One of the clearest available introductions to variational methods, this text requires only a minimal background in calculus and linear algebra. Its self-contained treatment explains the application of theoretic notions to the kinds of physical problems that engineers regularly encounter. The text’s first half concerns approximation theoretic notions, exploring the theory and computation of one- and two-dimensional polynomial and other spline functions. Later chapters examine variational methods in the solution of operator equations, focusing on boundary value problems in one and two dimensions. Additional topics include least squares and other Galerkin methods. Many helpful definitions, examples, and exercises appear throughout the book. A classic reference in spline theory, this volume will benefit experts as well as students of engineering and mathematics.
1.6APPROXIMATING FROM FINITE DIMENSIONAL SUBSPACES
1.1 A SIMPLY STATED PROBLEM
The need for good techniques for the approximation of functions arises in many settings; one of these is the numerical solution of differential equations. For example, suppose you are given the differential equation
subject to the boundary conditions
where α and ÎČ are constant and a(t), b(t), and f(t) are functions of t defined on the interval [a, b]. Moreover, suppose you know that this equation, subject to the boundary conditions, has a unique solution x(t) which you would like to find. Our first problem then is
PROBLEM 1
How do we find x(t)?
As those who have worked to any extent in ordinary differential equations know, the answer to this query is decidedly gloomy. In particular
Answer. For most choices of a(t), b(t), and f(t) we cannot find x(t) exactly.
This being the case, we compromise. Since we cannot find x(t) exactly, we can try to find it approximately. This leads us to a new problem.
PROBLEM 2
How do we find a good approximation
(t) to the solution x(t) of our differential equation?
This problem is far more tractable, and there are many ways of answering it. All possible solutions depend in some way on the answer to yet another problem.
PROBLEM 3
Given a function x(t), what kind of functions
(t) make good approximations to x(t)?
and to the companion problems
PROBLEM 4
What is meant by a good approximation?
and
PROBLEM 5
How does one compute a good approximation
(t) to a given function x(t)?
Providing some answers to these simple questions and their two-dimensional analogs is precisely what this book is all about. Since linear spaces, subspaces, norms, and basis are notions fundamental to all we do in the sequel, we start with a minor digression to define these entities.
1.2 LINEAR SPACES
A real linear space X is simply a set of mathematical objects called vectors which add according to the usual laws of arithmetic and can be multiplied by real numbers in accord with the usual laws of arithmetic. Specifically to qualify as a real linear space, elements of X must satisfy the following conditions or axioms.
For all x, y, and z, in X and for all real numbers α and ÎČ, αx is in X, x + y is in X, x + y = y + ...
