Consider the partial differential equation du du = for 0≤x≤1, t≥0, with (0, t) х (1, t) = 0. х du J²u = 2 Ət əx² These boundary conditions are called Neumann boundary conditions. You can think of the function u(x, t) as mod- elling the temperature distribution in a metal rod of length 1 which is completely insulated from its surroundings. a. Find all separated solutions which satisfy the given boundary conditions. b. A general solution of the equation can be obtained by superimposing the separated solutions: u(x, t) = Σ u₁(x, t) = ΣciXi(x)Ti(t) Show that any solution of this form also satisfies the given boundary conditions. c. Find a cosine series for the function f(x)= = x on the interval [0, 1], and use this to obtain a solution u(x, t) which satisfies the initial condition u(x,0) = f(x) d. Evaluate the following limit: lim u(x, t). t→[infinity] The result you obtain can be interpreted as follows: after a long time, the heat becomes uniformly distributed throughout the rod and the temperature is constant.
