9.5 Section 2 Background .1 Cosmology .11 Math Chapter 10

Book 9 Computational Mathematics Chapter 9 Ordinary Differential Equation Initial Value Problems

Section 6 The collocation method

Look for the solution of the boundary value problem (1.3) and (1.4) in the form

(6.1)

where ui(x). i = 0, 1, . . , n are linearly independent functions which satisfy (5.4) and (5.5). Now make the residual

(6.2)

vanish at points xi, i = 0, 1, . . , n of the interval [a, b], the so called collocation points (the number of which must equal the number of the coefficients ci.in (5.1). These coefficients will be determined by the system of equations

(6.3)

This method can also be used for the solution of boundary value problems for non-linear differential equations

(6.4)

with the the conditions (1.4), when the residual assumes the form

(6.5)

and (6.3) becomes a system of non-linear algebraic equations in the ci.(cf. Example 2).

Example 1: Find by collocation an approximate solution of the boundary value problem

(6.6)

Solution:.In general, the form of the equation and the boundary conditions will suggest whether the solution will be even or odd. Use the polynomials u0(x) = 0, u1(x) = 1 - x2 and u2(x) = x2(1 - x2), which obviously fulfill the boundary conditions in (6.6).

Now look for its solution in the form

with the collocation points

Substitution of these values into the residual

yields the equations

with the solution c1 = 0.957 and c2 = -0.022, whence the approximate solution of (6.6) is

Example 2. Solve by collocation the boundary value problem

Solution: Introduce the base functions

and seek the solution in the form

with the residual

where u"1 = -2 and u"2 = 2 - 6x. Select as collocation points x1 = 0.25 and x2 = 0.75 and find by computation of the residues at these points the system of non-linear equations

Rewrite this system in the form

and solve it by iteration using


with the subscript k denoting the iteration step.

First cycle:

Second cycle:


,

Third cycle:

Since the second and third approximations of cl and c2 coincide, one finds accurate within 10-4

Thus, the required approximation is

Note: As is clear from the above examples, the computations involved in collocations are simpler than in the Galerkin method.

Exercises

Use collocation to find approximate solutions of the boundary value problems: