How can I solve the following partial differential equation using DSolve or NDSolve:

equation= D[c[z, t], {z, 2}] - 1/d D[c[z, t], z] - r/d  
c[z, t] == 1/d D[c[z, t], t]

Initial condition c[z, 0] == 0

Boundary conditions:

c[0, t] == DiracDelta[z]; c[Infinity, t] == 0; 

You can take d = 0.1 and r=1

I need the solution for {t, 0, 30} and {z, 0, 1}

Using Laplace Transforms I find that the solution is:

c[z,t]->z/(E^((4*d*r*t^2 + t^2 - 2*t*z + z^2)/(4*d*t))*(2*Sqrt[Pi]*d^2*
      Sqrt[t^3/d^3])); 

but I find problem using DSolve even NDSolve

Any help?

Guillermo