I am given the heat equation with the following boundary conditions: $$u,_x = (K_0(x)u,_x),_x$$ $$u(t,x=0) = 0$$ $$u(t,x=1) = 1$$ Where $$K_0 = \frac{e^x}{cos(x)}$$
In a steady state solution $u,_t$ goes to 0 and the boundary conditions become $u(x=0) = 0$ and $u(x=1) = 1$. From this I obtain the following: $$(k_0(x)u,_x),_x = 0$$ $$K_0(x)u,_x,_x + K_0(x),_xu,_x = 0$$ My question is am I on the right track and if so where do I go from here?
Best Answer
You're first step is true, but not useful for finding the solution. Use integration to find $$(ku_x)_x=0\Rightarrow ku_x = \int 0\mathrm dx = C,$$ where $C$ is some constant. Then $$u_x = \frac{C}{k} = \frac{C\cos x}{e^x},$$ which can be integrated again. Use the boundary conditions to find the values of the integration constants.