Question

2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and eigenfunctions on(z) of the homogeneous problem.

Answer 1

"Given that A thin Rod of Length L = pi (so that 0 lessthanorequalto x lessthanorequalto pi) general source of heat Q(x, t) partial differential u (x, t)/partial differential t = K partial differential^2 u (x, t)/partial differential x^2 + Q (x, t) (1) partial differential u (0, t)/partial differential x = 0 partial differential u(pi, r)/partial differential x = 0 rightarrow (2) u(x,0) = 0 rightarrow (3) The positive constant �k� is the thermal diffusivity of root, integral dependents upon the thermal conductivity of the material it is made of the density of the rod and the specific that of the rod Assuming a specific solution to the homogeneous version that is taking Q(x, t) 20 of the PDf and boundary conditions (10) - (11) of the form u(x, t) = Q(x) G(t) rightarrow (4) then partial differential u(x, t)/partial differential t = k partial differential^2 u(x, t)/partial differential x^2 \r\n gives Q(u) G�(t) = K phivariant ""(x) G(t) (or) G�(t)/KG(t) = phivariant "" (x)/phivariant (x) rightarrow (5) thus G�(t)/KG(t) "