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PDE with a partial derivative calculated at a given value of an indipendent variable

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Old   November 30, 2023, 05:18
Default PDE with a partial derivative calculated at a given value of an indipendent variable
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roberto popov
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Hi, I am quite new in Comsol, I have just solved few problems using the model wizard so far.
I am just wondering if it is possible to solve numerically the following problem.

\frac{\partial u(x,t)}{\partial t}=\frac{\partial^2 u(x,t)}{\partial x^2}+S \left [ 1+\frac{\partial u(0,t)}{\partial x} \right ]\frac{\partial u(x,t)}{\partial x}
with
u(0,t)=1
u(\infty,t)=0
and
u(x,0) =\exp \left ( -\frac{x^{2}}{\pi } \right )-x \, \text{erf}\left ( \frac{x}{\sqrt{\pi}} \right )

Where S is a constant equal to 100. I am interested in integrating from t going from 0 to 5. As x=\infty we can simply use x=10 in this case. In particular, I do not know how to write the derivative of the dependent variable in x=0 inside the equation (it is the term in the square brackets).
Many thanks in advance.
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