Question

Find the solution of the heat conduction problem u(0,0, 11 (x, 0) =sinOxx)-sin (m), u(1,1)0, t0 0 1 x

Answer 1

"Given that, the heat equation 10 u_xx = u_t [given conditions 0 < x < 1, t > 0] so, 10 partial differential^2u/partial differential x^2 = partial differential u/partial differential t ..(1) now, taking Laplace transform on both sides. We get L {10 partial differential^2 u/partial differential x^2} = L { partial differential u/partial differential t} implies 10 partial differential^2 u /partial differential x^2 = s u (x, s) - u (x, 0) 10 partial differential^2 u/partial differential x^2 = s u (x, s) = -u (x, 0) the differential operation from is (10d^2 - s) u = -u (x, 0) given u (x, 0) = sin (2 pi x) - son (5 pi x) (10d^2 - s) u = sin (5 pi x) - sin (2 pi x) therefore the auxiliary equation is 10m^2 - s = 0 m&2 = s