# Approach to 'double convolution'?

1. Oct 22, 2006

### lutzB

Hello,

I am trying to examine the following differential equation:

diff(M(a), a) =
Lambda * ( 1 -
epsilon * int(M(b)*exp(-sigma*(a-b)), b = 0 .. a) ) -
mu * M(a)

I found out that this kind of equation can be solved for M(a) by applying Laplace-Transformation, solving for Laplace(M(a)) and then applying inverse Laplace-Transformation. This works quite fine e.g. in Maple.

Now, I would like to extend above equation by sort of adding a second convolution term, so that

diff(M(a), a) =
Lambda * ( 1 -
epsilon * int(M(b)*exp(-sigma*(a-b)), b = 0 .. a) ) +
gamma * int(M(b)*exp(-tau*(a-b)), b = 0 .. a) ) -
mu * M(a)

But apparently this is not solvable in the same way. At least, Maple won't find the inverse Laplace that I need for the final solution.

I would appreciate very much any hints regarding whether a differential equation like this should be solvable by this method, or if there are other methods I could try, or if it simply cannot be solved?

Many thanks and best wishes-
Lutz