3. The Leap-Frog scheme for the wave equation Suppose we are given with the problem (which is the classical wave equation) 321128271 5? baT2=O, 0:33:31. (4) Assume we have Dirichlet boundary conditions 1103,16) = 71(l,t) = O, t > U and initial conditions (5'71 815 (a) Explain why two initial conditions are needed. (b) Develop the LeapFrog scheme for (4), also known as CentralTimeCentralSpace (CTCS). What is the schemes order of accuracy? 71(:1:,O) = U.5:1:(1 :r), (;1: U) 0 U S :1: 51 (c) Analyze the stability of the scheme using Fourier Analysis and nd a criterion for it. (d) Show that the following is an analytical solution of the problem above with a = 1: 2 0 1 (:1: t) H21 [l cos (7171)] cos(7mt} sin(7171:1:) 71'3 \"1: 713 GuidanCe: for the initial conditions} you may read in https://mathworld.wolfram. com/FourierSeries.html about Fourier series, and may get help from integral solving websites for solving the integrals