Question

(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at x-0 and x- , are insulated as described by the boundary conditions au(0, t) ди(n, t) and Using the solution from part (a) (ii) above (i) Determine the value of B for which the solution satisfies the boundary condition at r-0 (ii) Determine the values of p for which the solution also satisfies the boundary condition at π. (c) Suppose that that the initial temperature distribution along the rod in part (b) is given by ux,0)1-/ and thatc-1. Use Fourier methods to determine an expression for the temperature u(r, t) at subsequent times t >0

Answer 1

0)(a+ b) 0Bab) i) The first two y into the last equation gives equations yield the same thing. Subutituting th Thus where Bn-2f(r)sin (nr)dr and () is given in. The form of (1) is already a sine series, with BI-3/2, l⅓--1/2 and Bh-0 for all other n. You can check this for yourself by computing integrals in (3) for f (r) given by (1), from the orthogonality of sin E. Therefore, sin (3xr) e where the Bn's are the Fourier coefficients of f()-Px), given hy Breaking the integral into three péeces and substituting for P () from (5) gaVes 1/2-๒)2 P (sin(nr) dr + 2 P (x)sin (mr) dr 1/2+u2 =0+2 sin (naz) du +0 We apply the cosine rule with r-2u2 to F. 7, When n is even (and nonzero), e. 2m for some integer m, 2M0 sin a sin -0 (c) Show that for some z, 0 1, u' (1.0) 125 positive and for others it is negative. How is the sign of (r,0) related to the shape of the initial temperature profile? Hom is the sin of (r,t), t0, related to subsequent temperature profiles? Graph the temperature profile for 0,0.2, 0.5.1 on the same axis (you may use Matlab). Solution: Dillerentiating u(x,t) in tie gives Setting t-0 gives u,03sin (3) Note that This at r-1/6, From the PDE İs positive ahd for r-1/2,4 is negative.