Mathematica Mathematica: Second order PDE

I am trying to solve the following second order PDE (copy that into mathematica):

\*SubscriptBox[\(\[PartialD]\), \(x, t\)]\(\[Delta][x, t]\)\) + b \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Delta][x, t]\)\) + a \!\(
\*SubscriptBox[\(\[PartialD]\), \(\(x\)\(\ \)\)]\(\[Delta][x,
t]\)\) == 0

where a and b are (real) constants and x and t both run from 0 to infinity

My bc is \[Delta][0, t] == k, where k is any (real) constant.
My ic is \[Delta][x, 0] == 0.

(Obviously there is a slight ambiguity going on there at x=0 at t=0.)

Well.. I can't seem to do it using either DSOLVE or NDSOLVE

Thanks in advance for any help,

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