Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

Hafez, Ramy M.; Zaky, Mahmoud A.; Abdelkawy, Mohamed

doi:10.3389/fphy.2019.00240

Cited by 34 publications

(10 citation statements)

References 45 publications

(57 reference statements)

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“…To transform the integral form into the multi-term form (first of the two-step process), two common quadrature rules are often used by researchers: (1) Gauss–Legendre quadrature rule and (2) Newton–Cotes quadrature rule. Based on the Gauss–Legendre quadrature rules [ 84 , 85 , 86 , 87 , 88 , 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 , 98 , 99 , 100 , 101 , 102 , 103 , 104 , 105 , 106 , 107 ], the DO derivative can be approximated using the following multi-term form,

where

are the weights at the Gauss points

chosen for this integration over the DO. Although the Gauss–Legendre quadrature schemes are known to achieve highly accurate results (particularly when dealing with integrands of specific type such as, for example, polynomials), an analysis of the numerical convergence and of the truncation error (including steps 1 and 2) becomes difficult when the integrand consists of fractional derivatives (like

, as shown in Equation ( 11 )).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…A few researchers combined the GLSM scheme with an alternating direction implicit (ADI) scheme to improve the accuracy to

[ 98 , 139 ]. Numerical studies based on the GJSM approach can be found in [ 85 , 91 , 149 ]. Some interesting conclusions were presented in [ 150 ], which combined a

-stage implicit Runge–Kutta method in time and the GJSM/GLSM in space to solve time-space-fractional DODEs.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

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Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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where

are the weights at the Gauss points

, as shown in Equation ( 11 )).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…A few researchers combined the GLSM scheme with an alternating direction implicit (ADI) scheme to improve the accuracy to

[ 98 , 139 ]. Numerical studies based on the GJSM approach can be found in [ 85 , 91 , 149 ]. Some interesting conclusions were presented in [ 150 ], which combined a

-stage implicit Runge–Kutta method in time and the GJSM/GLSM in space to solve time-space-fractional DODEs.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…The spectral method has become increasingly popular in numerical solutions of partial differential equations due to its high-order accuracy [29][30][31][32][33][34]. For time-dependent partial differential equations, if the spectral scheme is used in spatial, the difference scheme is usually adopted in time.…”

Section: Applications To First-order Hyperbolic Equationsmentioning

confidence: 99%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes. KEYWORDS differentiation matrices, finite orthogonal polynomials, Gauss-type quadrature, Romanovski-Jacobi polynomials, spectral methods 1 INTRODUCTION The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1-3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types

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“…Spectral methods have exponential convergence rates as well as a high accuracy level. The spectral method has been classified into four classes, collocation [14], tau [15], Galerkin [16], and Petrov-Galerkin [17] methods.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Technique for Riesz Fractional Chen-Lee-Liu Equation

Abdelkawy

Alyami

2021

Journal of Function Spaces

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This paper discusses the study of optical solitons that are modeled by Riesz fractional Chen-Lee-Liu model, one of the versions of the famous nonlinear Schrödinger equation. This model is solved by the assistance of consecutive spectral collocation technique with two independent approaches. The first is the approach of the spatial variable, while the other is the approach of the temporal variable. It is concluded that the method of the current paper is far more efficient and credible for the proposed problem. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model.

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Jacobi Spectral Galerkin Method for Distributed-Order Fractional Rayleigh–Stokes Problem for a Generalized Second Grade Fluid

Cited by 34 publications

References 45 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

On Romanovski–Jacobi polynomials and their related approximation results

A Spectral Collocation Technique for Riesz Fractional Chen-Lee-Liu Equation

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