Consider the evolution of the temperature distribution in a rod of the length L, the ends of which are kept at zero temperature. The ends are located at the points x = -L/2 and x = L/2. The evolution of the temperature, u = u(x, t), is given by the heat equation ut = KUxx,(the thermal diffusivity K is a positive constant) with the boundary conditions u(-L/2, t) = u(L/2, t) = 0. The initial condition is u(x, 0) = u0[(1 - 2X/L)2] Re-scale x, t, and u to get rid of letters in the equation, boundary conditions, and the initial condition. If you find it more convenient, you may also shift x. Make sure that the boundary conditions are canonical. Construct the orthonormal basis of the eigenfunctions of the Laplace operator in the space of functions obeying the boundary conditions. Find the solution u(x, t) of the problem (4)-(6) in the form of the Fourier series in terms of the constructed ONB. Restore the original variables x, t, and u.