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We investigate the existence and uniqueness of the solution and also the rate of convergence of a numerical method for a fractional differential equation in both q-calculus and (p; q)-calculus versions. We use the Banach and Schauder fixed point theorems in this study. We provide two examples, one by definition of the q-derivative and the other by (p; q)-derivative. We compare the rate of convergence of the numerical method. We like to clear some facts on (p; q)-calculus. The data from our numerical calculations show well that q-calculus works better than (p; q)-calculus in each case. [ABSTRACT FROM AUTHOR]

The most significant objective of this article is the adoption of a method with a free parameter known as "The Optimum Asymptotic Homotopy Method" which has been utilized in order to obtain solutions for integral differential equations of high-order non integer derivative. The process in this method is more favorable than "Homotopy Perturbation Method" as it has a more rapid convergence compared to the mentioned method or even the similar methods. Another advantage of this method is that the convergence rate is recognized as control area. It is worth mentioning that Caputo derivative is adopted in this article. A number of instances are provided to better understand the method and its level of efficiency compared to other same methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solution of the Riccati differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) using a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The linearization of the method involves splitting the problem into a linear and nonlinear part. While this approach is a semi-analytical iterative technique, there is a significant amount of analytical work where the computational times of integrals are needed to be carried out numerically. The novelty and the motivation behind this paper is the comparison of the time used in minutes which is given for three derivatives: CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. With the help of the software Mathematica, the computational results have been obtained. [ABSTRACT FROM AUTHOR]

This article investigates the existence of a solution for a singular fractional differential equation. For this, the researchers have changed the main differential equation into an integral equation, then through determining some assumptions that could control its main singular points; the existence of a solution has been proved for the equation by applying a fixed point theorem, as well. The significance of the proposed paper is regarded as the equation's novelty on its boundary condition which is a generalization of similar ones. Likewise the condition, is a generalization for the similar cases and it conduces to consider a singular equation with infinite singular points. Having two dealings on its dominate is of high significance for the equation which should be remarkable. [ABSTRACT FROM AUTHOR]