Question:
This question concerns a uniform heat-conducting rod. The rod is insulated along its length and also at one end(x=0), with the other end (x = L) in contact with ice. The temperature u(x,t), at position x and time t, satis?es the equations
∂u /∂t -k ∂2u/ ∂x2 =0 //(0 0),
∂u/∂x (0,t)=u(L,t)=0( t ≥ 0),
u(x,0) =1- x L (0
(a) By using separation of variables, with u(x,t)=X(x)T(t), ?nd an eigenvalue problem for X(x). Assuming that all of the eigenvalues are positive, ?nd these eigenvalues and show that the eigenfunctions of the problem are Xn(x) = cos'(n-1/2)πx/ L (n =1,2,...).
(b) Assume that any function f for which f and its derivative f! are piece wise continuous on the interval0
an = 2 L( L 0
f(x) cos'(n- 1 2)πx L/dx (n =1,2,...).
Find, in the form of an in?nite series, the solution to the heat conduction problem statedat the start of the question.
(c) Assuming that the series for u(x,0) converges point wise for 0 ≤ x ≤ L,show that 1+1 9+ 1 25 +···= 1
8π2. [2] (d)By applying appropriate results verify that:
(i) alleigen values of the problem for X(x)are positive,as for part (a);
(ii) the eigen functions form a basis for V [0,L], as assumed for part (b)