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We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid's surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles' behavior on the colloidal particle's surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better. [ABSTRACT FROM AUTHOR]

Viral infections pose significant threats to public health globally. Understanding the behavior, transmission, and epidemiology of viruses is essential for developing strategies to prevent, control, and manage outbreaks. Mathematical models help in identifying emerging viral pathogens, assessing their risks, and implementing effective public health measures to mitigate their impact. In this work, we formulate the dynamics of Covid-19 viral infection with the effect of vaccination in fractional framework. Our study is mainly concerned with the dynamical behavior and qualitative analysis of Covid-19 dynamics. The model is investigated for basic properties and the threshold of the system is determined. To scrutinize the solution of the recommend system, we use the fixed point theorems of Schaefer and Banach to evaluate the existence and uniqueness of solutions. Moreover, sufficient conditions for the Ulam–Hyers stability of system is established through mathematical skills. We examine the solution pathways of our model through a numerical scheme to show the importance different input parameters of the system. Our findings emphasize the pivotal role of asymptomatic carriers and losing rate of immunity as critical determinants that can heighten the challenge of controlling Covid-19. Vaccination rate and the fractional parameters are attractive parameters while asymptotic fraction poses a significant risk since it has the ability to spread the illness to uninfected areas. We suggest that effective control of the transmission rate can effectively regulate the intensity of Covid-19 transmission.

This research investigates the time-fractional dynamics of two interacting dipolar spins in the presence of the Dzyaloshinsky-Moriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions, alongside a uniform external magnetic field. The main subject is to employ the time-fractional Schrödinger equation in order to examine the time-fractional development of quantum resources within the dipolar spin system, initially prepared in either a separable or partially entangled state. The study employs the ℓ 1 norm of coherence, logarithmic negativity, and local quantum uncertainty to evaluate quantum coherence, entanglement and nonclassical correlations, respectively. This examination encompasses various parameters including the fractional derivative order, initial state configurations, strengths of DM and KSEA interactions, and dipolar coupling. Our main findings reveal that these quantifiers increase over time and evolve towards states of maximal coherence, regardless of whether the system is initially in a separable or partially entangled state. Lower values of the fractional derivative parameter are observed to enhance or steady the memory of the produced fractional state, which demonstrates the significant influence of fractional time dynamics. Additionally, strong DM and KSEA interactions significantly enhance the levels of quantum correlations and coherence of the system, a trend mirrored by the dipolar coupling constants. [ABSTRACT FROM AUTHOR]

Viral infections pose significant threats to public health globally. Understanding the behavior, transmission, and epidemiology of viruses is essential for developing strategies to prevent, control, and manage outbreaks. Mathematical models help in identifying emerging viral pathogens, assessing their risks, and implementing effective public health measures to mitigate their impact. In this work, we formulate the dynamics of Covid-19 viral infection with the effect of vaccination in fractional framework. Our study is mainly concerned with the dynamical behavior and qualitative analysis of Covid-19 dynamics. The model is investigated for basic properties and the threshold of the system is determined. To scrutinize the solution of the recommend system, we use the fixed point theorems of Schaefer and Banach to evaluate the existence and uniqueness of solutions. Moreover, sufficient conditions for the Ulam–Hyers stability of system is established through mathematical skills. We examine the solution pathways of our model through a numerical scheme to show the importance different input parameters of the system. Our findings emphasize the pivotal role of asymptomatic carriers and losing rate of immunity as critical determinants that can heighten the challenge of controlling Covid-19. Vaccination rate and the fractional parameters are attractive parameters while asymptotic fraction poses a significant risk since it has the ability to spread the illness to uninfected areas. We suggest that effective control of the transmission rate can effectively regulate the intensity of Covid-19 transmission. [ABSTRACT FROM AUTHOR]

The impact of power-law memory on the dynamics of the Caputo standard fractional map is investigated in this paper. The definition of a complexity measure for the Caputo standard fractional map is introduced. This measure evaluates both the average algebraic complexity of a trajectory and the distribution of the trajectories of different types in the phase space of the system. The interplay between the small-scale spatial chaos and the large-scale spatial behavior is observed and measured during the transition of the Caputo standard fractional map from the circle map to the classical standard map. It is demonstrated that the impact of the fractional derivative on the complexity of the fractional system is not straightforward and is predetermined by the physical properties governing the dynamics of that system. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of parametric GF operators are described. Some properties of the parametric GFIs and the parametric GFDs are proved. Representation of the parametric GFIs and parametric GFDs via the GF integrals on the finite interval (a, b). Fundamental theorems of parametric GFC are proved. Applied mathematical interpretations of parametric general fractional derivatives for processes with memory are given. [ABSTRACT FROM AUTHOR]

An acute bacterial infection called avian spirochetosis is spread by ticks to a variety of birds. Clinical symptoms can vary greatly and are frequently non-specific. To diagnose a condition, the infectious spirochete must be detected. Here, we structure an epidemic model for the transmission of avian spirochetosis to visualize the interaction between tick and bird populations. The recommended dynamics of avian spirochetosis is illustrated with the help of fractional framework. We inspected the steady-states of the system of the avian spirochetosis for the stability analysis. The next-generation technique is used to evaluate the model's reproduction parameter R 0. The infection-free and endemic steady-state of avian spirochetosis were shown to be locally asymptotically stable under the specified conditions. Through mathematical skills, the positivity of solutions is determined. Additionally, evidence supporting the existence and uniqueness of the avian spirochetosis framework solution has been shown. We conduct modified simulations of the suggested avian spirochetosis system with different input factors to study the complex phenomena of avian spirochetosis under the effect of numerous input parameters. Our outcomes illustrate the significance and plausibility of fractional parameter, and they also suggest that this input parameter may adequately account for these kinds of observations. [ABSTRACT FROM AUTHOR]

Currently, immunotherapy is seen to be the most effective cancer treatment. This is especially true while treating chronic lymphocytic leukemia (CLL), a slow-growing B-lymphocyte neoplasm that gradually compromises the immune system. Mathematical modeling is acknowledged as a key technique for analyzing theoretical and practical challenges in this field of cancer research and others. We were inspired to develop a mathematical model because of its dearth in investigations of chemotherapy-induced immunotherapy for CLL. This study effort formulates the dynamics of lymphocytic leukemia utilizing fractional calculus to conceptualize the complex processes of this viral illness. The basic idea of fractional calculus has been shown using the Atangana–Baleanu framework. For the suggested model's chaotic and dynamic behavior, a new numerical approach is described. We inspect the stability and convergence of the suggested numerical technique in our work. The fluctuation of various system input elements has demonstrated the oscillatory and chaotic behavior of the system. Moreover, we have demonstrated how the suggested mechanism of lymphocytic leukemia infection is affected by fractional order. Through numerical simulations, the most important input parameters are emphasized, and the policymakers are given control intervention suggestions. [ABSTRACT FROM AUTHOR]

Discretemaps with uniformly distributed memory fading parameter are proposed. These maps are derived from equations with fractional derivatives and fractional integrals of uniformly distributed orders and periodic sequence of kicks that are described by Dirac delta-functions. The fractional differential and fractional integral equations, for which the discrete maps are derived, are nonlinear equations with fractional operators proposed by Nakhushev and Pskhu. The solutions of these fractional differential and integral equations of distributed orders are derived. It is emphasized that the fractional integrals and derivatives with a uniformly distributed order cannot be considered as mutually inverse operators for which the fundamental theorems hold. Fundamental theorems of fractional calculus for fractional operators with uniformly distributed order, which are proved by Pshu, are used to obtain discrete maps with uniformly distributed memory fading parameter. In this paper, we also derive discrete maps with memory from equations with FDs and FIs that are inverse operators to FDs and FIs of a distributed order. The all suggested discrete maps are derived from fractional differential and integral equations without an approximation. [ABSTRACT FROM AUTHOR]

Typhoid fever poses a severe hazard to public health and affects at least 27 million people annually around the globe. It is significant to conceptualize the key factors of the transmission phenomena of this bacterial infection. We formulate a novel epidemic model for typhoid fever with vaccination and carriers via Caputo‐Fabrizio operator. The fundamental results of the model are inspected analytically and the basic reproduction value are then assessed. The Jacobian matrix approach has been used to demonstrate the local stability of the infection‐free steady‐state for R0<1$$ {R}_0<1 $$. We interrogated the uniqueness and existence of the solution of the recommended fractional dynamics of typhoid fever. We illustrated the solution pathways of the recommended system of typhoid fever numerically and demonstrated the dynamics of the infection with variation of different parameters. Our results demonstrated how the dynamical behavior of the typhoid fever infection is influenced by input parameters. The findings highlight the significance and persuasive behavior of fractional order and show that the model's fractional memory effects appear to be a good fit for these kinds of findings. We have shown the most attractive factors of the system for the control and prevention through our study. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u (0 , t) − 2 , our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher's equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. [ABSTRACT FROM AUTHOR]

We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid’s surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles’ behavior on the colloidal particle’s surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better.

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: "is there a more optimal way to optimize?". Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

We investigate a three-level system in the context of the fractional Schrödinger equation by considering fractional differential operators in time and space, which promote anomalous relaxations and spreading of the wave packet. We first consider the three-level system omitting the kinetic term, i.e., taking into account only the transition among the levels, to analyze the effect of the fractional time derivative. Afterward, we incorporate a kinetic term and the fractional derivative in space to analyze simultaneous wave packet transition and spreading among the levels. For these cases, we obtain analytical and numerical solutions. Our results show a wide variety of behaviors connected to the fractional operators, such as the non-conservation of probability and the anomalous spread of the wave packet.

In this research work, we construct an epidemic model to understand COVID-19 transmission vaccination and therapy considerations. The model's equilibria were examined, and the reproduction parameter was calculated via a next-generation matrix method, symbolized by $ \mathcal{R}_0 $. We have shown that the infection-free steady state of our system is locally asymptotically stable for $ \mathcal{R}_0 < 1 $. Also, the local asymptotic stability of the endemic steady state has been established for $ \mathcal{R}_0 > 1 $. We have used a partial rank correlation coefficient method for sensitivity analysis of the threshold parameter $ \mathcal{R}_0 $. The contribution of vaccination to the threshold parameter is explored through graphical results. In addition to this, the uniqueness and existence of the solution to the postulated model of COVID-19 infection is shown. We ran various simulations of the proposed COVID-19 dynamics with varied input parameters to scrutinize the complex dynamics of COVID-19 infection. We illustrated the variation in the dynamical behavior of the system with different values of the input parameters. The key factors of the system are visualized for the public health officials for the control of the infection.

Drinking or recreating water that has been polluted with disease-causing organisms or pathogens is what causes waterborne infections. It should be noted that many water-borne infections can also transmit from person to person, by contact with animals or their surroundings, or by ingesting tainted food or beverages. Schistosomiasis is a water-borne infection found in different areas of the globe. Mostly people with this viral infection live in Africa with limited resources and medications. Therefore, investigation of this infection is significant to reduce its economic burden on the society. We formulated a novel epidemic model for schistosomiasis water-borne infection with the help of the Atangana–Baleanu derivative. The rudimentary theory of fractional-calculus has been presented for the analysis of our system. We start by looking at the model solution's non-negativity and uniqueness. The basic reproduction number and equilibria of the hypothesized water-borne infection model are next evaluated. Local stability of the infection-free steady-state has been established through Jacobian matrix method for R 0 < 1. In addition, the suggested model's solution is calculated using an iterative technique. Finally, we give numerical simulations for various input values to illustrate the impact of memory index and other input factors of the system. Our findings showed the influence of input parameters on the dynamical behaviour of the schistosomiasis infection. The results demonstrate the importance and persuasive behaviour of fractional order, and reveal that fractional memory effects in the model seem to be a good fit for this type of findings. [ABSTRACT FROM AUTHOR]

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can yield inconsistent parameter values, as they are tailored to specific time series models. In our previous work, we explored order disbalance time series from the broader perspective of fractional L'evy stable motion, revealing a stable anti-correlation in the financial market order flow. However, a more detailed analysis of empirical data indicates the need for a more specific order flow model that incorporates the power-law distribution of limit order cancellation times. When considering a series in event time, the limit order cancellation times follow a discrete probability mass function derived from the Tsallis q-exponential distribution. The combination of power-law distributions for limit order volumes and cancellation times introduces a novel approach to modeling order disbalance in the financial markets. Moreover, this proposed model has the potential to serve as an example for modeling opinion dynamics in social systems. By tailoring the model to incorporate the unique statistical properties of financial market data, we can improve the accuracy of our predictions and gain deeper insights into the dynamics of these complex systems. [ABSTRACT FROM AUTHOR]

Dengue infection brings unimaginable damage in low-income countries across the world to public health, social, and economical fields. Scientists rely on modeling to preferable understand the transmission phenomena of dengue in order to forecast some preventive measures and to give better data for the development of vaccines and medications. In present work, we construct a compartmental model for dengue infection via fractional derivatives to show the influence of memory. The suggested system's solution is investigated for existence as well as uniqueness with the help of fractional calculus fundamental characteristics. The model is analyzed and the reproduction number is obtained through Next-generation Matrix technique, moreover, the sensitivity of the reproduction number is determined using the partial rank correlation coefficient (PRCC) method. We have shown that if $ R_0<1 $, the disease-free steady-state of the dengue model is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) in the absence of vaccine. In particular case, we have shown that the infection of dengue persists uniformly in the system for $ R_0 $ greater than one. Furthermore, we quantitatively showed memory's impact on $ R_0 $ by fluctuation of different factors. Our result predicted that the biting frequency, mosquito recruiting rate, and memory index are the most sensitive aspects in efficiently controlling new dengue illnesses, according to our findings. The effect of vaccination on the suggested system's threshold is also investigated. Novel numerical schemes are presented to conceptualize the time series of the system in different scenarios. The findings suggested that more accurate and precious results are obtained through non-integer derivative and that memory plays a beneficial role in the management of dengue disease. We demonstrated that manipulating the index of memory in the system may manage the system reproduction number and endemic level of dengue infection. [ABSTRACT FROM AUTHOR]

Complex problems in nonlinear dynamics foreground the critical support of artificial phenomena so that each domain of complex systems can generate applicable answers and solutions to the pressing challenges. This sort of view is capable of serving the needs of different aspects of complexity by minimizing the problems of complexity whose solutions are based on advanced mathematical foundations and analogous algorithmic models consisting of numerous applied aspects of complexity. Evolutionary processes, nonlinearity and all the other dimensions of complexity lie at the pedestal of time, reveal time and occur within time. In the ever-evolving landscape and variations, with causality breaking down, the idea of complexity can be stated to be a part of unifying and revolutionary scientific framework to expound complex systems whose behavior is perplexing to predict and control with the ultimate goal of attaining a global understanding related to many branches of possible states as well as high-dimensional manifolds, while at the same time keeping abreast with actuality along the evolutionary and historical path, which itself, has also been through different critical points on the manifold. In view of these, we put forth the features of complexity of varying phenomena, properties of evolution and adaptation, memory effects, nonlinear dynamic system qualities, the importance of chaos theory and applications of related aspects in this study. In addition, processes of fractional dynamics, differentiation and systems in complex systems as well as the dynamical processes and dynamical systems of fractional order with respect to natural and artificial phenomena are discussed in terms of their mathematical modeling. Fractional calculus and fractional-order calculus approach to provide novel models with fractional-order calculus as employed in machine learning algorithms to be able to attain optimized solutions are also set forth besides the justification of the need to develop analytical and numerical methods. Subsequently, algorithmic complexity and its goal towards ensuring a more effective handling of efficient algorithms in computational sciences is stated with regard to the classification of computational problems. We further point out the neural networks, as descriptive models, for providing the means to gather, store and use experiential knowledge as well as Artificial Neural Networks (ANNs) in relation to their employment for handling experimental data in different complex domains. Furthermore, the importance of generating applicable solutions to problems for various engineering areas, medicine, biology, mathematical science, applied disciplines and data science, among many others, is discussed in detail along with an emphasis on power of predictability, relying on mathematical sciences, with Artificial Intelligence (AI) and machine learning being at the pedestal and intersection with different fields which are characterized by complex, chaotic, nonlinear, dynamic and transient components to validate the significance of optimized approaches both in real systems and in related realms.

We investigated two different approaches, which can be used to extend the standard quantum statistical mechanics. One is based on fractional calculus, and the other considers the extension of the concept of entropy, i.e., the Tsallis statistics. We reviewed and discussed some of the main properties of these approaches and used the thermal Green function formalism to perform the developments, simultaneously allowing us to analyze each case's dynamics and thermodynamics aspects. In particular, the results allow us to understand how the extensions change the behavior of some quantities, particularly fluctuations related to the system. [ABSTRACT FROM AUTHOR]

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod 1 topology instead of Skorohod 1 topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the 1 topology. [ABSTRACT FROM AUTHOR]

We investigate a three-level system in the context of the fractional Schrödinger equation by considering fractional differential operators in time and space, which promote anomalous relaxations and spreading of the wave packet. We first consider the three-level system omitting the kinetic term, i.e., taking into account only the transition among the levels, to analyze the effect of the fractional time derivative. Afterward, we incorporate a kinetic term and the fractional derivative in space to analyze simultaneous wave packet transition and spreading among the levels. For these cases, we obtain analytical and numerical solutions. Our results show a wide variety of behaviors connected to the fractional operators, such as the non-conservation of probability and the anomalous spread of the wave packet. [ABSTRACT FROM AUTHOR]

In this paper, we extend the applications of difference operators and related sequence spaces for the study of the fractional calculus and its dynamics. In this context, some useful results on dynamics of fractional derivatives have been established and demonstrated with the help of existing formulas. Particularly, this study suggests some basic reasons for dynamic behaviors of Riemann‐Liouville fractional derivatives and provides certain idea for neutralizing them. [ABSTRACT FROM AUTHOR]

Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by R 0 . The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if R 0 < 1 and unstable otherwise. Furthermore, the Ulam–Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection. [ABSTRACT FROM AUTHOR]

We apply an extension of dual-phase-lag in thermal systems to predict the photoacoustic signal for transmission configuration and characteristics of the open photoacoustic cell technique. For this, we consider a particular case from Jeffrey's equation as an extension of the generalized Cattaneo equations. In this context, we obtain the temperature distribution under the effects of fractional differential operators, allowing the calculation of the Photoacoustic signal for the transmission set-up. The results show a rich class of behaviors related to the anomalous diffusion connected to these fractional operators. [ABSTRACT FROM AUTHOR]

In this research work, we construct an epidemic model to understand COVID-19 transmission vaccination and therapy considerations. The model's equilibria were examined, and the reproduction parameter was calculated via a next-generation matrix method, symbolized by R0. We have shown that the infection-free steady state of our system is locally asymptotically stable for R0 < 1. Also, the local asymptotic stability of the endemic steady state has been established for R0 > 1. We have used a partial rank correlation coefficient method for sensitivity analysis of the threshold parameter R0. The contribution of vaccination to the threshold parameter is explored through graphical results. In addition to this, the uniqueness and existence of the solution to the postulated model of COVID-19 infection is shown. We ran various simulations of the proposed COVID-19 dynamics with varied input parameters to scrutinize the complex dynamics of COVID-19 infection. We illustrated the variation in the dynamical behavior of the system with different values of the input parameters. The key factors of the system are visualized for the public health officials for the control of the infection. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green’s function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario.

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey's equation, an extension of the Generalized Cattaneo Equations (GCEs). Obtaining the temperature distribution by incorporating the effects of fractional differential operators enables us to determine the TE effects in solid samples accurately. This study contributes to understanding the mechanisms that contribute to the PA signal and highlights the importance of considering fractional differential operators in the analysis of thermoelastic bending. As a result, we can determine the PA signal's TE component. Our findings demonstrate that the fractional differential operators lead to a wide range of behaviors, including dissipative effects related to anomalous diffusion. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a generalized diffusion-like equation by considering a spatial and time fractional derivative and the presence of non-local terms, which can be related to reaction or adsorption–desorption processes. We use the Green function approach to obtain solutions and evaluate the spreading of the system to show a rich class of behaviors. We also connect the results obtained with the anomalous diffusion processes. [ABSTRACT FROM AUTHOR]

In this contribution, we investigate the effects of the immunological memory in the population against the strain mutation of a disease, assuming that this memory is enhanced by the dynamics that follow a multi-order fractional SIRC model. We use weekly infection data on Influenza H1N1 in the state of Rio Grande do Sul, reported in the year of 2010, as the guide for our simulations and parameter choices. The simulated results suggest that the best scenarios, regarding the Influenza H1N1 data fit and that have a long-term prevention of reinfection for mutated strains of a circulating disease is the one in which the compartment of the population has a distinct level of immunological memory. Hence, any immunization strategy should be applied as early as possible, allowing the individual to acquire immunological memory before the strain can undergo mutations. [ABSTRACT FROM AUTHOR]

Although the therapy of chronic myelogenous leukemia (CML) has progressed because of imatinib (IM) and other tyrosine kinase inhibitors (TKIs), the majority of patients still do not recover. To better regulate the remaining leukemic cell population, TKI combo therapy may be improved with a deeper understanding of the underlying mechanisms. We employed a mathematical system which incorporated the intricate phenomena of immune system to CML. We use a fractional derivative framework in this work to understand the dynamics of CML. Additionally, in our work, we concentrate on the qualitative characterization and dynamical behavior of CML interactions. For the proposed model, we examine the singularity and existence using fixed point theorems by Banach and Schaefer. We provide the necessary criteria for our suggested fractional model's Ulam–Hyers stability. The influence of the factors on the dynamics of CML is highlighted by closely examining the solution paths by using a numerical scheme. To be more precise, we emphasized how the suggested system's dynamic and chaotic behavior varied depending on the fractional order and other system factors. Policymakers are advised to consider the most crucial elements of CML dynamics. In order to inform policymakers and health authorities about the systems essential for control and treatment, it is crucial to investigate the dynamic characteristics of CML disease. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green's function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario. [ABSTRACT FROM AUTHOR]

In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing mechanics of media with general form of nonlocality in space and time. Balance equations for mass, momentum, and energy, which describe conservation laws for nonlocal continuum, are derived by using the fundamental theorems of the GFC. The general balance equation in the integral form are derived by using the second fundamental theorems of the GFC. The first fundamental theorems of GFC and the proposed fractional analogue of the Titchmarsh theorem are used to derive the differential form of general balance equations from the integral form of balance equations. Using the general fractional vector calculus, the equations of conservation of mass, momentum, and energy are also suggested for a wide class of regions and surfaces. [ABSTRACT FROM AUTHOR]

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a generalized diffusion-like equation by considering a spatial and time fractional derivative and the presence of non-local terms, which can be related to reaction or adsorption–desorption processes. We use the Green function approach to obtain solutions and evaluate the spreading of the system to show a rich class of behaviors. We also connect the results obtained with the anomalous diffusion processes.

In this article, we proposed new discrete maps with memory (DMM). These maps are derived from fractional differential equations (FDE) with the Hilfer fractional derivatives of non-integer orders and periodic sequence of kicks. The suggested DMM are obtained from these equations without any approximation, and they are a discrete form of the exact solutions. DMM are proposed for arbitrary positive orders of equations with the Hilfer fractional derivatives. As an example, the suggested maps are described for the orders lying in the intervals (0, 1) and (1, 2). The maps, which are derived from the equations with the Hilfer operators, allow us to consider a whole range of the maps derived from the FDE with the Caputo and Riemann–Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we proposed new predator‐prey models that take into account memory and kicks. Memory is understood as the dependence of current behavior on the history of past behavior. The equations of these proposed models are generalizations of the Lotka‐Volterra and Kolmogorov equations by using the Caputo fractional derivative of non‐integer order and periodic kicks. This fractional derivative allows us to take into account memory with power‐law fading. The periodic kicks, which are described by Dirac delta‐functions, take into account short duration of interaction between predators and prey. For the proposed equations, which are fractional differential equations with kicks, we obtain exact solutions that describe behaviors of predator and prey with power‐law fading memory. Using these exact solutions, we derive, without using any approximations, new discrete maps with memory that represent the proposed predator‐prey models with memory. [ABSTRACT FROM AUTHOR]

An approach, which is based on exact fractional differences, is used to formulate a lattice fractional field theories on unbounded lattice spacetime. An exact discretization of differential and integral operators of integer and non-integer orders is suggested. New type of fractional differences of integer and non-integer orders, which are represented by infinite series, are used in quantum field theory with non-locality. These exact differences have a property of universality, which means that these operators do not depend on the form of differential equations and the parameters of these equations. In addition, characteristic feature of the suggested differences is an implementation of the same algebraic properties that have the operator of differentiation (property of algebraic correspondence). Lattice analogs of the fractional-order N-dimensional differential operators are proposed. The continuum limit of the suggested lattice field theory gives a fractional field theory for the continuum four-dimensional spacetime. The fractional field equations, which are derived from equations for lattice spacetime with long-range properties of power-law type, contain the Riesz type derivatives on non-integer orders with respect to spacetime coordinates. [ABSTRACT FROM AUTHOR]

Novel dynamics are emerging in the power system due to the new Smart Grid (SG) environment. The high sampling rate of the Phasor Measurement Units (PMUs) enables them to capture the dynamic fluctuations in the power system measurements. Understanding the statistical and dynamic characteristics of the PMU data requires advanced data analytics techniques capable of performing accurate modeling of the power system variables (voltage, frequency, phase angle, and rate of change of frequency (ROCOF)). In this article, we provide evidence of the non-stationarity and fractality of PMU data collected from Europe. We adopt the Autoregressive Fractionally Integrated Moving Average (ARFIMA) models with non-integer differencing parameter to model the short-range and long-range correlations in the PMU data. Furthermore, the goodness-of-fit of the ARFIMA model is confirmed by analyzing the correlation and independence of the model residuals. Anomaly detection is among the promising applications of the PMU ARFIMA models. It is shown that the 2012 Indian blackout is accompanied by a change point in the differencing parameter opening the road to event (anomaly) detection by ARFIMA monitoring. [ABSTRACT FROM AUTHOR]

We study some fractal properties of the hereditary αω-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the α-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales.

• Obtaining mathematical discrete-time approach for fractional order models. • Rendering solution for utilizing actuators with fractional models. • Universal solution applicable to broad class of nonlinear systems. • Designing robust algorithm for discrete-time fractional control. This paper attempts to lay the theoretical foundations of the efficient employment of actuation systems with fractional order models. A discussion on the superiority of the discrete-time approaches, relative to continuous-time, is held and it is shown that discrete-time approaches demonstrate further compatibility to real world problems. Afterwards, a standard procedure is introduced that enables researchers and engineers to implement such actuation systems in practice. A general case of a nonlinear plant operated via an actuation system represented by Torvik's fractional three parameter model is delineated, analyzed and extensively discussed. The actuation system in discussion emerges frequently in numerous practical engineering problems and can be considered as a paragon of fractional actuation systems. To render the merits of the paper, the actuation system is simulated and it is illustrated that owing to the standard procedure offered in this paper, fractional actuation systems can be implemented both in simulation and practice hereafter to enhance the performance of numerous processes. [ABSTRACT FROM AUTHOR]

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.

By the rapid growth of available data, providing data‐driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive‐decision‐making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg‐Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible. [ABSTRACT FROM AUTHOR]