Fuzzy Caputo [formula omitted]-fractional linear equations on the time scale [formula omitted

Abstract

In this paper, we explore various concepts in q-fractional calculus and extend them to the fuzzy domain. Specifically, we define fuzzy q-exponential and fuzzy q-Mittag-Leffler functions using r-cuts. Additionally, we evaluate the fuzzy Caputo q-derivatives of the fuzzy q-exponential functions and introduce the concept of fuzzy q-Laplace transform. Furthermore, we provide a formula for the fuzzy q-Laplace transform of the fuzzy Caputo q-fractional derivative.We also tackle the problem of solving fuzzy Caputo q-fractional linear differential equations under the generalized Hukuhara differentiability conditions, with a focus on specific forcing functions. The key technique employed for solving these equations is the fuzzy q-Laplace transform. To illustrate the practicality of our theoretical results, we present several numerical examples.This paper contributes to the development of q-fractional calculus in the context of fuzzy mathematics and provides valuable insights into solving fuzzy differential equations using the proposed techniques.