Progress in Fractional Differentiation & Applications

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The Homotopy Analysis Method (HAM) is an approximate-analytical method for solving linear and nonlinear problems. HAM provides the auxiliary or convergence parameter, which considered as a powerful tool to examine and analyze the precision of the approximate series solution and ensure its convergence. In this article, fuzzy set theory properties is introduced to extend and reformulate HAM for the determination of approximate series solutions for fuzzy fractional differential equations involving initial value problems. The extension and reformulation of the method corresponding to Caputo’s derivative in the fuzzy domain and the fuzzification of the method followed by the convergence analysis are presented in detail. Consequently, a new HAM for the general fractional differential equation has been developed in fuzzy domain. The difference between other types of approximate-analytical approaches and HAM is that the proposed HAM offers a better way to track the convergence region of the series solution via the convergence control parameter. The capability and accuracy of the method are illustrated by solving two examples involving linear and nonlinear fuzzy fractional differential equations. The obtained results using HAM suggested HAM is effective and simple to use when solving first order initial value problem involving a fuzzy fractional differential equation.

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