Consider the equation $u_t = u_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t) = -7 , u(1,t) = 2$, and initial condition $u(x,0) = \cos\!\left(x\right)$.

Setup we wish to change the problem into homogeneous boundary conditions. What is the steady state solution:
$u_{\text{steady state}}(x) = {}$

Let $U(x,t)$ denote the transitent solution. That is, the solution of $U_t = U_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $U(0,t) = 0 , U(1,t) = 0$, and initial condition
$U(x,0) = {}$ ,
where the initial condition is such that
$u(x,t) = u_{\text{steady state}}(x) + U(x,t)$
solves the original problem above.

help (formulas)