Fractional Taylor Series Research Articles

Inverse coefficient problem by fractional Taylor series method

This study focus on determining the unknown function of time or space in space-time fractional differential equation by fractional Taylor series method. A significant advantage of this method is that over-measured data is not used unlike most inverse problems. This advantage allows us to determine the unknown function with less error. The presented examples illustrate that the obtained solutions are in a high agreement with the exact solutions of the corresponding inverse problems.

On the solutions of some nonlinear fractional partial differential equations using an innovative and direct procedure

In this article, a highly effective technique is implemented to obtain the approximate solutions of strongly nonlinear fractional order partial differential equations (NFPDEs). The findings of this study show the successful behavior of the fractional novel analytical method (FNAM), which can be used successfully for the solutions of common, severe NFPDEs. In the proposed method, the nonlinearity in each mathematical model is directly handled by using fractional Taylor series, which reduces the calculation effort. In this work, the method's strength is primarily demonstrated on NFPDEs, and the obtained results are displayed via graphs and tables. From the numerical simulations, it is evident that the suggested technique has greater accuracy despite smaller calculations. It is the most straightforward method for determining the formulaic solution to any type of NFPDE and is considered to be the unique numerical methodology.

Fractional distributional representation of gamma function and the generalized kinetic equation

Motivated by the recent research about non-integer order derivatives of Dirac delta function, we use a fractional Taylor series to explore a fractional distributional representation of gamma function. It proved significant to develop a novel fractional kinetic equation, which is an essential mathematical model to measure the degree of transformation in the chemical structure of stars (sun) as a key component of our environment. In addition to the classical solution of the formulated fractional kinetic equation, the distributional solution is also attained which is not possible without using fractional derivatives of Dirac delta function. Novel identities involving the gamma function are obtained using several fractional transforms. Convergence of new series is proved using the theory of distributions leading to further new applications.

A Simple Solution for the General Fractional Ambartsumian Equation

Fractionalisation and solution of the Ambartsumian equation is considered. The general approach to fractional calculus suitable for applications in physics and engineering is described. It is shown that Liouville-type derivatives are the necessary ones, because they fully preserve backward compatibility with classical results. Such derivatives are used to define and solve the fractional Ambartsumian equation. First, a solution in terms of a slowly convergent fractional Taylor series is obtained. Then, a simple solution expressed in terms of an infinite linear combination of Mittag–Leffler functions is deduced. A fast algorithm, based on a bilinear transformation and using the fast Fourier transform, is described and demonstrated for its approximate numerical realisation.

Identification of a time-dependent diffusivity coefficient in heat-like space-time fractional differential equations

The goal of this research is to reveal the unknown time dependent diffusion coefficient in space-time fractional differential equations by means of fractional Taylor series method. Unlike most methods used in inverse problems, using no over-measured data is a substantial advantage of this method.
 As a result, the unknown diffusion coefficient could be determined with high precision. Illustrative examples shows that the retrieved unknown coefficient and the solution of the problem are in a high agreement with the exact solution of the corresponding the inverse problems.

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

A Numerical Scheme for Fractional Mixed Convection Flow Over Flat and Oscillatory Plates

Abstract A fractional scheme is proposed to solve time-fractional partial differential equations. According to the considered fractional Taylor series, the scheme is compact in space and provides fourth-order accuracy in space and second-order accuracy in fractional time. The scheme is conditionally stable when applied to the scalar fractional parabolic equation. The convergence of the scheme is demonstrated for the system of fractional parabolic equations. Moreover, a fractional model for heat and mass transfer of mixed convection flow over the flat and oscillatory plate is given. The radiation effects and chemical reactions are also considered. The scheme is tested on this model and the nonlinear fractional Burgers equation. It is found that it is more accurate than considering existing schemes in most of the regions of the solution domain. The compact scheme with exact findings of spatial derivatives is better than considering linearized equations. The error obtained by the proposed scheme with the determination of exact spatial derivatives is better than that obtained by two explicit existing schemes. The main advantage of the proposed scheme is that it is capable of providing the solution for convection-diffusion equations with compact fourth-order accuracy. Still, the corresponding implicit compact scheme is unable to find the solution to convection-diffusion problems.

On Fractional Newton‐Type Method for Nonlinear Problems

The current manuscript is concerned with the development of the Newton–Raphson method, playing a significant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. The development and modification of the Newton–Raphson method allow us to establish two new methods, which are called first‐ and second‐order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of first‐ and second‐order fractional methods and give a general condition for the convergence of higher‐order FNR. Finally, some illustrative examples are considered to confirm the accuracy and effectiveness of both methods.

A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

<abstract> <p>This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.</p> </abstract>

A fractional taylor series-Based least mean square algorithm, and its application to power signal estimation

The definition of fractional derivative by Caputo and Riemann inspired the researchers to develop new adaptive algorithms having better convergence properties in comparison with integer-gradient based adaptive algorithms. As reported in many studies, the existing fractional gradient-based adaptive algorithms lack justification for using fractional derivative in addition to integer-gradient, may become inconsistent in the event of negative weights, and may yield more or less same performance as compared to the conventional integer-derivative-based algorithms by appropriate selection of step-size parameter. Accordingly, this paper presents a novel fractional adaptive algorithm based on Fractional Taylor Series. Unlike (most of) the existing fractional-derivative-based algorithms, the proposed algorithm only involves fractional-derivative and ensures convergence of mean square error (MSE) provided the step-size is chosen appropriately. Simulation results are presented in order to depict a scenario where exploitation of fractional-derivative in the weight-update equation yields better convergence as compared to LMS algorithm in the context of power signal parameters estimation.

Finite difference method for solving fractional differential equations at irregular meshes

This paper presents a novel meshless technique for solving a class of fractional differential equations based on moving least squares and on the existence of a fractional Taylor series for Caputo derivatives. A “Generalized Finite Difference” approach is followed in order to derive a simple discretization of the space fractional derivatives. Consistency, stability and convergence of the method are proved. Several examples illustrating the accuracy of the method are given.

Stable and convergent difference schemes for weakly singular convolution integrals

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in Cγ[0,T], where 0<γ≤5. A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be O(τγ), 0<γ≤5 for u∈Cγ[0,T], where τ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings.

From continuous-time random walks to the fractional Jeffreys equation: Solution and properties

Jeffreys equation provides an increasingly popular extension of the diffusive laws of Fourier and Fick for heat and particle transport. Similar to generalisations of the diffusion equation, we here investigate the connection between a time-fractional generalisation of the Jeffreys equation and a continuous-time random walk process based on a generalised waiting time density with diverging mean. We demonstrate that the mean squared displacement exhibits a variety of anomalous behaviors, such as retarding and accelerating subdiffusion, as well as a crossover from superdiffusion to subdiffusion. Moreover, we provide two alternative approaches, namely, a fractional Taylor series and distributed-order derivatives, that transform Fourier’s or Fick’s law into the time-fractional Jeffreys equation. Our discussion provides physics-based support for the fractional Jeffreys equation and shows its versatility for practical applications.

A Nonlocal Fractional Peridynamic Diffusion Model

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations.

Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating

In this paper, considering the heat flux and the temperature gradient approximated by the fractional Taylor's series expansions with different orders in dual-phase-lag (DPL) model, we propose the time fractional DPL heat conduction model for thin gold films heating by femtosecond laser pulses. The solutions of the model are investigated analytically and numerically. Firstly, the semi-analytical solution is presented by the Laplace transform method and the Fourier cosine transform method. With the L1 approximation of the Caputo fractional derivative, the finite difference scheme is derived to describe the effects of femtosecond laser pulses on the temperature distributions of the gold films. The effectiveness of the numerical algorithm is examined by the comparison with the semi-analytical solution. And then based on the numerical algorithm and the experimental datasets of femtosecond laser pulses heating on gold films of various thicknesses, the estimations of the model parameters are considered by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO). Using the optimal values of the parameters estimation, we verify the consistencies between the semi-analytical solutions of the model and the datasets. It proves the accuracy of the time fractional DPL heat conduction model in characterizing the temperature distributions of thin gold films. Finally, the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.

Derivation of fractional-derivative models of multiphase fluid flows in porous media

This paper is devoted to deriving several fractional-order models for multiphase flows in porous media, focusing on some special cases of the two-phase flow. We derive the mass and momentum conservation laws of multiphase flow in porous media. The mass conservation-law has been developed based on the flux variation using Taylor series approximation. The fractional Taylor series's advantage is that it can represent the non-linear flux with more accuracy than the first-order linear Taylor series. The divergence term in the mass conservation equation becomes of a fractional type. The model has been developed for the general compressible flow, and the incompressible case is highlighted as a particular case. As a verification, the model can easily collapse to the traditional mass conservation equation once we select the integer-order. To complete the flow model, we present Darcy’s law (momentum conservation law in porous media) with time/space fractional memory. The modified Darcy’s law with time memory has also been considered. This version of Darcy’s law assumes that the permeability diminishes with time, which has a delay effect on the flow; therefore, the flow seems to have a time memory. The fractional Darcy’s law with space memory based on Caputo's fractional derivative is also considered to represent the nonlinear momentum flux. Then, we focus on some cases of fractional time memory of two-phase flows with countercurrent-imbibition mechanisms. Five cases are considered, namely, traditional mass equation and fractional Darcy’s law with time memory; fractional mass equation with conventional Darcy’s law; fractional mass equation and fractional Darcy’s law with space memory; fractional mass equation and fractional Darcy’s law with time memory; and traditional mass equation and fractional Darcy’s law with spatial memory.

A Study on Fractional Taylor Series and Variations

A Study on Fractional Taylor Series and Variations

On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

We drive efficient and reliable finite difference methods for fractional differential equations (FDEs) based on recently defined conformable fractional derivative. We first derive fractional Euler and fractional Taylor methods based on the fractional Taylor expansion. This fractional Taylor series are the generalized fractional Taylor series that are independent of initial point. We show that the proposed methods are more efficient and faster by applying these methods on first order FDEs and second order oscillatory FDEs. Our second approach is based on inverting FDEs to a weakly singular integral equation that is approximated by product integration rule. This new definition has no special functions and thus the proposed numerical methods will be more accurate and easier to implement than existing methods for FDEs. We prove the stability and convergence of the proposed methods. Numerical examples are given to support the theoretical results.

On fractional and fractal formulations of gradient linear and nonlinear elasticity

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.