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The general solution of the heat equation by separation of the variables is $$X(x)=a \cos\omega x + b \sin \omega x$$ Dirichlet

with the Dirichlet boundary conditions $$u_t = \kappa u_{xx}(x,t)\\ u(0,t)=T_0, u(L,t)=T_L\\ u(x,0)=f(x)$$ Since the steady state codition is linear between two Dirichlet boundary conditions, we have $$u(x,t)=u_s(x)+\nu(x,t)$$ and then we can apply new boundary conditions in the general solution $$\nu(0,t)=\nu(L,t)=0$$


Neumann

with Nwumann homogenous boundary conditions, we already have $$u_x(0,t)=u_x(L,t)=0$$ and then the steady state condition is not required, we can apply the boundary conditions to the general solution.


Dirichlet-Neumann

Consider the boundary conditions for a metal bar with an end at a fixed temperature and the end is insulated: $$u(0,t)=T_1\\ u_x(L,t)=0$$

How can we apply these boundary conditions to the general solution to eliminate one term and obtain the coefficient?

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How about defining $u = T_1 + v$ and demanding $v(0,t) = v_x(L,t) = 0$? Notice that the problem for $v$ becomes homogenous (PDE and BCs) with initial condition $v(x,0) = f(x) - T_1$.

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