Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation

Zheng, Rumeng; Liu, Fawang; Jiang, Xiaoyun; Turner, Ian

doi:10.1016/j.camwa.2020.06.017

Cited by 15 publications

(7 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…Collocation methods require that the approximate solution satisfies the DODE at specific locations known as the collocation points. Similar to the Galerkin spectral method, various collocation methods have been developed starting from (1) Legendre basis (LCM) [ 100 , 134 ] and (2) Jacobi basis (JCM) [ 105 , 152 ]. Zaky constructed a LCM to solve both linear and nonlinear boundary value problems [ 100 ], and later extended this method to simulate initial value DODEs [ 99 , 153 ].…”

Section: Mathematical Backgroundmentioning

confidence: 99%

See 1 more Smart Citation

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

View full text Add to dashboard Cite

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.

show abstract

Section: Mathematical Backgroundmentioning

confidence: 99%

Section: Mathematical Backgroundmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

View full text Add to dashboard Cite

show abstract

“…In the past few decades, fractional partial differential equations have provided very effective alternatives for modeling complex phenomena 16–20 . There are extensive mathematical and numerical investigations on the distributed‐order fractional PDEs 21–35 . However, most existing analysis results are proved for problems with regular probability density functions

\rho

; for example,

\rho \left(\alpha \right)

is assumed to be at least continuous on

\left[0,1\right]

.…”

Section: Introductionmentioning

confidence: 99%

A distributed‐order fractional diffusion equation with a singular density function: Analysis and approximation

Yang

Wang

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

We prove the well-posedness and smoothing properties of a distributed-order time-fractional diffusion equation with a singular density function in multiple space dimensions, which could model the ultraslow subdiffusion processes. We accordingly derive a finite element approximation to the problem and prove its optimal-order error estimate. Numerical results are presented to support the mathematical and numerical analysis.

show abstract

“…Liu et al 13 have proposed second‐order θ schemes combined with Galerkin finite element method for the numerical solution of FC equation with time‐fractional derivative. In 2020, Zheng et al 14 have proposed a finite difference method in time direction and Legendre spectral method in space direction for the numerical solutions of two‐dimensional distributed‐order time‐FC equation. Kumar and Baleanu 15 have proposed the ABC operational matrix of shifted Legendre method for the numerical solutions of a two‐dimensional time FC equation with Mittag–Leffler kernel.…”

Section: Introductionmentioning

confidence: 99%

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Mittal

Balyan

Panda

et al. 2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

show abstract

Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation

Cited by 15 publications

References 25 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

A distributed‐order fractional diffusion equation with a singular density function: Analysis and approximation

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Contact Info

Product

Resources

About