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This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L 1 space is presented for the studied models using the proposed scheme. [ABSTRACT FROM AUTHOR]

The purpose of this study is to propose a computationally effective algorithm for the numerical evaluation of a fractional-order system of singular Lane-Emden type equations arising in physical problems. The fractional operator considered is in the sense of the Liouville-Caputo derivative. The presented matrix collocation method is based upon a combination of the quasilinearization method (QLM) and the shifted Legendre functions (SLFs) and is called QLM-SLFs method. By applying first the QLM to the nonlinear underlying system, we get a family of linear equations. Hence, a spectral matrix collocation scheme relied on the SLFs is designed to solve the resulting sequence of linear system of equations at very few iterations. The uniform convergence of the shifted Legendre expansion series solution is established. To illustrate the effectiveness of the proposed QLM-SLFs technique in the present paper, three test examples are carried out. The applicability and validity of the proposed method are testified through comparisons with the outcomes of other existing procedures in the literature. The proposed QLM-SLFs method is efficient and easy to implement. The approximation obtained by the method also converges quickly to the solutions of the underlying model problem. In comparison with available existing computational procedures, the QLM-SLFs approach shows that the use of Legendre functions together with QLM provides solutions with high accuracy and exponential convergence rate. [ABSTRACT FROM AUTHOR]

• Providing an inclusive, up-to-date, and organized review of the published research. • Taxonomy of microgrid based on control structures, specifications and components. • Energy management summation and classification based on several significant factors. • Discussing some of the skilled methodologies and techniques developed or adopted. • Taxonomy of the different modeling and handling methods of the uncertainty. Energy is one of the essential foundations for the sustainable development of human society, so its management is necessary. Energy management system (EMS) can be explained as the procedure of optimizing, planning, controlling, monitoring, and saving energy to maximize operations and efficiency and minimize consumption. Microgrid (MG) requires EMS as an efficient and optimal tool owing to the stochastic nature of electrical loads and renewable sources. Moreover, energy management system is responsible for operation of a MG in reliable, secure and economical manner in either states of grid-connected or disconnected. Many literatures have recently focused on the expansion of advanced strategies of the MG energy management for establishing a self-sustained MG in both industrial and academic research. Thus, a comparative research is needed for having a 360° viewpoint of the energy management domain in MGs. In this regard, this research investigates a comparative and critical analysis of the developed strategies of the energy management for the MGs from different views and aspects from 2009 to 2022. The review strategy systematically adopted by the author includes: (i) Extracting research papers relevant to energy management in MGs; (ii) Filtering the significant papers to prepare a database of related research papers (iii) Classifying the used methods for EMS based on the technique, control strategies, and structure; (iv) Discussing potential directions for future studies. In a wider outlook, this research provides a systematic and updated review of energy management strategies for MGs developed by different researchers. The author hopes that academicians and practitioners can use the suggested framework as well as the offers presented for further studies on this significant yet sophisticated issue. [ABSTRACT FROM AUTHOR]

We introduce a new generalized Caputo-type fractional derivative which generalizes Caputo fractional derivative. Some characteristics were derived to display the new generalized derivative features. Then, we present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. The proposed algorithm can be considered as a fractional extension of the classical Adams-Bashforth-Moulton method. Dynamic behaviors of some fractional derivative models are numerically discussed. We believe that the presented generalized Caputo-type fractional derivative and the proposed algorithm are expected to be further used to formulate and simulate many generalized Caputo type fractional models. [ABSTRACT FROM AUTHOR]

Abstract We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class of partial differential equations of fractional order. We demonstrate the applicability of the method by implementing it to solve a model fractional problem. [ABSTRACT FROM AUTHOR]

By using the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio (CF) and the Atangana–Baleanu (AB) fractional derivative, firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non-square matrices, or square & singular. We study existence of solutions and provide formulas for the case that there do exist solutions. Then, we study the existence of unique solution for given initial conditions. Several numerical examples are given to justify our theory. [ABSTRACT FROM AUTHOR]

In this paper we construct the Weyl–Titchmarsh theory for the fractional Sturm–Liouville equation. For this purpose we used the Caputo and Riemann–Liouville fractional operators having the order is between zero and one. [ABSTRACT FROM AUTHOR]

This study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete-time equations. With theories of r -cut set, fuzzy Caputo and Riemann–Liouville fractional differences are defined on a isolated time scale. Discrete Leibniz integral law is given by use of w -monotonicity conditions. Furthermore, equivalent fractional sum equations are established. Fuzzy discrete Mittag-Leffler functions are obtained by the Picard approximation. Finally, fractional discrete-time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete Mittag-Leffler functions. This paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty. [ABSTRACT FROM AUTHOR]

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered. [ABSTRACT FROM AUTHOR]

An image encryption technique based on the fractional logistic map is designed in this work. A novel shuffling technique is established by use of fractional chaotic signals. Then it is used to scramble pixel positions. The results are analyzed in comparison with the classical logistic map. Since the employed fractional chaotic map holds complicated dynamics behavior, the encryption result is highly secure. Moreover, by experimental and statistical analysis, we demonstrate that the encryption performance is better than the results in literature. [ABSTRACT FROM AUTHOR]

In this letter we propose a class of linear fractional difference equations with discrete-time delay and impulse effects. The exact solutions are obtained by use of a discrete Mittag-Leffler function with delay and impulse. Besides, we provide comparison principle, stability results and numerical illustration. [ABSTRACT FROM AUTHOR]

The first integral method (FIM) is applied to get the different type optical solitons of Kundu–Eckhaus equation (KE). A class of optical solitons of this equation is presented, and some of which are acquired for the first time. Constraint conditions guarantees existence of these solitons. It is illustrated that FIM is very effective method to reach the various type of the soliton solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a numerical method for solving generalized fractional differential equation of Caputo–Katugampola derivative. A corresponding discretization technique is proposed. Numerical solutions are obtained and convergence of numerical formulae is discussed. The convergence speed arrives at O ( Δ T 1 − α ) . Numerical examples are given to test the accuracy. [ABSTRACT FROM AUTHOR]

Discrete memory effects are introduced by fractional difference operators. Asymptotic stabilities of nonlinear fractional difference equations are investigated in this paper. A linear scalar fractional difference equality is utilized. Lyapunov second direct method is proposed for nonlinear discrete fractional systems. Asymptotic stability conditions are provided and some examples are given. [ABSTRACT FROM AUTHOR]

In this work, we investigate the soliton solutions to the resonant nonlinear Shrödinger's equation (R-NSE) with Kerr law nonlinearity. By adopting the Riccati–Bernoulli sub-ODE technique, we present the exact dark optical, dark-singular and periodic singular soliton solutions to the model. The soliton solutions appear with all necessary constraint conditions that are necessary for them to exist. We studied the R-NSE by analyzing a system of nonlinear partial differential equations (NPDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system, then we apply the general conservation theorem to establish a set of nontrivial and nonlocal conservation laws (Cls). Some interesting figures for the acquired solutions are Cls also presented. [ABSTRACT FROM AUTHOR]

In this study, the optimum q -homotopic analysis method is employed to solve fourth order partial integro-differential equations with high-order non-integer derivatives. Several specific and clear examples are also given to illustrate the simplicity and capacity of the proposed approach. All of the computations were performed using M a t h e m a t i c a . [ABSTRACT FROM AUTHOR]

In this paper we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. We provide the integration by parts formula and we use the Q -operator to confirm our results. The related Euler—Lagrange equations are obtained and one example is reported. Moreover, we formulate and discuss the discrete counterparts of our results. [ABSTRACT FROM AUTHOR]

The Backlund transformation is used to obtain optical soliton for a type of the Schrödinger equation. Kink-type and dark-optical soliton solutions are acquired of the Schrödinger equation. It is illustrated that the examined equation is integrable because if an equation has a Backlund transformation it could be integrable. Several constraint conditions for the parameters are derived that establish the existence of the soliton solutions. The numerical simulations supplement the analytical schemes. [ABSTRACT FROM AUTHOR]

In the fractional calculus, one of the main challenges is to find suitable models which are properly described by discrete derivatives with memory. Fractional Logistic map and fractional Lorenz maps of Riemann–Liouville type are proposed in this paper. The general chaotic behaviors are investigated in comparison with the Caputo one. Chaos synchronization is designed according to the stability results. The numerical results show the method’s effectiveness and fractional chaotic map’s potential role for secure communication. [ABSTRACT FROM AUTHOR]

The main purpose of this research is to present a generalization of Ψ –Hilfer fractional derivative, called as regularized Ψ –Hilfer, and study some of its basic characteristics. In this direction, we show that the ψ –Riemann–Liouville integral is the inverse operation of the presented regularized differentiation by means of the same function ψ. In addition, we consider an initial-value problem comprising this generalization and analyze the existence as well as the uniqueness of its solution. To do so, we first present an approximation sequence via a successive substitution approach; then we prove that this sequence converges uniformly to the unique solution of the regularized Ψ –Hilfer fractional differential equation (FDE). To solve this FDE, we suggest an efficient numerical scheme and show its accuracy and efficacy via some real-world applications. Simulation results verify the theoretical consequences and show that the regularized Ψ –Hilfer fractional mathematical system provides a more accurate model than the other kinds of integer- and fractional-order differential equations. [ABSTRACT FROM AUTHOR]

A fractional difference is defined by the use of the right and the left Caputo fractional differences. The definition is a two-sided operator of Riesz type and introduces back and forward memory effects in space difference. Then, a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains. A lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders. The result shows that the Riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media. [ABSTRACT FROM AUTHOR]

In this letter, the local fractional similarity solution is addressed for the non-differentiable diffusion equation. Structuring the similarity transformations via the rule of the local fractional partial derivative operators, we transform the diffusive operator into a similarity ordinary differential equation. The obtained result shows the non-differentiability of the solution suitable to describe the properties and behaviors of the fractal content. [ABSTRACT FROM AUTHOR]

The variational iteration method newly plays a crucial role in establishing new integral equations. The Lagrange multipliers of the method serve kernel functions of the Volterra integral equations. A concept of an optimal integral equation is proposed. Then nonlinear examples are used to show the strategy’s efficiency. [ABSTRACT FROM AUTHOR]

[Display omitted] • A new kind of piecewise fractional derivative is defined. • The one- and two dimensional piecewise fractional Galilei invariant advection–diffusion equations are defined. • The orthonormal piecewise Vieta-Lucas (VL) functions as a new family of basis functions are defined. • Fractional derivatives in the Caputo and ABC senses of these functions are computed. • Two hybrid methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for the introduced problems. • The accuracy of the proposed methods is shown in several numerical examples. Recently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously. This study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations are defined using this piecewise fractional derivative. A new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations. The accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically. This study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

Hidden Markov Model (HMM) is a stochastic process where implicit or latent stochastic processes can be inferred indirectly through a sequence of observed states. HMM as a mathematical model for uncertain phenomena is applicable for the description and computation of complex dynamical behaviours enabling the mathematical formulation of neural dynamics across spatial and temporal scales. The human brain with its fractal structure demonstrates complex dynamics and fractals in the brain are characterized by irregularity, singularity and self-similarity in terms of form at different observation levels, making detection difficult as observations in real-time occurrences can be time variant, discrete, continuous or noisy. Multiple Sclerosis (MS) is an autoimmune degenerative disease with time and space related dissemination, leading to neuronal apoptosis, coupled with some subtle features that could be overlooked by physicians. This study, through the proposed integrated approach with multi-source complex spatial data, aims to attain accurate prediction, diagnosis and prognosis of MS subgroups by HMM with Viterbi algorithm and Forward–Backward algorithm as the dynamic and efficient products of knowledge-based and Artificial Intelligence (AI)-based systems within the framework of precision medicine. Multifractal Bayesian method (MFM) accordingly applied to identify and eliminate "insignificant" irregularities while maintaining "significant" singularities. An efficient modelling of HMM is proposed to diagnose and predict the course of MS while using MFM method. Unlike the methods employed in previous studies, our proposed integrated novel method encompasses the subsequent approaches based on reliable MS dataset (X) collected: (i) MFM method was applied (X) to MS dataset to characterize the irregular, self-similar and significant attributes, thus, attributes with "insignificant" irregularities were eliminated and "significant" singularities were maintained. MFM-MS dataset (X ˆ) was generated. (ii) The continuous values in the MS dataset (X) and MFM-MS dataset (X ˆ) were converted into discrete values through vector quantization method of the HMM (iii) Through transitional matrices, different observation matrices were computed from the both datasets. (v) Computational complexity has been computed for both datasets. (vi) The results of the HMM models based on observation matrices obtained from both datasets were compared. In terms of the integrated HMM model proposed and the MS dataset handled, no earlier work exists in the literature. The experimental results demonstrate the applicability and accuracy of our novel proposed integrated method, HMM and Multifractal (HMM-MFM) method, for the application to the MS dataset (X). Compared with conventional methods, our novel method has achieved more superiority regarding extracting subtle and hidden details, which are significant for distinguishing different dynamic and complex systems including engineering and other related applied sciences. Thus, we have aimed at pointing a new frontier by providing a novel alternative mathematical model to facilitate the critical decision-making, management and prediction processes among the related areas in chaotic, dynamic complex systems with intricate and transient states. • Novel HMM-MFM model reveals critical significance of predictive quantization in dynamic complexity. • Predictive quantization by HMM-MFM model for dynamic and transient states in varying complex systems. • Viterbi algorithm's recursion enables maximization and uncovering of the most probable hidden state sequence. • Computational complexity and reliability of Forward–Backward procedure, guaranteeing local maxima and maximizing the objective function φ (N 2 T). • Multifarious knowledge-based approach with a facilitating function in precision medicine ensuring personalized treatment tailoring. [ABSTRACT FROM AUTHOR]

The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of fractional nabla difference equations, its proper dual system and its transposed dual system. By taking into consideration the case that the coefficients are square constant matrices with the leading coefficient singular, we study the prime system and by using the invariants of its pencil we give necessary and sufficient conditions for existence and uniqueness of solutions. After we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of the proper dual system and the transposed dual system. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, numerical examples are given based on a singular fractional nabla real dynamical system to justify our theory. [ABSTRACT FROM AUTHOR]

The goal of the manuscript is to analyze the existence of solutions for the Caputo fractional differential inclusion c D q x ( t ) ∈ F ( t , x ( t ) , c D β x ( t ) ) with the boundary value conditions x ( 0 ) = 0 and x ( 1 ) + x ′ ( 1 ) = ∫ 0 η x ( s ) ds , such that 0 < η < 1 , 1 < q ≤ 2 , 0 < β < 1 and q - β > 1 . Also, we investigate the existence of solutions for the Caputo fractional differential inclusion c D q x ( t ) ∈ F ( t , x ( t ) ) such that x ( 0 ) = a ∫ 0 ν x ( s ) ds and x ( 1 ) = b ∫ 0 η x ( s ) ds , where 0 < ν , η < 1 , 1 < q ≤ 2 and a , b ∈ R . [ABSTRACT FROM AUTHOR]

The model describing the mitigation of contamination through ventilation inside a moving waterway polluted via dispersed bases together with connected reduction of liquefied oxygen was investigated within the scope of fractional derivatives. The steady-state cases were investigated using some Caputo derivatives properties. The steady-state solutions in presence and absence of the dispersion were derived in terms of the Mittag–Leffler function. In the case of non-steady state, we derived the solution of the first equation in terms of the α -stable error function via the Laplace transform method. To solve the second equation, we constructed the fractional Green function via the Laplace, Fourier and Mellin transforms. The fractional Green function was expressed by mean of the H-function. Particularly, we presented the selected numerical results a function of distance and α . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, master–slave synchronization for the fractional difference equation is studied with a nonlinear coupling method. The numerical simulation shows that the designed synchronization method can effectively synchronize the fractional logistic map. The Caputo-like delta derivative is adopted as the difference operator. [Copyright &y& Elsevier]

Abstract: Fractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively. [Copyright &y& Elsevier]

Abstract: In this article we study the initial value problem of a class of non-homogeneous singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, non-square and square with a matrix pencil which has an identically zero determinant, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically we study the conditions under which the system has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory. [Copyright &y& Elsevier]

Abstract: In this manuscript, we consider a new fractional order chaotic system which exhibits interesting behavior such as two, three, and four scrolls. Such systems can be found extensively in mechatronics and power electronic systems which exhibit self-sustained oscillations. Synchronization between two such systems is an interesting problem either theoretically or practically. Using a sliding mode control methodology, we synchronize a unidirectional coupling structure for the two chaotic systems. Numerical simulations are used to verify the theoretical analysis. Additionally, we report the robustness of the system in the presence of a noise in simulation. [Copyright &y& Elsevier]

In this paper we discuss an application of fractional variational calculus to the Basset-type fractional equations. It is well known that the unsteady motion of a sphere immersed in a Stokes fluid is described by an integro-differential equation involving derivative of real order. Here we study the inverse problem, i.e. we consider the problem from a Lagrangian point of view in the framework of fractional variational calculus. In this way we find an application of fractional variational methods to a classical physical model, finding a Basset-type fractional equation starting from a Lagrangian depending on derivatives of fractional order. [ABSTRACT FROM AUTHOR]