Question

* Exercise 4: Let k,l 〉 0. The temperature of a rod insulated at the ends with an expo- nentially decreasing heat source in it satisfies the following problem: ux (0, t) u(z,0) 0 = ux (1, t), φ(z) Find the solution to this problem by writing u as a cosine series: Ao(t) u(x, t) An(t) cos and determine limt-Hou(z, t

Answer 1

Let u (x, t) = A_0 (t)/2 + Sigma_n = 1^infinity A_n (t) cos (n pi n/l) given partial differential u/ given partial differential t - k given partial differential^2 u/ given partial differential x^2 = e^-2t h (x) rightarrow d A_0/2 dt + Sigma_n cos n pi x/l - k Sigma_n = 1^infinity - (n pi/l)^2 cos (n pi x/l) = e^-2t h (x) rightarrow 1/2 d A_0/dt + Sigma_n d A_n/dt cos (n pi x/l) + k Sigma_n = 1^infinity (n pi/l)^2 cos (n pi x/l) = e^-2t h (x) rightarrow (1) Integrating above eqn w r t from o to l rightarrow 1/2 dA_0/dt = e^-2t integral_0^l h (x) dx or, d A_0 = alpha e^-2t/l dt H_1 (x) where H_1 (l) = integral_0^l h (x) dx therefore A_o = 2 e^-2t/(-2l) H (l) + const A_0 = -e^-2t.l (H_1 (l) + cosnt A_0 = -e^-2t/l H_1 (l) + const A_0 = -e^-2t/l H_1 (l) + a integrating equ (1) by multiplying by cos (n pi x/l)