Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

This paper analyses a coupled system of generalized coupled system of fractional Jaulent–Miodek equations, including uncertain initial conditions with fuzzy extension. In this regard, an extension of the homotopy with a generalized integral algorithm is adopted for a class of time-fractional fuzzy Jaulent–Miodek models by mixing the fuzzy q-homotopy analysis algorithm with a generalized integral transform and Caputo fractional derivative. The triangular fuzzy numbers (TFNs)are expressed in double parametric form using κ-cut and r-cut and utilized to explain the uncertainties arising in the initial conditions of highly nonlinear differential equations with generalized Hukuhara differentiability (gH-differentiability). The TFNs are controlled by the κ-cut and r-cut, and the variability of uncertainty is examined using a “triangular membership function” (TMF). The results are analyzed by finding the solutions for different spatial coordinate values of time with κ-cut and r-cut for both lower and upper bounds and validated through numerical and graphical representations in crisp cases. Finally, it can be seen that the uncertain probability density function rapidly decreases at the left and right edges when the fractional order is increased, and it is observed that the obtained solutions are more accurate than the existing results through the Hermite wavelet method in the literature.