Need help!! For part a,c and d, please provide a step-by-step solution along with a detailed explanation!!!! For part b, e, f, g, h, i, j, please provide a step-by-step solution through the use of MALAB code along with a detailed explanation!!!! It is alright if you handwrite the step-by-step solution for part b, e, f, g, h, i, j but I would prefer the solution to be shown in MATLAB code. This is very URGENT!!! PLEASE reply as soon as possible!! Thank you!!

Problem 2. Consider the wave propagation PDE you solved in HW 3, exp(-30(x-?/2)2) 0) u(t,r) = 0 and initial conditions u(0,x) with boundary conditions u( and P(0,2) = 0 (a) Show that the eigenvalues of the differential operator are integers in the imaginary axis, i.e 0,tli, 2i,?3i.. (b) Consider the finite-difference discretization used in HW3 for the spatial derivatives. That is, discretize u at the gridpoints ri = jh (j = 0, N) and p at the nodes zj+1/2 = (j + 1/2)a (j = 0, . . . N-1), where h = ?/N. The eigenvectors of the discrete FD matrix. -1 1 00 (21 u(32 u(r2 u(N- p(z1/2) p(r3/2) p(r1/2) p(x3/2) | h|?1 0 0 0 0 EN-1/2 0-1?Nx(N-1) EN-1/2