The concentration \( S(x, t) \) of pollutants in a one-dimensional reservoir at position \\(( x \text{ and time } t \text{ satisfies the diffusion equation}\)
\[\frac{\partial s(x, t)}{\partial t} = D \frac{\partial^2 s(x, t)}{\partial x^2}\] on the domain \( 0 \le x \le L \), where \( D \) is the diffusion coefficient of the pollutants. The initial condition \( S(x, 0) \) is defined by the step-function shown in the figure.
The boundary conditions of the problem are given by \( \frac{\partial s(x, t)}{\partial x} = 0 \) at the boundary points \(x = 0 \text{ and }\) \(x = L \text{ at all times}\). Consider \(D = 0.1 \ \text{m}^2/ \text{s}, S_0 = 5 \ \mu \text{mol/m}\), \(\text{and} L = 10 \ \text{m}.\) The steady-state concentration} \(\hat{s} \left( \frac{L}{2} \right) = s \left( \frac{L}{2}, \infty \right)\) at the center \(x = \frac{L}{2} \)\( \text{ of the reservoir is}\) _________ (in \(\mu \text{mol/m}) \)(in integer).
