Fuzzy enhanced numerical modeling of advection–diffusion equation for pollutant dispersion in water

Purpose Water quality is critical for maintaining ecological balance and supporting human health. The study aims to model and analyze the dispersion of pollutants in water using the Advection-Diffusion Equation (ADE), with a particular focus on addressing uncertainties in environmental parameters such as flow velocity. Design/methodology/approach This research investigates the numerical solutions of the ADE under both crisp and uncertain (fuzzy) scenarios. Two numerical schemes are employed: the Fuzzy Explicit Finite Difference Scheme (FEFDS) and the Fuzzy Crank–Nicolson Scheme (FCNS). The study models pollutant dispersion in a one-dimensional channel, incorporating uncertainty in the flow velocity using fuzzy set theory. A double parametric approach is used to obtain the lower and upper bound solutions of the fuzzy ADE. Findings Both the EFDS and FEFDS yield results that closely approximate the exact solution, particularly at higher pollutant concentrations. The FCNS method, however, demonstrates superior accuracy and stability in representing pollutant transport under uncertainty. Three-dimensional visualizations further illustrate the role of fuzzy modeling in capturing the effects of parameter uncertainty on pollutant dispersion. Originality/value This work brings a new way of solving ADE by using fuzzy set theory to deal with uncertain parameter. The combined use of FEFDS and FCNS gives better understanding and solution of pollution in water. This can help in managing water quality and planning pollution control methods in real life.