Expansion methods

The finite-difference methods of previous chapters have been based essentially on the idea of approximating functions by polynomials, and in fact such methods clearly give exact solutions when these are polynomials of appropriate degree. For more general functions, local truncation errors depend on the accuracy with which the solution can be approximated by a polynomial. It is sometimes more convenient to make quite explicit this relation with polynomials, and we may then be able to calculate the coefficients b r of an approximating expansion (7.1) $$y(x) = \sum\limits_{r = 0}^\infty {b_r x^r }, $$ rather than working with a representation of the approximate solution at a set of discrete points. For some simple classes of problems these methods are particularly useful for both initial-value and boundary-value problems.