Consider a uniform bar of length L having an initial

Chapter 10, Problem 15

(choose chapter or problem)

Consider a uniform bar of length L having an initial temperature distribution given byf(x), 0 x L. Assume that the temperature at the end x = 0 is held at 0C, while theend x = L is insulated so that no heat passes through it.(a) Show that the fundamental solutions of the partial differential equation and boundaryconditions areun(x, t) = e(2n1)222t/4L2sin[(2n 1)x/2L], n = 1, 2, 3, ....(b) Find a formal series expansion for the temperature u(x, t)u(x, t) = n=1cnun(x, t)that also satisfies the initial condition u(x, 0) = f(x).Hint: Even though the fundamental solutions involve only the odd sines, it is still possibleto represent f by a Fourier series involving only these functions. See ofSection 10.4.