= 2. Let be the initial value problem for the heat equation in R ut = Uxx, u(x,0) = f(x), x R, t > 0. The initial condition satisfiesf(x) = 1 if x [0.1] and f(x) = 0 if x /[0.1]. a) Write a formula for the solution. Show that u(x,t) > 0 for all real x and t> 0. b) Write the solution of the initial condition problem for the inhomogeneous heat equation u, = Uxx +h(x, t), u(x,0) = f(x), XER, t > 0, with f as (a) and h(x,t) = e -21 for x (-1,1],t > 0, y h(x, t) = 0 for x /[-1,1],t > 0. = 2. Let be the initial value problem for the heat equation in R ut = Uxx, u(x,0) = f(x), x R, t > 0. The initial condition satisfiesf(x) = 1 if x [0.1] and f(x) = 0 if x /[0.1]. a) Write a formula for the solution. Show that u(x,t) > 0 for all real x and t> 0. b) Write the solution of the initial condition problem for the inhomogeneous heat equation u, = Uxx +h(x, t), u(x,0) = f(x), XER, t > 0, with f as (a) and h(x,t) = e -21 for x (-1,1],t > 0, y h(x, t) = 0 for x /[-1,1],t > 0