Consider the problem $u_t = u_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t) = 0 , u(1,t) = \cos(3 t)$, and initial condition $u(x,0) = x$.

Solve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write $u = S + U$, where $S(x,t) = A(t) (1-x) + B(t) x$ is (affine) linear in $x$, that is
$S(x,t) = {}$

The initial boundary value problem for $U$ becomes
PDE: $U_t = U_{xx} + f(x,t)$
where $f(x,t) = {}$
BC: $U(0,t) = 0, U(1,t) = 0$
IC: $U(x,0) = {}$

Now solve for $U$.

The eigenfunctions to use are
$X_n(x) = {}$
First decompose the inhomogeneity in the PDE as
$\displaystyle f(x,t) = \sum_{j=1}^n f_n(t) X_n(x)$ ,
where $f_n(t) = {}$
You find that
$\displaystyle U(x,t) = \sum_{j=1}^\infty T_n(t) X_n(x)$ ,
where
$T_n = {}$ $\displaystyle \int_0^t$ $ds$

Then $u(x,t) = S(x,t) + U(x,t)$

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