singularity problems with Chebyshev approximations
- From: "danjel" <danlu025@xxxxxxxxx>
- Date: 2 Nov 2005 06:36:07 -0800
Math Group,
I'm trying to solve a system of differential equations using Chebyshev
approximation. The problem is that I'm using cylindrical geometry and
therefore the factor 1/r turns up in a number of places and causes
singularity problems.
A basic example:
du(t,r)/dt = (1/r)*u
u(0,r) = 1
would result in the solution u(t,r) = exp(t/r)
In my Maple code I've designed routines for integrating,
differentiating, multiplying.. Chebyshev polynomials (or the Cheb.
coefficients to be more exact).
The problem is that I can't think of way to solve the problem above,
since I can't Cheb. approximate the (1/r) factor and then Cheb.multiply
it with u.
Without this factor the problem would simply have been solved by
du(t,r)/dt = u
u(0,r) = 1
solution u(t,r) = exp(t)
1. Cheb.integration of the rhs, say getting the Cheb.coeff A[i,j]
2. Applying the initial condition
3. Solving the set of equations generated when setting the lhs
cheb.coefficients, say a[i,j] equals the A[i,j]:s
Done!
I've tried to substitute the (1/r)*u to a new variable, say ru, solved
the problem and then related the new variable to u by cheb.mult r*ru =
u, but I don't get the right answer!
The routines I've used are derived from Numerical Recipes, and I've
done the programming in Maple.
What do you think would work?
Cheers,
Daniel
.
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