Scientific Computing

This module illustrates the collocation method for numerically
solving boundary value problems for ordinary differential equations.
A general boundary value problem (BVP) consists of an ordinary
differential *u*″ =
*f*(*t*, *u*, *u*′)*a*, *b*]*u*(*a*) = *α**u*(*b*) = *β**collocation
method* approximates the solution to the BVP by a linear combination
of basis functions determined by requiring that the ODE be satisfied at
each of a discrete set of mesh points within *a*,
*b*]*t*_{i},
*i* = 1,…,*n**t*_{1} = *a**t*_{n} = *b**φ*_{i}, *i* =
1,…,*n**v*(*t*,
** x**)

The user begins by selecting from the menu provided an ODE and a
specific solution to be sought (if there is more than one). Boundary
values *u*(*a*) = *α**u*(*b*) = *β*

After the parameters for collocation have been specified, the user next
clicks *Initialize* to begin the process of solving the BVP. A
predetermined starting guess *x*_{0} is used to
initialize Broyden's method for solving the system of equations, and
the corresponding approximate solution to the BVP,
*v*(*t*, *x*_{0})*Iterate* repeatedly to execute
successive iterations of Broyden's method to solve the system of
equations. For each iterate *x*_{k}, the
corresponding function *v*(*t*,
*x*_{k})

To compare approximate solutions obtained using different basis functions or different numbers or distribution of collocation points, see the alternative Collocation Method module.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 10.5.

**Developers:** Evan VanderZee and Michael Heath