Page 447 - Advanced engineering mathematics

WHY FOURIER SERIES? THE FOURIER
SERIES OF A FUNCTION SINE AND
CHAPTER 13 COSINE SERIES INTEGRATION AND
DIFFERENTIATION

Fourier Series

In 1807, Joseph Fourier submitted a paper to the French Academy of Sciences in competition for
a prize offered for the best mathematical treatment of heat conduction. In the course of this work
Fourier shocked his contemporaries by asserting that “arbitrary” functions (such as might specify
initial temperatures) could be expanded in series of sines and cosines. Consequences of Fourier’s
work have had an enormous impact on such diverse areas as engineering, music, medicine, and
the analysis of data.

13.1 Why Fourier Series?

A Fourier series is a representation of a function as a series of constant multiples of sine and/or
cosine functions of different frequencies. To see how such a series might arise, we will look at a
problem of the type that concerned Fourier.
Consider a thin homogeneous bar of metal of length π, constant density and uniform cross
section. Let u(x,t) be the temperature in the bar at time t in the cross section at x. Then (see
Section 12.8.2) u satisfies the heat equation
2
∂u ∂ u
= k
∂t ∂x 2
for 0 < x <π and t > 0. Here k is a constant depending on the material of the bar. If the left and
right ends are kept at temperature zero, then
u(0,t) = u(π,t) = 0for t > 0.

These are the boundary conditions. Further, assume that the initial temperature has been
specified, say

u(x,0) = f (x) = x(π − x).
This is the initial condition.

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October 14, 2010 14:57 THM/NEIL Page-427 27410_13_ch13_p425-464