Incorporating Danckwerts’ boundary conditions into the solution of the stochastic differential equation

The one-dimensional dispersion model has been solved analytically as well as numerically to describe flow in continuous “closed” boundary systems using the celebrated Danckwerts boundary conditions. Nevertheless, a continuous state stochastic approach can sometimes be more appropriate especially in cases when input fluctuations are of the same order as the time scale of the system and in such cases an accurate treatment of the boundary conditions is indispensable for the successful application of the method. A deterministic approach was carried out in which the differential equation was solved using Fourier's method and the Laplace transform. These solutions were used as a yardstick to assess the precision of the stochastic solution with its proposed boundary conditions conforming to Danckwerts’ boundary conditions. Our problem is somehow simplified if we assume that the convection term and the dispersion term are constants independent of space and time. A stochastic differential equation was thus employed, governed by the Wiener process and solved using the Euler-Maruyama method.