Partial Differential Equations, Autumn 2021 Worksheet 2 The starred problems are for credit, and should be submitted by Wednesday 03/11/2021, 2pm through the submission point on Canvas. 1.* Use the Fourier transform to solve the initial value problem Uh - Uxx + Ux = 0, x ER, t> 0, u(x, 0) = 9(x), TER, where g E L'(R). 2. Consider the fundamental solution of the heat equation in R", (0) ABOUT . H(x, 1) := (ARKze , 1> 0, 0 , 1 co.) 3.* Consider the following initial and boundary-value problem Uht - Uxx = e"+ 2cos(x), -T0 u(x, 0) = x, - IT 0 Ux (-7, t) = Ux(7, 1), t> 0. Suppose we know that the homogeneous solution of the above problem is given by un(x,t) = Lent(An cos(nx) + Bn, sin(nx)) 1 = 0 with An = 0 and Bn = 2(-D) for any integer n 2 0. Use the Duhamel principle to find the inhomogeneous solution. [9 marks] 4. Let f(x) := H(x) cos x, with H(x ) := 0 *