Fuzzy modeling of critical flow over irregular bottom topography using the time-fractional forced Korteweg–de Vries equation

The analysis of the impact of force induced by uneven bottom topography on the free surface is a fascinating area of fluid dynamics. This research is motivated by the need to investigate the effect of the force generated by various shapes of bottom topography on the free surface elevation (FSE). The phenomenon is modeled using the time-fractional forced Korteweg–de Vries (fKdV) equation, which extends the classical Korteweg–de Vries (KdV) equation to fractional form by incorporating an external forcing term arising from bottom topography. The series solution of the model is obtained using the fractional reduced differential transform method (FRDTM). The impact of hole shapes identified in the bottom topography for n=2 and n=8 on the FSE is investigated under different critical flow regimes based on Froude number. The study reveals that the amplitude of FSE is significantly influenced by both the hole shape and the critical flow nature. To account for the uncertainty involved in the wave speed parameter c due to different conditions, the time-fractional fKdV equation is extended to a fuzzy environment by modeling c as a triangular fuzzy number (TFN). The solution of the fuzzified time-fractional fKdV equation is obtained using the fuzzified fractional reduced differential transform method, an extension of FRDTM to solve fuzzy nonlinear models. The fuzzy solution bounds for each type of critical flow are analyzed at different fractional order values. The fuzzy results demonstrate the impact of hole shape on the amplitude of the FSE in each critical flow regime.