= - 2. Consider the following problem on the propagation of heat in a laterally insulated 1D bar of length L = 1m, with thermal diffusivity a = 0.23 cm/s. At time t = 0) the bar has a temperature u(x,0) = x (L x), where x is the coordinate along the bar axis. The extremities of the bar are kept at temperature u(0,t) = u(L,t) = 0. It is well known that the distribution of temperature in this problem is = = u(x,t) = Bn sin , ("). le ( ont) t (1) n=1 [2 marks] Show that the distribution of temperature (1) satisfies the heat equation au = a at (2) ax2 To this end, substitute the solution (1) into (2) and check that the identity is satisfied. [6 marks] Find the coefficients Bn in (1) by using the prescribed conditions. [2 marks] A good approximation to the problem above is provided by the function Uappr(x, t) = x (L x)e-(E)'. By considering this approximate solution, how long does it take in order that the temperature in x = L/2 is halved with respect to its initial value? = - 2. Consider the following problem on the propagation of heat in a laterally insulated 1D bar of length L = 1m, with thermal diffusivity a = 0.23 cm/s. At time t = 0) the bar has a temperature u(x,0) = x (L x), where x is the coordinate along the bar axis. The extremities of the bar are kept at temperature u(0,t) = u(L,t) = 0. It is well known that the distribution of temperature in this problem is = = u(x,t) = Bn sin , ("). le ( ont) t (1) n=1 [2 marks] Show that the distribution of temperature (1) satisfies the heat equation au = a at (2) ax2 To this end, substitute the solution (1) into (2) and check that the identity is satisfied. [6 marks] Find the coefficients Bn in (1) by using the prescribed conditions. [2 marks] A good approximation to the problem above is provided by the function Uappr(x, t) = x (L x)e-(E)'. By considering this approximate solution, how long does it take in order that the temperature in x = L/2 is halved with respect to its initial value