Discrete Fractional Difference Research Articles

Novel aspects of discrete dynamical type inequalities within fractional operators having generalizedℏ-discrete Mittag-Leffler kernels and application

Discrete fractional calculus (DFC) has had significant advances in the last few decades, being successfully employed in the time scale domain ℏZ. Understanding of DFC has demonstrated a valuable improvement in neural networks and modeling in other terrains. In the context of Riemann form (ABR), we discuss the discrete fractional operator influencing discrete Atangana-Baleanu (AB)-fractional operator having ℏ-discrete generalized Mittag-Leffler kernels. In the approach being presented, some new Pólya-Szegö and Chebyshev type inequalities introduced within discrete AB-fractional operators having ℏ-discrete generalized Mittag-Leffler kernels. By analyzing discrete AB-fractional operators in the time scale domain Z, we can perform a comparison basis for notable outcomes derived from the aforesaid operators. This type of discretization generates novel outcomes for synchronous functions. The specification of this proposed strategy simply demonstrates its efficiency, precision, and accessibility in terms of the methodology of qualitative approach of discrete fractional difference equation solutions, including its stability, consistency, and continual reliance on the initial value for the solutions of many fractional difference equation initial value problems. The repercussions of the discrete AB-fractional operators can depict new presentations for various particular cases. Finally, applications concerning bounding mappings are also illustrated.

Discrete Prabhakar fractional difference and sum operators

The Prabhakar fractional operator is commonly acclaimed as the queen model of fractional calculus . Our aim in this article is to introduce the notion of the discrete Prabhakar fractional operator with discrete generalized Mittag-Leffler function in the kernel, in the context of discrete fractional calculus. Also, we examine some relationships between our new model with the discrete Atangana–Baleanu fractional model implemented by Abdeljawad. By doing these relationships, we can find a few interesting properties of both, as well as of the original discrete Atangana–Baleanu fractional models and their iterated forms. We can confirm that this is the first paper introducing and studying the discrete Prabhakar fractional operators in the context of discrete fractional calculus.

An analysis of polynomial sequences and their application to discrete fractional operators

We consider sequences of polynomials , where , together with both additive and multiplicative perturbations of them under rational functions. After demonstrating several algebraic and analytical properties of these sequences, we show that they can be used to prove new theorems in the qualitative analysis of discrete fractional operators, which are a class of nonlocal operators. Consequently, as an application of the algebraic results, we prove several theorems regarding conditions under which the positivity of the discrete fractional difference of f implies the monotonicity of f.

IMAGE ENCRYPTION BASED ON TWO-DIMENSIONAL FRACTIONAL QUADRIC POLYNOMIAL MAP

A new fractional two-dimensional quadric polynomial discrete chaotic map (2D-QPDM) with the discrete fractional difference is proposed. Afterwards, the new dynamical behaviors are observed, so that the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits of the proposed map are given, respectively. The new discrete fractional map is exploited into color image encryption algorithm and it is illustrated with several examples. The proposed image encryption algorithm is analyzed in six aspects which indicates that the proposed algorithm is superior to other known algorithms as a conclusion.

IMAGE ENCRYPTION TECHNOLOGY BASED ON FRACTIONAL TWO-DIMENSIONAL DISCRETE CHAOTIC MAP ACCOMPANIED WITH MENEZES-VANSTONE ELLIPTIC CURVE CRYPTOSYSTEM

A new fractional two-dimensional triangle function combination discrete chaotic map (2DTFCDM) with the discrete fractional difference is proposed in this paper. The chaos behaviors are observed through the bifurcation diagrams, the largest Lyapunov exponent plot and the phase portraits. The proposed map is applied in color image encryption with the secret keys generated by Menezes–Vanstone Elliptic Curve Cryptosystem. The image encryption system is analyzed using six aspects indicating the superiority of the proposed algorithm compared to the other algorithms.

Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound

We investigate relationships between the sign of the discrete fractional sequential difference and the convexity of the function . In particular, we consider the case in which the bound for some and where , is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation.

QUALITATIVE PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENTIAL AND DIFFERENCE EQUATIONS ARISING IN PHYSICAL MODELS

Discrete fractional calculus (DFC) is suggested to interpret neural schemes with memory impacts. This study seeks to formulate some discrete fractional nonlinear inequalities with [Formula: see text] fractional sum operators that are used with some conventional and forthright inequalities. Taking into consideration, we recreate the explicit bounds of Gronwall-type inequalities by observing the principle of DFC for unknown functions here. Such inequalities are of new version relative to the current literature findings so far and can be used as a helpful method to evaluate the numerical solutions of discrete fractional differential equations. We show a few uses of the rewarded inequalities to mirror the advantages of our work. Regarding applications, we can apply the acquainted results to discuss boundedness, uniqueness, and continuous dependency on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations. The leading consequences may be proven by the usage of the analysis process and the methodology of the mean value theorem. These variations can be used as an advantageous device in the subjective examination solutions of discrete fractional difference equations.

Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

<p style='text-indent:20px;'>We investigate the relationship between the sign of the discrete fractional sequential difference <inline-formula><tex-math id="M1">\begin{document}$ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $\end{document}</tex-math></inline-formula> and the monotonicity of the function <inline-formula><tex-math id="M2">\begin{document}$ t\mapsto f(t) $\end{document}</tex-math></inline-formula>. More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>for some <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula>. We prove that even though the fractional difference can be negative, the monotonicity of the function <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula>, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

On Discrete Fractional Integral Inequalities for a Class of Functions

Discrete fractional calculus ℱ C is proposed to depict neural systems with memory impacts. This research article aims to investigate the consequences in the frame of the discrete proportional fractional operator. ℏ -discrete exponential functions are assumed in the kernel of the novel generalized fractional sum defined on the time scale ℏ ℤ . The nabla ℏ -fractional sums are accounted in particular. The governing high discretization of problems is an advanced version of the existing forms that can be transformed into linear and nonlinear difference equations using appropriately adjusted transformations invoking property of observing the new chaotic behaviors of the logistic map. Based on the theory of discrete fractional calculus, explicit bounds for a class of positive functions n n ∈ ℕ concerned are established. These variants can be utilized as a convenient apparatus in the qualitative analysis of solutions of discrete fractional difference equations. With respect to applications, we can apply the introduced outcomes to explore boundedness, uniqueness, and continuous reliance on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations.

The Nehari manifold method for discrete fractional p-Laplacian equations

The aim of this paper is to investigate the multiplicity of homoclinic solutions for a discrete fractional difference equation. First, we give a variational framework to a discrete fractional p-Laplacian equation. Then two nontrivial and nonnegative homoclinic solutions are obtained by using the Nehari manifold method.

Positivity and monotonicity results for triple sequential fractional differences via convolution

AbstractWe investigate the relationship between the discrete fractional difference(\Delta^{\gamma}\circ\Delta^{\beta}\circ\Delta^{\alpha}f)(t)and the positivity or monotonicity of the functionf. Our approach relies on interpreting the fractional difference as an appropriate convolution operator. The results we provide demonstrate that when compared to the double sequential case, i.e.,{(\Delta^{\beta}\circ\Delta^{\alpha}f)(t)}, there is relatively more complexity observed.

A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity

We utilize a new definition for the fractional delta operator and prove that it is equivalent by translation to the more commonly used operator. By means of the convolution operation we demonstrate that this new operator is strongly connected to the positivity, monotonicity, and convexity of the functions on which it operates. We also analyze the case of compositions of discrete fractional operators. Finally, since the operator we study here is translationally related to the more commonly used discrete fractional operators, we are able to establish many new results for all types of discrete fractional differences, and we explicitly demonstrate that our results improve all known existing results in the literature.

Existence and finite-time stability of discrete fractional-order complex-valued neural networks with time delays

Without decomposing complex-valued systems into real-valued systems, the existence and finite-time stability for discrete fractional-order complex-valued neural networks with time delays are discussed in this paper. First of all, in order to obtain the main results, a new discrete Caputo fractional difference equation is proposed in complex field based on the theory of discrete fractional calculus, which generalizes the fractional-order neural networks in the real domain. Additionally, by utilizing Arzela–Ascoli’s theorem, inequality scaling skills and fixed point theorem, some sufficient criteria of delay-dependent are deduced to ensure the existence and finite-time stability of solutions for proposed networks. Finally, the validity and feasibility of the derived theoretical results are indicated by two numerical examples with simulations. Furthermore, we have drawn the following facts: with the lower order, the discrete fractional-order complex-valued neural networks will achieve the finite-time stability more easily.

A uniformly sharp convexity result for discrete fractional sequential differences

We prove that a class of convexity-type results for sequential fractional delta differences is uniformly sharp. More precisely, we consider the sequential difference $\Delta _{1-\mu +a}^{\nu }\Delta _{a}^{\mu }f(t)$, for $t\in \mathbb {N}_{3+a-\mu -\nu }$, and demonstrate that there is a strong connection between the sign of this function and the convexity or concavity of $f$ if and only if the pair $(\mu ,\nu )$ lives in a particular subregion of the parameter space $(0,1)\times (1,2)$.

Different type kernel [formula omitted]fractional differences and their fractional [formula omitted]sums

The aim of this article is to recall and study fractional derivatives with singular kernels on hZ and define fractional derivatives with non-singular exponential and Mittag–Leffler kernels on hZ and study some of their properties. We shall follow the nabla time scale analysis and relate the h−nabla classical discrete fractional derivatives to the delta existing ones studied before by some authors. Some dual identities between left and right and delta and nabla, left and right h−fractional difference types will be investigated. The nabla h− discrete versions of the Mittag-Leffler functions will be recalled by means of the nabla h−fatorial functions and nabla h−Taylor polynomials. The discrete Laplace on hZ and its convolution theory are used often to proceed in our investigation. The obtained results will generalize the nabla classical discrete fractional differences and the nabla discrete fractional differences with discrete exponential and ML−kernels studied recently by Abdeljawad and Baleanu by setting h=1.

Image encryption technique based on new two-dimensional fractional-order discrete chaotic map and Menezes–Vanstone elliptic curve cryptosystem**Project supported by the National Natural Science Foundation of China (Grant Nos. 61072147 and 11271008).

We propose a new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference. Moreover, the chaos behaviors of the proposed map are observed and the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits are derived, respectively. Finally, with the secret keys generated by Menezes–Vanstone elliptic curve cryptosystem, we apply the discrete fractional map into color image encryption. After that, the image encryption algorithm is analyzed in four aspects and the result indicates that the proposed algorithm is more superior than the other algorithms.

Image Encryption Technology Based on Fractional Two-Dimensional Triangle Function Combination Discrete Chaotic Map Coupled with Menezes-Vanstone Elliptic Curve Cryptosystem

A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM) with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.

Existence of solutions for discrete fractional difference inclusions with boundary conditions

This paper is mainly concerned with the existence of solutions for a certain class of discrete fractional difference inclusions with boundary conditions.Under certain suitable conditions, the existence results are established by using fixed point theory for multi-valued upper semicontinuous maps. Also, an example is presented to illustrate the possible applications of the obtained results.

A uniformly sharp monotonicity result for discrete fractional sequential differences

A monotonicity-type result for functions \(f\ : \ \mathbb {N}_a\rightarrow \mathbb {R}\) satisfying the sequential fractional difference inequality $$\begin{aligned} \Delta _{1+a-\mu }^{\nu }\Delta _{a}^{\mu }f(t)\ge 0, \end{aligned}$$ for \(t\in \mathbb {N}_{2+a-\mu -\nu }\), where \(0<\mu <1\), \(0<\nu <1\), and \(1<\mu +\nu <2\), is proved, subject to the restriction that $$\begin{aligned} \mu <2(1-\nu ). \end{aligned}$$ We demonstrate that this result is sharp in the sense that the restriction \(\mu <2(1-\nu )\) cannot be improved.

Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality

In this article, we studied the Caputo and Riemann–Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall’s inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall’s inequality under a new defined α -Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.