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book   Mobk070    March 22, 2007  11:7








                                                 THE FOURIER METHOD: SEPARATION OF VARIABLES         29
                   so that
                                                   ∞
                                               4h     (−1) n+1
                                           y =               sin(nπξ)cos(nπτ)                  (2.109)
                                               π 2      n 2
                                                  n=1
                   2.2.4  Lessons
                   The solutions are in the form of infinite series. The coefficients of the terms of the series
                   are determined by using the fact that the solutions of at least one of the ordinary differential
                   equations are orthogonal functions. The orthogonality condition allows us to calculate these
                   coefficients.

                   Problem
                      1. Solve the boundary value problem

                                                       u tt = u xx
                                                       u(t, 0) = u(t, 1) = 0
                                                       u(0, x) = 0

                                                       u t (0, x) = f (x)
                          Find the special case when f (x) = sin(πx).


                   FURTHER READING
                   V. Arpaci, Conduction Heat Transfer. Reading, MA: Addison-Wesley, 1966.
                   J. W. Brown and R. V. Churchill, Fourier Series and Boundary Value Problems. 6th edition. New
                      York: McGraw-Hill, 2001.
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