Question

2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<<t (a) Set up the initial-boundary value problem modeling this scenario. (b) Set up and solve the steady-state problem for (a) (c) Set up and solve the transient problem for (a) d) Solve the problem in (a). (e) Take k. Plot the solution surface from (d) using the first 5 terms of the Fourier series. On a single coordinate plane, plot the initial temperature distribution and the steady-state solution and animate the dynamics of the solution with an appropriately chosen set of time slices. (f) Is the "steady-state" solution a physical steady-state here? Explain.

PDE. Please show all steps in detail.

Answer 1

W (x, t) is a transient solution of w_t = kw_xx with w (0, t) = u (0, t) - v (0) = 100 - 100 = 0 w (l, t) = u (l, t) - v (x) = 0 - 0 = 0 w (x, 0) = u (x, 0) - v (x) = 1000 x^3 (1 - x) + 100 x/l = f (x) The suitable solution is w (x, t) = (c_1 cos rho x + c_2 sin P_) e^-k rho^2 t we know that \r\n w (0, t) = 0 Rightarrow c_1 e^-k p^2 t = 0 Rightarrow c_1 = 0 Therefore w (x, t) = c_1 p x e^-k P^2 t we know that w (l, t) = 0 Rightarrow c_2 sin Pl e^-k p^a t = 0 sin Pl = 0 Rightarrow P = h pi/l for n = 1, 2, 3 Therefore w (x, t) = sigma_n = 1^infinity c_n sin n pi x/l e^-k h^2 pi^/l^2 t