Interactive Educational Modules in
Scientific Computing

Collocation Method for Boundary Value Problems

This module enables the user to compare different approximate solutions computed using the collocation method for boundary value problems for ordinary differential equations. A general boundary value problem (BVP) consists of an ordinary differential equation (ODE) with side conditions specified at more than one point. This module illustrates the solution of second-order scalar ODEs of the form u″ = f(t, u, u′) on an interval [a, b] with boundary conditions u(a) = α and u(b) = β. The 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], and that the boundary conditions be satisfied. Denote the fixed mesh points by t_i, i = 1,…,n, where t₁ = a, and t_n = b. Denote the basis functions by φ_i, i = 1,…,n. Let v(t, x) denote a linear combination of basis functions φ_i with coefficients x_i. The collocation method seeks a solution x to the system of equations v(t₁, x) = α, v(t_n, x) = β, v″(t_i, x) = f(t_i, v(t_i, x), v′(t_i, x)) for i = 2,…,n − 1. Note that the basis functions must be twice differentiable.

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 and u(b) = β are indicated by black dots on the graph. Next the user chooses either polynomials (Chebyshev polynomials in this module) or cubic B-splines as the set of basis functions. The user also chooses the number and distribution of mesh points. If the cubic spline basis is used, the selected distribution determines the set of knots defining the B-splines as well as the set of collocation points.

When the user clicks Solve the resulting approximate solution to the BVP is drawn on the graph. To compare this solution with other approximate solutions, the user can make additional choices of basis functions and the number and distribution of collocation points, then click Solve to add each new approximate solution to the graph. The solutions for different parameters are color coded so that the color changes from blue to red as the number of collocation points increases and from light to dark as the selected basis and point distribution change. To clear all of the solutions from the graph, click Reset.

To illustrate the details of the individual steps of the collocation method for a particular choice of parameters, 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