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This paper presents a state-of-art survey of the research on fractional-order (FO) modeling with parameter identification, and FO estimation methods for state of charge (SOC), state of health (SOH), and remaining usage life (RUL) of lithium-ion batteries (LIBs) mainly in recent five years. FO electrochemical models and six different types of FO equivalent circuit models (ECMs) are introduced in detail. Then, the corresponding tuning algorithm for parameters of these FO models are also provided in brief. Moreover, FO estimation methods for SOC are listed and analyzed, mainly including FO observers, and FO Kalman filters (FO-KFs). SOH and RUL estimation is another vital aspect for LIBs ageing and degradation monitoring, thus FO estimation methods proposed in recent research within five years are all listed. Finally, some suggestions that may be helpful for further research are proposed in conclusion.

Bedload particles in the armoring layer may experience a multi-scale effect and multiple mass transfer rates between mobile and immobile domains. Anomalous transport behaviors and retarded space evolution plume cannot be described by the normal diffusion equation. In this paper, we apply the continuous time random walk model with different distributions of waiting times to capture bedload transport behavior under different conditions. Experimental data indicate that fluctuations of diffusive rates for bedload transport can be captured by the truncated power law (TPL). The retarded plume evolution can be well characterized by an exponential distribution of waiting times and advection-diffusion equation with a retarded kernel. The heavy-tailed snapshots of bedload transport are interpreted in terms of mobile and immobile states.

A new approach to modeling the geometric structure of real materials based on a new type of functions – porous functions – is introduced. The general concept, basic operations, operations of calculus, visualization and interpretations, and potential applications of porous functions are outlined.

This paper presents a time-space Hausdorff derivative model for depicting solute transport in aquifers or water flow in heterogeneous porous media. In this model, the time and space Hausdorff derivatives are defined on non-Euclidean fractal metrics with power law scaling transform which, respectively, connect the temporal and spatial complexity during transport. The Hausdorff derivative model can be transformed to an advection-dispersion equation with time- and space-dependent dispersion and convection coefficients. This model is a fractal partial differential equation (PDE) defined on a fractal space and differs from the fractional PDE which is derived for non-local transport of particles on a non-fractal Euclidean space. As an example of applications of this model, an explicit solution with a constant diffusion coefficient and flow velocity subject to an instantaneous source is derived and fitted to the breakthrough curves of tritium as a tracer in porous media. These results are compared with those of a scale-dependent dispersion model and a time-scale dependent dispersion model. Overall, it is found that the fractal PDE based on the Hausdorff derivatives better captures the early arrival and heavy tail in the scaled breakthrough curves for variable transport distances. The estimated parameters in the fractal Hausrdorff model represent clear mechanisms such as linear relationships between the orders of Hausdorff derivatives and the transport distance. The mathematical formulation is applicable to both solute transport and water flow in porous media.

The main goal of this article is to establish a new 4th-order numerical differential formula to approximate Riesz derivatives which is obtained by means of a newly established generating function. Then the derived formula is used to solve the Riesz space fractional advection-dispersion equation. Meanwhile, by the energy method, it is proved that the difference scheme is unconditionally stable and convergent with order 𝓞( τ 2 + h 4 ). Finally, several numerical examples are given to show that the numerical convergence orders of the numerical differential formulas and the finite difference scheme are in line with the theoretical analysis.

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.

Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Laplacian derivative, but only the latter satisfies this relation. There also exists several seemingly different expressions for the phase velocity for these equations and it is shown here that they are approximately equivalent. Causality of the Chen–Holm equation has also been a topic of some discussion and it is found that despite the lack of agreement with the Kramers–Kronig relation, it is still causal.

Fractional-derivative models (FDMs) are promising tools for characterizing non-Fickian transport in natural geological media. Hydrologic applications of FDMs, however, have been limited in the last two decades, due to the lack of feasible models and solvers to quantify multi-dimensional anomalous diffusion for pollutants in bounded aquifers. This study develops and applies FDM tools to capture vector fractional dispersion for both conservative and reactive pollutants in fractional Brownian motion (fBm) random fields with bounded domains. A d -dimensional anisotropic fBm field for hydraulic conductivity ( K ) is first generated numerically. A particle-tracking based, fully Lagrangian solver is then developed to approximate particle dynamics in the fBm K fields under various boundary conditions, where the governing equation is the vector FDM subordination to regional flow. Numerical experiments show that the Lagrangian solver can combine nonlocal anomalous transport and local aquifer properties to quantify pollutant transport in bounded aquifers. Application analyses further reveal that the K correlation can significantly enhance the spreading of conservative pollutant particles, and increase the reaction rate by enhancing the mobility and mixing of reactant particles undergoing bimolecular reactions. Extension of the Lagrangian solver is also discussed, including modeling transient flow, generalizing boundary conditions, and capturing complex chemical reactions. This study therefore provides the hydrologic community an efficient Lagrangian solver to model reactive anomalous transport in bounded anisotropic aquifers with any dimension, size, and boundary conditions.

The Laplace transform is important for exact solutions of linear differential equations and frequency response analysis methods. In comparison with the continuous–time systems, less results can be available for fractional difference equations. This study provides some fundamental results of two kinds of fractional difference equations by use of the Laplace transform. Some discrete Mittag–Leffler functions are defined and their Laplace transforms are given. Furthermore, a class of variable–order and short memory linear fractional difference equations are proposed and the exact solutions are obtained.

A novel linear time-invariant (LTI) system model with fractional degree generalized frequency variables (FDGFVs) is proposed in this paper. This model can provide a unified form for many complex systems, including fractional-order systems, distributed-order systems, multi-agent systems and so on. This study mainly investigates the stability and robust stability problems of LTI systems with FDGFVs. By characterizing the relationship between generalized frequency variable and system matrix, a necessary and sufficient stability condition is firstly presented for such systems. Then for LTI systems with uncertain FDGFVs, we present a robust stability method in virtue of zero exclusion principle. Finally, the effectiveness of the method proposed in this paper is demonstrated by analyzing the robust stability of gene regulatory networks.

The first fractional model for Reynolds stresses in wall-bounded turbulent flows was proposed by Wen Chen [2]. Here, we extend this formulation by allowing the fractional order α ( y ) of the model to vary with the distance from the wall ( y ) for turbulent Couette flow. Using available direct numerical simulation (DNS) data, we formulate an inverse problem for α ( y ) and design a physics-informed neural network (PINN) to obtain the fractional order. Surprisingly, we found a universal scaling law for α ( y + ), where y + is the non-dimensional distance from the wall in wall units. Therefore, we obtain a variable-order fractional model that can be used at any Reynolds number to predict the mean velocity profile and Reynolds stresses with accuracy better than 1%.