We consider a scaled diffusion PDE without source terms, Suppose we manage to find two exact solutions V (xt)  and w(xt)  to this equation. Show that the linear combination is also a solution of the...

We consider a scaled diffusion PDE without source terms,

Suppose we manage to find two exact solutions V (xt)  and w(xt)  to this equation. Show that the linear combination

is also a solution of the equation for arbitrary values of the constants a and b. Set up boundary and initial conditions that uO fulfills (hint: simply evaluate v and w at the boundary and at t = 0)

Sketch the function u(xo) from an understanding of how a small-amplitude, rapidly oscillating sine function w(xt) is added to the half-wave sine function v(xt)Set up the complete initial-boundary value problem for u, and adjust a program for the diffusion equation such that the program can compute this u. Choose
t =

²
Find the time t = T when there are no more visible tracks of the rapidly oscillating part of the initial condition. Explain that the solution you observe as graphical output of the program is in accordance with the analytical expression for u.