Class Of Derivatives Research Articles

A Newton interpolation based predictor–corrector numerical method for fractional differential equations with an activator–inhibitor case study

This paper presents a new predictor–corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana–Seda formula is obtained by considering the neglected terms and used as the predictor stage of the proposed method. Numerical formulas are presented that approximate the classical first derivative as well as the Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Simulation results are used to assess the approximation error of the new method for various differential equations. In addition, a case study is considered where the proposed scheme is used to obtain numerical solutions of the Gierer–Meinhardt activator–inhibitor model with the aim of assessing the system’s dynamics.

Earthquake dynamic induced by the magma up flow with fractional power law and fractional-order friction.

In this work, we examine the dynamical behaviour of the “single mass-springs” model for earthquake subjected to the strength due to the up flow of magma for the period of volcanism, considering the fractional viscous damping force, the fractional weakening friction and fractional power law of elastic force. The numerical simulation method used in this paper is that of Grünwald-Letnikov based on the generalization of the classical derivative, and the approximately analytical solution obtained by the harmonic balance method. The results have shown that the fractional-order derivative can affect the dynamical properties of fault rock, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude-frequency equation for the steady-state solution was established. It appears that the resonant amplitude and resonant frequency are strongly dependent on the fractional-order damping r, fractional-order friction q, the fractional deflection 𝛼, the nonlinear stiffness coefficient and the fractional viscous coefficient. We have also shown that, the recurrence time of an event, the duration time of an event and the slip size of an earthquake can be controlled by the fractional-order derivative, the fractional-order deflection and the magnitude of the magma strength. The model allowed us to better interpret the earthquake as a stick-slip motion.

Fractional synchronization involving fractional derivatives with nonsingular kernels: Application to chaotic systems

We present a novel master‐slave fractional synchronization in chaotic systems by using fractional derivatives with nonlocal and nonsingular kernel. The master system is the fractional‐order chaotic system, and then, we designed a fractional‐order high gain observer, which is the slave system, based on the chaotic system model to achieve the synchronization. We used three different oscillator systems, a novel Chen‐Burke‐Shaw chaotic attractor, a novel fractional‐order chaotic system, and the fractional‐order model of a simple autonomous Jerk circuit. Our research followed to test the performance of the fractional synchronization. We use two performance indices, the integral of the square error (ISE) and the integral of the square error multiplied by time (ITSE). We showed that by using the fractional‐order approach, we can reduce the values on the ISE and ITSE indices; hence, we guaranteed a full synchronization between the master and slave system. Our analysis shows that when using fractional derivatives with variable order, the ISE and ITSE indices have a lower value than when using the classical derivatives. We used the definition of Atangana‐Baleanu‐Caputo and Liouville‐Caputo derivatives for the three examples mentioned in this work. We numerically solved the equations using the Adams method. Our results show that when using the fractional‐order approach, the ISE and ITSE indices are lower than the integer‐order case.

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.

Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

General Cluster Sorption Isotherm

Abstract Adsorption isotherms are an essential tool in chemical physics of surfaces. However, several approaches based on a different theoretical basis exist and for some isotherms existing approaches can fail. Here, the sorbate-sorbate interaction is not limited to an orientation vertical to the adsorbent surface. Instead, a formal orientation-independent clustering, also comprising the possibility of multilayers as a special case, is underlain. The flexible clusters are described by classical thermodynamics including a free energy relationship depending on the cluster size. In this paper a rigorous unified treatment of the adsorbate-adsorbate-adsorbent interaction is shown to result in a general isotherm equation which is applied to literature data both concerning type IV isotherms of argon and nitrogen in ordered mesoporous silica, and type II isotherms of disordered macroporous silica. The new isotherm covers the full range of partial pressure (10−6 - 0.7). The determination of surface areas is not possible by this isotherm because the cross-sectional area of a cluster is unknown. Based on the full description of type IV isotherms, most known isotherms including BET are accessible by respective simplifications. The presented model is an extension of the classical derivation of the ζ-isotherm which was shown to describe type IV isotherms restricted to adsorbates with very strong tendency of clustering such as water on hydrophobic surface (Phys. Chem. Chem. Phys. 21 (2019) 5614).

An impulse-based derivation of the Kutta–Joukowsky equation for wind turbine thrust

Abstract. Using the concept of impulse in control volume analysis, we derive general expressions for wind turbine thrust in a constant, spatially uniform wind. The absence of pressure in the impulse equations allows for their application in the near wake, where the flow is assumed to be steady in the frame of reference rotating with the blades. The assumption of circumferential uniformity in the near wake – as applies when the number of blades or the tip speed ratio tends to infinity – is needed to reduce these general expressions to the Kutta–Joukowsky (KJ) equation for blade-element thrust. The present derivation improves upon the classical derivation based on the Bernoulli equation by allowing the flow to be rotational in the near wake. The present derivation also yields intermediate expressions for thrust that are valid for a finite number of blades and trailing vortex sheets of finite thickness. For the circumferentially uniform case, our analysis suggests that the magnitudes of the radial velocity and the axial induction factor must be equal somewhere on the plane containing the rotor, and we cite previous studies that show this to occur near the rotor tip across a wide range of thrust coefficients. The derivation reveals one further complication; when deriving the KJ equations using annular control volumes, the existence of vorticity on the lateral control surfaces may cause the local blade loading to differ from the KJ equation, but the magnitude of these deviations is not explored. This complication is not visible to the classical derivation due to its neglect of vorticity.

Size of National Assemblies: The Classic Derivation of the Cube-Root Law is Conceptually Flawed

For half a century, the analysis of the size of national assemblies was dominated by the famous cube-root relation with the population. However, a revisitation of that historical work with a physicist’s approach reveals basic conceptual problems that fatally undermine its conclusions. Furthermore, the assembly size evaluation exceeds the accuracy of all power equations, which cannot be reliably used for political analysis.

Dynamics of fractional order COVID-19 model with a case study of Saudi Arabia

The novel coronavirus disease or COVID-19 is still posing an alarming situation around the globe. The whole world is facing the second wave of this novel pandemic. Recently, the researchers are focused to study the complex dynamics and possible control of this global infection. Mathematical modeling is a useful tool and gains much interest in this regard. In this paper, a fractional-order transmission model is considered to study its dynamical behavior using the real cases reported in Saudia Arabia. The classical Caputo type derivative of fractional order is used in order to formulate the model. The transmission of the infection through the environment is taken into consideration. The documented data since March 02, 2020 up to July 31, 2020 are considered for estimation of parameters of system. We have the estimated basic reproduction number (R0) for the data is 1.2937. The Banach fixed point analysis has been used for the existence and uniqueness of the solution. The stability analysis at infection free equilibrium and at the endemic state are presented in details via a nonlinear Lyapunov function in conjunction with LaSalle Invariance Principle. An efficient numerical scheme of Adams-Molten type is implemented for the iterative solution of the model, which plays an important role in determining the impact of control measures and also sensitive parameters that can reduce the infection in the general public and thereby reduce the spread of pandemic as shown graphically. We present some graphical results for the model and the effect of the important sensitive parameters for possible infection minimization in the population.

πτερόν, πτέρυξ, πτερύγιον. A Cultural-Historical Study of a Classical Greek Architectural Term and Derivatives, and their Post-Classical Meaning

πτeρόν, πτέρυξ and πτeρύγιον all stem from the same root, and their primary meaning is connected to birds: “feather”,“wing”, “shoulder blade”. The history of these words is worthy of note for they came into use in the description of Graeco-Egyptian temple complexes and in the nomenclature of Classical architecture. In order to observe the semantic import of theseand related words and expressions across various cultural contexts (Graeco-Egyptian, Classical and Byzantine) this articlepresents and discusses a range of ancient, late antique and Byzantine literary, erudite, archaeological and iconographic sources.This study aims at giving the reader a glimpse into the Greek lexical domain of architecture, by providing insights into some ofthe general mechanisms of its construction, and specifically to illuminate, by way of a cultural-historical investigation of theseterms, ideas of appearance and significance inherent in the Graeco-Egyptian and Classical and Byzantine worlds. This surveysuggests lexical forms as a key research domain for the understanding of the intellectual, social and cultural life of societies.

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations. Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

Extremal solutions of $ \varphi- $Caputo fractional evolution equations involving integral kernels

This paper deals with the existence and uniqueness of solution for the Cauchy problem of $ \varphi- $Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.

Exciton coupling chirality in helicene-porphyrin conjugates.

Enantiopure helicene-porphyrin conjugates were prepared. They show strong changes in their circular dichroic response as compared to classical helicene derivatives, with highly intense bisignate Exciton Coupling (EC) signal and Δε values up to 680 M-1 cm-1 for the Soret band. They also display circularly polarized fluorescence in the (far-)red region, with dissymmetry factors up to 7 × 10-4.

Compartmental Models with Application to Pharmacokinetics

In this paper, a comprehensive study on different forms of a system of fractional differential equations (SFDE) on the compartmental system is investigated, distinguishing the characteristics surrounding these types: (1) commensurable SFDE; (2) non-commensurable SFDE; (3) implicit non-commensurable SFDE. With further classification, we illustrate the types and their characteristics by first modeling a classical derivative two-compartmental system with a single intravenous (I.V) dose through which the three models would be constructed. It is shown that the properties surrounding a non-commensurable SFDE type violate the idea of mass balance. It however, survives the theory of fractional calculus in modeling anomalous kinetics especially in an area where compartmental analysis is applicable such as Pharmacokinetics. We may affirm this by fitting the various forms of the proposed models to an experimental dataset with parameters determined by the least square method. In addition, the results show that the non-commensurable model predicts a good fit as compare to that of the other two models nevertheless, these fractional models explain the anomalous behavior better than the classical models. The fractional derivative used in this work is described in the Caputo sense. Since analytic solutions are sometimes difficult to be obtained, we implement the use of fractional finite difference method (FFDM) to simulate solutions to the fractional compartmental models. Numerical results also illustrate that the proposed numerical method is equally efficient to solve this and any complicated compartmental models since they perform well when the simulations are done for the classical case of the model.

New Aspects of Bloch Model Associated with Fractal Fractional Derivatives

Abstract To model complex real world problems, the novel concept of non-local fractal-fractional differential and integral operators with two orders (fractional order and fractal dimension) have been used as mathematical tools in contrast to classical derivatives and integrals. In this paper, we consider Bloch equations with fractal-fractional derivatives. We find the general solutions for components of magnetization ℳ = (Mu , Mv , Mw ) by using descritization and Lagrange's two step polynomial interpolation. We analyze the model with three different kernels namely power function, exponential decay function and Mittag-Leffler type function. We provide graphical behaviour of magnetization components ℳ = (Mu , Mv , Mw ) on different orders. The examination of Bloch equations with fractal-fractional derivatives show new aspects of Bloch equations.

Characterization of the Quality Factor Due to the Static Prestress in Classical Caputo and Caputo-Fabrizio Fractional Thermoelastic Silicon Microbeam.

The thermal quality factor is the most significant parameter of the micro/nanobeam resonator. Less energy is released by vibration and low damping, which results in greater efficiency. Thus, for a simply supported microbeam resonator made of silicon (Si), a thermal analysis of the thermal quality factor was introduced. A force due to static prestress was considered. The governing equations were constructed in a unified system. This system generates six different models of heat conduction; the traditional Lord–Shulman, Lord–Shulman based on classical Caputo fractional derivative, Lord–Shulman based on the Caputo–Fabrizio fractional derivative, traditional Tzou, Tzou based on the classical Caputo fractional derivative, and Tzou based on the Caputo–Fabrizio fractional derivative. The results show that the force due to static prestress, the fractional order parameter, the isothermal value of natural frequency, and the beam’s length significantly affect the thermal quality factor. The two types of fractional derivatives applied have different and significant effects on the thermal quality factor.

Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative

The new emerged infectious disease that is known the coronavirus disease (COVID-19), which is a high contagious viral infection that started in December 2019 in China city Wuhan and spread very fast to the rest of the world. This infection caused million of infected cases globally and still pose an alarming situation for human lives. Pakistan in Asian countries is considered the third country with higher number of cases of coronavirus with more than 200,000. Recently, many mathematical models have been considered to better understand the coronavirus infection. Most of these models are based on classical integer-order derivative which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using the non-integer Caputo derivative. In the absence of vaccine or therapy, the role of non-pharmaceutical interventions (NPIs) is examined on the dynamics of theCOVID-19 outbreak in Pakistan. First, we construct the model in integer sense and then apply the fractional operator to have a generalized model. The generalized model is then used to present the detailed theoretical results. We investigate the stability of the model for the case of fractional model using a nonlinear fractional Lyapunov function of Goh-Voltera type. Furthermore, we estimate the values of parameters with the help of least square curve fitting tool for the COVID-19 data recorded in Pakistan since March 1 till June 30, 2020 and show that our considered model give an accurate prediction to the real COVID-19 statistical cases. Finally, numerical simulations are presented using estimated parameters for various values of the fractional order of the Caputo derivative. From the simulation results it is found that the fractional order provides more insights about the disease dynamics.

Dirac equation perspective on higher-order topological insulators

In this Tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focusing on the low-energy description of higher-order topological insulators in terms of the Dirac equation. Our aim is to give a mostly self-contained treatment. After introducing the Dirac approximation of topological crystalline band structures, we use it to derive the anomalous end and corner states of first- and higher-order topological insulators in one and two spatial dimensions. In particular, we recast the classical derivation of domain wall bound states of the Su–Schrieffer–Heeger (SSH) chain in terms of crystalline symmetry. The edge of a two-dimensional higher-order topological insulator can then be viewed as a single crystalline symmetry-protected SSH chain, whose domain wall bound states become the corner states. We never explicitly solve for the full symmetric boundary of the two-dimensional system but instead argue by adiabatic continuity. Our approach captures all salient features of higher-order topology while remaining analytically tractable.

Orthonormal shifted discrete Legendre polynomials for solving a coupled system of nonlinear variable-order time fractional reaction-advection-diffusion equations

In this paper, we generalize a coupled system of nonlinear reaction-advection-diffusion equations to a variable-order fractional one by using the Caputo-Fabrizio fractional derivative, which is a non-singular fractional derivative operator. In order to establish an appropriate method for this system, we introduce a new formulation of the discrete Legendre polynomials namely the orthonormal shifted discrete Legendre polynomials. The operational matrices of classical and fractional derivatives of these basis functions are extracted. The devised method uses these polynomials and their operational matrices together with the collocation technique to transform the system under consideration into a system of algebraic equations which is uncomplicated for solving. Two numerical examples are analyzed to examine the accuracy of the method.