The Contents is available at: https://www.degruyter.com/view/product/506187#?tdsourcetag=s_pctim_aiomsg

Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations

Tinggang Zhao, Zhiping Mao, George Em Karniadakis

A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

Younes Shekari, Ali Tayebi, Mohammad Hossein Heydari

Fractional Gray–Scott model: Well-posedness, discretization, and simulations

Tingting Wang, Fangying Song, Hong Wang, George Em Karniadakis

High-order central difference scheme for Caputo fractional derivative

Yuping Ying, Yanping Lian, Shaoqiang Tang, Wing Kam Liu

A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

Yanping Lian, Yuping Ying, Shaoqiang Tang, Stephen Lin, Wing Kam Liu

Fractional-order uniaxial visco-elasto-plastic models for structural analysis

J. L. Suzuki, M. Zayernouri, M. L. Bittencourt, G. E. Karniadakis

Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview

Bangti Jin, Raytcho Lazarov, Zhi Zhou

Adaptive finite element method for fractional differential equations using hierarchical matrices

Xuan Zhao, Xiaozhe Hu, Wei Cai, George Em Karniadakis

A tunable finite difference method for fractional differential equations with non-smooth solutions

Xuejuan Chen, Fanhai Zeng, George Em Karniadakis

A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations

Fangying Song, Chuanju Xu, George Em Karniadakis

Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver

Mark Ainsworth, Christian Glusa

Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

Fanhai Zeng, Zhongqiang Zhang, George Em Karniadakis

A unified Petrov–Galerkin spectral method for fractional PDEs

Mohsen Zayernouri, Mark Ainsworth, George Em Karniadakis

[Back]

https://www.sciencedirect.com/science/article/pii/S0022509619300766/pdfft?md5=a73022abf527d972dbc08583fd8e63bc&pid=1-s2.0-S0022509619300766-main.pdf

In this work, a physical connection between the fractional time derivative and fractal geometry of fractal media is developed and applied to viscoelasticity and thermal diffusion in elastomers. Integral to this formulation is the application of both the fractal dimension and the spectral dimension which characterizes diffusion in fractal media. The methodology extends the generalized molecular theory of Rouse and Zimm where generalized Gaussian structures (GGSs) replace the Rouse matrix with the generalized Gaussian Rouse matrix (GRM). Importantly, the Zimm model is extended to fractal media where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. Through the use of thermodynamic laws in fractal media, we derive the linear fractional model of viscoelasticity based on both spectral and fractal dimensions. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is a rate dependent material property that is strongly correlated with fractal and spectral dimensions in fractal media. To validate the correlation between fractional rates and fractal material structure, we measure the viscoelasticity and thermal diffusion of two different dielectric elastomers: Very High Bond (VHB) 4910 and VHB 4949. Using Bayesian uncertainty quantification (UQ) based on uniaxial stress–strain measurements, the fractional order of the derivative in the linear fractional model of viscoelasticity is quantified. Two dimensional fractal dimensions are also independently quantified using the box counting method. Lastly, the diffusion equation in fractal media is inferred from experiments using Bayesian UQ to quantify the spectral dimension by heating the polymer locally with a laser beam and quantifying thermal diffusion. Comparing theory to experiments, a strong correlation is found between the viscoelastic fractional order obtained from stress–strain measurements in comparisons with independent predictions of fractional viscoelasticity based on the fractal structure and fractional thermal diffusion rates.


　

Many theoretical and experimental results show that anomalous diffusion/dispersion occurs in porous media. To exactly solve anomalously fast dispersion, we introduce a structural derivative diffusion model to present superfast diffusion via a logarithmic structural function in space. The fundamental solution of the diffusion model is a form of log-normal distribution, and the corresponding analytical mean squared displacement grows like et/β2, 0<β<1. Compared with the existing models, the proposed model is more effective and accurate in fitting experimental data.