Fuzzy fractional shallow water wave equations: analysis, convergence of solutions, and comparative study with depth as triangular fuzzy number

Abstract We explore the integration of fuzzy fractional calculus into the modeling framework, recognizing its significance in capturing the inherent uncertainties and complexities present in Shallow Water Wave (ffSWW) dynamics. By incorporating fuzzy fractional calculus, we aim to enhance the accuracy and robustness of ffSWW equations, particularly in representing vague or imprecise parameters such as seabed topography, initial wave conditions, and material properties. In this article, we consider the time derivative as a fractional order instead of the traditional integer order, which allows us to interpret the behavior of the solution for different orders. Further, the sea depth has been considered as a Triangular Fuzzy Number (TFN). We employ the Homotopy Perturbation Transform Method (HPTM) to obtain the solution of ffSWW equations. The convergence of the obtained series solutions has been investigated theoretically and numerically. Also, the acquired results using the current method are validated through the comparison with pre-existing findings concerning integer order. Furthermore, simulation results for various fractional orders, as well as fuzzy lower and upper solutions of depth-averaged velocity and water surface elevation, are provided for triangular fuzzy numbers.