Class Of Derivatives Research Articles

A mathematical model for precise predicting microbial propagation based on solving variable‐order fractional diffusion equation

Continued mortality from Covid‐19 disease and environmental considerations resulting from the cemetery, landfill, wastewater treatment plants, and mass personal protective equipment (PPE) disposal sites demonstrate the importance of carefully studying how microorganisms spread in and across the soil, surface water, and groundwater. In this research, fractional diffusion equation for different functions has been solved using fractional derivative form with variable‐order (VO) to predict the diffusion of the SARS‐CoV‐2 virus. Comparison of the obtained results with the available experimental data and mathematical calculations shows that the use of VO form can predict the viral behavior more accurately. The main advantage of considering a VO fractional derivatives instead of a constant‐order fractional derivative or classical derivative is accurately predicting the behavior of propagation of microorganisms over large time and dimensional scales. In addition, to the specified SARS‐CoV‐2 virus, the method presented in this study will be able to investigate the diffusion pattern of all environmental pollutants at the nanoscale/microscale, including carcinogenic compounds, such as biogenic amines in aqueous and porous media. The obtained results manifest a high accuracy of mathematical approach which provides helpful information to urban and health planners/managers worldwide to manage other possible outbreaks.

Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomials

In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples.

Theoretical derivation of particle collision kernels and its enhancement at a high concentration from a first-time-passage approach in the diffusive regime

A theoretical explanation for obtaining collision kernels from the first-time passage approach proposed by [Gopalakrishnan, R., & Hogan Jr, C. J. (2011). Aerosol Science and Technology, 45(12), 1499-1509] is developed in this work. This new approach leads to an interesting correction for particle coagulation at high concentrations and enlights the limitations of classical derivations by [Fuchs, N. A. (1964). Dover Publications] and [Friedlander, S. K. (2000). (Vol. 198). New York: Oxford university press] where the transient part of the diffusive collision kernel is typically neglected.

Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever R0<1 and when R0≤1 at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald–Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population’s risk of infection can be reduced by adhering to the WHO’s suggestions, which include keeping social distances, wearing facemasks, washing one’s hands, avoiding crowds, etc.

Weighted Gagliardo-Nirenberg interpolation inequalities

In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives, generalizing the celebrated result with classical derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupt-type conditions. We also obtain improved results for some particular families of weights, including power-law weights |x|α. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights 〈x〉α=(1+|x|2)α2.

Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis

In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results.

Spectral method for solving fractal fractional differential equations

Lately introduced fractal fractional Caputo - Fabrizio operator. By substituting the single kernel at the classic derivative of fractal fractional Caputo with the ordinary kernel This modern operator was derived. We introduce some beneficial characteristics relied on the qualifier of fractal fractional Caputo - Fabrizio. Here, we extend Caputo-Fabrizio for nonlinear fractal fractional differential equations. We apply Legendre operational matrix relied on this modern operator and then, we employ it to solve the differential equations determined in the sense of fractal fractional Caputo-Fabrizio. To show the simplicity and precision of the suggested technicality Some numerical examples are given.

Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory

This article aims to investigate the existence and uniqueness of solutions to impulsive fractional differential equations. Firstly, we show a formula for solutions to an impulsive fractional problem involving a generalization of the classical Caputo derivative with a nonlocal condition in a Banach space. Secondly, some sufficient conditions for the existence of solutions are established by applying fixed point methods. Furthermore, we reconsider some of operators for deriving the existence and uniqueness of the solution to impulsive fractional problem using topological degree techniques. The results support this in one example.

New perspective on the creep characteristic of fiber–dependent shape memory polymers: variable–order fractional constitutive model

With the aim of describing and explaining the creep tensile property of shape memory polymers (SMPs) better, a variable--order fractional creep constitutive model with fewer parameters and good simulation performance was constructed on the basis of existing constitutive models. The consists of a Hookean spring connected in series with a Newtonian dashpot was employed to propose a new variable--order fractional Maxwell model for SMPs. Drawing support from the Laplace transform, we derive the analytical solution of this novel model. The verification of the obtained theoretical results with the creep test data for different fiber weight fractions and stress levels available from the literature reveals their satisfactory agreement. It can be concluded that the developed variable--order fractional constitutive model can better describe the complex viscoelastic relationship of shape memory polymers which is difficult to be described by the classical integer order derivative, and obtain a more accurate constitutive model of shape memory polymers with fewer parameters.

Optimization of Energy Consumption of Industrial Robots Using Classical PID and MPC Controllers

Industrial robots have a key role in the concept of Industry 4.0. On the one hand, these systems improve quality and productivity, but on the other hand, they require a huge amount of energy. Energy saving solutions have to be developed and applied to provide sustainable production. The purpose of this research is to develop the optimal control strategy for industrial robots in order to minimize energy consumption. Therefore, a case study was conducted for the development of two control strategies to be applied to the RV-2AJ Mitsubishi robot arm with 5 DOF, where the system is a nonlinear one. The first examined controller is the classical linear proportional integral derivative (PID) controller, while the second one is the linear model predictive control (MPC) controller. In our study, the performances of both the classical PID model and the linear MPC controller were compared. As a result, it was found that the MPC controller in the execution of the three defined reference trajectories [(1) curve motion, (2) N-shaped motion, and (3) circle motion] was always faster and required less energy consumption, whereas in terms of precision the PID succeeded in executing the trajectory more precisely than the MPC but with higher energy consumption. The main contribution of the research is that the performances of the two control strategies with regard to a complex dynamic system were compared in the case of the execution of three different trajectories. The evaluations show that the MPC controller is, on the one hand, more energy efficient; on the other hand, it provides a shorter cycle time compared to the PID controller.

Agitation of SARS‐CoV‐2 disease (COVID‐19) using ABC fractional‐order modified SEIR model

The present article studies the agitation scenario of SARS‐CoV‐2 (COVID‐19), the current pandemic around the globe, by applying Atangana–Baleanu–Caputo derivative operator where . Using classical notions, we study various qualitative features, like existence, uniqueness and investigate Hyers–Ulam stability analysis of the model under consideration. Lagrange's polynomial approach is used for the approximation of nonlinear terms of the system. We carry out numerical simulations for different values of the fractional‐order . The results obtained are compared with those of the classic order derivatives. It is observed that the results obtained with fractional order are better as compared to the classical order.

Commutators on Fock spaces

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

Generalized versus classical normal derivative

"Given a bounded domain with Lipschitz boundary, the general Green formula permits to justify that the weak solutions of a Neumann elliptic problem satisfy the Neumann boundary condition in a weak sense. The formula involves a generalized normal derivative. We prove a general result which establishes that the generalized normal derivative of an operator coincides with the classical one, provided that the operator is continuous. This result allows to deduce that, under usual regularity assumptions, the weak solutions of a Neumann problem satisfy the Neumann boundary condition in the classical sense. This information is nec- essary in particular for applying the strong maximum principle."

Investigation of Fractional Order Dynamics of Tuberculosis under Caputo Operator

In this article, a new deterministic disease system is constructed to study the influence of treatment adherence as well as awareness on the spread of tuberculosis (TB). The suggested model is composed of six various classes, whose dynamics are discussed in the sense of the Caputo fractional operator. Firstly the model existence of a solution along with a unique solution is checked to determine whether the system has a solution or not. The stability of a solution is also important, so we use the Ulam–Hyers concept of stability. The approximate solution analysis is checked by the technique of Laplace transformation using the Adomian decomposition concept. Such a solution is in series form which is decomposed into smaller terms and the next term is obtained from the previous one. The numerical simulation is established for the obtained schemes using different fractional orders along with a comparison of classical derivatives. Such an analysis will be helpful for testing more dynamics instead of only one type of integer order discussion.

Prabhakar fractional simulation for inspection of CMC-based nanofluid flowing through a poured vertical channel

Nanofluids can be used in different solar thermal systems. Solar energy devices utilized in industries and nanofluids as a source of solar energy in thermal engineering are two other sources of solar energy, according to the article. This work inspects a viscous, incompressible nanofluid made of different (graphene oxide (GO) and molybdenum disulfide (MoS2)) nanoparticles suspended in carboxymethyl cellulose (CMC). The governed model also considers permeability and angled magnetic effects. Due to memory effects, classical derivatives cannot explore and estimate the physical significance of several fluid limitations. We have addressed the issues with nanofluid suspension using the Prabhakar fractional derivative definition, the quickest and latest modern fractional technique. First, the governed leading equations are transferred to a fractional version using the integral transform technique and Laplace transform, and then it is resolved with various numerical approaches. Each constraint visual impact and significance are analyzed by varying their considered values. The numerical influences of the heat and flow rate are observed at multiple time values. Therefore, we deduce that the momentum and heat profiles are slowed as the Prabhakar fractional restraints increase. While with the increment in the radiation parameter, the velocity profile shows an increasing behavior. Furthermore, the impact of graphene oxide-based suspension is more significant than molybdenum disulfide nanofluid on all governed equations due to the physical features of the considered nanoparticles.

Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter α. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shell have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development.

Piecewise Business Bubble System under Classical and Nonsingular Kernel of Mittag-Leffler Law.

This study aims to investigate the dynamics of three agents in the emerging business bubble model based on the Mittag-Leffler law pertaining to the piecewise classical derivative and non-singular kernel. By generalizing the business bubble dynamics in terms of fractional operators and the piecewise concept, this study presents a new perspective to the field. The entire set of intervals is partitioned into two piecewise intervals to analyse the classical order and conformable order derivatives of an Atangana-Baleanu operator. The subinterval analysis is critical for removing discontinuities in each sub-partition. The existence and uniqueness of the solution based on a piecewise global derivative are tested for the considered model. The approximate root of the system is determined using the piecewise numerically iterative technique of the Newton polynomial. Under the classical order and non-singular law, the approximate root scheme is applied to the piecewise derivative. The curve representation for the piece-wise globalised system is tested by applying the data for the classical and different conformable orders. This establishes the entire density of each compartment and shows a continuous spectrum instead of discrete dynamics. The concept of this study can also be applied to investigate crossover behaviours or abrupt changes in the dynamics of the values of each market.

Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations

This work deals with the distributed-order time fractional nonlinear diffusion-wave equations. These equations are generated by replacing the first- and second-order time derivative terms with the distributed-order fractional derivative terms. The distributed-order fractional derivatives used in these problems are in the Caputo sense. The Chelyshkov polynomials as a well-known family of basis functions are used to develop a spectral collocation method for these problems. Through the way, some operational matrices regarding the classical and distribute-order fractional derivatives for these polynomials are extracted. In the proposed method, after approximating the solution of the problem in terms of the Chelyshkov polynomials and employing the expressed matrices, solving the primary equations transforms into solving algebraic systems which can be easily solved. Some numerical examples are studied to show the adequacy of the approach.

Chaos in fractional order financial model with fractal–fractional derivatives

Recently, a new differential operator which combines fractal differentiation and fractional differentiation with different kernels such as power law, exponential decay, and the Mittag–Leffler function has been introduced. We apply fractal–fractional derivative operators to study chaos in financial chaotic model and implement a numerical procedure to obtain their graphical results. In order to determine the existence of chaos for the chosen value of fractional order, we examine the effects of the saving rate, the per-investment cost, the elasticity of demand, and the Lyapunov exponent. In the case of the classical derivative with power law the obtained attractors presented no similarities. While the attractors obtained via fractal–fractional derivative show some crossover effects. The outcomes of this study are novel and extremely important in dealing with financial issues.

Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells.

A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models in fuzzy conditions. In this paper, an explicit finite difference method has been developed and applied to solve a fuzzy time-fractional cancer tumor model. The impact of using the fuzzy time-fractional derivative has been examined under the double parametric form of fuzzy numbers rather than using classical time derivatives in fuzzy cancer tumor models. In addition, the stability of the proposed model has been investigated by applying the Fourier method, where the net killing rate of the cancer cells is only time-dependent, and the time-fractional derivative is Caputo's derivative. Moreover, certain numerical experiments are discussed to examine the feasibility of the new approach and to check the related aspects. Over and above, certain needs in studying the fuzzy fractional cancer tumor model are detected to provide a better comprehensive understanding of the behavior of the tumor by utilizing several fuzzy cases on the initial conditions of the proposed model.