Fuzzy uncertainty modeling of generalized Hirota–Satsuma coupled Korteweg–de Vries equation

Abstract

This article explores the solitary wave solutions of a generalized Hirota–Satsuma Coupled Korteweg–de Vries (HSCKdV) equation. The HSCKdV equation is a mathematical model that describes certain types of long waves, particularly those found in shallow water. The generalized HSCKdV equation is solved exactly using the Homotopy Perturbation Transform Method (HPTM). By applying this technique, the authors obtain solutions in the form of a convergent power series. These solutions offer an understanding of the characteristics of solitary waves within the domain of shallow water waves. The HSCKdV equation has been solved using the adomian decomposition method, and the results have been compared with those obtained from the HPTM. This comparison demonstrates the effectiveness of the HPTM in solving such nonlinear equations. Further, the HSCKdV equation is extended to a fuzzy version considering the initial condition as a fuzzy parameter. Uncertainty in the initial condition is addressed by representing it using triangular and trapezoidal fuzzy numbers. The generalized fuzzy HSCKdV equation is subsequently tackled using the fuzzy HPTM (FHPTM) providing fuzzy bound solutions. Using the FHPTM, we explain the fuzzy results, highlighting how the solitary wave splits into two solitary waves and noting that the lower and upper bound solutions are interchanged due to negative fuzzy results.