Use the Laplace transform to solve the initial value problem for the advection-diffusion equa- tion ut(x, t)+2ux(x, t) = uxx(x, t) Vx R, t> 0 subject to u(x, 0) = sin x Vx R. You may use the shift theorem and table of standard transforms without proof. Verify your solution by direct substitution into the problem. [7] [1] Describe the evolution of the solution in time in terms of the initial data, including its behaviour as t . [2] Total marks available: [10] N.B. Full working/explanation must be given.