PDE: Initial Conditions Contradicting Boundary Conditions

[Arkuski]

Suppose we have the following IBVP:

PDE: $u_{t}=α^{2}u_{xx}$ $0<x<1$ $0<t<∞$
BCs: $u(0,t)=0, u_{x}(1,t)=1$ $0<t<∞$
IC: $u(x,0)=sin(πx)$ $0≤x≤1$

It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1 while with the initial condition, the derivative is equal to -π. Do I conclude that the problem is incorrect or is there another way to reconcile this error?

 

[LCKurtz]

I don't think there is an error. You can think of a bar with the given temperature distribution inserted into a situation with those bc's. I would begin by using a substitution like $u(x,t) = v(x,t)+\psi(x)$ to make a homogeneous system in $v(x,t)$ and let the $\psi(x)$ take care of the $u_x(1,t)=1$ nonhomogeneous term.