Numerical approach for a class of distributed order time fractional partial differential equations

Moghaddam, Behrouz Parsa; Machado, J. A. Tenreiro; Morgado, Maria Luísa

doi:10.1016/j.apnum.2018.09.019

Cited by 40 publications

(24 citation statements)

References 32 publications

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“…All these schemes are also associated with different orders of convergence. Following the open Newton–Cotes approach, the mid-point rule is widely used [ 107 , 123 , 124 , 125 , 126 , 127 , 128 , 129 , 130 , 131 , 132 , 133 , 134 , 135 , 136 , 137 , 138 , 139 , 140 , 141 , 142 , 143 ]. The truncation error at the end of step 1, when employing the Newton–Cotes approach, simply follows the classical results.…”

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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Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…Ding [19] developed a high-order finite difference algorithm for solving the time-space tempered fractional diffusion-wave. Moghaddam, et al [36] presented a numerical solution based on the midpoint quadrature rule and linear B-spline interpolation for solving distributed order time fractional partial differential equations. Li, et al [32] studied the properties of the time tempered fractional derivative and the well-posedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation.…”

Section: Introductionmentioning

confidence: 99%

Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation

Abbaszadeh

Amjadian

2020

Commun. Appl. Math. Comput.

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The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equation. The time variable has been discretized by a second-order finite difference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confirm the theoretical results.

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“…In consequence, fractional differential equations (FDEs) are gaining significant importance and have been analyzed by several researchers, for example, see References . Some of the latest works related with the high‐order numerical methods for FDEs have been worked in References .…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of difference methods for two‐dimensional Riesz space fractional advection‐dispersion equations with delay

Heris

Javidi

Ahmad

2020

Comp and Math Methods

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In this article, the Riesz space fractional advection‐dispersion equations with delay in two‐dimensional (RFADED in 2D) are considered. The Riesz space fractional derivative is approximated with the aid of backward differential formulas method of second order and shifted Grünwald difference operators. We develop the Crank‐Nicolson scheme using the finite difference method for the RFADED in 2D and show that it is conditionally stable and convergent with the accuracy order O(κ2+h2+k2). Finally, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical method.

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Numerical approach for a class of distributed order time fractional partial differential equations

Cited by 40 publications

References 32 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation

Stability and convergence of difference methods for two‐dimensional Riesz space fractional advection‐dispersion equations with delay

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