In Task 1 we will study a combination of the transport and heat equation called the advection-diffusion equation ou (x, t) + voou (x, t) = vozu (x, t) at x x (1) For all numerical examples in task 1 please use 0 x 2, a final time of tend=2.0 and a value of vo = 1.5. Task 1. (40 marks) 1. (a) (4 marks) Apply the continuous Fourier transform in x to equation (1). Write down the resulting equation in spectral space. Make sure to clearly explain what identities from Fourier analysis you have used to obtain your result. (b) (2 marks) Write down the general solution of the differential equation arising from Task 1.1a. (c) (4 marks) Given that the Fourier transform of cos(kx), where k is a real number, is given by k k - ($ ( - 2) + 8 ( + =)) 1 (5 2 2 cos (kx) where & is the Dirac delta function, use this expression to obtain the solution of the differential equation from Task 1.1b with initial value u(x,0) = cos(kx). 2. Derive the initial value problem in spectral space that arises when solving equation (1) using the pseudo-spectral method. Proceed along the following steps: a) (1 mark) How does the equation for the residual R(u(x,t)) look like that comes out of plugging the truncated Fourier series Nx-1 1 u(x, t) = Ak (t) elkx Nx k=0