Fuzzy integro-differential equations are used for modeling real-life phenomena that involve uncertain (fuzzy) parameters or variables. The combination of the fuzzy Elzaki transform and homotopy perturbation method provides a powerful hybrid technique for solving fuzzy integro-differential equations. Therefore, the aim of this paper is to modify and apply a new hybrid method called fuzzy Elzaki transform homotopy perturbation method for the first time in literature to solve fuzzy integro-differential equations. In particular, the fuzzy Elzaki transform homotopy perturbation method is developed and applied for solving linear and non-linear second-kind fuzzy Volterra integro-differential equations, and non-linear second kind fuzzy mixed Fredholm- Volterra integro-differential equations. Finally, several examples are presented to show that the fuzzy Elzaki transform homotopy perturbation method is efficient for solving wide types of fuzzy integro-differential equations with high accuracy. The novelty of this work lies in its ease of use and its high efficiency, which allows mathematicians to obtain reliable results under fuzzy Hukuhara differentiability aspects in a short time.

In this paper, a modified method based on artificial neural network and Newton-Cotes methods with positive coefficients was developed and applied for the first time in the literature to solve the intuitionistic fuzzy integro-differential equations (IFFIDEs), where the parameters and variables are considered intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy number (TIFN) was used for expressing intuitionistic fuzzy variables and parameters. The fuzzification of the deterministic r-cut and β-cut solutions leads to the artificial neural intuitionistic numerical solution. The main reason for using neural networks is their applicability in handling the intuitionistic fuzzy variables and parameters of IFFIDEs. A numerical example was given to demonstrate the proposed method. The results agree with the theoretical prediction and highlight how the combination of artificial neural networks and Newton-Cotes method enhances the accuracy and efficiency of solving IFFIDEs, making our proposed method valuable for application in fields such as engineering, medicine, and physics.