# Collocation method

A projection method for solving integral and differential equations in which the approximate solution is determined from the condition that the equation be satisfied at certain given points. For example, for the approximate solution of the integral equation one chooses a certain -parameter family of functions and certain points (collocation knots, collocation nodes) on the interval . The approximate solution is determined from the conditions which represent a system of equations in the unknowns . If the equation is linear and the approximate solution is sought in the form of a linear combination of the given (so-called coordinate) functions , then the system of equations in will also be linear.

## Convergence of the collocation method for linear boundary value problems.

Suppose that for on the following boundary value problem is posed: (1) (2)

One looks for an approximate solution of this problem in the form where the are polynomials of degree satisfying the boundary conditions (2). The coefficients are determined from the linear system (3)

with Chebyshev nodes , . The following theorem  holds. Suppose that the functions and , , are continuous on and that the boundary value problem (1), (2) has a unique solution . Then there exists an such that for the system (3) is uniquely solvable and   where , Similar results hold (see ) if the nodes are roots of orthogonal polynomials with respect to some weight function. For equally-spaced nodes the above method diverges. Effective computational schemes with coordinate spline functions have also been developed (see , , ).