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

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 *u*_{i}(*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 *x*_{i}, *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 *c*_{i}.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 *c*_{i}.(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 *u*_{0}(*x*) = 0, *u*_{1}(*x*) = 1 - *x*^{2} and *u*_{2}(*x*)
= *x*^{2}(1 - *x*^{2}), 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 *c*_{1} = 0.957 and *c*_{2} = -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 - 6*x*. Select as collocation
points *x*_{1} = 0.25 and *x*_{2} = 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 *c*_{l} and *c*_{2} 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: