Definition Of Fractional Derivative Research Articles

Generalized fractional differential ring

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.

THEORETICAL STUDY OF MHD MAXWELL FLUID WITH COMBINED EFFECT OF HEAT AND MASS TRANSFER VIA LOCAL AND NONLOCAL TIME DERIVATIVES

This study highlights the combined effect of heat and mass transfer on MHD Maxwell fluid under time-dependent generalized boundary conditions for velocity, temperature, and concentration. The classical calculus due to the fact that it is assumed as the instant rate of change of the output when the input level changes. Therefore, it is not able to include the previous state of the system called the memory effect. But in the fractional calculus (FC), the rate of change is affected by all points of the considered interval, so it can incorporate the previous history/memory effects of any system. Due to this reason, we applied the modern definition of fractional derivatives (local and nonlocals kernels). Here, the order of fractional derivative will be treated as an index of memory. The exact and semi-analytical solutions are obtained using the integral transform and inversion algorithm. Several important properties of different parameters are analyzed by graphs. Interesting results are revealed by this investigation due to their vast applications in engineering and applied sciences.

Mathematical modeling of cancer treatments with fractional derivatives: An Overview

This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.

Fractional-order simulations for heat and mass transfer analysis confined by elliptic inclined plate with slip effects: A comparative fractional analysis

In this investigation, the novel fractional order simulations are performed in order to inspect the characteristics of mass and heat transfer over porous inclined surface subject to the magnetic force impact. The radiative phenomenon and slip wall constraints are utilized to make the model more comprehensive. The convective thermal mechanism of heat transfer is predicted with assumptions of ramped temperature. The recent definitions of fractional derivatives are followed. The famous technique of fractional approach namely Atangana-Baleanu (AB) and Caputo-time fractional derivative are employed for the solution procedure. Various relations expressing the mathematical functions for the fractional derivatives are encountered. The major outcomes are listed for parameters govern for the model. This obtained result reflects many engineering and industrial applications like thermal processes, processing industries, nuclear reactions, solar systems, energy production etc.

The Bouncing Ball and the Grünwald-Letnikov Definition of Fractional Derivative

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

On fractional approaches to the dynamics of a SARS-CoV-2 infection model including singular and non-singular kernels

Covid-19 (2019-nCoV) disease has been spreading in China since late 2019 and has spread to various countries around the world. With the spread of the disease around the world, much attention has been paid to epidemiological knowledge. This knowledge plays a key role in understanding the pattern of disease transmission and how to prevent a larger population from contracting it. In the meantime, one should not overlook the significant role that mathematical descriptions play in epidemiology. In this paper, using some known definitions of fractional derivatives, which is a relatively new definition in differential calculus, and then by employing them in a mathematical framework, the effects of these tools in a better description of the epidemic of a SARS-CoV-2 infection is investigated. To solve these problems, efficient numerical methods have been used which can provide a very good approximation of the solution of the problem. In addition, numerical simulations related to each method will be provided in solving these models. The results obtained in each case indicate that the used approximate methods have been able to provide a good description of the problem situation.

Sensitive behavior and optical solitons of complex fractional Ginzburg–Landau equation: A comparative paradigm

This article obtains the optical solitons of the complex fractional Ginzburg–Landau equation by the hypothesis of traveling wave and generalized projective Riccati equation scheme. There are four conditions, Kerr law, parabolic law, power law and dual power law of nonlinearity associated with the model. The constraint conditions for the existence of these solutions have also been discussed. Moreover, the physical significance of the constructed solutions has been provided using graphical representation. A comparative study is made by using two distinct definitions of fractional derivatives namely as Beta and M-truncated. Furthermore, a quantitative overview is also included, which involves solutions to the model under discussion. The complex Ginzburg–Landau equation is subjected to a comprehensive sensitivity analysis. Finally, the modulation instability (MI) analysis of proposed model is also carried out on the basis of linear stability analysis. A dispersion relation is obtained between the wave number and frequency.

Analytic solutions of a (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Analytic solutions of fractional order physical equations are very significant to explain the behavior of mathematical models expressing complex phenomena in engineering and natural sciences. The modified extended tanh-function (METHF) method is an especially capable and highly effective mathematical technique to attain analytic traveling wave solutions. This research proposes to examine the analytic solutions of the time-fractional (2+1)-dimensional non-linear Heisenberg ferromagnetic spin chain (HFSC) equation that describes electromagnetic waves in modern magnet theory by using the suggested method and the definition of conformable fractional derivative. We obtain some new analytic solutions of the proposed equation in terms of hyperbolic, trigonometric, and rational functions. The validity and precision of these solutions are also examined. The 2D, 3D, and contour graphs of solutions are given to manifest the physical behavior of the waves with the aid of the Mathematica package program.

Generalized thermal investigation of unsteady MHD flow of Oldroyd-B fluid with slip effects and Newtonian heating; a Caputo-Fabrizio fractional model

This article critically examines the influences of Newtonian heating and slip effects on unsteady magnetohydrodynamic (MHD) flow of an Oldroyd-B fluid near an infinitely long plate. The nonlinear thermal radiation influence is considered in normal direction to the vertical plate. A modern definition of Caputo-Fabrizio fractional derivative is applied to generalize the constitutive mass and energy equations. The Laplace transformation technique and Durbin’s numerical algorithm are served to develop the solutions of proposed problem. The solutions are determined for both fractional and ordinary cases and Stehfest’s and Zakian’s numerical algorithms are employed to secure the authenticity of these solutions. The relations for Nusselt number and skin friction coefficient are efficiently computed to precisely estimate the shear stress and rate of heat transfer at boundary. The respective numerical computations are accessible through tables. To comprehensively analyze the dynamics of the proposed problem, physical influence of various parameters is studied and repercussions are graphically highlighted and discussed. Furthermore, two limiting models named as second grade model and Maxwell model are deduced to compare the pertinent flow characteristics. A comparative analysis between fractional and classical models depicts that the Caputo-Fabrizio fractional model explains the memory effects more adequately.

Thermal activity of conventional Casson nanoparticles with ramped temperature due to an infinite vertical plate via fractional derivative approach

With inspired thermal characteristics and dynamic applications of nanoparticles, the researchers are motivated to suggest multidisciplinary significances of nano-materials in various thermal engineering and industrial processes. In this analysis, a mixed free convection nanofluid flowing on a vertical plate is discussed with heat transfer in the drilling of nanofluid. The Casson liquid is assumed as a base fluid with uniform suspension of clay nanoparticles. The leading partial differential governing equations with physical properties of nanoparticles are demonstrated with corporal flow phenomenon and imposed thermal conditions. In the governing partial differential equations, the partial derivative with respect to time is replaced by the most recent definition of fractional derivatives i.e. Atangana-Baleanu time-fractional derivative, and then the solution of temperature and velocity field is found by utilizing the Laplace transformation. The effects of different parameters with different values are deliberated and plotted graphically and numerically for temperature and momentum equations. The main results and conclusions are accomplished at the end of this exertion and compared with existing literature. It is concluded that the velocity profile boost up with Grashof number. The results of temperature and momentum equations show a more decaying tendency as compared to Caputo-Fabrizio fractional derivative.

A Caputo-Fabrizio Fractional-Order Model of HIV/AIDS with a Treatment Compartment: Sensitivity Analysis and Optimal Control Strategies.

Although most of the early research studies on fractional-order systems were based on the Caputo or Riemann–Liouville fractional-order derivatives, it has recently been proven that these methods have some drawbacks. For instance, kernels of these methods have a singularity that occurs at the endpoint of an interval of definition. Thus, to overcome this issue, several new definitions of fractional derivatives have been introduced. The Caputo–Fabrizio fractional order is one of these nonsingular definitions. This paper is concerned with the analyses and design of an optimal control strategy for a Caputo–Fabrizio fractional-order model of the HIV/AIDS epidemic. The Caputo–Fabrizio fractional-order model of HIV/AIDS is considered to prevent the singularity problem, which is a real concern in the modeling of real-world systems and phenomena. Firstly, in order to find out how the population of each compartment can be controlled, sensitivity analyses were conducted. Based on the sensitivity analyses, the most effective agents in disease transmission and prevalence were selected as control inputs. In this way, a modified Caputo–Fabrizio fractional-order model of the HIV/AIDS epidemic is proposed. By changing the contact rate of susceptible and infectious people, the atraumatic restorative treatment rate of the treated compartment individuals, and the sexual habits of susceptible people, optimal control was designed. Lastly, simulation results that demonstrate the appropriate performance of the Caputo–Fabrizio fractional-order model and proposed control scheme are illustrated.

FRACTIONAL ORDER THERMOELASTICITY FOR PIEZOELECTRIC MATERIALS

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered Caputo fractional derivative. Then, thermoelastic model of fractional order is formulated for piezoelastic materials by combining the unified heat conduction law and the governing equations of elastic and electric fields. Numerically, the present theoretical model is applied to study the transient responses of piezoelectric medium that is subjected to a thermal shock. The governing equations are analytically derived and numerically solved with the aids of Laplace transform method. The obtained results are graphically illustrated, and the influences of different definitions of fractional derivative and different fractional order are revealed. This work may be helpful for understanding the multi-coupling effect of elastic, thermal and electric fields, and for inspiring further developments of fractional calculus.

OPIAL-TYPE INEQUALITY ABOUT CONFORMABLE FRACTIONAL INTEGRALS AND THE APPLICATION

In this paper, we define partial conformable fractional derivative (PCFD) which is based on the fractional derivative definition proposed by Abdeljawad. At the same time, we present some simple properties about the definition, the properties are useful for us when we carry out further research. In addition, taking advantage of Hölder’s inequality, we establish Opial-type inequalities for the partial conformable fractional (PCF) integral. As an application, the uniqueness of the partial system with initial value is proved.

Semi-Analytic Technique for Solving Fractional Partial Differential Equations with Conformable Derivatives

In this work, we present a semi-analytical technique to find an approximate result of the conformable fractional partial differential equations (CFPDEs). The fractional order derivative will be in the conformable (CFD) sense. This definition is effective and simple in the solution of the fractional differential equations that have intricate solution with classical fractional derivative definition like Riemann-Liouville and Caputo. Furthermore, the result obtained by the proposed technique is like those in previous studies that used other types of approximate methods like (Homotopy analysis method) but it has the advantage of being simpler than the rest of these methods. In addition, results demonstrate obtained the Precision and effectiveness of the suggested technique.

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.

Neural network controller design for fractional-order systems with input nonlinearities and asymmetric time-varying Pseudo-state constraints

Abstract This article considers the neural adaptive control issues of a category of non-integer-order non-square plants with actuator Nonlinearities and Asymmetric Time-Varying pseudo-State Constraints. First, the original non-square non-affine system with input nonlinearities is transformed into an equivalent affine-in-control square model by defining a set of auxiliary variables and by employing the mean-value theorem. Second, Neural networks and Nussbaum functions are exploited to obviate the requirement of a complete knowledge of the system dynamics and the control directions, respectively. Third, a novel adaptive dynamic surface control method based on Caputo fractional derivative definitions and fractional order filters is developed to overcome the “explosion of complexity” problem in the traditional backstepping design process and to determine the parameter update laws and control signals, concurrently. Then, Asymmetric Barrier Lyapunov Functions with error variables are adopted to ensure the uniform stability of the closed-loop system and to prevent the violation of the full pseudo-State constraints. The novelties and contributions of this article are: (1) through the introduction of new technical Lemmas and corollaries, existing control design and stability theories linked to integer-order square systems are developed and extended to non-square non-integer-order ones. (2) all signals, including variables and errors in the closed-loop system are semi-global practical finite-time stability whereas the the tracking errors are asymptotically driven to zero without transgression of the constraints. Finally, the effectiveness and potential of the proposed control approach are substantiated by two example simulations.

Comparative study of DST interpolation approach of fractional orders

In this paper, comparative study of DST interpolation approach of various order by using different fractional derivatives are presented. First the definition of different fractional order derivatives like Grunwald-Letnikov, Weyl’s and Conformable are reviewed. Next, Fractional derivative of a discrete signal is determined after applying the DST interpolation approach. Next, the DST-IV method approach transfer function are obtained with the help of index-mapping technique. Lastly, some computative problems are discussed for checking the effectiveness of digital fractional order differentiators for design of proposed method using the integral square error formula. Error values of various fractional order derivatives have been presented in the form of table.

Generalized fractional viscothermoelastic nanobeam under the classical Caputo and the new Caputo–Fabrizio definitions of fractional derivatives

In this paper, we introduce a new unified formula that governs two different definitions of fractional derivative; the classical Caputo definition and Caputo–Fabrizio's new definition. Hence, an analysis has been constructed for fractional viscothermoelastic, isotropic, and homogeneous nanobeams. The governing equations of the viscothermoelastic nanobeam have been constructed in the context of the non-Fourier heat conduction law with one relaxation of Lord and Shulman (L-S). Laplace transform has been applied, and its inversions have been calculated by using the Tzou method of approximation. The numerical results have been validated for a thermoelastic rectangular nanobeam of silicon as a case when it is subjected to ramp-type heating and simply supported. The fractional-order parameter based on the two types of fractional derivatives has significant impacts on all the studied functions except the temperature increment function. The results based on the two types of fractional derivatives are different, although they generate the same behavior of the thermomechanical waves. The ramp-time heat parameter has significant effects on all the studied functions.

Explicit tight frames for simulating a new system of fractional nonlinear partial differential equation model of Alzheimer disease

This paper is devoted to develop a new mathematical model for Alzheimer disease based on a system of fractional-order partial differential equations. The system of Alzheimer disease includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. We consider the Caputo fractional derivative definition to analyze the formulated system by simulating the effect of drugs that either failed or currently in clinical trials. To simulate the model, we use tight frame (framelet) systems generated using the unitaryand oblique extension principle. According to the simulation results, and based on using such new direction of fractional modeling, the progression of Alzheimer disease and its consequences will be slowing down. Which may give clinical insights on intervention measures against the disease and its effective therapies.

A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A <i>ψ</i>-HILFER FRACTIONAL OPERATOR

Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable definition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general definition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the $\psi-$fractional spaces $\mathbb{H}_{p}^{\alpha, \beta; \psi}([0, T], \mathbb{R})$. As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.