Overview

The differential equations that arise in practice are usually of first, second, third, or fourth order. In this chapter we consider some fourth-order problems, both because they are interesting in their own right and because this will give us further practice with the kinds of manipulations of differential operators and boundary conditions that arise ubiquitously in spectral methods.

We begin with a one-dimensional example. Suppose that we wish to solve the biharmonic problem

Physically, u( x) might represent the transverse displacement of a beam subject to a force f(x). The conditions at x = 1 are known as clamped boundary conditions, corresponding to holding both the position and the slope of a beam fixed at the ends.

How shall we compute the spectral approximation to u _xxxx? Our standard design philosophy gives an answer. Let { v ^j} be the ( N ? 1)-vector of values of u sampled at x ₁, ,x _N. The "method (I)" strategy of the last chapter for imposing boundary conditions suggests the following:

• Let p be the unique polynomial of degree ? N + 2 with p( 1) = p _x( 1) = 0 and p(x _j) = v _j, j =1, , N ? 1.

• Set w _j = p _xxxx(x _j).

Now one might think that implementing this procedure will necessitate some new mathematics to derive the...