Well I thought I had a simple problem to solve using maple: a ﬁrst order linear pde of a function of two variables: n(x,t). This pde has the form:
dn/dt + v*dn/dx+n/tau=G0*Heaviside(t)
(the derivitives are in fact partial derivitives)
with boundary conditions: n(x,0)=0 and n(0,t)=0. Note v and G0 are positive constants.
When I invoke pdsolve in R5 (Mac) I get a solution with an unknown function of \(x\) and \(t\):
The form is not useful to me as I do not know _F1. How do I insert the above boundry conditions into pdsolve or is pdsolve strictly for symbolic solutions?
Should I avoid using pdsolve and use Laplace transforms instead?
pdsolve doesn’t always ﬁnd enough solutions for arbitrary boundary values, but in this case it does work. I assume you’re interested in the solution for t > 0 and x > 0.
For the boundary condition at t=0:
Note that this determines _F1(s) for s <= 0.
For the boundary condition at x=0:
And this tells you _F1(t) for t >= 0.
PDEtools is in shareware if you can’t ﬁnd it elsewhere (eg, in R4).
Yes, substitute your BC in the solution and solve for _F1. Basically you have two equations for it and both must be satisﬁed.
The general solution to your diﬀerential equation is indeed
Using the ﬁrst boundary condition n(x,0) = 0 implies _F1(x) = 0 for arbitrary \(x\), so that _F1=0.
This gives the solution G0*tau*(exp(t/tau)-1)*Heaviside(t).
Then you add another boundary condition n(0,t) = 0 , which would lead to an impossibility, unless you assume \(t<0\).
I am wondering about your boundary conditions: aren’t they too many for a ﬁrst-order diﬀerential equation?