3. Consider the heat conduction in a semi infinite rod 0 < x <co. At x = 0, the temperature gradient is -Q, whilst u(x, t) → 0°C when xco. If the initial temperature of the rod is zero, the heat conduction problem in the rod can be given as PDE: = ди at 0 < x < ∞, t> 0. (0,t) =Q, u(x, t) 0 when x∞o, t> 0. ди BC: IC: u(x, 0) = 0, 0<x<∞o. where k and Q are constants. (a) Define the Laplace transform of u(x, t). (2 marks) (b) Using the Laplace transform method, show the temperature of the rod at any point x, at time t, is (c) u(x,t) = -x²/(4k²t) √T dT 0 Hence, show that the temperature in part (b) can be written as u(x,t) = Q x e-x²/(4k²)-x erfc 2k√√t (8 marks) (8 marks)