Answer to Question #297057 in Differential Equations for Raghu

Question

<e> Q.(a): Construct the solution of the heat equation using separation of variables method:

u_xx=4u_t , 0<x<40, t>0

u(0, t)=0, u(40, t)=0, t>0

u(x,0)= x, 0 less than or equal x less than or equal 20, u(x,0)=40-x, 20 less than or equal x less than or equal 40

(b): Find the steady-state solution of the heat equation u_xx=4u_t , 0<x<40, t>0 that satisfies the Bcs:

u(0, t)=10 , u(40, t)=40

(c): Plot u versus x for several values of t.

(d): Plot u versus t for several values of x.

(e): Determine how much time must elapse before the temperature at x=40 comes within 1 centigrade of its steady state.smi