Fractional Derivative Formulas Research Articles

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Generalized Taylor’s formula for power fractional derivatives

We establish a new generalized Taylor’s formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor’s formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor’s expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.

Fractional‐type integral operators and their applications to trend estimation of COVID‐19

In this paper, we construct a novel family of fractional‐type integral operators of a function by replacing sample values with the fractional mean values of that function. We give some explicit formulas for higher order moments of the proposed operators and investigate some approximation properties. We also define the fractional variants of Mirakyan–Favard–Szász and Baskakov‐type operators and calculate the higher order moments of these operators. We give an explicit formula for fractional derivatives of proposed operators with the help of the Caputo‐type fractional derivative Furthermore, several graphical and numerical results are presented in detail to demonstrate the accuracy, applicability, and validity of the proposed operators. Finally, an illustrative real‐world example associated with the recent trend of Covid‐19 has been investigated to demonstrate the modeling capabilities of fractional‐type integral operators.

A fractionally magnetized flow of force fields and Fermi–Walker conformable derivative on the unit sphere

ABSTRACT In this study, we aim to investigate the differential geometric aspect of the motion of the charged particle, when it is exposed to particular force fields on the unit sphere , by considering the effect of the fractional calculus. We describe the conformable fractional derivative formula of the spherical vector fields moving along with the charged particle along with its trajectory. Moreover, we characterize the magnetic flows of the charged particle associated with the dynamical spherical conformable curve by defining Lorentz conformable force on the unit sphere Finally, we obtain the parallel transportation rule and uniformness of the motion occurring along with the spherical conformable curve in view of fractional derivatives.

Numerical investigation of unconditionally stable spline function for three-dimensional time-fractional telegraph equations

Telegraph equations are hyperbolic partial differential equations that may be used to represent reaction–diffusion processes in a variety of engineering and biological disciplines. The development of numerical techniques for telegraph-type equations has received a lot of interest in the literature in recent years. The main objective of this article is to introduce and evaluate a new approach for approximating the time-fractional telegraph equation utilizing spline functions. Initially, an operational matrix method based on the consolidation of block pulse functions and Fibonacci wavelets is presented to derive the solutions to Time-Fractional Telegraph Equations (TFTs). The suggested approach converts the fractional model into an algebraic equation system that can be solved utilizing the Newton iteration method. The Crank Nicolson approach is also offered for the solution of three-dimensional time-fractional telegraph equations using the Trigonometric Quintic B-spline (TQBS). The rationale behind using the collocation method is to select specific collocation spots where the differential equation is fulfilled exactly. The suggested technique combats nonlinearity by employing a quasi-linearization procedure. The time-fractional derivative’s discretization is done using the Caputo fractional derivative formula. The calculated solutions are obtained using a combination of a trigonometric Quintic B-spline and the Caputo fractional derivative. The primary goal is to verify the well-posedness and produce a numerical solution for an initial–boundary value issue for a hyperbolic equation using finite-difference methods. Accordingly, the research developed the exponentially fitted approach for solving initial boundary value problems utilizing finite difference formulae and temporal frequencies. The scheme’s convergence is demonstrated using normal analytical approaches, demonstrating that the method is unconditionally stable and has an order of convergence. MATLAB software is used to run the numerical simulations. Two model examples with boundary layer behaviour are investigated to support the theoretical conclusion. Furthermore, the graphs show that exact and numerical solutions are near together, demonstrating the method’s precision.

Integral transforms and probability distributions for a certain class of Fox–Wright type functions and its applications

Main focus of this paper is to introduce a new class of Fox–Wright type functions and obtain several new integral representations and various integral transforms such as Mellin transform, Whittaker transform, Jacobi transform, Gegenbauer transform, Laplace transform, Euler (Beta) transform, Pδ, Rν and Hankel transform. Moreover, a new probability density function is also introduced and several properties are studied. As application, fractional derivative and fractional integration formula are established. The results obtained in this paper, are presumably new and their importance is illustrated by several interesting consequences and examples. Furthermore, numerical computation and graphical representations have been expressed to validate the results.

A Fast Galerkin Approach for Solving the Fractional Rayleigh–Stokes Problem via Sixth-Kind Chebyshev Polynomials

Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. Some specific integer and fractional derivative formulas are used to introduce our proposed numerical algorithm. Moreover, the stability and convergence accuracy are derived in detail. As a final validation of our theoretical results, we present a few numerical examples.

New Fractional Derivative Expression of the Shifted Third-Kind Chebyshev Polynomials: Application to a Type of Nonlinear Fractional Pantograph Differential Equations

The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain 4 F 3 1 terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations are included. The proposed numerical method is demonstrated by numerical results to ensure its applicability and efficiency.

Associated Conformable Fractional Legendre Polynomials

Along with the work of Abul-Ez et al. [37], we introduce the associated conformable fractional Legendre polynomials (ACFLPs), from which the fractional differential equation of ACFLPs is established. Subsequently, some of interesting properties are derived such as generating function, hypergeometric representation, analytical formula, besides various of recurrence relations. Also, orthogonal properties of ACFLPs are developed in conformable context. We append our study by presenting the shifted ACFLPs and driving some of important properties such as Rodrigues’ type representation formula of fractional order derivative and explicit formula. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. This result is easily generalized for integrands involving products of an arbitrary number of shifted associated Legendre polynomials in conformable sense.

Stock price forecasting based on Hausdorff fractional grey model with convolution and neural network.

Forecast of stock prices can guide investors' investment decisions. Due to the high-dimensional and long-memory characteristics of stock data, it is difficult to predict. The fractional grey model with convolution (FGMC (1, m)) can be used to predict time series, because of its memory and ability to process high-dimensional data. However, the FGMC (1, m) model has some disadvantages, including complex calculation, loss of information, and approximate background values. In this paper, Hausdorff fractional derivative and Newton-Cotes formula are used to optimize these shortcomings and can get a Hausdorff fractional grey model with convolution (HFGMC (1, m)) model. The HFGMC (1, m)-LLE-BP model is proposed in this paper. HFGMC (1, m) provides a solution that can reduce the complexity of the cumulative generator matrix calculation and preserve the global information of the sequence. Newton-Cotes formula is used to calculate the background value, which can solve the shortcomings of approximate background values. The HFGMC (1, m) model is used to predict the linear component of the sequence, and the BP neural network is used to predict the nonlinear component of the sequence. In addition, because of the high-dimensional and nonlinear characteristics of stock data, a local linear embedding (LLE) algorithm is used to remove redundant information in high-dimensional non-linear data. The experimental results show that the HFGMC (1, m)-LLE-BP model is effective for predicting the stock price in different trends.

Generalized Wright Function and Its Properties Using Extended Beta Function

Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extentions (and generalizations) have been investigated and presented. The purpose and design of the paper is intended to study and come up with a new extention of the genralized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also we present the Mellin transform of this function in the form of Fox Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann-Liouville fractional derivative, we present a fractional derivative formula for the extended Wright function.

Unified fractional integral and derivative formulas, integral transforms of incomplete $$\tau $$-hypergeometric function

The objective of this article is to evaluate unified fractional integrals and derivative formulas involving the incomplete $$\tau $$ -hypergeometric function. These integrals and derivatives are further applied in proving theorems on Marichev–Saigo–Maeda operators of fractional integration and differentiation. The results are expressed in terms of the generalized Gauss hypergeometric functions (Fox–Wright function). Corresponding assertions in terms of Saigo, Erdelyi–Kober, Riemann–Liouville, and Weyl type of fractional integrals and derivatives are presented. Also, we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

On a $$\psi $$-Generalized Fractional Derivative Operator of Riemann–Liouville with Some Applications

Various extended beta functions have been investigated by many researchers (Mubeen et al. in FJMS 102:1545–1557, 2017; Ozergin et al. in J Comput Appl Math 235:4601–4610, 2011; Parmar et al. in J Class Anal 11:91–106, 2017). Our aim in this paper is to develop $$\psi $$ -generalized Riemann Liouville fractional derivative operator by using $$\psi $$ -generalized beta function. We also present some new results on $$\psi $$ -generalized beta function and introduce some integral representations of $$\psi $$ -generalized hypergeometric functions. Moreover, we establish fractional derivative formulas of some known functions and discuss the properties of Mellin transform. Further we discuss some applications based on generating relation for the $$\psi $$ -generalized hypergeometric functions with the help of new $$\psi $$ -generalized Riemann–Liouville fractional derivative operator.

Spectrally accurate approximate solutions and convergence analysis of fractional Burgers\u2019 equation

In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

Numerical solutions of time and space fractional coupled Burgers equations using time–space Chebyshev pseudospectral method

The aim of the paper is to develop and analyze a spectrally accurate time–space pseudospectral method to the approximate solution of nonlinear time and space fractional coupled Burgers equations. Liouville–Caputo fractional derivative formula is used to evaluate the fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivative matrices, the given problem is reduced to a system of nonlinear algebraic equations. A mapping is used to transform the nonhomogeneous initial–boundary values to homogeneous initial–boundary values. Error analysis of the proposed method for the equation is presented. A model example of fractional coupled Burgers equations is tested for a set of fractional‐order derivatives. For the proposed method, highly accurate numerical results are obtained, which confirm the accuracy and efficiency of the proposed method.

Some properties concerning the analysis of generalized Wright function

Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extensions (and generalizations) have been investigated and presented. The purpose and design of the paper are intended to study and come up with a new extension of the generalized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also, we present the Mellin transform of this function in the form of Fox–Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann–Liouville fractional derivative, we present a fractional derivative formula for the generalized Wright function.

ДРОБНАЯ ПРОИЗВОДНАЯ КОШИ

In this paper, we introduce a new sort of fractional derivative. For this, we consider the Cauchy's integral formula for derivatives and modify it by using Laplace transform. So, we obtain the fractional derivative formula F(α)(s) = L{(–1)(α)L–1{F(s)}}. Also, we find a relation between Weyl's fractional derivative and the formula above. Finally, we give some examples for fractional derivative of some elementary functions.

Fractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Function

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

Certain generalized fractional calculus formulas and integral transforms involving (p,q)-Mathieu-type series

In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p,q)-Mathieu-type series.

Fractional calculus of generalized p-k-Mittag-Leffler function using Marichev–Saigo–Maeda operators

In this paper, we establish fractional integral and derivative formulas involving the generalized p-k-Mittag-Leffler function by using Marichev–Saigo–Maeda type fractional integral and derivative operators. We also consider some special cases of derived results by considering specific values of the parameters of the generalized p-k-Mittag-Leffler function to give the application of our main results.