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Compute derivative and integral weight matrices for orthogonal collocation.
Reference: J. Villadsen, M. L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation.
Here is an example of using
colloc to generate weight matrices
for solving the second order differential equation
u’ - alpha * u” = 0 with the boundary conditions
u(0) = 0 and u(1) = 1.
First, we can generate the weight matrices for n points (including the endpoints of the interval), and incorporate the boundary conditions in the right hand side (for a specific value of alpha).
n = 7; alpha = 0.1; [r, a, b] = colloc (n-2, "left", "right"); at = a(2:n-1,2:n-1); bt = b(2:n-1,2:n-1); rhs = alpha * b(2:n-1,n) - a(2:n-1,n);
Then the solution at the roots r is
u = [ 0; (at - alpha * bt) \ rhs; 1] ⇒ [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]