In this article, we study a new nonlinear Langevin equation of two fractional orders with fractional boundary value conditions which is a generalization of previous Langevin equations. Based on Banach and Schauder fixed point theorems, the existence and uniqueness of solutions of this equation are investigated. Moreover, our hypotheses are simpler than similar works.

In this paper, we propose a new type of Fuzzy fractional differential equations under Fuzzy Kolwankar- Gangal local fractional derivatives (for short, Fuzzy KG-LFD) using Fuzzy Riemann-Liouville differentiability. Then, we prove some basic results in this area like, the relation between different types of Fuzzy KG-LFD and their r-cut representations, composition of the Fuzzy KG-LFD and the Fuzzy local fractional integral. Also, an application is provided in details such that the explicit solutions are expressed through Mittag-Leffler function.

Fuzzy fractional heat equations (FFHEs) are utilized to an-alyze the behaviour of the certain phenomena in various mathematical and scienti, c models. The main goal of this paper is to construct the solution of Fuzzy fractional heat equations by taking a reliable recipe of Sumudu transformation method and homotopy analysis method into account. These method allow us to remove the di, culties and restric-tions confronted in other methods. The feasibility of this method is con, rmed by given numerical examples. The result presented that the proposed method is suitable, powerful and reliable for obtaining the solution of Fuzzy fractional problems with FFHEs.

The aim of the current paper is to construct the shifted fractional-order Jacobi functions (SFJFs) based on the Jacobi polynomials to numerically solve the fractional-order pantograph differential equations. To achieve this purpose,  rst the operational matrices of integration, product, and panto-graph, related to the fractional-order basis, are derived (operational matrix of integration is derived in Riemann{Liouville fractional sense). Then, these matrices are utilized to reduce the main problem to a set of algebraic equa-tions. Finally, the reliability and efficiency of the proposed scheme are demon-strated by some numerical experiments. Also, some theorems are presented on existence of solution of the problem under study and convergence of our method.

This article is devoted to investigate the anlysis for the solution for a class of linear and nonlinear Caputo fractional FredholmVolterra integro-differential equations with multiple delays. The convergence and stability analysis for these equations are investigated.