Numerical solutions for fractional reaction–diffusion equations

Baeumer, Boris; Kovács, Mihály; Meerschaert, Mark M.

doi:10.1016/j.camwa.2007.11.012

Cited by 118 publications

(66 citation statements)

References 62 publications

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“…Here the term R * /[L/N A (t)] in (35) approximates the proportion of reactant A within R * , and the factor "2" in (35) accounts for the fact that the reactant B can be located on both sides of a A particle (hence doubling the reaction probability). During the 2nd sub-step (see Subsec.…”

Section: Discussionmentioning

confidence: 99%

“…The product of the two sub-probabilities defined by (35) and (36) (which are independent) leads to the average forward reaction probability…”

Section: Discussionmentioning

confidence: 99%

“…Although there is no rigorous derivation of (1) for describing reactive transport combined with anomalous diffusion in porous media, similar governing equations were proposed and applied by various researchers [22,35]. The choice of this type of model is also motivated by the work of hydrologists [36,37,38], who developed a similar temporally nonlocal model in the multi-rate mass transfer formulation.…”

Section: Governing Equation Of the Dynamic Processmentioning

confidence: 99%

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Particle-tracking simulation of fractional diffusion-reaction processes

Zhang

Papelis

2011

Phys. Rev. E

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Abstract. Computer simulation of reactive transport in heterogeneous systems remains a challenge, due to the multi-scale nature of reactive dynamics and the non-Fickian behavior of transport. This study develops a fully Lagrangian approach via particle tracking to describe the reactive transport controlled by the tempered super-or sub-diffusion. In the particle-tracking algorithm, the local-scale reaction is affected by the interaction radius between adjacent reactants, whose motion can be simulated by the Langevin equations corresponding to the tempered stable models. Lagrangian simulation results show that the transient superdiffusion enhances reaction by enhancing the degree of mixing of reactants. The proposed particle-tracking scheme can also be extended conveniently to multi-scaling super-diffusion. For the case of transient sub-diffusion, the trapping of solutes in the immobile phase can either decrease or accelerate the reaction rate, depending on the initial condition of reactant particles. Further practical applications show that the new solver efficiently captures bimolecular reactions observed in laboratories.

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Section: Discussionmentioning

confidence: 99%

“…The product of the two sub-probabilities defined by (35) and (36) (which are independent) leads to the average forward reaction probability…”

Section: Discussionmentioning

confidence: 99%

Section: Governing Equation Of the Dynamic Processmentioning

confidence: 99%

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Particle-tracking simulation of fractional diffusion-reaction processes

Zhang

Papelis

2011

Phys. Rev. E

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“…The case of non-integer  is used to model the super diffusive or anomalous diffusive flow in which a cloud of particles spreads at a faster rate than in the classical model [5]- [8].…”

Section: Introductionmentioning

confidence: 99%

A Second Order Finite Difference Approximation for the Fractional Diffusion Equation

Nasir¹,

Gunawardana²,

Abeyrathna³

2013

IJAPM

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Abstract-We consider an approximation of one-dimensional fractional diffusion equation. We claim and show that the finite difference approximation obtained from the Grünwald-Letnikov formulation, often claimed to be of first order accuracy, is in fact a second order approximation of the fractional derivative at a point away from the grid points. We use this fact to device a second order accurate finite difference approximation for the fractional diffusion equation. The proposed method is also shown to be unconditionally stable. By this approach, we treat three cases of difference approximations in a unified setting. The results obtained are justified by numerical examples.Index Terms-Fractional derivative, diffusion equation, grunwald approximation, crank-nicolson method.

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“…Equations (1.4) and (1.5) directly lead to the stretched exponential expression described by equation (1.3). At present, a growing number of works in science and engineering deal with dynamical systems described by fractional-order equations that involve derivatives and integrals of non-integer order [16][17][18][19][20]. These new models are more adequate than the previously used integer-order models, because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances [21].…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

Liu

Turner

et al. 2013

Phil. Trans. R. Soc. A.

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Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch–Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch–Torrey equation are still limited. In this paper, the space and time fractional Bloch–Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

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Numerical solutions for fractional reaction–diffusion equations

Cited by 118 publications

References 62 publications

Particle-tracking simulation of fractional diffusion-reaction processes

Particle-tracking simulation of fractional diffusion-reaction processes

A Second Order Finite Difference Approximation for the Fractional Diffusion Equation

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

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