Numerical approach of riemann-liouville fractional derivative operator

This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings

A new RSA public key encryption scheme with chaotic maps

Public key cryptography has received great attention in the field of information exchange through insecure channels. In this paper, we combine the Dependent-RSA (DRSA) and chaotic maps (CM) to get a new secure cryptosystem, which depends on both integer factorization and chaotic maps discrete logarithm (CMDL). Using this new system, the scammer has to go through two levels of reverse engineering, concurrently, so as to perform the recovery of original text from the cipher-text has been received. Thus, this new system is supposed to be more sophisticated and more secure than other systems. We prove that our new cryptosystem does not increase the overhead in performing the encryption process or the decryption process considering that it requires minimum operations in both. We show that this new cryptosystem is more efficient in terms of performance compared with other encryption systems, which makes it more suitable for nodes with limited computational ability

Analytic solution of nonlinear singular BVP with multi-order fractional derivatives in electrohydrodynamic flows

In this study, a power series formula is proposed in order to introduce a new innovated numerical method called a newly Power Series Method (NPSM), beside with a construction of its error bound, to obtain approximate solutions of the standard fractional counterpart for a Boundary Value Problem (BVP) that appears in ElectroHydroDynamic (EHD) flows of the fluid. The solution for numerous fractional derivatives of both rational and irrational orders are numerically computed. Based on the residual error computation, the validity of the obtained results is verified. A high accuracy and a clear efficiency of the proposed method are revealed by discussing several numerical comparisons between such method and others.Publisher's Versio

On the Stability of Incommensurate <i>h</i>-Nabla Fractional-Order Difference Systems

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy&ndash;Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand