Type-2 fuzzy fractional derivatives

• Defining type-2 fuzzy fractional derivative in the sense of Caputo and deriving the related theorem.
• Defining type-2 fuzzy fractional derivative and integral in the sense of Riemann–Liouville and deriving the related theorem.
• Obtaining fractional derivatives of the triangular perfect QT2FN-valued functions.
• Presenting existence and uniqueness of the solutions of type-2 fuzzy fractional differential equations.
• Solving type-2 fuzzy fractional differential equations numerically, using the PECE method.

In this paper, we introduce two definitions of the differentiability of type-2 fuzzy number-valued functions of fractional order. The definitions are in the sense of Riemann–Liouville and Caputo derivative of order β ∊ (0, 1), and based on type-2 Hukuhara difference and H2-differentiability. The existence and uniqueness of the solutions of type-2 fuzzy fractional differential equations (T2FFDEs) under Caputo type-2 fuzzy fractional derivative and the definition of Laplace transform of type-2 fuzzy number-valued functions are also given. Moreover, the approximate solution to T2FFDE by a Predictor-Evaluate–Corrector-Evaluate (PECE) method is presented. Finally, the approximate solutions of two examples of linear and nonlinear T2FFDEs are obtained using the PECE method, and some cases of T2FFDEs applications in some sciences are presented.