Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

Morgado, Maria Luísa; Rebelo, Magda; Ferrás, Luís Jorge Lima; Ford, Neville J.

doi:10.1016/j.apnum.2016.11.001

Cited by 48 publications

(32 citation statements)

References 56 publications

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“…-fulfillment of equations specific for the problem, e.g., basic laws, such as Kirchhoff's laws or the power balance for a circuit problem [39]; -comparison with results obtained through another method, preferably one operating on a very different basis (e.g., the results obtained through the application of numerical methods are compared with results from analytical solutions [35] and vice versa [53]). …”

Section: Motivationmentioning

confidence: 99%

A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

Sowa

2018

Circuits Syst Signal Process

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This paper discusses the ability to obtain periodic steady-state solutions for fractional nonlinear circuit problems. For a class of nonlinear problems with fractional derivatives (based on the Caputo or Riemann-Liouville definitions), a methodology is proposed to derive equations representing the dependencies between the harmonics of the sought variables. Two approaches are considered for how to address the apparent nonlinear dependencies: one based on symbolic computation and the other a numerical approach based on the analysis of time functions. An example problem with fractional and nonlinear elements is presented to illustrate the usefulness of the proposed methodology. Two error criteria are introduced to verify the accuracy of the obtained results. The methodology is mainly designed to provide referential solutions in analyses of the numerical method called SubIval (the subinterval-based method for computation of the fractional derivative in initial value problems).

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Section: Motivationmentioning

confidence: 99%

A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

Sowa

2018

Circuits Syst Signal Process

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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%

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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“…One of the many challenges in solving distributed order differential equations is the huge computational cost involved compared to fixed order fractional differential equations. Among the few studies that have been dedicated to obtaining numerical solutions of time distributed order fractional partial differential equations are [9,12,14,15]. In Hu et al [10], an implicit difference method was used to solve a one-dimensional differential equation of the form (1.1).…”

Section: Introductionmentioning

confidence: 99%

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

2021

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Time distributed order two-sided space differential equations of arbitrary order offer a robust approach to modelling complex dynamical systems. In this study, we describe a scheme for obtaining the numerical solutions of time distributed order multidimensional two-sided space fractional differential equations. The numerical discretization scheme is a hybrid scheme, comprising a Newton-Cotes quadrature formula and a spectral collocation method. The time distributed order fractional differential operator is approximated using the composite Simpson's rule, and the solution of the resulting differential equation is expressed as a linear combination of shifted Chebyshev polynomials in all variables. Convergence analysis of the numerical scheme is presented. Some one-and two-dimensional time distributed order two-sided space fractional differential equations, such as the fractional advection-dispersion and diffusion equations, are presented to demonstrate the accuracy and computational efficiency of the numerical scheme, and numerical solutions are compared with the exact solutions, where these are available.

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Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

Cited by 48 publications

References 56 publications

A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

A Harmonic Balance Methodology for Circuits with Fractional and Nonlinear Elements

Applications of Distributed-Order Fractional Operators: A Review

A Pseudo-spectral Method for Time Distributed Order Two-sided Space Fractional Differential Equations

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