Question

I having some trouble solving this boundary value problem. Thank you for your assistance.

A thin wire coinciding with the x-axis on the interval [-L. L] is bent into the shape of a circle so that the ends x =-1 and x = L are joined. Under certain conditions, the temperature u(r, t) in the wire satisfies the boundary- value problem a au ot 11(-L. ) = u(Lt), t > 0 du ou .tso 11(x, 0) =fo), -L < x < L. Find the temperature ux, t).

Answer 1

Any query in any step then commem tbelow...i will explain you..

"u_t = K u_xx let u (x_1 x) = x (x) T (x) T'/K^T = y""/x = -lambda^2 T^2 + lambda^2 KT = 0, T^(+) = C_1 e^-k lambda^2 t, y"" + lambda^2 x = 0 x (x) = C_2 cos (lambda x) + c_3 sin (lambda x) x (-l) = x (L) implies 2 c_3 sin (AL) = 0 x' (- L) = x' (L) 2 c_2 sin (lambda L) = 0 implies lambda = n pi/L So, u (x + 1) = sigma^infinity_n = 1 [An cos (n pi x/L) + ln Sin (n pi x/L)] e^- K x^2 lambda^2/L^2 t u (x + 0) = f (x) \r\n sigma^infinity_n = 1 An cos (n pi x/L) + ln sin (n pi x/L) = f (x) So, by using orthogonelity An = 2/L integral^L_0 f (x) cos (n pi x/L) dx B_n = 2/L integral^L_0 f (x) sin (n pi x/L) dx"