3. Let H : R - R be a function such that H(x) = 1 if x e [-1, 1] and H(x) = 0 otherwise. Consider the following IBVP: au at arttH(x) on R x R. with u(x, 0) = 0 for r ER. (3) (a) Show that HE L'(R). [1 mark] (b) State fully the Fourier and Laplace transforms. [4 marks] (c) By using a suitable integral transform from (b), find an equivalent first order ODE and boundary conditions to problem (3). [6 marks] (d) Solve fully the ODE problem obtained in (c). [7 marks] (e) Hence write down a solution to (3), leaving it in integral form. [2 marks]