I am having a lot of problems with these sorts of questions..help please!

#1

u_t = alpha^2*u_xx + sin(2x)

u(−pi, t) = u(pi, t), u_x(−pi, t) = u_x(pi, t)

u(x, 0) = cos(x)

Solve the problem by eigenfunction expansion

I don't know how to deal with the boundary conditions. I am used to seeing the BC being u(x,t)=0, u(L,t)=4 or similar but never having the BC equal to each other.

#2

I don't know how to deal with the r. Do i treat both the r and k as constants?

Help is greatly appreciated. Please be as detailed as possible, I am not very comfortable with these types of problems.