Pde problem: inspiration needed

[Hyperreality]

$$u_{a}+u_{t}=-\mu t_{u}$$
$$u(a,0)=u_{0}(a)$$
$$u(0,t)=b\int_0^\infty \left u(a,t)da$$

Solve $$u(a,t)$$ for the region $$a<t$$

Got this question from assignment. My solution is incomplete though, need some inspirations! I have shown that the general solution is

$$u(a,t)=F(a-t)e^{-1/2{\mu}t^{2}}$$

So for u(0, t) we have

$$u(0,t)=F(-t)^{-1/2{\mu}t^{2}}=b\int_0^\infty \left u(a,t)da$$

Substitute the general solution u(a,t) we get

$$F(-t)=b\int_0^\infty \left F(a-t)da$$

Now -t -> a-t, and we obtain F(a-t) to be

$$F(a-t)=b \int_0^\infty \left F(2a-t)da$$

Hence our solution u(a,t) for a<t is

$$u(a,t)=be^{-1/2{\mu}t^{2}}\int_0^\infty\left F(2a-t)da$$

However, when I talked to my lecturer, he told me I can simplify this further. I need some inspirations

 

[JustinLevy]

Your general solution has an arbitrary function F.
But as you can see in your steps there, the boundary conditions on u have given you quite a bit of information on F. So you can restrict it further.

So here is some inspiration:
-What is F(0) equal to?
- Note you obtained one equation with F on both sides. Can you solve for F given what F(0) is?

Good luck with your homework.