Fractional Derivative Operator Research Articles

Effects of Damage and Fractional Derivative Operator on Creep Model of Fractured Rock

Effects of Damage and Fractional Derivative Operator on Creep Model of Fractured Rock

Analytical solutions and physical interpretation of a predator–prey system with Allee effect using fractional derivative operators

This study investigates the dynamics of a predator–prey system by incorporating the Allee effect. The exact analytical solutions of the system are explored using the generalized exponential function method and the Bäcklund transformation approach. The system is discussed using Conformable, M-truncated, and Beta time fractional derivative operators to provide a physical interpretation. The effects of these derivative operators on the solutions are demonstrated through various 2D and 3D graphical simulations. Furthermore, the impacts of physical parameters on the model are analysed, providing valuable insights into the dynamics of Predator–prey systems with the Allee effect. Our obtained solutions differ from those in the literature, owing to the methodologies employed and the choice of differential operators utilized. Nevertheless, we believe that the comparisons presented for the differential operators are effective and beneficial.

On a fractional operator of adjoint hybrid fractional derivative operator

The achievement of this paper is to propose a new kind of fractional derivative which is called New Constant Proportional Caputo (NCPC) operator and to construct the solution of time-fractional initial value problem (TFIVPs) with NCPC derivative by taking the combination of Laplace transform (LT) and Homotopy Analysis method (HAM) into account. Later, the obtained solution is compared with the solutions of TFIVPs with Caputo and Constant Proportional Caputo (CPC) derivatives. The gained results reveal that the combination of LT and HAM together form an efficient method to build the approximate results of TFIVPs in NCPC sense.

Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

A geometric framework for distributed frequency models

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Some Applications of Fractional Derivative Operator

Introduction: The aim of this paper is to introduce a new subclass of univalent functions with negative coefficients related to fractional derivative operator in the unit disk We obtain basic properties like coefficient inequality, distortion and covering theorem, radii of starlikeness, convexity and close-to-convexity, extreme points, Hadamard product, and closure theorems for functions belonging to our class

A powerful tool for dealing with high-dimensional fractional-order systems with applications to fractional Emden–Fowler systems

In this study, an approximation solution for a high-dimensional system in terms of the Caputo fractional derivative operator is obtained using the improved modified fractional Euler method, or IMFEM for short. To accomplish this aim, a result that can transform such a system into a double-equation, one-dimensional fractional-order system, is provided theoretically. Some physical applications, including fractional-order systems of equations of Emden–Fowler type, are discussed, and their graphs are plotted using MATLAB to demonstrate the IMFEM schema’s validity.

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Analytical Solution of Cancer Cells Interaction with Virotherapy in the Framework of Fractional Derivative Operator

Analytical Solution of Cancer Cells Interaction with Virotherapy in the Framework of Fractional Derivative Operator

Fractional Ito calculus

We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero p p th variation along a sequence of partitions, where p ≥ 1 p \geq 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with ‘fractional’ regularity. In the case where p p is not an integer, we show that the change of variable formula may sometimes contain a nonzero ‘fractional’ Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero ϕ \phi -variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of ϕ \phi -variation.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

On the comparative study for Klein–Fock–Gordon equation: Fractional exact solutions, bifurcation and sensitivity analysis

In this study, we consider the Klein–Fock–Gordon equation model that emerges in the fields of quantum field theory, condensed matter physics and theory of relativity etc. The exact solutions of the model with various physical properties were systematically created in terms of Beta, Atangana–Baleanu meaning Caputo and M-fractional fractional derivatives. 3-D graphic analysis, physical interpretations of solutions and comparisons of derivative operators are presented. The fact that the solutions to be presented are different from those in the literature and examining the effects of different fractional derivative operators on the solutions are among the novelties of the study. In the last part of the article, bifurcation analysis and sensitivity analysis were performed and phase portraits shown. At the same time we examine the effects of some parameters on sensitivity.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

Influence of induced magnetic field and chemically reacting on hydromagnetic Couette flow of Jeffrey fluid in an inclined channel with variable viscosity and convective cooling: A Caputo derivative approach

In this study, the influence of induced magnetic field and chemically reacting on hydromagnetic generalized Couette flow of Jeffrey fluid in an inclined channel through a porous medium with variable viscosity and convective cooling has been investigated using the Caputo fractional order derivative operator. The mathematical formulation used for the hydromagnetic Couette flow of Jeffrey fluid takes into account the effects of viscous dissipation, Soret, and Dufour. The system of nonlinear partial differential equations governing the flow were solved numerically using the explicit finite difference method. The numerical results for the behavior of various physical parameter on the flow variables are obtained and represented graphically. Moreover, effects of the flow parameters on heat and mass transfer rates are obtained and discussed numerically through tabular forms. The graphical findings show that velocity, concentration, induced magnetic field, and thermal field profiles decline with progressively increment of Jeffrey parameter. The velocity, and temperature of the fluid decline with higher values of fluid viscosity parameter. An increase in the chemical reaction parameter recede the concentration field profiles while increase with raises the values of Soret number. Increasing Biot, and Dufour numbers significantly grows the thermal field profiles. Induced magnetic field grows with larger values of fluid viscosity parameter. The findings of this study are important due to its application in magnetohydrodynamics pumps, polymer manufacturing, fins designs, and food processing.

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

A new solution of the nonlinear fractional logistic differential equations utilizing efficient techniques

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.

Comparative study of fractional Newell–Whitehead–Segel equation using optimal auxiliary function method and a novel iterative approach

This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations.

Comparison principles for a class of general integro-differential inequalities with applications

Comparison principles for fractional differential equations have been investigated in many papers using different types of fractional integral and derivative operators. We here prove the strongest such results so far, for a very broad class of operators that is even more general than those with Sonine kernels. Starting from inequalities valid at global extrema, we obtain comparison principles for these general operators, which are applied to prove bounds on solutions to related integro-differential equations. Many results in the literature will be considered as particular cases of the current study.

On generalized fractional differential equation with Sonine kernel on a function space

Sonine kernel is a special type of the general kernels, satisfying some integrability and monotonicity conditions and properties. Sonine kernels generate some convolution polynomials (power functions) used in the construction of the general fractional calculus. The general fractional calculus with Sonine kernel is applied in the setting up and development of operational calculus.Consequently, we research on a kind of a generalized fractional differential equation with a Sonine type kernel: (∗D(κ)Cμ)(t)=ϱϑ(t,μ(t)),t∈(0,T],T<∞,with μ(0)=ω a non-negative and bounded initial data. The function ϑ:(0,T]×C−11(0,T]→R is assumed to be Lipschitz continuous on its second variable, ϱ>0 is a constant, ∗D(κ)C is a generalized Caputo fractional derivative operator and κ is a Sonine kernel belonging to a function space with an integrable singularity at the point of origin. Given some defined properties of the kernel, we establish evidence for the well-posedness of the solution via Banach fixed point theorem and reveal that the solution indicates an exponential growth bound. Moreso, we show that the solution is uniformly continuous in time and has continuous dependence on the initial condition.