Fractional Differential Model Research Articles

Influence of the finiteness of particle velocity on the energy spectrum of cosmic rays in an anomalous diffusion model with Lévy flights

An estimate of the influence the finiteness of particle velocity has on the results of a fractional differential (anomalous) model of cosmic ray propagation in the Galaxy with Levy flights developed by the authors is considered. The results from Monte Carlo simulations of particle diffusion in random walk models with finite and infinite velocities are presented. It is shown that considering particle velocity finiteness has almost no effect on the cosmic ray energy spectrum obtained for E > 1 GeV in the anomalous diffusion model with Levy flights for nearby young sources.

A fractional model for heat transfer in Mongolian yurt

A yurt is a portable tent-like dwelling structure favored by Mongolian nomads for more than three millennia and it can be favorably used even at a harsh environment as low as ?50 degrees. The paper concludes that the multi-layer structure of the felt cover is the key for weatherproofing. A fractional differential model with He?s fractional derivative is established to find an optimal thickness of the fractal hierarchy of the felt cover. A better understanding of the yurt mechanism could help the further design of yurt-like space suits and other protective clothing for extreme cold region.

Mathematical modelling of fractional order circuit elements and bioimpedance applications

In this work a classical derivation of fractional order circuits models is presented. Generalised constitutive equations in terms of fractional Riemann–Liouville derivatives are introduced in the Maxwell’s equations for each circuit element. Next the Kirchhoff voltage law is applied in a RCL circuit configuration. It is shown that from basic properties of Fractional Calculus, a fractional differential equation model with Caputo derivatives is obtained. Thus standard initial conditions apply. Finally, models for bioimpedance are revisited.

Fractional order integration and fuzzy logic based filter for denoising of echocardiographic image

BackgroundUltrasound is widely used for imaging due to its cost effectiveness and safety feature. However, ultrasound images are inherently corrupted with speckle noise which severely affects the quality of these images and create difficulty for physicians in diagnosis. To get maximum benefit from ultrasound imaging, image denoising is an essential requirement. MethodTo perform image denoising, a two stage methodology using fuzzy weighted mean and fractional integration filter has been proposed in this research work. In stage-1, image pixels are processed by applying a 3 × 3 window around each pixel and fuzzy logic is used to assign weights to the pixels in each window, replacing central pixel of the window with weighted mean of all neighboring pixels present in the same window. Noise suppression is achieved by assigning weights to the pixels while preserving edges and other important features of an image. In stage-2, the resultant image is further improved by fractional order integration filter. ResultsEffectiveness of the proposed methodology has been analyzed for standard test images artificially corrupted with speckle noise and real ultrasound B-mode images. Results of the proposed technique have been compared with different state-of-the-art techniques including Lsmv, Wiener, Geometric filter, Bilateral, Non-local means, Wavelet, Perona et al., Total variation (TV), Global Adaptive Fractional Integral Algorithm (GAFIA) and Improved Fractional Order Differential (IFD) model. Comparison has been done on quantitative and qualitative basis. For quantitative analysis different metrics like Peak Signal to Noise Ratio (PSNR), Speckle Suppression Index (SSI), Structural Similarity (SSIM), Edge Preservation Index (β) and Correlation Coefficient (ρ) have been used. Simulations have been done using Matlab. Simulation results of artificially corrupted standard test images and two real Echocardiographic images reveal that the proposed method outperforms existing image denoising techniques reported in the literature. ConclusionThe proposed method for denoising of Echocardiographic images is effective in noise suppression/removal. It not only removes noise from an image but also preserves edges and other important structure.

Twin iterative solutions for a fractional differential turbulent flow model

We investigate the existence of twin iterative solutions for a fractional p-Laplacian equation with nonlocal boundary conditions. Using the monotone iterative technique, we establish a new existence result on the maximal and minimal solutions under suitable nonlinear growth conditions. We also consider some interesting particular cases and give an example to illustrate our main results.

Research on viscoelastic behavior and rheological constitutive parameters of metallic glasses based on fractional-differential rheological model

Metallic glasses offer novel physical, chemical and mechanical properties and have promising potential applications. Recently, exploring the structure and deformation mechanism of metallic glasses according to the rheological mechanical behavior in the nominal elastic region has been the object of intensive research. Physically the mechanical analogues of fractional elements can be represented by self-similarity spring-dashpot fractal networks. In light of the fractal distribution features of the structural heterogeneities, a fractional differential rheological model is introduced to explore the viscoelastic a behavior of metallic glasses in this paper. To investigate the viscoelastic deformation mechanism, carefully designed nanoindentation tests at ambient temperature are proposed in this study. Three kinds of metallic glasses with different Poisson's ratio and glass transition temperature, which have the chemical compositions of Pd40Cu30Ni10P20, Zr48Cu34Pd2Al8Ag8, and (Fe0.432Co0.288B0.192Si0.048Nb0.04) 96Cr4 are selected as the model materials. Experimental and theoretical results clearly indicate that in the nominal elastic region, these metallic glasses exhibit linear viscoelasticity, implying a loading rate-dependent character. Based on the fractional calculus and Riemann-Liouville definition, experimental results are analyzed by the fractional-differential and integer order rheology models respectively. According to the stability of the fitting parameters, here we show that the fractional-differential Kelvin model, which consists of a spring and a fractional dashpot element, can fit the experimental viscoelastic deformation data of the investigated metallic glasses better than that with integer order rheological model. The extracted elastic modulis E1 of the spring in the fractional-differential Kelvin model are comparable to those of samples measured by traditional methods. Such a similarity can be well explained by the mechanical analogue of fractal model proposed for describing the distribution features of the structural heterogeneities in metallic glasses. The rheological parameters obtained here including viscosity index A and fractional order are capable of reflecting the rheological features and the flowing tendency of the above-mentioned metallic glasses. It is found that there exists a clear relationship between the rheological parameters and the reduced glass transition temperature as well as Poisson's ratio, which is helpful for understanding the correlation between plasticity and Poisson's ratio from the micro-structural point of view. The current work provides a rheological model-structure-property relation that may be applicable to a wide range of glassy materials.

Fractional differential and variational method for image fusion and super-resolution

This paper introduces a novel fractional differential and variational model that includes the terms of fusion and super-resolution, edge enhancement and noise suppression. In image fusion and super-resolution term, the structure tensor is employed to describe the geometry of all the input images. According to the fact that the fused image and the source inputs should have the same or similar structure tensor, the energy functional of the image fusion and super-resolution is established combining with the down-sampling operator. For edge enhancement, the bidirectional diffusion term is incorporated into the image fusion and super-resolution model to enhance the visualization of the fused image. In the noise suppression term, a new variational model is developed based on the fractional differential and fractional total variation. Thanks to the above three terms, the proposed model can realize the image fusion, super-resolution, and the edge information enhancement simultaneously. To search for the optimal solution, a gradient descent iteration scheme derived from the Euler–Lagrange equation of the proposed model is employed. The numerical results indicate that the proposed method is feasible and effective.

Unified fluid flow model for pressure transient analysis in naturally fractured media

Naturally fractured reservoirs present special challenges for flow modeling with regards to their internal geometrical structure. The shape and distribution of matrix porous blocks and the geometry of fractures play key roles in the formulation of transient interporosity flow models. Although these models have been formulated for several typical geometries of the fracture networks, they appeared to be very dissimilar for different shapes of matrix blocks, and their analysis presents many technical challenges. The aim of this paper is to derive and analyze a unified approach to transient interporosity flow models for slightly compressible fluids that can be used for any matrix geometry and fracture network. A unified fractional differential transient interporosity flow model is derived using asymptotic analysis for singularly perturbed problems with small parameters arising from the assumption of a much smaller permeability of the matrix blocks compared to that of the fractures. This methodology allowed us to unify existing transient interporosity flow models formulated for different shapes of matrix blocks including bounded matrix blocks, unbounded matrix cylinders with any orthogonal crossection, and matrix slabs. The model is formulated using a fractional order diffusion equation for fluid pressure that involves Caputo derivative of order 1/2 with respect to time. Analysis of the unified fractional derivative model revealed that the surface area-to-volume ratio is the key parameter in the description of the flow through naturally fractured media. Expressions of this parameter are presented for matrix blocks of the same geometrical shape as well as combinations of different shapes with constant and random sizes. Numerical comparisons between the predictions of the unified model and those obtained from existing transient interporosity ones for matrix blocks in the form of slabs, spheres and cylinders are presented for linear, radial and spherical flow types for both fixed boundary pressure and rate.

Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection

Fractional order derivative is nonlocal which exhibits a long time memory behavior. With advantage of these, fractional order dynamic system models are more accurate than integer order ones in understanding the dynamic behavior of bioprocesses such as HIV infection. In this paper, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HIV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.

Stability analysis for HIV infection of CD4+ T-cells by a fractional differential time-delay model with cure rate

In this paper, a fractional differential model of HIV infection of CD4+ T-cells is investigated. We shall consider this model, which includes full logistic growth terms of both healthy and infected CD4+ T-cells, time delay items, and cure rate items. A more appropriate method is given to ensure that both equilibria are asymptotically stable for under some conditions. Furthermore, the dynamic behaviors of the fractional HIV models are described by applying an Amads-type predictor-corrector method algorithm.

Unified fractional differential approach for transient interporosity flow in naturally fractured media

A unified approach to modeling flows of slightly compressible fluids through naturally fractured media is presented. The unified fractional differential model is derived by combining the flow at micro scale for matrix blocks and macro scale for fractures, using the transient interporosity flow behavior at the interface between matrix blocks and fractures. The derived model is able to unify existing transient interporosity flow models formulated for different shapes of matrix blocks in any medium dimensions. The model is formulated in the form of a fractional order partial differential equation that involves Caputo derivative of order 1/2 with respect to time. Explicit solutions for the unified model are derived for different axisymmetrical spatial domains using Hankel or Hankel–Weber finite or infinite transforms. Comparisons between the predictions of the unified model and those obtained from existing transient interporosity flow models for matrix blocks in the form of slabs, spheres and cylinders are presented. It is shown that the unified fractional derivative model leads to solutions that are very close to those of transient interporosity flow models for fracture-dominant and transitional fracture-to-matrix dominant flow regimes. An analysis of the results of the unified model reveals that the pressure varies linearly with the logarithm of time for different flow regimes, with half slope for the transitional fracture-to-matrix dominant flow regime vs. the fracture and matrix dominant flow regimes. In addition, a new re-scaling that involves the characteristic length in the form of matrix block volume to surface area ratio is derived for the transient interporosity flow models for matrix blocks of different shapes. It is shown that the re-scaled transient interporosity flow models are governed by two dimensionless parameters Θ and Λ compared to only one dimensionless parameter Θ for the unified model. It is shown that the solutions of the transient interporosity flow models for different shapes of matrix blocks are almost identical for the re-scaled variables. Furthermore, the driving parameters for solution behavior are identified based on asymptotic approximations for different flow regimes. It is found that the matrix diffusion and the matrix area-to-volume ratio affect the solution behavior only for the transitional fracture-to-matrix dominant flow regime, that the capacitance ratio affects the solution behavior only for transitional and matrix dominant flow regimes and that the fracture diffusion is involved in all three flow regimes. Similar identification of the driving parameters is also presented in the re-scaled case.

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress intensity factors (DSIFs) for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractionalwave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation (ODE). The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional differential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.

Numerical Analysis for a Fractional Differential Time-Delay Model of HIV Infection of CD4+T-Cell Proliferation under Antiretroviral Therapy

We study a fractional differential model of HIV infection ofCD4+T-cell, in which theCD4+T-cell proliferation plays an important role in HIV infection under antiretroviral therapy. An appropriate method is given to ensure that both the equilibria are asymptotically stable forτ≥0. We calculate the basic reproduction numberR0, the IFEE0, two IPEsE1*andE2*, and so on, and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells.

Numerical study of differential equation governing speech gestures with caputo derivative

This paper presents approximate solutions for the fractional differential dynamical model of speech gestures. The fractional derivatives are described in the Caputo sense. Using the ADM method, we obtain new approximate results when the equation parameters are changed. The behaviour of Adomian solutions are shown graphically for some illustrative examples.

Numerical Solution of Fractional Diffusion Equation Model for Freezing in Finite Media

Phase change problems play very important role in engineering sciences including casting of nuclear waste materials, vivo freezing of biological tissues, solar collectors and so forth. In present paper, we propose fractional diffusion equation model for alloy solidification. A transient heat transfer analysis is carried out to study the anomalous diffusion. Finite difference method is used to solve the fractional differential equation model. The temperature profiles, the motion of interface, and interface velocity have been evaluated for space fractional diffusion equation.

Cosmic ray anisotropy in fractional differential models of anomalous diffusion

The problem of galactic cosmic ray anisotropy is considered in two versions of the fractional differential model for anomalous diffusion. The simplest problem of cosmic ray propagation from a point instantaneous source in an unbounded medium is used as an example to show that the transition from the standard diffusion model to the Lagutin-Uchaikin fractional differential model (with characteristic exponent α = 3/5 and a finite velocity of free particle motion), which gives rise to a knee in the energy spectrum at 106 GeV, increases the anisotropy coefficient only by 20%, while the anisotropy coefficient in the Lagutin-Tyumentsev model (with exponents α = 0.3 and β = 0.8, a long stay of particles in traps, and an infinite velocity of their jumps) is close to one. This is because the parameters of the Lagutin-Tyumentsev model have been chosen improperly.

Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

One Generalization of the Fractional Differential Geoinformation Model of Research of Locally-nonequilibrium Geomigration Processes

A generalized fractional differential mathematical model of studying the dynamics of the locally-nonequilibrium in time geomigration processes is considered. The nonlinear boundary value problem, corresponding to this model, is formulated, the technique of its numerical solution is developed and the results of numerical experiments are given.

Some features of quantum statistical systems with an energy spectrum of the fractional-power type

We derive and investigate the equation of state for a system with a fractional-power spectrum and also derive an expression for the density of states in such systems. We consider a fractional-differential model of electron-phonon interaction.

Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay

In this paper, we introduce fractional-order derivatives into a model of HIV infection of CD4+ T-cells with time delay. We deal with the stability of both the viral free equilibrium and the infected equilibrium. Criteria are given to ensure that both the equilibria are asymptotically stable for all delay under some conditions. Numerical simulations are carried out to illustrate the results.