Heat equation with two boundary conditions on one side

Question

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!

Answer 1

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 $v_x(0,t)=0$