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Convolution in time
Posted Feb 5, 2010, 7:07 p.m. EST 0 Replies
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I'm looking at a first order PDE where the source term has a convolution in time, not space,
u_t + u_x = int_0^t u(s) Q(t - s) ds
where Q is a given, positive, function. Any recomendations (other than differentiating everything in time and solving a second order equation). To make life a little more tricky, actually I want,
u_t + u_x = int_0^t u_s Q(t - s) ds
where there's a derivative on the u before convolution. Any suggestions?
I've seen a neat trick posted on the forum to solve u(x) = int_0^t f(x,y) dy, namely solve the pde 0 = u_y+f but I don't see how that works when you have a convolution.
I've attached my first pass, Geom1 solves the 1D transient problem as a 2D first order. It seems to give the right answer when I give it some simple force functions. Then I've also created Geom5 which is a prism with axes x,s and t. I've got an extrusion of u_s from Geom1 into Geom5 and then to come back agin I do a projection integral and I can put that as the force term for the PDE.
Answer looks distinctly wrong, so I've got some more debugging to do, but either way, the program is really, really slow because of all the extrusion and projection, I guess.
Regards, John
Hello John
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