Type Fractional Derivative Research Articles

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

Multidimensional Fractional Calculus: Theory and Applications

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications.

Dynamic J-A model improved by waveform scale parameters and R-L type fractional derivatives

PurposeThe purpose of this study is to model the global dynamic hysteresis properties with an improved Jiles–Atherton (J-A) model through a unified set of parameters.Design/methodology/approachFirst, the waveform scaling parameters β, λk and λc are used to improve the calculation accuracy of hysteresis loops at low magnetic flux density. Second, the Riemann–Liouville (R-L) type fractional derivatives technique is applied to modified static inverse J-A model to compute the dynamic magnetic field considering the skin effect in wideband frequency magnetization conditions.FindingsThe proposed model is identified and verified by modeling the hysteresis loops whose maximum magnetic flux densities vary from 0.3 to 1.4 T up to 800 Hz using B30P105 electrical steel. Compared with the conventional J-A model, the global simulation ability of the proposed dynamic model is much improved.Originality/valueAccurate modeling of the hysteresis properties of electrical steels is essential for analyzing the loss behavior of electrical equipment in finite element analysis (FEA). Nevertheless, the existing inverse Jiles–Atherton (J-A) model can only guarantee the simulation accuracy with higher magnetic flux densities, which cannot guarantee the analysis requirements of considering both low magnetic flux density and high magnetic flux density in FEA. This paper modifies the dynamic J-A model by introducing waveform scaling parameters and the R-L fractional derivative to improve the hysteresis loops’ simulation accuracy from low to high magnetic flux densities with the same set of parameters in a wide frequency range.

Three-dimensional fractional plastic models for saturated sand using Caputo derivative and R-L derivative: A comparative study

The strength and dilatancy of sand are mainly influenced by the void ratio, confining pressure and stress path. In engineering, the stress-strain relationship of sand in three-dimensional stress state is complicated, where the state-dependent properties of sand are difficult to be described by traditional constitutive models. Thus, fractional plastic models based on different fractional derivatives were developed to capture such state-dependent behavior of sand. This paper attempts to make a comparative study of the fractional models based on two typical fractional derivatives, i.e., the R-L derivative and Caputo derivative. The results show that the different types of fractional derivatives are mainly reflected in the different order of differentiation and integration, which will have a significant impact on the calculation of plastic flow direction, stress-dilatancy ratio and fractional order parameter βin the constitutive model. By determining the constitutive parameters of the two models, the constitutive behaviors predicted by different models were compared with the corresponding test results of saturated sands under different initial mean pressures and different stress paths. The models can reasonably simulate the test results of saturated sands, where the dilatancy characteristic can be captured. Compared with R-L model, Caputo model can predict more volumetric strain in the dilatancy process, and has better prediction effect. Overall, the predicted effects of the two models are close, with a maximum difference of about 7 %.

Application of Newton’s polynomial interpolation scheme for variable order fractional derivative with power-law kernel

This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo–Fabrizio derivative, the Atangana–Baleanu fractal and fractional derivative, and the Atangana–Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton’s polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang–Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach’s significance and effectiveness through numerical examples.

New solutions of time-fractional cancer tumor models using modified He-Laplace algorithm

Cancer develops through cells when mutations build up in different genes that control cell proliferation. To treat these abnormal cells and minimize their growth, various cancer tumor samples have been modeled and analyzed in literature. The current study is focused on the investigation of more generalized cancer tumor model in fractional environment, where net killing rate is taken into account in different domains. Three types of killing rates are considered in the current study including time and position dependent killing rates, and concentration of cells based killing rate. A hybrid mechanism is proposed in which different homotopies are used with perturbation technique and Laplace transform. This leads to a convenient algorithm to tackle all types of fractional derivatives efficiently. The convergence and error bounds of the proposed scheme are computed theoretically by proving related theorems. In the next phase, convergence and validity is analyzed numerically by calculating residual errors round the fractional domain. It is observed that computed errors are very less in the entire fractional domain. Moreover, comparative analysis of Caputo, Caputo-Fabrizio (CF), and Atangana–Baleanu (AB) fractional derivatives is also performed graphically to discern the effect of different fractional approaches on the solution profile. Analysis asserts the reliability of proposed methodology in the matter of intricate fractional tumor models, and hence can be used to other complex physical phenomena.

A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications.

Mild solution for (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem

Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional integral and (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the (ρ,α,β,k,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,\\alpha ,\\beta ,k,r)$\\end{document}-resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.

The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only

Through duality, it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators. The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulation and the corresponding necessary optimality conditions. As a practical application, we present a new Lagrangian that relies solely on left-hand side fractional derivatives. The fractional variational principle derived from this Lagrangian leads us to the equation of motion for a dissipative/damped system.

Solving the general form of the fractional Black–Scholes with two assets through Reconstruction Variational Iteration Method

The objective of this study is to examine the dynamic components of option pricing in the European put option market by utilizing the two-dimensional time fractional-order Black–Scholes equation. To enhance the classical Black–Scholes equation, we utilize the Caputo type of the Katugampola fractional derivative. The Reconstruction of Variational Iteration Method is employed as a powerful tool for analyzing option price behavior in the European-style market. In our investigation, we utilize this method to obtain an exact solution for fractional Black–Scholes with two assets. Moreover, the findings demonstrate the impressive effectiveness of the Reconstruction of Variational Iteration Method in addressing two-dimensional fractional-order differential equations, thereby highlighting its potential as a valuable numerical solution technique.

Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana – Baleanu derivative

In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense.

Analysis of the convective heat transfer through straight fin by using the Riemann-Liouville type fractional derivative: Probed by machine learning

This work aims to analyze the transfer of heat through new fractional-order convective straight fins by using the Riemann-Liouville type fractional derivatives. The convection through the fins is considered in such a way that the thermal conductivity depends on the temperature. The transformed fractional-order problems are constituted through an optimization problem in such a way that the L2 norm remains minimal. The objective functions are further analyzed with the hybrid Cuckoo search (HCS) algorithm that use the artificial neural network (ANN) mechanism. The impacts of the fractional parameter β, the thermo-geometric parameter of fin ψ, and dimensionless thermal conductivity α are explained through figures and tables. The fin efficiency during the whole process is explained with larger values of ψ. It is found that the larger values of ψ decline the fin efficacy. The fractional parameter declines the thermal profile as we approach the integer order. The convergence of HCS algorithm is performed in each case study. The residual error touches E−14 for the integer order of α. The present results are validated through Table 6 by comparing with HPM, VIM and LHPM, while the error for HCS-ANN touches E−13. This proves that the proposed HCS is efficient.

Extended Caputo [formula omitted]-type fractional derivative operator and its properties

In this paper, we present some extensions of the k-hypergeometric functions and then develop the extended Caputo k-type fractional derivative operator by using two parameters k-Mittag–Leffler function. We also discuss some properties like generating functions and Mellin transform of the new extended Caputo k-type fractional derivative operator.

Porous Elastic Soils with Fluid Saturation and Boundary Dissipation of Fractional Derivative Type

Porous Elastic Soils with Fluid Saturation and Boundary Dissipation of Fractional Derivative Type

A New Mixed Fractional Derivative with Applications in Computational Biology

This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented.

Tritrophic fractional model with Holling III functional response

<abstract><p>In this paper, we analyzed the local stability of three species in two fractional tritrophic systems, with Caputo's fractional derivative and Holling type Ⅱ and Ⅲ functional responses, when the prey density has a linear growth. To begin, we obtained the equilibria in the first octant under certain conditions for the parameters. Subsequently, through linearization and applying the Routh-Hurwitz Criterion, we concluded that only the system with Holling type Ⅲ exhibits an asymptotically stable equilibrium point, where the fractional derivative order belongs to the interval $ (0, 1] $. Finally, we obtained the solution of the system with the Holling type Ⅲ functional response, using the multistage homotopic perturbation method, and presented an example that shows the dynamics of the solutions around the stable equilibrium point.</p></abstract>

Caputo-fabrizio fractional-order systems: periodic solution and stabilization of non-periodic solution with application to gunn diode oscillator

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

General Fractional Calculus Operators of Distributed Order

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

An incommensurate fractional order model for complex dynamics of viral infection with immunity

This paper deals with an incommensurate fractional order mathematical model for dynamic analysis of viral infection with immunity. The primary focus of the work is to explore stability analysis of this version of incommensurate fractional order model with harmonic mean type incidence function and fractional derivative in Caputo sense. First, well-posed ness of the model has been established by analyzing existence and uniqueness of the solution. In the next section, stability analysis of the equilibrium points has been caried out based on the basic reproduction number. Sensitivity analysis of the threshold parameter have been performed in the following sections. Finally, rigorous numerical simulations have been performed to support the theoretical findings as well as to observe the effect of various fractional orders and incidence function.