Consider heat flow through a thin insulated circular ring described by: u_r = ku

Question

Consider heat flow through a thin insulated circular ring described by: u_r = ku_xx, x elementof (-L, L), t > 0, u(-L, t) = u(L, t), t > 0, u_x(-L, t) = u_x(L, t), t > 0, u(x, 0) = f(x), x elementof (-L, L). Write all separated solutions u_n(x, t) = phi_n(x)G_n(t). (State and solve the corresponding eigenvalue problem for phi(x) and an ODE for G(t).) Write the general solution u(x, t) defined by the separated solutions u_n(x, t). Determine the particular solution satisfying the initial data u(x, 0) = f(x). By using what you've learned above, find the unique solution of the following initial boundary value problem for heat condition in a thin insulated circular ring: u_r = ku_xx, x elementof (-L, L), t > 0, u(-L, t) = u(L, t), t > 0, u_x(-L, t) = u_x(L, t), t > 0, u(x, 0) = 2 + cos 3 pi x/L + sin pi x/L, x elementof (-L, L).

Explanation / Answer

ANSWER Consider a thin wire (with lateral sides insulated) of length 2L which is bent into the shape of a circle.

u/ t = k 2u/ x2 , L < x < L

with the boundary conditions u(L, t) = u(L, t), t > 0

u /x(L, t) = u/ x(L, t), t > 0

and the initial condition

u(x, 0) = f(x), L x L

apply the method of separation of variables and seek for product solutions

u(x, t) = (x)G(t)

thus,

G(t) = cekt

d 2 /dx2 =

(L) = (L)

d dx (L) = d dx (L)

these are required periodic boundary conditions for the above problem.

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