Question

7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a

Answer 1

100 u_xx = u_t - (1) o < x < 1 t > 0 u(0, t) = u(1, t) = 0 t > o u(x, 0) = sin 2 pi x - sin 5 pi x 0 lessthanorequalto x < 1} - (2) Let u(x, t) = x(x, T(t) From eq^n(1) 100 x'' T = xt' Rightarrow x''/x = 1/100 T''/T = -lambda(say) Rightarrow x'' + lambda x = 0 - (3) T' + 100 lambda T = 0 - (4) u(0,t) = 0 Rightarrow x(0) = 0 u(1,t) = 0 Rightarrow x(1) = 0} - (5) If lambda = 0 then x'' = 0 Rightarrow x(x) = Ax + B x(0) = 0 Rightarrow B = 0 x(1) = 0 Rightarrow A = 0 Rightarrow x)x) = 0 Rightarrow u(x, t) = 0 (Trivial solution) If lambda = -mu^2, mu notequalto 0 then x'' - mu^2 x = 0