This manuscript is devoted to the investigation of the existence results of fractional Cauchy problem forsome nonlinear ψ−Caputo fractional differential equations with non local conditions. By applying fixedpoint theorems, some results of topological degree theory for condensing maps and some fractional analysistechniques, we establish some new existence theorems. As application, a nontrivial example is given toillustrate our theoretical results.

In this paper, we study the fuzzy Laplace transforms introduced by the authors in (Allahviranloo and Ahmadi in Soft Comput. 14:235-243, 2010) to solve only first-order fuzzy linear differential equations. We extend and use this method to solve second-order fuzzy linear differential equations under generalized Hukuhara differentiability.

We give a new definition of fuzzy fractional derivative called fuzzy conformable fractional derivative. Using this definition, we prove some results and we introduce new definition of generalized fuzzy conformable fractional derivative.

The aim of the present study is to analyze and find a solution for the model of nonlinear ordinary differential equations (ODEs) describing the so-called coronavirus (COVID-19), a deadly and most parlous virus. The mathematical model based on four nonlinear ODEs is presented, and the corresponding numerical results are studied by applying the variational iteration method (VIM) and differential transformation method (DTM).

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.