Fractional Differential Operator Research Articles

Some Results of R-Matrix Functions and Their Fractional Calculus

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the θ-integral operator. Additionally, we investigate the compositional properties of θ-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics.

Dynamic analysis and optimal control of a hybrid fractional monkeypox disease model in terms of external factors

The monkeypox virus (MPXV), which is a member of the Orthopoxvirus genus in the class Poxviridae, is the causative agent of the zoonotic viral infection MPXV. The disease is similar to smallpox, but it is usually less dangerous. This study examines the evolution of the MPXV epidemic in Canada with an emphasis on the effects of control employing actual data. The main challenge is accurately estimating the virus’s rate of transmission and assessing the effectiveness of vaccination campaigns. By taking into account the modified Atangana–Baleanu–Caputo (mABC) fractional difference operator, we develop an analytical framework for an outbreak caused by MPXV and broaden it to accommodate the fractional scenario. The non-negativity and boundedness are guaranteed by the computation of the fractional-order MPXV system. At the disease-free equilibrium (DFE) , we perform a local asymptotical stability analysis (LAS) and display the outcome for . When 1$$\\end{document}]]> and , the single endemic equilibrium point (EEP) of infectious rodents is globally asymptotically stable (GAS). Lyapunov’s approach and LaSalle’s invariant criterion demonstrate that the GAS in terms of EEP for infectious persons occurs when and 1$$\\end{document}]]>. Through the application of nonlinear least squares, we determine the parameter values applying actual cases collected from Canada. To further bolster the operator’s effectiveness, a number of tests of this novel kind of operator were conducted. We remark that in various time scale domains , the investigated discrete formulations will be -nonincreasing or -nondecreasing by examining -monotonicity formulations and the basic properties of the suggested operator. Algorithms are constructed in the discrete generalized Mittag–Leffler (GML) kernel for mathematical simulations, emphasizing the effects of the infection resulting from multiple factors. The dynamical technique used to build the MPXV framework was significantly impacted by fractional-order. In order to lessen the infections, severity, five time-dependent control factors are also implemented. The optimality criteria are produced by applying Pontryagin’s maximal argument to prove the validity of the most effective control. Various combos of control factors are offered to reduce the incidence of MPXV. The results of the current research provide governmental officials and healthcare professionals with practical steps to take in order to establish efficient and ideal preventive approaches to reduce the MPXV outbreaks.

Fractal Continuum Maxwell Creep Model

In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain ε(t), creep modulus J(t), and relaxation compliance G(t) in materials with fractal linear viscoelasticity can be described by their generalized forms, εβ(t),Jβ(t) and Gβ(t), where β=dimS/dimH represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are dimS and dimH for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study.

Fractional-order Sliding Mode Control of Manipulator Combined with Disturbance and State Observer

A fractional-order sliding mode control (FSMC) method for a manipulator based on disturbance and state observers is proposed. First, a state estimator is designed that can estimate the velocity and acceleration, and only joint position feedback and the mathematical model of the manipulator are needed. The state estimator converges in finite time. Then, the disturbance observer is designed. By designing the nominal system model of the manipulator, a disturbance observation error is introduced into the closed-loop control so that the performance of the manipulator can track the nominal system. Finally, a sliding mode controller (SMC) based on the fractional differential operator theory is also designed. The value of the sliding mode variable in the derivation of the controller is composed of the fractional derivative of the trajectory tracking error of the manipulator, whereas the fractional differentiation operation uses integration in its realization, and the integration is a low-pass filter, thus, high-frequency noise is suppressed. In the experimental section, the method designed is compared with the conventional sliding mode, which further reveals the rapidity and control accuracy of FSMC.

An Inverse Inequality for Fractional Sobolev Norms in Unbounded Domains

The nonlocal operators have found applications in various areas of contemporary science. The anomalous diffusion phenomena have been modeled by the fractional Poisson boundary-value problem. Electromagnetic fluids have been described by fractional differential equations. The fractional differential operators have found applications in material sciences, planar and space elasticity, probabilistic theory, harmonic analysis, and even in finance. The inverse inequality plays an important role in Numerical Analysis. The well-known results on inverse inequalities have been obtained in bounded domains and finite-dimensional spaces. Naturally, a new challenge arises to obtain inverse inequalities in the fractional Sobolev spaces. This paper is devoted to differential inequalities between fractional Sobolev norms. We expand the notion of a monotone function into a new notion supermonotone function and rigorously prove an inverse inequality for a class of differentiable functions in unbounded domains. Examples that demonstrate the theory are presented.

On existence results of boundary value problems of Caputo fractional difference equations for weak-form efficient market hypothesis

The weak-form efficient market hypothesis (EMH) raises profound issues in the theory of emerging phenomena. One fundamental question is whether log-returns and log-prices exhibit closely integer-valued dynamics when considering a set of fractional difference operators. To address these complex problems, we investigate the sufficient conditions for positive solutions to boundary value problems of Caputo fractional difference equations with nonlocal boundary conditions. These results have significant applications in determining positive rates and stock prices in finance. We focus on the boundary value problem with the Caputo fractional difference operator, derive the Green’s function, and analyze it to establish existence results using Krasnosel’skii fixed-point theorem. To account for data availability, we also reformulate the problem in a more flexible topological framework. We demonstrate the methodology using log-returns on Amazon stock prices and provide insights into EMH theory and its topological aspects.

Online public opinion prediction based on a novel conformable fractional discrete grey model

PurposeAs social networks have developed to be a ubiquitous platform of public opinion spreading, it becomes more and more crucial for maintaining social security and stability by accurately predicting various trends of public opinion dissemination in social networks. Considering the fact that the dissemination of online public opinion is a dynamic process full of uncertainty and complexity, this study establishes a novel conformable fractional discrete grey model with linear time-varying parameters, namely the CFTDGM(1,1) model, for more accurate prediction of online public opinion trends.Design/methodology/approachFirst, the conformable fractional accumulation and difference operators are employed to build the CFTDGM(1,1) model for enhancing the traditional integer-order discrete grey model with linear time-varying parameters. Then, to improve forecasting accuracy, a base value correction term is introduced to optimize the iterative base value of the CFTDGM(1,1) model. Next, the differential evolution algorithm is selected to determine the optimal order of the proposed model through a comparison with the whale optimization algorithm and the particle swarm optimization algorithm. The least squares method is utilized to estimate the parameter values of the CFTDGM(1,1) model. In addition, the effectiveness of the CFTDGM(1,1) model is tested through a public opinion event about “IG team winning the championship”. Finally, we conduct empirical analysis on two hot online public opinion events regarding “Chengdu toddler mauled by Rottweiler” and “Mayday band suspected of lip-syncing,” to further assess the prediction ability and applicability of the CFTDGM(1,1) model by comparison with seven other existing grey models.FindingsThe test case and empirical analysis on two recent hot events reveal that the CFTDGM(1,1) model outperforms most of the existing grey models in terms of prediction performance. Therefore, the CFTDGM(1,1) model is chosen to forecast the development trends of these two hot events. The prediction results indicate that public attention to both events will decline slowly over the next three days.Originality/valueA conformable fractional discrete grey model is proposed with the help of conformable fractional operators and a base value correction term to improve the traditional discrete grey model. The test case and empirical analysis on two recent hot events indicate that this novel model has higher accuracy and feasibility in online public opinion trend prediction.

Geometric Features of a Multivalent Function Pertaining to Fractional Operators

The Prabhakar fractional operator is commonly acclaimed as the queen model of fractional calculus. The distinction between univalent and multivalent functions became more formalized as part of the broader field of geometric function theory. This area of mathematics focuses on the study of analytic functions with specific geometric properties, such as injectivity, and their applications in various domains, including conformal mapping and potential theory. This paper’s goal is to discover new results of the harmonic multivalent functions defined in the open unit disc . Let present , the class of multivalent harmonic functions of the form in the open unit disc. Analyzing convolution with prabhahar fractional differential operator with multivalent harmonic function to be in the class . The coefficient inequality, growth rates, distortion properties, closure characteristics, neighborhood behaviors, and extreme points, all pertinent to this class were explored.

The time-fractional Allen–Cahn equation on geometric computational domains

Phase separation, the formation of two distinct phases from a single homogeneous mixture, has been extensively studied and observed in classical systems, typically as temperature changes. These separation phenomena depend on temperature and vary with different materials. We explore the numerical model, which shows various aspects of phase separation, such as time delay, the influence of heterogeneous materials on each domain, and the separation behavior in polyhedrons for industrial potentials.This work investigates the time-fractional Allen–Cahn model with the Caputo fractional differential operator, which is particularly suited for capturing long waiting times in highly inhomogeneous porous media. We numerically examine the behavior of the order parameter u on the surfaces of polyhedral and conical computational domains. We achieve the numerical simulation using an explicit Predictor–Corrector method in conjunction with a collocation approach, where Non-Uniform Rational B-Spline (NURBS) geometrical mapping is applied. Through this numerical method, we simulate the behavior of the order parameter u on various surfaces of solids, investigating the effects of varying time-fractional derivative orders on each face of these geometries. We introduce the Schwarz alternating collocation method to handle discrepancies in boundary values when different time-fractional orders are applied to each subdomain.Certain crystalline solids can be formed in materials science by joining different materials along their folding curves, resulting in varying material properties across boundaries. Observing phase separation in interconnected regions with varying material properties is challenging. Furthermore, phase separation that reflects the distinct characteristics of materials represented by time-fractional order differentiation in such interconnected regions with varying material properties has yet to be studied, making it a challenging problem, as far as our knowledge is concerned.To address the aforementioned challenge, this study conducts a comprehensive series of numerical tests to investigate phase-field properties on surfaces of varying materials. Our thorough approach effectively captures how material properties are reflected across different geometries and how time-fractional behaviors vary on each surface. The numerical results illustrate critical phenomena, such as Ostwald ripening under varying time-fractional orders on the surface of geometries like cubes, tetrahedrons, prisms, and cones. Furthermore, we observe the curve shortening flow on the flattened sector of a circular conical domain, providing a complete picture of the behavior of the order parameter u.

Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality

In this paper, we present a Wolfe-type dual model containing the Caputo-Fabrizio fractional derivative, weak and strong duality results, number of Kuhn-Tucker type sufficient optimality conditions and duality results for variational problems (VPs) with Caputo-Fabrizio (CF) fractional derivative operator under weaker invexity assumptions. This newly developed fractional derivative operator delivers an exponential type kernel of nonsingular nature which characterizes the dynamics of physical systems and engineering processes with memory characteristics in a better way. This derivative operator is a convolution of first-order derivative and an exponential function. The proposed work also derives the global optimality criterion of the primal problem, Mond-Weir type duality results, and Mangasarian type strict converse duality theorem in view of this fractional differential operator possessing an exponential type kernel. The derived theorems investigate the global optimal solution of the primal problem. The main results of the present work are duality theorems and sufficient optimality conditions for VPs possessing the CF derivative. In view of applications of the derived optimality theorems, Mond-Weir type duality results have been established subjected to invexity assumptions. These applications and results generalize other important duality results of VPs and also provide results connected to duality with generalized invexity in mathematical programming. Several conventional results can also be seen as a special case of the obtained results in this work. 2010 Mathematics Subject Classification90C29; 90C46; 26A33

The impact of the Caputo fractional difference operator on the dynamical behavior of a discrete-time SIR model for influenza A virus

Abstract Since there are few studies that deal with the fractional-order discrete-time epidemic models, this paper presents a new fractional-order discrete-time SIR epidemic model that is constructed based on the Caputo fractional difference operator. The effect of the fractional orders on the global dynamics of the SIR model is analyzed. In particular, the existence and stability of equilibrium points of the model are presented. Furthermore, we investigate the qualitative dynamical properties of the SIR model for both commensurate and incommensurate fractional orders using powerful nonlinear tools such as phase attractors, bifurcation diagrams, maximum Lyapunov exponent, chaos diagrams, and 0-1 test. In addition, the complexity of the discrete model is measured via the spectral entropy complexity algorithm. Further, an active controller is designed to stabilize the chaotic dynamics of the fractional-order SIR model. Finally, the suggested model is fitted with real data to show the accuracy of the current stability study. Our goal was achieved by confirming that the proposed SIR model can display a variety of epidiomologically observed states, including stable, periodic, and chaotic behaviors. The findings suggest that any change in parameter values or fractional orders could lead to unpredictable behavior. As a result, there is a need for additional research on this topic.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Time delayed fractional diabetes mellitus model and consistent numerical algorithm

The diabetes mellitus model (DMM) is explored in this study. Many health issues are caused by this disease. For this reason, the integer order DMM is converted into the time delayed fractional order model by fitting the fractional order Caputo differential operator and delay factor in the model. It is proved that the generalized model has the advantage of a unique solution for every time t. Moreover, every solution of the system is positive and bounded. Two equilibrium states of the fractional model are worked out i.e. disease free equilibrium state and the endemic equilibrium state. The risk factor indicator, R0 is computed for the system. The stability analysis is carried out for the underlying system at both the equilibrium states. The key role of R0 is investigated for the disease dynamics and stability of the system. The hybridized finite difference numerical method is formulated for obtaining the numerical solutions of the delayed fractional DMM. The physical features of the numerical method are examined. Simulated graphs are presented to assess the biological behavior of the numerical method. Lastly, the outcomes of the study are furnished in the conclusion section.

Efficient pure qP-wave modeling and reverse time migration in tilted transversely isotropic media calculated by a finite-difference approach

Subsurface anisotropy is commonly induced by shale layers, aligned cracks, and fine bedding and has a significant impact on seismic wave propagation. Ignoring anisotropic effects during seismic migration degrades image quality. Therefore, we derive a pure qP-wave equation with high accuracy for modeling seismic wave propagation in tilted transversely isotropic (TTI) media. However, the derived pure qP-wave equation requires a computationally expensive spectral-based method for performing numerical simulations. This is unsuitable for large-scale industrial applications, particularly 3D applications. For numerical efficiency, we first decompose the newly derived wave equation into some fractional differential operators and spatial derivatives. The spatial derivatives can be directly solved by conventional finite-difference (FD) approaches. Then, we use an asymptotic approximation to find an equivalent form of fractional differential operators, obtaining scalar operators that we can discretize with the FD method. Numerical tests show that our TTI pure qP-wave equation with an FD discretization can produce accurate and highly efficient wavefield simulations in TTI media. We also use our TTI pure qP-wave equation with an FD discretization as a forward engine to implement TTI reverse time migration (RTM). Synthetic examples and a field data test demonstrate that our TTI RTM can effectively correct the anisotropic effects, providing high-quality imaging results while maintaining good computational efficiency.

Sinusoidal heating on convective heat transfer of nanofluid under differential technique

AbstractSinusoidal convection describes a smooth and repetitive heat transfer oscillation that reasonably calculates the optimal heat transfer for nanoparticle's intermolecular interactions. This study aims to investigate the heat transfer effects of Stokes second problem on the performance of novel fractional differential operators based on rheological characteristics of nanofluid. For the sake of augmentation in thermal characteristics of the nanofluid, the fractionalized governing equations based on the suspension of nanoparticles namely copper and silver in ethylene‐glycol have been developed. The fractional Stokes second problem is simulated with the help of transformation techniques and statistical data analysis. The analytic optimization of mathematical solutions of velocity and temperature have been established in terms of newly defined different Mittag‐Leffler functions. The statistical data for velocity field is generated by varying Hartman number, Prandtl number and Eckert number. The comparative analysis for two types of fractional differential operators of temperature distribution is perceived through bar column label, three‐dimensional color pie chart, three‐dimensional color map surface with projection, three‐dimensional bars and linear fitting of data. The comparative results revealed that temperature distribution has a significant effect and great influence on the stability of nanofluid by the embedded parameters.

3rd-Order Differential Subordination and Superordination Results for Univalent Analytic Functions Associated with Zeta-Riemann Fractional Differential Operator

Using the results of 3rd order differential subordination, we introduce certain families of admissible functions and discuss some applications of 3rd order differential subordination for normalized analytic functions associated with novel fractional operator namely Zeta-Riemann Fractional differential operator. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted.

Numerical Analysis of the Discrete, Fractional Order PID Controller Using FOBD Approximation

This paper proposes a methodology of the numerical testing of the discrete, approximated Fractional Order PID Controller (FOPID). The fractional parts of the controller are approximated using the Fractional Order Backward Difference (FOBD) operator. The goal of the analysis is to find the memory length optimum from point of view both accuracy and duration of computations. To do it new cost functions describing both accuracy and numerical complexity were proposed and applied. Results of tests indicate that the optimum memory length lies between 200 and 400. The proposed approach can be also useful to examine of another discrete implementations of a fractional order operator using FOBD.

On α-Fractional Bregman Divergence to study α-Fractional Minty’s Lemma

In this paper fractional variational inequality problems (FVIP) and dual fractional variational inequality problems (DFVIP), Fractional minimization problems are defined with the help of fractional Caputo differential operator and Reimann-Liouville differential operator. The equivalence theorem of Caputo FVIP and fractional minimization problem is established. The equivalence between FVIP and DFVIP is studied with the help of fractional Bregman divergence to establish the fractional Minty’s lemma. Introduction: The theory of variational inequality problems is one of the best tool to solve the equilibrium and non-equilibrium problems arises in the branches of Applied Mathematics, Applied Physics, Engineering, Medical, Financial and many more. On the other hand, fractional calculus manages to study various types of dynamic problems arises in real and complex analysis It is certain for the development of fractional calculus because the fractional derivative has global correlation, which can reflect the historical process of the systematic function.With regard to the definition of fractional derivatives, many mathematicians studied fractional derivatives from different aspects and advanced different definitions for them, for example Riemann-Liouville (RL) derivative, Caputo derivative and other derivatives. Conclusions: The concept of -fractionally differential inequality problem is defined and studied its existence theorem with the help of -fractionally convex function. The concept of -fractionally Bregman divergence is developed and using it the existence of -fractionally minimization problem and -fractionally differential inequality problem are studied. Equivalence theorem in between -fractionally differential inequality problem and -fractionally dual differential inequality problem is studied to establish the Minty’s lemma in fractional calculus.

Weighted estimates of certain variable kernel operators on generalized Morrey spaces with variable exponent

Let T be the singular integral operator with variable kernel and Dγ(0≤γ≤1) be the fractional differentiation operator. Simultaneously, let T⁎ and T♯ be the adjoint and pseudo-adjoint of T, T1T2 and T1∘T2 be the product and pseudo-product of T1 and T2 respectively. In this article, we show that, TDγ−DγT, (T⁎−T♯)Dγ and (T1∘T2−T1T2)Dγ on the variable exponent generalized weighted Morrey spaces are bounded.