Consider a uniform bar of length L having an initial temperature distribution given by

Chapter 10, Problem 15

(choose chapter or problem)

Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 x L. Assume that the temperature at the end x = 0 is held at 0C, while the end x = L is insulated so that no heat passes through it. (a) Show that the fundamental solutions of the partial differential equation and boundary conditions are un(x, t) = e(2n1)222t/4L2 sin[(2n 1)x/2L], n = 1, 2, 3, .... (b) Find a formal series expansion for the temperature u(x, t) u(x, t) = n=1 cnun(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 possible to represent f by a Fourier series involving only these functions. See of Section 10.4.