q-fractional differential equations with uncertainty

• Published in: Soft computing (Berlin, Germany) Vol. 23; no. 19; pp. 9507 - 9524
• Main Authors: Noeiaghdam, Z., Allahviranloo, T., Nieto, Juan J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2019; Springer Nature B.V
• Subjects: Approximation; Artificial Intelligence; Boundary value problems; Calculus; Computational Intelligence; Control; Derivatives; Differential equations; Engineering; Existence theorems; Foundations; Fractional calculus; Mathematical functions; Mathematical Logic and Foundations; Mechatronics; Robotics; Uniqueness theorems; Generalized Hukuhara difference; Mittag-Leffler function; fractional derivative; derivative; fractional initial value problems; Fuzzy; Krasnoselskii–Krein-type conditions; Fuzzy Caputo
• ISSN: 1432-7643; 1433-7479
• DOI: 10.1007/s00500-019-03830-w

In this paper, first we are going to introduce the fuzzy q -derivative and fuzzy q -fractional derivative in Caputo sense by using generalized Hukuhara difference, and then provide the related theorems and properties in detail. Moreover, the characterization theorem between the solutions of fuzzy Caputo q -fractional initial value problem (for short FCqF-IVP), which allows us to translate a FCqF-IVP and system of ordinary Caputo q -fractional differential equations (for short OCqF-DEs), is presented. In detail, the existence and uniqueness theorem is proved for the solution of FCqF-IVP. Finally, we restrict our attention to explain our idea for solving the FCqF-IVP and introducing its numerical solution by means of q -Mittag-Leffler function. The numerical examples demonstrate that the proposed idea is quite reasonable.