Fractional Derivative Modeling Research Articles

Fractional derivative modeling of double-diffusive free convection with von Neumann stability analysis

ABSTRACT This paper focuses on the problem of fractional time derivative of fluid flow and convective heat and mass transfer from a heated semi-infinite wall immersed. We provided two cases of study, one is free convective heat transfer and the other is a free double-convective heat and mass transfer. The time-derivative terms in the equations of momentum, energy and concentration are assumed to be fractional using the Grunwald-Letnikov (GL) model. A finite difference scheme has been developed for each case of study and followed by a von Neumann stability analysis. Therefore, a stability condition has been derived for each case of study. It was found that the derived stability condition collapses to the traditional one as the derivative order equals one, which reflects an accurate implementation. Also, selected physical results have been presented for the velocity, temperature and concentration profiles as well as variations of Nusselt/Sherwood number and friction coefficient.

Fractional Derivative Modeling on Solute Non-Fickian Transport in a Single Vertical Fracture

Solute transport in a single vertical fracture (SVF) cannot be reliably described by the classical advection-dispersion equation (ADE) model, due to the heterogeneity nature of fracture. This study conducted a group of experiments to investigate chloride ion transport in the SVFs under different rough-walled conditions, and then applied a time fractional advection-dispersion equation (F-ADE) model to offer an accurate description. A comparison between F-ADE model and a classical ADE model in describing experimental data, was also carried out. Results show that the FADE model is better than the ADE model in describing the breakthrough curve and heavy-tail phenomenon of solute transport in the fracture. Especially in the experiments with lower flow rate and higher roughness fracture, the FADE model can offer an accurate description, indicating that it is a promising tool for characterizing solute transport heterogeneous vertical fracture.

Solutions to ramp-hold dynamic oscillation indentation tests for assessing the viscoelasticity of hydrogel by Kelvin-Voigt fractional derivative modeling

Indentation type dynamic mechanical analysis (IT-DMA) is an attractive approach to characterizing viscoelastic properties of soft matter under both quasi-static and dynamic loadings. Frequency-dependent viscoelastic properties are usually characterized by storage and loss moduli whereas extracting inherent viscoelastic model parameters from dynamic loadings remains unsolved. In this study, closed-form solutions to dynamic oscillation loadings were first derived and adapted to a variety of excitations for arbitrary shape indenters. A uniform theoretical framework for ramp-hold-oscillation-relaxation/creep indentation tests are established and Kelvin-Voigt fractional derivative (KVFD) solutions are presented to predict the viscoelastic behavior for both quasi-static loadings (creep, relaxation) and dynamic loadings (sinusoidal displacement/force cycles at 5,10,15,20,30 Hz). Four gelatin-cream hydrogel samples were investigated under ramp-hold-oscillation-relaxation and ramp-hold-oscillation-creep testing conditions. Viscoelastic quantities of elasticity E0, fluidity α, and viscosity τ were extracted by fitting the experimental curves to KVFD predictions. The results show that KVFD models can accurately depict the viscoelastic behavior under both dynamic and quasi-static loadings with high goodness of fit (R2>0.936). To assess the proposed method, comparison of the viscoelastic quantities from different experimental protocols were conducted. Statistical results revealed viscoelastic parameters from different loading are in close agreement. For dynamic loadings, significant differences were found for [E0, α] estimates at different frequencies. For different experimental protocols, estimations were consistent in that E0 showed maximum value on Gel5Cream15 sample, and α was increased with increasing cream percentage. Moreover, viscoelastic quantities extracted from dynamic relaxation were correlated with those of the dynamic creep (r = 0.977, p < 0.001 for E0, and r = 0.969, p < 0.001 for α). Viscoelastic quantities from quasi-static relaxation were highly correlated with those from quasi-static creep (r = 0. 910, p < 0.001 for E0, and r = 0.979, p < 0.001 for α). The complex modulus from KVFD estimations was found to be correlated with that of the DMA measurement (for dynamic relaxation, r = 0.603, p < 0.001 for E′ and r = 0.818, p < 0.001 for E″; for dynamic creep, r = 0.756, p < 0.001 for E′ and r = 0.762, p < 0.001 for E″). Cross validation and assessment by alternative DMA method confirmed that KVFD solutions provided a method for relating viscoelastic measurements at load frequencies from quasi-static to dynamic for soft viscoelastic tissue-like materials.

Fluidity and elasticity form a concise set of viscoelastic biomarkers for breast cancer diagnosis based on Kelvin-Voigt fractional derivative modeling.

Cancer progression involves biomechanical changes within transformed cells and the surrounding extracellular matrix (ECM). The viscoelastic features of fluidity and elasticity that are based on a novel Kelvin-Voigt fractional derivative (KVFD) model were found capable of discriminating normal, benign and malignant breast biopsy tissues on the cellular scale. The improved specificity of KVFD model parameters derives from greater accuracy of fitting the entire approaching force-indentation measurement curve ([Formula: see text] > 0.99) compared with traditional elastic models ([Formula: see text] < 0.86). Moreover, model parameters can be interpreted in terms of histopathological features. First, statistical comparisons reveal there are significant differences (p < 0.001) in elasticity E0, fluidity [Formula: see text], and viscosity [Formula: see text] among healthy, benign, and malignant groups. Malignant breast tissues show low-value, broad-distributions in E0 and with high fluidity [Formula: see text] as compared with healthy and benign tissues. Second, histograms of E0 and [Formula: see text] provide distinctive features by fitting to Gaussian mixture (GM) models. The histograms of E0 and [Formula: see text] are best fit by two kernels GM for malignant tissues, indicating that the cells are soft but with high fluidity and the ECM is stiff but with low fluidity. However, the data suggest one-kernel GM model for benign tissue and a patched uniform distribution for healthy tissue. Third, using fluidity [Formula: see text] as the test statistic, the area under the receiver operator characteristiccurve (AUC) is 0.701 ± 0.012 (p < 0.0001) for control versus malignant and 0.706 ± 0.013 (p < 0.0001) for benign versus malignant group. Variations in tissue fluidity and elasticity offer a concise set of viscoelastic biomarkers that correlate well with histopathological features.

A Fractional Derivative Modeling of Heating and Cooling of LED Luminaires

In the context of energy efficient lighting, we present a mathematical study of the heating and cooling processes of a common type of luminaires, consisting of a single light-emitting diode source in thermal contact with an aluminum passive heat sink. First, we study stationary temperature distributions by addressing the appropriate system of partial differential equations with a commercial finite element solver. Then, we study the temporal evolution of the temperature of the chip and find that it is well approximated with a fractional derivative generalization of Newton’s cooling law. The mathematical results are compared and shown to largely agree with our laboratory measurements.

Fractional derivative heat conduction modeling based on thermal elements combination

This paper proposes a general theoretical framework to establish the time‐space fractional derivative models for non‐Fourier heat conduction. Similar to viscoelastic modeling in mechanics, some new heat conduction elements are proposed instead of mechanical ones. With different combinations of the thermal elements, the time‐space fractional generalizations, such as the Kelvin‐Voigt model, the Maxwell model, and the Zener model, are presented. In addition, it is possible to obtain the fractional derivative heat conduction equations from the corresponding transport equations. While incorporating the existing non‐Fourier heat conduction models into the proposed theoretical framework, some new time‐space fractional derivative models are presented. Finally, the nonlocal models considering the long‐range heat fluxes are investigated and confirmed to be consistent with the proposed framework.

Fractional derivative modeling for axisymmetric consolidation of multilayered cross-anisotropic viscoelastic porous media

This paper presents a fractional derivative modeling for axisymmetric consolidation of multilayered cross-anisotropic viscoelastic porous media. The elastic solution for multilayered porous media is acquired by the extended precise integration method. The flexibility coefficient of the fractional Merchant viscoelastic model is deduced in the Laplace transformed domain, and the viscoelastic solution of the problem is further obtained by the elastic–viscoelastic correspondence principle. The numerical results are compared with those of published literatures to verify the proposed method, and the influences of fractional derivative order, viscoelasticity of solid skeleton and layering of materials on the axisymmetric consolidation of multilayered viscoelastic porous media are investigated by several examples.

Fractional derivative modeling for suspended sediment in unsteady flows

This paper makes an attempt to develop a fractional model for describing the distribution of suspended sediment in unsteady flows and study nonlinear dynamic phenomenon of fluids. This model shows the dynamic process of suspended sediment transport. The continuous-time random walk (CTRW) framework provides reasonable physical meaning for fractional order. By solving the equations with different orders and analyzing the results, we get the changing laws of concentration of suspended sediment and find some interesting phenomenon. The above results prove that fractional derivative can well describe the non-local properties of suspended sediment transport, including the non-local properties of time and space. Thus, the fractional derivative model can be serve as a candidate to describe the distribution of suspended sediment in unsteady flows.

A Survey on Fractional Derivative Modeling of Power-Law Frequency-Dependent Viscous Dissipative and Scattering Attenuation in Acoustic Wave Propagation

This paper aims at presenting a survey of the fractional derivative acoustic wave equations, which have been developed in recent decades to describe the observed frequency-dependent attenuation and scattering of acoustic wave propagating through complex media. The derivation of these models and their underlying elastoviscous constitutive relationships are reviewed, and the successful applications and numerical simulations are also highlighted. The different fractional derivative acoustic wave equations characterizing viscous dissipation are analyzed and compared with each other, along with the connections and differences between these models. These model equations are mainly classified into two categories: temporal and spatial fractional derivative models. The statistical interpretation for the range of power-law indices is presented with the help of Lévy stable distribution. In addition, the fractional derivative biharmonic wave equations governing scattering attenuation are introduced and can be viewed as a generalization of viscous dissipative attenuation models.

Whey protein gelation induced by enzymatic hydrolysis and heat treatment: Comparison of creep and recovery behavior

Whey proteins can form different types of gels depending on the method used for their preparation. The purpose of this study was to characterize and compare gels obtained by heating and enzymatic (Bacillus Licheniformis protease) treatments in terms of gelation kinetic, gel mechanical and microstructural properties, and conformational changes of the protein secondary structure due to protein hydrolysis/denaturation. By comparing heat- and enzyme-induced gelation it was observed that although G′ and G″ profiles looked fairly similar, heat-induced WPI gels exhibited higher G′ and G″ values than enzyme-induced gels. G′ values at 25 °C for heat-induced gels were approximately 3500 Pa, while those for enzyme-induced gels were about 850 Pa. For both gels and during temperature sweep, storage modulus G′ slightly decreased as temperature increased. Frequency sweep in both gels showed that heat-induced gels had slight frequency dependence in comparison with enzyme-induced gels that were practically independent of frequency. These results indicated that although enzyme-induced gels exhibit smaller strength they are significantly more elastic than heat-induced gels. Same behavior was found during creep and recovery test, where the macromolecular network formed by enzyme-induced gels exhibited high flexibility: 82.6% of recovery strain in the first cycle, and 79% in the second one, being more easily stretched than heat-induced WPI gels, which had 49.8% and 40.8% in the first and second cycle, respectively. The fractional derivative modeling approach applied to the creep data showed that enzyme-induced gels were more elastic than thermally generated gels. These characteristics were also observed by microcopy analysis, showing that heat-induced gels had smaller pore sizes and a denser network with more interaction zones than enzyme-induced gels. These observations may explain the larger strength and the lower elasticity of heat-induced gels which are likely due to less specific interactions among denatured whey proteins.

Anomalous diffusion: fractional derivative equation models and applications in environmental flows

Heterogeneity embedded in natural media and flow field challenge the application of Ficks 1st Law in anomalous diffusion well documented in many disciplines. Anomalous diffusion is one of the major topics in theoretical physics and statistical mechanics, and it is also the fundamental physical process with good potential application in environmental and hydrologic sciences and engineering. As a novel modeling tool in mathematics and physics, the fractional-order derivative diffusion equation models characterize anomalous diffusion with history-dependence and spatial non-locality, accurately describe the tailing in breakthrough curves of solute transport. We summarize the recent progresses and discuss the key challenges of fractional derivative diffusion equation models including the existed research and current development, fractional derivative modeling, numerical algorithms, and related applications in the field of environmental fluid mechanics. Here also made some preliminary discussions on issues of fractional derivative diffusion equation model, such as statistical description, model parameter determination and dimensional analysis, which may contribute to the further study of anomalous diffusion.

Fractional derivative anomalous diffusion equation modeling prime number distribution

Abstract This study suggests that the power law decay of prime number distribution can be considered a sub-diffusion process, one of typical anomalous diffusions, and could be described by the fractional derivative equation model, whose solution is the statistical density function of Mittag-Leffler distribution. It is observed that the Mittag-Leffler distribution of the fractional derivative diffusion equation agrees well with the prime number distribution and performs far better than the prime number theory. Compared with the Riemann’s method, the fractional diffusion model is less accurate but has clear physical significance and appears more stable. We also find that the Shannon entropies of the Riemann’s description and the fractional diffusion models are both very close to the original entropy of prime numbers. The proposed model appears an attractive physical description of the power law decay of prime number distribution and opens a new methodology in this field.

A Matlab toolbox for positive fractional time derivative modeling of arbitrarily frequency-dependent viscosity

This paper introduces a new Matlab toolbox for the numerical solution of power law frequency-dependent damped vibration and dissipative wave equations involving the positive fractional time derivative. The classical integer-order derivative models have long encountered huge difficulty in describing such complex thermoviscous behaviors, particularly if a broadband excitation is involved. Recent decades have seen a growing interest in the fractional derivative modeling of such anomalous viscosity. Among various time fractional derivative models, the positive fractional derivative model has clear advantages to hold causality of wave problems thanks to its positive definition. However, the numerical methods and software available today are mostly for the standard fractional derivative equations. This paper will present a finite difference method for positive fractional vibration and acoustic equations, and then focus on a new Matlab toolbox of its implementations, which is very easy to use with a friendly graphical user interface. The toolbox is freely available as an open source software and will help promote the application of the positive fractional derivative models to diverse dynamic problems where the viscosity plays an essential role.

Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity

The viscoelastic response of hydropolymers, which include glandular breast tissues, may be accurately characterized for some applications with as few as 3 rheological parameters by applying the Kelvin-Voigt fractional derivative (KVFD) modeling approach. We describe a technique for ultrasonic imaging of KVFD parameters in media undergoing unconfined, quasi-static, uniaxial compression. We analyze the KVFD parameter values in simulated and experimental echo data acquired from phantoms and show that the KVFD parameters may concisely characterize the viscoelastic properties of hydropolymers. We then interpret the KVFD parameter values for normal and cancerous breast tissues and hypothesize that this modeling approach may ultimately be applied to tumor differentiation.

An Intuitive Study of Fractional Derivative Modeling and Fractional Quantum in Soft Matter

Soft matter (such as biomaterials, polymers, sediments, oil, emulsions, etc) has become an important bridge between theoretical physics and a range of applied disciplines. Its fundamental physical mechanism, however, is largely obscure. This study makes a first attempt to connect the fractional Schrodinger equation and soft matter physics within a consistent framework from empirical power scaling to phenomenological kinetics and macromechanics to mesoscopic quantum mechanics. The new perspectives of this study are the fractional quantum relationships in soft matter, which show that Lévy statistics and fractional Brownian motion are essentially related to momentum and energy, respectively. Fractional quantum theory has been conjectured as underlying fractal mesostructures and many-body interactions of macromolecules in soft matter, and is experimentally testable.

New Methods for Identifying Rheological Parameter for Fractional Derivative Modeling of Viscoelastic Behavior

The modelling of viscoelastic behaviour by fractional derivatives, the graphical method called the TIBI Diagram that is developed in this article, is a new approach, completely different from the classical methods, which are somewhat inspired by the BODE diagram technique. The TIBI technique permits the reduction to a strict minimum of two break frequencies, which is the minimum number necessary, while the other graphical methods of identification require up to ten parameters to determine linear viscoelastic behaviour. The second method developed in this paper is a numerical approach which is very flexible with regard to the experimental complex modulus permitting the determination of rheological characteristics. One needs a group of frequencies limited only to the transition region of the material. These two original methods are combined in certain cases to obtain semi-graphical techniques. Experimental and simulation results are presented for each method with comparative analysis.