A time fractional convection–diffusion equation to model gas transport through heterogeneous soil and gas reservoirs

Chang, Ailian; Sun, HongGuang; Chen, Kewei; Lu, Bingqing; Lu, Chengpeng; Ma, Rui; Zhang, Yong

doi:10.1016/j.physa.2018.02.080

“…The first formalism presented in this review was related to the fractional walker, this method has a great focus of investigation in the present day, because daily new techniques, theorems and theories continue to be developed by the mathematicians who investigate the fractional calculus [112,113,114,115,116,117,118,119,120,121,122,123,124]. With all this progress of the fractional calculus, physics advances together, since it allows the modelling of a series of interesting problems in physics, such as diffusion equation with tempered derivatives [125,126,127,128,129,130], memory systems [131,132,133,134,135], non-homogeneous systems [60,136,137,138], etc [139,140,141]. One of the most important roles of fractional calculus in physics has been to introduce ever more sophisticated memory kernels, which capture more precisely the processes observed in the real world [142].…”

Section: Brief Discussion and Some Considerationsmentioning

confidence: 99%

Analytic approaches of the anomalous diffusion: A review

Santos

¹

2019

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

show abstract

“…Many authors analyzed time-fractional partial differential equations (PDEs), for example, Wyss [14], Agrawal [21 ], Liu et al [22], Jiang and Ma. [23], and Chang et al [24].…”

Section: Introductionmentioning

confidence: 96%

Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

Shakeel

¹

,

Hussain

²

,

Ahmad

³

et al. 2020

Journal of Function Spaces

View full text Add to dashboard Cite

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

show abstract

“…Nowadays, the non‐integer‐order derivative plays a vital role in modeling many processes in physics, engineering, and science such as optimal control problems, 2 heat transfer model, 3,4 convection–diffusion reaction equations, 5 quantum mechanics, 6–8 fractional predator–prey biological model, 9 fractional tumor‐immune models, 10,11 and the reference therein. The fractional nonlinear reaction–diffusion equations (FNRDEs) appear in many applications such as gas transport model, 12 gas dynamics system, 13 the Lotka–Volterra type, 14,15 fractional telegraph equation, 16 chaotic dynamical systems, 17 diffusion with reaction terms, 18,19 Gray–Scott model, 20,21 reaction–diffusion system arising in biology, 22 Belousov–Zhabotinskii reaction systems, 23 Gierer–Meinhardt model, 24 Lengyel–Epstein system, 25 and dynamics of coronavirus (2019‐nCov) 26 . Also, the new fractional derivatives that have nonsingular kernel are given in Kumar et al, 27,28 fractional Navier–Stokes equation, 29 Boussinesq–Burger's equation, 30 and the nonlinear Kaup–Kupershmidt equation 31 …”

Section: Introductionmentioning

confidence: 99%

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

Zahra

¹

,

Hikal

²

,

Băleanu

³

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In this article, two nonstandard high‐order schemes on a uniform and nonuniform time stepping combined with the multi‐parameter exponential fitting technique (MPEF) have been developed to solve the time‐fractional nonlinear reaction–diffusion system. The first method based on the MPEF combined with the 3‐weighted shifted‐Grünwald–Letnikov approximation with uniform time stepping, this scheme leads to a numerical solution that suffers from the singularity near t = 0. In order to frustrate this singularity, a nonstandard higher‐order L1‐approximation for a nonuniform time‐stepping scheme is developed. The developed scheme's convergence and unconditionally stability analysis have been verified. Numerical results effectively validate the theoretical aspects.

show abstract

A time fractional convection–diffusion equation to model gas transport through heterogeneous soil and gas reservoirs

Cited by 57 publications

References 48 publications

Analytic approaches of the anomalous diffusion: A review

Analytic approaches of the anomalous diffusion: A review

Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

Contact Info

Product

Resources

About