$u_{xx}+u_{yy}=0 \quad in \quad the \quad rectangle \quad 0<x<a \quad 0<y<1$
$u=0 \quad on \quad y=1$
$u=j(y) \quad on \quad x=0 $
$u_y +u=0 \quad on \quad y=0$
$u_x=0 \quad on \quad x=a$
I tried variable separation method. And the result is that this problem is not solved by this method.
Variable separation works... the Sturm–Liouville problem in $y$-variable has solutions
$$\phi_n(y)=\sin \lambda_n y -\lambda_n \cos \lambda_n y$$
where $\lambda_n>0$ are the roots of the equation $\tan \lambda=\lambda$.
The solution is obtained in the form $$u(x,y)=\sum_n c_n \sinh [\lambda_n (x-a)]\, \phi_n(y)$$ where the coefficients $c_n$ come from expanding the function $j$ in the basis $(\phi_n)$.
