I am trying to solve heat diffusion equation: $$\frac{\partial^{2} T}{\partial x^{2}}+\frac{g(t)}{k}=\frac{1}{α}\frac{\partial T}{\partial t}$$ With initial condition $$T(x,t=0)=T_0$$ Boundary conditions: At $x=0$
$$-k\frac{\partial T}{\partial x}+h_1(t)T=q(t)$$
At $x=L$
$$-k\frac{\partial T}{\partial x}+h_2(t)T=0$$
I cannot use separation of variables to solve this because of the time dependent coefficients. Can anyone suggest how to approach this problem?