Analytical solutions of advection–dispersion equation using fuzzy theory
Christos D. Tzimopoulos, Kyriakos Papadopoulos, Christos Evangelides, Basil Papadopoulos · Desalination and Water Treatment · 2020
The aim of this research is to apply mathematical models for the solution of the one-dimensional convective-dispersive solute transport equation. For the derivation of the advection-dispersion equation it is assumed that the flow in the medium is unidirectional and that the average velocity is constant throughout the length of the flow field. Moreover, it is assumed that the porous medium is homogeneous and isotropic and that no mass transfer occurs between the solid and liquid phases. Unfortunately, the boundary conditions of the problem are not always intuitively apparent and in many cases convey uncertainties. For that reason, the problem is solved by utilizing fuzzy systems and fuzzy logic. The significance and the main advantage of this research is the introduction of fuzzy logic in order to solve similar problems presenting uncertainties. Since the aforementioned problem involves differential equations, a method of generalized Hukuhara derivative for total derivatives was applied, as well as the extension of the corresponding theory, concerning partial derivatives. So the fuzzy problem was transformed in a system of two classical differential equations, which were resolved with a Laplace transformation. The development of the fuzzy concentration profile is presented, as well as the membership functions of concentration. The results have given some beneficial conclusions for the effects of the uncertainties. In conclusion, it is expected that this conception will help the researchers and the engineers to take the right decision in similar problems. It is a special effort in this research to solve an advection-dispersion equation presenting uncertainties in boundaries and to follow the effect of these uncertainties in time.