Graduate Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution

[umby]

TL;DR

Numerical solution of a partial differential equation containing the derivative of the unknown at a point

What is the best way to solve numerically the following equation using Comsol 5.3.

$\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}$
$T(0,t)=1$
$T(\infty ,t)=0$
$T(x,0)=\exp \left(-\frac{x^2}{\pi }\right)-x \text{erfc}\left(\frac{x}{\sqrt{\pi }}\right)$

where $\text{St}$ is a parameter which can varies from 0.01 to 100.
I am particularly interested in following the evolution of $\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]$ with time.
Thanks in advance.

 

[wrobel]

Be care. This is not a PDE at all. Even the correctness of such a problem needs for a separate study.

 

[umby]

Be care. This is not a PDE at all. Even the correctness of such a problem needs for a separate study.

You mean because of the coefficient of the derivative of $T$ with respect to $x$? Maybe differential relation is better? Can you help me please in determining the correctness of this problem?

 

[wrobel]

I mean the term
$$\frac{\partial T}{\partial x}\Big|_{x=0}$$

 

[umby]

Exsactly what I was mentioning, thank you for point it out. Coefficients cannot be function of the unknown, only of the indipendent variables, in my case $x$ and $t$.
Can you please tell me more about the matter of its correctness?

 

[hunt_mat]

You can probably write a finite difference code easier than using comsol. Then you can insert the nonlinear part of the PDE with ease.