I have inhomogeneous parabolic PDE (heat equation) with boundary and initial condition: $$\frac{\partial u(x,t)}{\partial t} = \frac{\partial ^ 2 u(x, t)}{\partial x^2 } + 2t^3$$ $$ x \in [0, 1], \space t > 0$$ $$Initial \space condition: $$ $$ u(x, 0) = 1 + \sin{\frac{5\pi}{2}x}$$ $$Boundary \space conditions: $$ $$ u(0, t) = 1, \space u_x(1, t) = 2t$$ As you can see the boundary conditions are different: the first is a function, and the second is a derrivative.
I can solve this eq. with Fourier method in case if both boundary conditions would be $u(t)$ or both bondary condition would be $u_x(t)$. So what I should do in this case?