Consider the following problem Given a positive constant c and function B, find u such that du = cozu in (0, 1) for t > 0, (1) u(0, t) = u(1, t) = 0, (2) u(x, 0) = B(x) for r E (0, 1). (3) Suppose the eigenfunctions are denoted by Xx. Suppose also that B is Rie- mann integrable and the Fourier coefficients c are defined by Ck / (Xx ( 2 ) )2 do = / B(x)Xx(z) da. (4) The aim is to prove that ck exp(-cAxt) Xk(x) converges uniformly on [0, 1] x [to, ti], * = 1 where to > 0. (The notation can be simplified if one uses the fact that Xk(0) = Xk(1) = 0.) Prove the following steps 1. Provide an example or examples such that lol s 2 / 18 ( 2) | de s 21 31 . 2. Prove that for any positive constant d, ER, Ck exp(-dAk) converges. 3. Prove that [ ck exp(-cAxt) XX(x) converges uniformly on [0, 1] x [to, ti], k= 1 where to > 0