Numerical simulation of nonlinear wave propagation under Type-1 and Type-2 fuzzy scenarios

Purpose This research aims to create and evaluate fuzzy numerical methods to solve the coupled Whitham–Broer–Kaup equations (cWBKEs). These equations are used to study nonlinear wave movement through shallow water waves (SSW) and related models. This research aims to improve uncertainty analysis for real-world measurements of wave height and flow velocity. Design/methodology/approach The research introduces fuzzy extensions of two numerical techniques: the fully implicit Crank–Nicolson scheme (FICNS) and the Elzaki Adomian decomposition method (EADM), referred to as FFICNS and FEADM. The researchers first solved crisp equations which they confirmed through exact solutions. The Von Neumann analysis method was used to evaluate the stability of the numerical scheme. The system uses both Type-1 and interval Type-2 fuzzy numbers to effectively model uncertainty. Furthermore, the double parametric approach is applied to generate the resulting upper and lower bound solutions. Findings The results show that introducing uncertainty significantly influences the behavior of wave solutions over space and time. Both FFICNS and FEADM demonstrate strong accuracy and reliability in capturing these effects. The numerical outcomes align well with exact solutions in the crisp case and provide meaningful insights under fuzzy conditions. Originality/value This work develops a new fuzzy extension which combines Type-1 and Type-2 fuzzy frameworks to solve cWBKEs through established numerical techniques. The system provides an effective method to model and study uncertainty which affects nonlinear wave equations, making these models suitable for actual physical scenarios.