I have been given a standard heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ with $u(t,x)$ and the intial condition $u(0,x) = 0$ and the boundary condtions $\frac{\partial u}{\partial x}(t,0)=0, u(t,1)= \sin(\omega t)$.
Due to having an inhomogeneous boundary condition separation of variables method fails, however, the method I was taught for inhomogeneous boundary conditions is for $$u(t,0)=f(t) \\ u(t,1)=g(t)$$ and then using $$u(t,x) = u_p(t,x)+v(t,x)$$ where $$u_p(t,x)=xg(t)+(1-x)f(t)$$ How do I go about modifying this so that I can solve the initial-boundary value problem?