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Solve the heat equation $$ u_t = u_{xx} $$ in $0\le x\le 1$ and $t\ge0$ with the initial condition $u(x,0)=u_0(x)$ and boundary conditions only on the left boundary $x=0$ $$ u(0,t)=f(t), \quad u_x(0,t)=g(t). $$

I do not know this problem is well-posed or not. Thanks!

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  • $\begingroup$ To solve it, you may try the method of separation of variables. I don't think this problem is well-posed since you do not have a boundary condition on $x=1$. $\endgroup$ – Chee Han Jan 10 '17 at 2:34
  • $\begingroup$ @CheeHan You are right. It is not well-posed. $\endgroup$ – Michael Jan 12 '17 at 7:52
  • $\begingroup$ Why is it not well-posed!? $\endgroup$ – Ktree Jul 12 '18 at 9:38
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Hint:

Apply the method similar to diffusion equation, inhomogenous boundary conditions (the subtraction method) , i.e. let $u(x,t)=v(x,t)+f(t)+xg(t)$ ,

Then $u_t(x,t)=v_t(x,t)+f_t(t)+xg_t(t)$

$u_x(x,t)=v_x(x,t)+g(t)$

$u_{xx}(x,t)=v_{xx}(x,t)$

$\therefore v_t+f_t(t)+xg_t(t)=v_{xx}$ with $v(0,t)=0$ and $v_x(0,t)=0$

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  • $\begingroup$ How do I see at yout hint that the problem is not well-posed? I have the same problem right now and intuitively I would suppose the problem is well-posed.. $\endgroup$ – Ktree Jul 12 '18 at 9:33

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