Fuzzy Triangular Finite Elements Solution for Solving the Nonlinear Boussinesq Equation

In this work, the fuzzy solution of the nonlinear Boussinesq equation is investigated for an unconfined aquifer bordering a lake, where the lake water level undergoes a sudden rise (recharge event) from 4 to 6 meters, followed by stabilization. The aquifer conditions are considered crisp, while the hydraulic parameters (hydraulic conductivity and specific yield) are treated as fuzzy due to uncertainties arising from measurement imprecision, data limitations, etc. To represent these uncertainties, fuzzy estimators of maximal specificity are employed. This novel methodology incorporates parameter uncertainty directly into the nonlinear Boussinesq equation, thereby advancing the reliability of groundwater flow modelling under uncertain conditions. The proposed approach: (a) enables the construction of fuzzy numbers directly from statistical samples rather than relying on subjective expert judgment, and (b) allows efficient fuzzy arithmetic operations through well-defined formulae involving only statistical parameters of the samples. To solve the problem, the theory of fuzzy differential equations is applied, reformulating the fuzzy system into a set of second-order crisp boundary value problems, referred to as the corresponding system. A triangular fuzzy finite element method (FEM) based on the Galerkin approach is then developed and solved using the α-level method. The objectives of this study are threefold: (1) to design and implement a fuzzy FEM numerical scheme based on triangular elements tailored to the nonlinear Boussinesq problem; (2) to determine the range in which the semi-analytical solution approximates the new triangular fuzzy FEM solution, and to perform comparative analysis with both semi-analytical and previously developed orthogonal fuzzy FEM approaches; (3) to provide engineers and water resource planners with improved tools for quantifying hydraulic properties under uncertainty. The findings show that the triangular fuzzy FEM solution is strongly consistent with the orthogonal fuzzy FEM method and demonstrates good agreement with the semi-analytical solution, validating its effectiveness for recharge volume estimation in practical hydraulic applications.