Temporal Fractional Derivatives Research Articles

Second-order approximation scheme combined with [formula omitted]-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

In this article, a second-order approximation scheme combined with an H1-Galerkin mixed finite element (MFE) method for solving nonlinear convection–diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H1-Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second-order weighted and shifted Grünwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L2 and H1-norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used L1-approximation with (2−α)th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis.

Fractional diffusion modeling of heat transfer in porous and fractured media

Fracture–matrix interactions strongly affect anomalous heat transfer in geological sites. This study investigates effects of the interactions between fractures and rock matrix by using the method of multiple interacting continua (MINC). The MINC generates different temperature histories for varied fracture spacings. Two analytical solutions of each porous model and fracture model are used to fit the numerical results for temperature histories due to cold-water injection. The porous model is good agreement with the result for small fracture spacing, while a solution of the fracture model fits the result for large fracture spacing. The MINC yields intermediate behaviors in between a porous medium and a single fracture. A fractional heat transfer equation (fHTE) has been developed to describe anomalous thermal diffusion in a fractured reservoir. The fHTE accounts for heat flux from fracture into matrix by using a temporal fractional derivative. The fHTE can capture numerical results for temperature histories with different fracture spacings. The fracture spacing has correlations to the fHTE best-fit parameters (i.e., the orders of fractional derivatives and the retardation parameters). The fHTE with varying time fractional derivatives can cover descriptions of subdiffusion, Fickian diffusion, and superdiffusion.

Finite volume element method for two-dimensional fractional subdiffusion problems

In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $\alpha \in (0,1)$ in a two-dimensional convex polygonal domain. Optimal error estimates in $L^\infty(L^2)$- norm is shown to hold. Superconvergence result is proved and as a consequence, it is established that quasi-optimal order of convergence in $L^{\infty}(L^{\infty})$ holds. We also consider a fully discrete scheme that employs FV method in space, and a piecewise linear discontinuous Galerkin method to discretize in temporal direction. It is, further, shown that convergence rate is of order $O(h^2+k^{1+\alpha}),$ where $h$ denotes the space discretizing parameter and $k$ represents the temporal discretizing parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics

This paper studies a few nonlinear evolution equations that appear with fractional temporal evolution and fractional spatial derivatives. These are Benjamin-Bona-Mahoney equation, dispersive long wave equation and Nizhnik-Novikov-Veselov equation. The extended Jacobi’s elliptic function expansion method is implemented to obtain soliton and other periodic singular solutions to these equations. In the limiting case, when the modulus of ellipticity approaches zero or unity, these doubly periodic functions approach solitary waves or shock waves or periodic singular solutions emerge.

Meshless analyses for time-fractional heat diffusion in functionally graded materials

The meshless local radial basis function method is applied to solve stationary and transient heat conduction problems in 2-D and 3-D bodies with functionally graded material properties. Time fractional derivative using Caputo definition is considered to describe anomalous diffusion phenomena. For temporal discretization, the Caputo time fractional derivative is approximated within each time interval 〈tk,tk+1〉 by series of derivatives of integer order. The spatial discretization is performed by using the local radial basis collocation method. Numerical analyses are given on square (2D) and cubic (3D) domains to show the influence of the temporal fractional derivative parameter and gradation material parameter on the temperature distribution and temperature evolution in transient heat conduction problem.

Modeling and simulation of the fractional space-time diffusion equation

In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.

Universal character of the fractional space-time electromagnetic waves in dielectric media

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

Transient flow in a linear reservoir for space–time fractional diffusion

Abstract A one-dimensional, fractional-order, transient diffusion equation is constructed to model diffusion in complex geological media. Such a conceptual model permits for the incorporation of a wide range of velocities as fluid particles in high and low permeability paths perform complex motions. The transient diffusion equation is non-local in character with both spatial and temporal fractional derivatives. The pressure distribution is derived in terms of the Laplace transformation and the Mittag–Leffler function. Results are used to deduce expectations in the early-time response of a fractured well producing complex reservoirs such as unconventional shales. The flux law considered here allows for declines in rate that are faster or slower than models based on classical diffusion. A brief survey of the Mittag–Leffler function and its computation is provided. We apply the results derived to obtain solutions for the ‘trilinear’ model that is often used to evaluate horizontal well performance.

Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure

We obtain an analytical solution for the pressure-transient behavior of a vertically hydraulic fractured well in a heterogeneous reservoir. The heterogeneity of the reservoir is modeled by using the concept of fractal geometry. Such reservoirs are called fractal reservoirs. According to the theory of fractional calculus, a temporal fractional derivative is applied to incorporate the memory properties of the fractal reservoir. The effect of different parameters on the computed wellbore pressure is fully investigated by various synthetic examples.

Finite difference schemes for two-dimensional time-space fractional differential equations

In this paper, finite difference schemes for differential equations with both temporal and spatial fractional derivatives are studied. When the order of the time fractional derivative is in , an alternating direction implicit (ADI) scheme with second-order accuracy in both space and time is constructed. For equations with time fractional derivatives of order lying in , a scheme is derived and solved by the generalized minimal residual method. We also propose a preconditioner to improve the efficiency for the implementation of the scheme in this situation.

Fractional thermal diffusion and the heat equation

Abstract Fractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 &lt; β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σx and σt with dimensions of meters and seconds respectively. The fractional derivative of Caputo type is considered and the analytical solutions are given in terms of the Mittag-Leffler function. The generalization of the equations in spacetime exhibit different cases of anomalous behavior and Non-Fourier heat conduction processes. An illustrative example is presented.

Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian.

The absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law. It is now well established that this type of loss behavior can be modeled using fractional derivatives. However, previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives. These operators are non-local in time, which makes them difficult to compute in a memory efficient manner. Here, a fractional Kelvin-Voigt model is derived based on the fractional Laplacian. This is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor. This allows the temporal fractional derivatives in the Kelvin-Voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed. The model is discretized using the Fourier collocation spectral method, which allows the fractional operators to be efficiently computed. The field splitting also allows the use of a k-space corrected finite difference scheme for time integration to minimize numerical dispersion. The absorption and dispersion behavior of the fractional Laplacian model is analyzed for both high and low loss materials. The accuracy and utility of the model is then demonstrated through several numerical experiments, including the transmission of focused ultrasound waves through the skull.

Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration

This paper presents mass-time fractional partial differential equations (fPDEs) formulated in a material coordinate for swelling–shrinking soils, and space-time fPDEs formulated in Cartesian coordinates for non-swelling soils. The fPDEs are capable of incorporating mobile and immobile zones or without immobile zones. As an example of the applications, the solutions of the fPDEs are derived and used to construct equations of infiltration. The new equation of cumulative infiltration into soils with mobile and immobile zones is I(t)=At+Stβ2+β1S1tβ2+S2tβ11/(2λ-1), where A is the final infiltration rate, S is the sorptivity which differs between swelling and non-swelling soils, β2 and β1 are orders of temporal fractional derivatives for mobile and immobile zones, respectively, λ is the order of spatial fractional derivatives, and S2 and S1 are parameters incorporating relative porosities and β2 and β1, respectively. The equation of cumulative infiltration without an immobile zone is I(t)=At+Stβ/(2λ-1), where β is the order of temporal fractional derivatives. Published data are used to demonstrate the use of the new equations and derive the parameters. The transport exponent for soils with mobile and immobile zones is μ=2(β2+β1)/λ, and μ=2β/λ for soils without an immobile zone. The transport exponent is the criteria for defining flow patterns: for μ < 1, the flow process is sub-diffusion as compared to μ = 1 for classic diffusion and μ > 1 for super-diffusion.

Radiating subdispersive fractional optical solitons.

It was recently found [Fujioka et al., Phys. Lett. A 374, 1126 (2010)] that the propagation of solitary waves can be described by a fractional extension of the nonlinear Schrödinger (NLS) equation which involves a temporal fractional derivative (TFD) of order α > 2. In the present paper, we show that there is also another fractional extension of the NLS equation which contains a TFD with α < 2, and in this case, the new equation describes the propagation of radiating solitons. We show that the emission of the radiation (when α < 2) is explained by resonances at various frequencies between the pulses and the linear modes of the system. It is found that the new fractional NLS equation can be derived from a suitable Lagrangian density, and a fractional Noether's theorem can be applied to it, thus predicting the conservation of the Hamiltonian, momentum and energy.

Classical non-Markovian Boltzmann equation

The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 &amp;lt; α &amp;lt; 1 and superdiffusive transport to α &amp;gt; 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.

Analysis of Water Injection in Fractured Reservoirs Using a Fractional-Derivative-Based Mass and Heat Transfer Model

This research proposes a numerical scheme for evaluating the effect of cold-water injection into a geothermal reservoir. A fractional heat transfer equation (fHTE) is derived based on the fractional advection–dispersion equation (fADE) that describes non-Fickian dispersion in a fractured reservoir. Numerical simulations are conducted to examine the applicability of the fADE and the fHTE in interpreting tracer and thermal responses in a fault-related subsidiary structure associated with fractal geometry. A double-peak is exhibited when the surrounding rocks have a constant permeability. On the other hand, the peak in the tracer response gradually decreases when the permeability varies with distance from the fault zone according to a power law, which can be described by the fADE. The temperature decline is more gradual when the permeability of surrounding rocks varies spatially than when they have a constant permeability. The fHTE demonstrates good agreement with the temperature profiles for the different permeabilities of surrounding rocks. The retardation parameters in the fADE and the fHTE increase with the permeability of the surrounding rocks. The orders of the temporal fractional derivatives in the fADE and the fHTE vary with the permeability patterns.

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Abstract Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.

Homotopy analysis method for solving a class of fractional partial differential equations

In this paper, the homotopy analysis method is applied to obtain the solution of fractional partial differential equations with spatial and temporal fractional derivatives in Riesz and Caputo senses, respectively. Some properties of Riesz fractional derivative utilized in obtaining the series solution are proved. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter ℏ on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders α and β on the solution behavior.

Decline-Curve Analysis of Fractured Reservoirs With Fractal Geometry

Summary Evaluation of reservoir parameters through well-test and decline-curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely, but it can be found in the distribution of fractures within a naturally fractured reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected. However, evidence from outcrops, well logging, production-behavior studies, and the dynamic behavior observed in these systems, in general, indicate the above assumptions are not representative of these systems. Fractal theory considers a nonuniform distribution of fractures and the presence of fractures at different scales; thus, it can contribute to explain the behavior of many fractured reservoirs. The objective of this paper is to investigate the production-decline behavior in an NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal-continuity expressions presented in previous studies in the literature and a more recent generalization of these equations, which includes a temporal fractional derivative. The second objective is to present a combined analysis methodology, which uses transient-well-test and bound-ary-dominated-decline production data to characterize an NFR exhibiting fractures, depending on scale. Several analytical solutions for different diffusion equations in fractal systems are presented in Laplace space for both constant-wellbore-pressure and pressure-variable-rate inner-boundary conditions. Both single- and dual-porosity systems are considered. For the case of single-porosity reservoirs, analytical solutions for different diffusion equations in fractal systems are presented. For the dual-porosity case, an approximate analytical solution, which uses a pseudosteady-state matrix-to-fractal fracture-transfer function, is introduced. This solution is compared with a finite-difference solution, and good agreement is found for both rate and cumulative production. Short- and long-time approximations are used to obtain practical procedures in time for determining some fractal parameters. Thus, this paper demonstrates the importance of analyzing both transient and boundary-dominated flow-rate data for a single-well situation to fully characterize an NFR exhibiting fractal geometry. Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation consistently explains the peculiar behavior observed in some real production-decline curves.

Necessary conditions for local and global existence to a reaction-diffusion system with fractional derivatives

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large|x|.