I'm trying to apply a boundary condition to the following equation:
$$c_A = {K_o}\int_{0}^{u} e^{-u^2} du + K_1$$
where $$u = \frac{x}{\sqrt{4Dt}}$$
and ${K_o}$ and $K_1$ are constants.
The boundary condition is:
$$\frac{\partial c_A}{\partial x} = 0$$
However, when I try to take the derivative with respect to both sides of the equation, I just seem to get $0 = 0$.
In other words, I'm not getting a value for one of the constants; so I think I must be doing something wrong.
Taking the derivative of the constant, $K_1$ with respect to x yields zero. Taking the derivative of the integral (error function) term, I think results in:
$$\frac{\partial \operatorname{erf}(Ax) }{\partial x}=\frac{2A}{\sqrt\pi}e^{-u^2}$$
where A is equal to:
$$A = \frac{1}{\sqrt{4Dt}}$$
If I'm right then the only way the derivative of the erf expression in the original equation is equal to zero is if the entire derivative of the erf expression equals zero, so I wind up with $0 = 0$ and no solutions for either constant.
What am I doing wrong? Thanks.