Riesz fractional derivative Elite-guided sine cosine algorithm

2019 ◽  
Vol 81 ◽  
pp. 105481 ◽  
Author(s):  
Wenyan Guo ◽  
Yuan Wang ◽  
Fengqun Zhao ◽  
Fang Dai
Keyword(s):  
Fractional Derivative ◽  
Riesz Fractional Derivative ◽  
Sine Cosine Algorithm

scholarly journals Integral transforms of the κ-Hilfer fractional derivative

2021 ◽  
Author(s):  
Felix Costa ◽  
Junior Cesar Alves Soares ◽  
Stefânia Jarosz
Keyword(s):  
Fractional Derivative ◽  
Gamma Function ◽  
Fundamental Theorem ◽  
Beta Function ◽  
Integral Transforms ◽  
Fractional Operators ◽  
Riesz Fractional Derivative ◽  
Hilfer Fractional Derivative ◽  
Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

Generalized GL Fractional Derivative and Its Laplace and Fourier Transform

2009 ◽  
Author(s):  
Manuel D. Ortigueira ◽  
Juan J. Trujillo
Keyword(s):  
Fourier Transform ◽  
Linear System ◽  
Fractional Derivative ◽  
Frequency Response ◽  
Fourier Transforms ◽  
Laplace And Fourier Transforms ◽  
Riesz Fractional Derivative ◽  
Laplace And Fourier Transform

It is well known the difficulties that the Riesz fractional derivative present, as the spatial fractional derivative involved in many models of the dynamics of anomalous processes. The generalized Gru¨nwal-Letnikov fractional derivative is analysed in this paper. Its Laplace and Fourier Transforms are computed and some current results criticized. It is shown that only the forward derivative of a sinusoid exists. This result is used to define the frequency response of a fractional linear system.

scholarly journals Toward fractional gradient elasticity

2014 ◽  
Vol 23 (1-2) ◽  
pp. 41-46 ◽  
Author(s):  
Vasily E. Tarasov ◽  
Elias C. Aifantis
Keyword(s):  
Fractional Derivative ◽  
Gradient Elasticity ◽  
Point Load ◽  
Three Dimensions ◽  
Riesz Fractional Derivative ◽  
Caputo Fractional Derivatives ◽  
Stress Equilibrium ◽  
First Case ◽  
Elastic Bar ◽  
Derivatives Of

AbstractThe use of an extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe the power law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one dimension. The second involves the Riesz fractional derivative in three dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case, stress equilibrium in a Caputo elastic bar requires the existence of a nonzero internal body force to equilibrate it. In the second case, in a Riesz-type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

Modelling and numerical synchronization of chaotic system with fractional-order operator

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Kolade M. Owolabi
Keyword(s):  
Fractional Derivative ◽  
Chaotic System ◽  
Reaction Diffusion ◽  
Domain Size ◽  
Two Dimensions ◽  
Second Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Riesz Fractional Derivative ◽  
Definition Of

Abstract Numerical solution of nonlinear chaotic fractional in space reaction–diffusion system is considered in this paper on a large but finite spatial domain size x ∈ [0, L] for L ≫ 0, x = x(x, y) and t ∈ [0, T]. The classical order chaotic ordinary differential equation is formulated by introducing the second-order spatial fractional derivative with order β ∈ (1, 2]. This second order spatial derivative is modelled by using the definition of the Riesz fractional derivative. The method of approximation combines the Fourier spectral method with the novel exponential time difference schemes. The proposed technique is known to have gained spectral accuracy over finite difference schemes. Applicability and suitability of the suggested methods are tested on Rössler chaotic system of recurring interests in one and two dimensions.

A High-Order Difference Scheme for the Space and Time Fractional Bloch–Torrey Equation

2018 ◽  
Vol 18 (1) ◽  
pp. 147-164 ◽  
Author(s):  
Yun Zhu ◽  
Zhi-Zhong Sun
Keyword(s):  
Fractional Derivative ◽  
Difference Scheme ◽  
Dimensional Space ◽  
High Order ◽  
Stability And Convergence ◽  
Difference Operators ◽  
Riesz Fractional Derivative ◽  
Central Difference ◽  
Order Difference ◽  
Space And Time

AbstractIn this paper, a high-order difference scheme is proposed for an one-dimensional space and time fractional Bloch–Torrey equation. A third-order accurate formula, based on the weighted and shifted Grünwald–Letnikov difference operators, is used to approximate the Caputo fractional derivative in temporal direction. For the discretization of the spatial Riesz fractional derivative, we approximate the weighed values of the Riesz fractional derivative at three points by the fractional central difference operator. The unique solvability, unconditional stability and convergence of the scheme are rigorously proved by the discrete energy method. The convergence order is 3 in time and 4 in space in {L_{1}(L_{2})}-norm. Two numerical examples are implemented to testify the accuracy of the numerical solution and the efficiency of the difference scheme.

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