Page 578 - Modelling in Transport Phenomena A Conceptual Approach

558 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

5. Solve the resulting differential equation and express the solution in the form

6. Continue the process as long as desired. The nth approximation to the small-
est permissible value of X is given by

(B.3-23)

B.3.5 Fourier Series

Let f (x) be an arbitrary function defined on a I x 5 b and let be an
orthogonal set of functions over the same interval with weight function ~(x). Let
us assume that f(x) can be represented by an infinite series of the form

(B.3-24)

The series CCnq5,(s) is called the Fourier series of f(x), and the coefficients Cn
axe called the Fourier coeficients of f(z) with respect to the orthogonal functions

472 (XI.
To determine the Fourier coefficients, multiply both sides of Eq. (B.3-24) by
w(x)q5,(z) and integrate from z = a to x = b,

00
(B.3-25)

Because of the orthogonality, all the integrals on the righbside of Eq. (B.3-25) are
zero except when n = m. Therefore, the summation drops and Eq. (B.3-25) takes
the form

(B.3-27)

Example B.14 Let f(x) = x for 0 I x 5 T. Find the Fourier series of f(x)
with respect to the simply orthogonal set {sin(nx)}r=l.