Various Fractional differential equations have been examined during the last decades. Among them, singular equations are more notable. In this article, by using control functions, the existence of a solution for a bi-singular Fractional differential equation with multi-point initial value conditions is considered. In the following, some examples elucidate our main result. In this paper by using control functions method, we prove the existence of the solution.

This work studies the existence and the uniqueness of the solution to a kind of high-order nonlinear Fractional integro-differential equations involving Rieman-Liouville Fractional derivative. The boundary condition is of integral type which entangles ending point of the domain. First, the unique exact solution is extracted in terms of Green's function for the linear Fractional differential equation and then Banach contraction mapping theorem is applied to prove the main result in the case of general nonlinear source term. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability.

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.

We consider two simple non-resonant boundary value problems for a nabla Fractional difference equation. First, we construct associated Green's functions and obtain some of their properties. Under suitable constraints on the nonlinear part of the nabla Fractional difference equation, we deduce sufficient conditions for the existence of solutions to the considered problems through an appropriate fixed point theorem. We also provide two examples to demonstrate the applicability of the established results.

In this paper, we are concerned with positive solutions for higher order m{point nonlinear Fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m{point nonlinear Fractional boundary value problems with integral boundary conditions by using some results from the theory of  xed point index, Avery{Henderson  xed point theorem and the Legget{Williams  xed point theorem, respectively.

The aim of this research is to establish the analytic solution of partial differential equations with homogenous initial boundary conditions. Having hybrid Fractional order derivative allows us to have classical boundary and initial conditions. The solution of the problem is obtained in terms of bivariate Mittag-Leffler function as a Fourier series by utilizing separation of variables method (SVM) and the inner product defined on L2 [0, l]. The presented examples illustrate the accuracy and effectiveness of the SVM for the Fractional diffusion problems. The accuracy of the obtained solution can also be seen from the observation that as the Fractional order α,tends to 1, the solution of the Fractional diffusion problem tends to the solution of the diffusion problem.