Here we state the main properties of the Caputo, Riemann-Liouville and the Caputo via Riemann-Liouville fractional derivatives and give some general notes on these properties. Some properties given in some recent literatures and used to solve fractional nonlinear partial differential equations will be proved that they are incorrect by giving some counter examples.

This paper investigates the solution to one-dimensional fractional differential equations with two types of derivative operators orders in range (1,2). Two linearized schemes numerical method are constructed. The considered FDEs equivalently transformed by Riemann–Liouville integral into their integro-partial problems reduce requirement for smoothness time. analysis stability and convergence is...

In this work, a Lie group reduction for (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the symmetry method with Riemann Liouville derivative. After reducing into two-dimensional nonlinear partial differential (NLFPDEs), power series (PS) applied to obtain exact solution. Further obtained solution analyzed convergence. Then, new conservation theorem gene...

We study a coupled system of implicit differential equations with fractional-order boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, at least one solution are established by applying Banach contraction Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities discussed. An example is presented to illustrate our main result. suggested general...

In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, discuss the existence solutions for mixed nonlinear differential equation involving right Caputo fractional derivative operator, and left Riemann–Liouville integral operators different orders. We apply decent fruitful approach fixed point theory to establish desired results. Examples a...

In this paper, we study two stochastic problems for time-fractional Rayleigh-Stokes equation including the initial value problem and terminal problem. Here, are perturbed by Wiener process, fractional derivative taken in sense of Riemann-Liouville, source function time-spatial noise nonlinear satisfy globally Lipschitz conditions. We attempt to give some existence r...