Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Hou, Yaxin; Cao, Wen; Li, Hong; Liu, Yang; Fang, Zhichao; Yang, Yining

doi:10.3390/math8020187

Cited by 7 publications

(2 citation statements)

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“…Similarly, several researchers first developed FEMs to solve multi-term FDEs and later extended them to solve DODEs [ 87 , 123 , 188 ]. Few researchers [ 112 , 189 ] developed the

-Galerkin FEM for DO sub-diffusion equations which allowed the estimation of the diffusive field variable as well as its spatial derivative. By using locally discontinuous Galerkin FEM, Aboelenen [ 137 ] and Wei [ 190 ] developed highly accurate numerical schemes with spatial convergence

( k is the degree of basis polynomials).…”

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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“…Similarly, several researchers first developed FEMs to solve multi-term FDEs and later extended them to solve DODEs [ 87 , 123 , 188 ]. Few researchers [ 112 , 189 ] developed the

( k is the degree of basis polynomials).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…While several numerical techniques have been proposed to solve some different problems (see, for instance, earlier researches 52–62 and references therein), there have been few research studies that developed numerical methods to solve the NTFSDEs of distributed order (see earlier studies, 39,63,64 ).…”

Section: Introductionmentioning

confidence: 99%

A new operational vector approach for time‐fractional subdiffusion equations of distributed order based on hybrid functions

Eftekhari

Rashidinia

2022

Math Methods in App Sciences

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This research study deals with the numerical solutions of linear and nonlinear time‐fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block‐pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.

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