Solving fuzzy system of Fredholm integro-differential equations of the second kind by using homotopy analysis method

Certain phenomena with uncertain properties that take the shape of intricate mathematical modeling are known to have fuzzy system integro-differential equations (FSIDEs). The methods used to roughly solve FSIDEs seek to provide open-form solutions that are regarded as solutions for polynomial series. However, for many FSIDEs, the polynomial series solutions are not easily derived, especially in nonlinear forms. Meanwhile, some existing approximate techniques cannot guarantee convergence of the series solution. Nevertheless, to solve second-kind fuzzy Fredholm integro-differential equations (FFSIDEs), there exist perturbation techniques based on the standard Homotopy Analysis Method (HAM) that have the ability to control and rate solution convergence. Therefore, this study focused on modifying new approximate techniques, fuzzy Fredholm HAM (HAMFF), for solving second-kind FFSIDEs subject to initial and boundary value problems. In the theoretical framework modification, the establishment of the series solution convergence was done based on combining some fuzzy sets theory concepts and convergence-control parameters from standard HAM. The HAMFF was not only able to solve linear systems but also difficult nonlinear systems with proper accuracy. The demonstration of the modified technique's performance was shown in comparison to other methods, where HAMFF was individually superior in terms of accuracy for solving linear and nonlinear test problems of FFSIDEs.