# Solve parabolic PDE with non-zero boundary and initial conditions

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?

• Your initial condition $u(x,0)=\ldots$ uses $t=0$ which is outside the allowed $t\in[1,10]$. – TZakrevskiy Jun 7 '17 at 15:52
• The idea of the solution would be to take a basis of, say $L^2[0,1]$ such that every function $\phi$ in this basis satisfies $\partial_x^2\phi(x) =0$, $\phi(0)=1$, and $\phi'(1)=1$. Then take a scalar product (in the sense of $L^2$) of these basis functions with the equation. – TZakrevskiy Jun 7 '17 at 15:55
• Let $u(x, t)=\frac{t^{4}}{2}+v(x, t)$ Then $$\partial_{t}v=\partial_{x}^{2}v$$ – Kiryl Pesotski Jun 7 '17 at 16:29
• @TZakrevskiy, I don't see where the OP constrained $t$ to be $\in [1,10]$. – Sharat V Chandrasekhar Jun 7 '17 at 17:12
• I would solve this problem by first setting up the auxiliary problem with homogeneous boundary conditions and then using Duhamel's Theorem. Refer to "Conduction of Heat in Solids" by Carslaw and Jaeger or "Heat Conduction" by M.N. Ozisik – Sharat V Chandrasekhar Jun 7 '17 at 17:13