Fractional Differential Calculus Research Articles

Conformal and Non-Minimal Couplings in Fractional Cosmology

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Studies on a new K-symbol analytic functions generated by a modified K-symbol Riemann-Liouville fractional calculus

Analytic functions are very helpful in many mathematical and scientific uses, such as complex integration, potential theory, and fluid dynamics, due to their geometric features. Particularly conformal mappings are widely used in physics and engineering because they make it possible to convert complex physical issues into simpler ones with simpler answers. We investigate a novel family of analytic functions in the open unit disk using the K-symbol fractional differential operator type Riemann-Liouville fractional calculus of a complex variable. For the analysis and solution of differential equations containing many fractional orders, it offers a potent mathematical framework. There are ongoing determinations to strengthen the mathematical underpinnings of K-symbol fractional calculus theory and investigate its applications in various fields.•Normalization is presented for the K-symbol fractional differential operator. Geometric properties are offered of the proposed K-symbol fractional differential operator, such as the starlikeness property and hence univalency in the open unit disk.•The formula of the Alexander integral involving the proposed operator is suggested and studied its geometric properties such as convexity.•Examples are illustrated to fit our pure result. Here, the technique is based on the concepts of geometric function theory in the open unit disk, such as the subordination and Jack lemma.

Some new Grüss inequalities associated with generalized fractional derivative

<abstract><p>In this paper, we prove several new integral inequalities for the $ k $-Hilfer fractional derivative operator, which is a fractional calculus operator. As a result, we have a whole new set of fractional integral inequalities. For the generalized fractional derivative, we also use Young's inequality to find new forms of inequalities. Such conclusions for this novel and generalized fractional derivative are extremely useful and valuable in the domains of differential equations and fractional differential calculus, both of which have a strong connections to real-world situations. These findings may stimulate additional research in a variety of fields of pure and applied sciences.</p></abstract>

NEURAL NETWORK MODELING OF THE PROCESS MOISTURE TRANSFER ON THE BASIS OF DERIVATIVES FRACTIONAL ORDER

In this paper, a mathematical model of moisture transfer in capillary-porous media in one- and two-dimensional space is presented and investigated, for its description the apparatus of fractional integrodifferentiation was used. This approach made it possible to take into account such properties of a system with a fractal structure as memory, self-organization, temporal and spatial correlations. The complexity of this mathematical model complicates its application and requires significant computing power. To calculate the numerical solution of the differential equation and speed up the calculations, the fractal neural network method, which is based on the fPINN architecture, is used. This method uses a loss function that takes into account physical information about the process under study. Formulas from fractional differential calculus were applied to express fractional derivatives and a difference scheme for the loss function was developed. The software for the implementation of the neural network method was developed and the applied approach was justified by comparing the obtained numerical results with the results of experiments by other scientists and the results obtained using finite difference numerical methods. The reliability check of the investigated indicators indicates the adequacy of the mathematical model and the prospects for further application of the numerical fractal neural network method.

On the dynamics of a class of difference equations with continuous arguments and its singular perturbation

The dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of the considered dynamical systems. The local stability of the fixed points is studied. It is proved that the systems exhibit Hopf bifurcation which means that periodic orbits can be created from a fixed point by varying the delays. We compare the results of the singularly perturbed equation with those of the associated difference equation with continuous arguments when the perturbation parameter ∊⟶0 and with those of the corresponding delay differential equation when ∊⟶1. By letting the perturbation parameter ∊⟶0, we find that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. Furthermore, the singularly perturbed equation behaves qualitatively the same as its corresponding delay differential equation when ∊⟶1. Finally, we discuss that how this work can be generalized in the fractional differential calculus sense.

On Λ-Fractional Differential Equations

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed.

Linearized Fraction Order Control and Stability Analyses for Switched Reluctance Motor Drives

The nonlinearity and double salient pole of the switched reluctance motor make the control difficult and the torque fluctuation is large. To achieve good control performance, this study introduces a novel linearized fractional-order controller with the disturbance observer to regulate the speed and torque of a switched reluctance motor. Based on fractional differential calculus theory, the fractional-order proportional integral controller is designed. The disturbance observer is introduced to overcome the influence of nonlinearity on the switched reluctance motor and reduce the torque fluctuation. By using the extended state in the disturbance observer, the disturbances are estimated and compensated to achieve linearization and determinism. The parameter values of the proposed controller are obtained by employing the trial-and-error method. The frequency-domain analysis of the proposed controller shows that it has a good suppression performance for medium- and high-frequency disturbances. Through simulation, the performance of the system is analyzed. The results show good performance of the controller in speed change, current, and torque fluctuation reduction.

Fractional order differential calculus applied on decision making system to smart grid management via decision trees

This article portrays the relationship between fractional order differential calculus and the computational intelligence method, applying it to the improvement of intelligent systems. The Kirchhoff Laws, represented by second order differential equations, were solved via non-integer order differential calculus. The results obtained were used in the implementation of decision trees, which allowed the decision rules to be incorporated into the controllers. The results obtained by mathematical modeling did magnify the information extracted from Kirchhoff's Laws. Due to the gain magnitude of this information, the decision trees were obtained with greater precision and accuracy. In this way, it was achieved to build a hybrid system capable of being used in the development of controllers automata that has the lower response time and highest efficiency.

A new theory of fractional differential calculus

This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works.

The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional (porous media) orders are widely used, since analytical solving methods for solving are impossible.In the paper, we study the initial-boundary value problem for the loaded differential heat equation with a fractional Caputo derivative and conditions of the third kind. To solve the problem on the assumption that there is an exact solution in the class of sufficiently smooth functions by the method of energy inequalities, a priori estimates are obtained both in the differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. Due to the linearity of the problem under consideration, these inequalities allow us to state the convergence of the approximate solution to the exact solution at a rate equal to the approximation order of the difference scheme. An algorithm for the numerical solution of the problem is constructed.

Some new edge detecting techniques based on fractional derivatives with non-local and non-singular kernels

Computers and electronics play an enormous role in today’s society, impacting everything from communication and medicine to science. The development of computer-related technologies has led to the emergence of many new important interdisciplinary fields, including the field of image processing. Image processing tries to find new ways to access and extract information from digital images or videos. Due to this great importance, many researchers have tried to utilize new and powerful tools introduced in pure and applied mathematics to develop new concepts in imaging science. One of these valuable research areas is the contents of fractional differential calculus. In recent years, extensive applications to the new fractional operators have been employed in real-world problems. This article attempts to address a practical aspect of this era of research in the edge detecting of an image. For this purpose, two general structures are first proposed for making new fractional masks. Then the components in these two structures are evaluated using the fractional integral Atangana–Baleanu operator. The performance and effectiveness of these proposed designs are illustrated by several numerical simulations. A comparison of the results with the results of several well-known masks in the literature indicates that the results presented in this article are much more accurate and efficient. This is the main achievement of this article. These fractional masks are all novel and have been introduced for the first time in this contribution. Moreover, in terms of computational cost, the proposed fractional masks require almost the same amount of computations as the existing conventional ones. By observing the numerical simulations presented in the paper, it is easily understood that with proper adjustment for the fractional-order parameter, the accuracy of the obtained results can be significantly improved. Each of the new suggested structures in this article can be regarded as a valid and effective alternative for the well-known existing kernels in identifying the edges of an image.

Optimal Control Based on Neuro Estimator for Fractional Order Uncertain Non-linear Continuous-Time Systems

In this paper, a novel method is presented for optimal control of fractional order systems in the presence of an unknown term in system dynamic where fractional order derivative is considered to be between zero and one. In this method, neural network is used to estimate the unknown term in system dynamic. Neural network coefficients are updated adaptively and online. Updating laws are presented considering system requirements to achieve a homogeneous fractional order system. Another problem is formulating optimal control laws for fractional order system which is solved through fractional differential calculus. Since optimal fractional order control is non-causal and does not have a online solution, step-by-step progression and predictive control idea are used to obtain control signal and combine optimal controller and estimator. This method results in an optimal run-time control and resolves unknown terms in system dynamic. In addition, the closed loop system being uniform ultimate bounded is proved through direct Lyapunov method. Finally, simulation results are given to show efficiency of the proposed method.

Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness

Non-local time evolution of material stress/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus. The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear springpot possess an equivalent mechanical hierarchy in terms of a functionally-graded elastic column resting on viscous dashpots with power-law decay of the material properties. Some numerical applications are reported to show the capabilities of the proposed model.

Efecto del patrón de flujo sobre las pérdidas de presión para flujo bifásico a través de lechos porosos bajo perspectiva fractal

La descripción del comportamiento de un flujo bifásico, a través de lechos porosos, por medio de modelos, basados en las ecuaciones de fenómenos de transporte, se dificulta debido a la irregularidad geométrica de los canales que se forman entre las partículas sólidas que constituyen el lecho. Los modelos deterministas desarrollados para flujos monofásicos requieren del ajuste de constantes empíricas y no pueden extrapolarse a flujos bifásicos, donde el patrón de flujo generado en el sistema influye significativamente sobre el comportamiento del flujo total y las pérdidas de presión por fricción. Por lo anterior, en este trabajo se presenta un modelo para describir el comportamiento del flujo bifásico en función del patrón de flujo y de la morfología, las dimensiones y las condiciones de operación del lecho poroso, cuya obtención se basó en un formalismo, empleando conjuntamente las ecuaciones de conservación de la cantidad de movimiento, la geometría fractal y el cálculo diferencial fraccional. El modelo predice que, para una misma composición del flujo bifásico, el patrón de flujo influye significativamente sobre las pérdidas de presión por fricción, habiendo un incremento cuando una de las fases se encuentra dispersa en el seno de la otra. Por otro lado, el incremento de la dimensión fractal de los poros causa a su vez un incremento de las pérdidas de presión por fricción. El modelo presenta limitaciones asociadas a las consideraciones establecidas durante su obtención, de tal manera que sólo es válido cuando los efectos de la tensión superficial son más significativos que los gravitatorios, despreciándose los efectos de esos últimos sobre el patrón de flujo, así como para el estado estacionario.

A generalized Mittag-Leffler function to describe nonexponential chemical effects

In this paper a differential equation with noninteger order was used to model an anomalous luminescence decay process. Although this process is in principle an exponential decaying process, recent data indicates that is not the case for longer observation time. The theoretical fractional differential calculus applied in the present work was able to describe this process at short and long time, explaining, in a single equation, both exponential and nonexponential decay process. The exact solution found by fractional model is given by an infinite series, the Mittag-Leffler function, with two adjusting parameters. To further illustrate this nonexponential behavior and the fractional calculus framework, an stochastic analysis is also proposed.

Elasticity for economic processes with memory: fractional differential calculus approach

Elasticity for economic processes with memory: fractional differential calculus approach

Fractional Order EOQ Model with Linear Trend of Time-Dependent Demand

In this paper we introduce the classical EOQ model with a linear trend of time-dependent demand having no shortages using the concept of fractional calculus. The application of fractional calculus has been already used in classical EOQ model where the demand is assumed to be constant. In this present article fractional differential calculus can be used to describe EOQ model with time-dependent linear trend of demand to develop more generalized EOQ model. Here, we want to discuss more deeply its role as a tool for describing the traditional classical EOQ model with time dependent demand.

Informational entropy of non-random non-differentiable functions. An approach via fractional calculus

By combining the explicit formula of the Shannon informational entropy ) , ( Y X H for two random variables X and Y , with the entropy )) ( ( X f H of ) (X f where (.) f is a real-valued differentiable function, we have shown that the density of the amount of information in Shannon sense involved in a non-random differentiable function is defined by the logarithm of the absolute value of its derivative (Maximum entropy, information without probability and complex fractals, Springer, 2009). Here, we shall show that in the case when the function ) (x f is non-differentiable, then it is possible to extend this result, but now, by using fractional derivative. The article contains three main parts. In the first one, one gives the essential of the fractional differential calculus as we re-visited it (Fractional differential calculus for non-differentiable function, LAP-LAMBERT, 2013), then we shall bear in mind some basic results on the Shannon informational entropy of non-random functions, and then we shall arrive at the entropy of functions which are non-differential but have fractional derivative of order lower than the unity. It is shown that the density of entropy of a given function ) (x f is c x f ) ( ) ( where c denotes a positive constant. As a result, it will be possible to meaningfully replace )) ( ( X f H by (.)) ( ) ( f H X H  and to consider some problems like Jaynes’ maximum entropy principle for instance with new points of view.

Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia

The aim of this study is to promote a model based on the fractional differential calculus related to the pharmacokinetic individualization of high dose methotrexate treatment in children with acute lymphoblastic leukaemia, especially in high risk patients.We applied two-compartment fractional model on 8 selected cases with the largest number (4–19) of measured concentrations, among 43 pediatric patients received 24-h methotrexate 2–5g/m2 infusions. The plasma concentrations were determined by fluorescence polarization immunoassay. Our mathematical procedure, designed by combining Post’s and Newton’s method, was coded in Mathematica 8.0 and performed on Fujicu Celsius M470-2 PC.Experimental data show that most of the measured values of methotrexate were in decreasing order. However, in certain treatments local maximums were detected. On the other hand, integer order compartmental models do not give values which fit well with the observed data. By the use of our model, we obtained better results, since it gives more accurate behavior of the transmission, as well as the local maximums which were recognized in methotrexate monitoring. It follows from our method that an additional test with a small methotrexate dose can be suggested for the fractional system parameter identification and the prediction of a possible pattern with a full dose in the case of high risk patients.A special feature of the fractional model is that it can also recognize and better fit an observed non-monotonic behavior. A new parameter determination procedure can be successfully used.

A class of degenerate fractional evolution systems in banach spaces

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.