula.
As expected, the formal expression includes two arbitrary functions, namely alpha() and beta() with our natations.
The main problem will be to find what are thoses functions alpha() and beta() in order to fulfil the given boundary conditions. As far as the formula contains some functions, f(t) and g(t) which aren't specified, this is impossible. And even if they were specified, this would be probably very difficult, except by chance in some particular cases of f(t) and g(t).
Attached Thumbnails