For the heat equation u_t = c*u_xx with c = 1 and an initial condition u(x, 0) = 4 sin x + 2 sin 3x + sin 10x
in a periodic domain of extent [0, 4π]. Using a mesh of 100 points (i.e., use ∆x = 4π/99),
a) Compute the numerical solution using FTCS and BTCS.
b) Plot the computed vs. the exact solution (which is u(x, t) = 4 sin (x)e^(−ct) + 2 sin (3x)e^(−9ct) + sin (10x)e^(−100ct)) at time t = 3, when ∆t = 0.0025 and ∆t = 0.025 in MATLAB