Problems solving a Partial differential equation using DSolve or NDSolve

Questions
1

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

2

Perhaps you meant c[0, t] == DiracDelta[t]? Your boundary data are singular at z=t=0, as well as your explicit solution. The following numeric variant with nonsingular and consistent boundary data works:

With[{d = 1, r = 1}, 
 NDSolve[{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], c[z, 0] == 1/(1 + z^2), c[0, t] == 1/(1 + t^2),
    c[10, t] == 1/(1 + 10^2)}, c, {z, 0, 10}, {t, 0, 10}]]
3

Gianluca, the solution doen´t match with I have obtained using Laplace Method.

Additional information:

One of the boundary contions is c[0, t] == DiracDelta[z], at least so it is shown in the original paper (Find attached in jpf, option: “mixed” boundary condition).