Studies on Nonlinear Fuzzy Fractional Volterra Integro Differential Equations

Abstract

The idea of fractional derivatives was coined by Leibniz in 1695 and numerous mathematicians have contributed to its development from its inception. Fractional integral equations are commonly used in physics and engineering to address various problems. Consequently, researchers are interested in investigating both analytical and numerical methods for fractional order integro-differential equations. In recent years, fuzzy fractional calculus has attracted considerable attention by the researchers. Fuzzy concepts aid in solving the system involving uncertainty and vagueness effectively. This investigation focuses on studying the Caputo type fuzzy fractional nonlinear Volterra integro-differential equations with initial and boundary values. Using the Leibniz rule, the integral equation of the first kind transformed into the second kind. To address the existence and uniqueness of solutions for the proposed problem, fixed point theories and the contraction mapping theorem are utilized. Additionally, a novel numerical method known as the Adomian decomposition technique is employed to solve it numerically. This method produces a series solution based on a recursive relation using Adomian polynomials. The approach significantly reduces the number of numerical computations required as it does not rely on discretization or restrictive assumptions. The computational results are obtained using MATLAB and the $\xi$-cut representations of the fuzzy solutions are demonstrated. Overall, this research contributes to advancing the understanding and application of fuzzy fractional calculus in addressing complex problems involving uncertainty and vagueness.