Spatial Fractional Derivatives Research Articles

Effect of adsorption, radioactive decay and fractal structure of matrix on solute transport in fracture.

Reservoir contamination by various contaminants including radioactive elements is an actual environmental problem for all developed countries. Analysis of mass transport in a complex environment shows that the conventional diffusion equation based on Fick's Law fails to model the anomalous character of the diffusive mass transport observed in the field and laboratory experiments. These complex processes can be modelled by non-local advection-diffusion equations with temporal and spatial fractional derivatives. In the present paper, fractional differential equations are used for modelling the transport of radioactive materials in a fracture surrounded by the porous matrix of fractal structure. A new form of fractional differential equation for modelling migration of the radioactive contaminant in the fracture is derived and justified. Solutions of particular boundary value problems for this equation were found by application of the Laplace transform. Through the use of fractional derivatives, the model accounts for contaminant exchange between fracture and surrounding porous matrix of fractal geometry. For the case of an arbitrary time-dependent source of radioactive contamination located at the inlet of the fracture, the exact solutions for solute concentration in the fracture and surrounding porous medium are obtained. Using the concept of a short memory, an approximate solution of the problem of radioactive contaminant transport along the fracture surrounded by the fractal type porous medium is also obtained and compared with the exact solution. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

The impact of spatial and temporal variability of anomalous diffusion flow mechanisms on reservoir performance in structurally complex porous media

The main objective of this paper is understanding the phenomenal anomalous diffusion flow mechanisms in unconventional fractured porous media. The targets are hydraulically fractured unconventional reservoirs characterized by porous media with scomplex structures. These media are consisted of a matrix and naturally induces fractures embedded in the matrix as well as hydraulic fractures.Several analytical models for pressure drop and decline rate as wells productivity index in ultralow permeability reservoirs are presented in this study for the two inner wellbore conditions. Numerical solution is also presented in this study for pressure behavior using a linearized implicit finite difference method. The analytical models are developed from trilinear flow models presented in the literature with a consideration given to the temporal and spatial fractional pressure derivative for the anomalous diffusion flow that could be the dominant flow mechanism in the stimulated reservoir volume between hydraulic fractures. Mittag-Leffler functions are used for solving fractional derivatives of pressure and flow rate considering that temporal and spatial fractional exponents are less than one. Analytically, two solutions are developed in this study for the two inner wellbore conditions. The first represents the transient state condition that controls fluid flow in unconventional reservoirs for a very long production time. The second is the solution of pseudo-steady state condition that might be observed after transient state flow. While numerically, the temporal and spatial domains are discretized into several time steps and block-centered grids respectively.The study has pointed out: 1) Temporal and spatial diffusion flows have a significant impact on pressure drop, flow rate, and productivity index. 2) Wellbore pressure drop for constant Sandface flow rate declines rapidly as the temporal diffusion flow mechanism is the dominant flow pattern in the porous media. 3) Wellbore pressure drop for constant Sandface flow rate slightly increases during transient state flow as the spatial diffusion flow mechanisms increase and rapidly increases during pseudo-steady state flow. 4) Productivity index of diffusion flow is higher than the index of normal diffusion flow during transient and pseudo-steady state conditions. 5) The linear flow regime is most affected by anomalous diffusing flow and can be used to characterize the type of diffusion flow.

Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order $${\mathcal {O}}(\delta \tau ^{2})$$ is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.

A regime switching fractional Black–Scholes model and European option pricing

In this paper, we investigate the European option pricing problem under a regime switching FMLS (finite moment log-stable) model. This model is not only able to capture the main characteristics of asset returns, it also incorporates the effect of regime switching being consistent with market observations. However, option prices under this model are governed by a coupled FPDE (fractional partial differential equation) system, and the difficulty in seeking for analytical solution arises from the combination of the coupled system and the spatial-fractional derivative. To deal with this difficulty, we develop a two-step solution procedure; we firstly assume that the future information of the Markov chain is known, and we derive the conditional option price by analytically solving a time dependent FPDE, based on which an exact and explicit pricing formula for the unconditioned price is successfully worked out by using the Fourier cosine series expansion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice.

时空分数阶的广义b-方程组的精确解

时空分数阶的广义b-方程组的精确解

Dispersion analysis for wave equations with fractional Laplacian loss operators

Abstract 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.

Boundary layer flows of viscoelastic fluids over a non-uniform permeable surface

This study investigates viscoelastic fluid, which possesses both viscous and elastics properties, by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance. The spatial-fractional derivative used in the momentum conservation equation is the Riemann–Liouville type derivative. Further, we use non-uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi-infinite permeable flat surface. Owing to the fractional derivative model and non-uniform boundary conditions, this problem is complex. Thus, a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized. The accuracy, convergence, and stability of the numerical method are presented. It is shown that non-uniform mass transfer through a permeable surface considerably affects the velocity boundary layer. By illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface, this paper predicts the velocity distributions with varying permeable surface, and also provides the possibility of changing the velocity fields by altering the permeable sheet. The results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering.

An Avant-Garde Handling of Temporal-Spatial Fractional Physical Models

Abstract In the present study, we dilate the differential transform scheme to develop a reliable scheme for studying analytically the mutual impact of temporal and spatial fractional derivatives in Caputo’s sense. We also provide a mathematical framework for the transformed equations of some fundamental functional forms in fractal 2-dimensional space. To demonstrate the effectiveness of our proposed scheme, we first provide an elegant scheme to estimate the (mixed-higher) Caputo-fractional derivatives, and then we give an analytical treatment for several (non)linear physical case studies in fractal 2-dimensional space. The study concluded that the proposed scheme is very efficacious and convenient in extracting solutions for wide physical applications endowed with two different memory parameters as well as in approximating fractional derivatives.

Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order and the fractional spatial derivative (fractional Laplacian) of order . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension if and additionally for in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders and of the time and spatial derivatives and on the space dimension n and is given by the expression , t being the time variable. Even if it is an increasing function in , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition , i.e., only the slow and the conventional diffusion can be described by this equation.

Dynamics of Duffing-Holmes oscillator with fractional order nonlinearity

In this work, the dynamics of Duffing-Holmes oscillator with fractional order nonlinearity is explored. Basically, a fractional spatial derivative is introduced to the cubic term, and the order of the derivative α is varied between zero and two. The evolution of the dynamics of the system from nonlinear behavior to linear behavior is investigated using multiple tools such as phase portraits, Poincare maps, and bifurcation diagrams. We have demonstrated that as α increases the system can alternate between chaotic and periodic states depending on the parameters setting. However, the overall impact transforms the system into simpler dynamics and eventually causes the chaotic regions to fade out regardless of the system settings. The largest α at which the system still exhibits chaotic behavior is estimated to be around 1.17 and for transient chaos is estimated to be 1.25.

Riemann Liouville Fractional Spatial Derivative Stabilization of Bilinear Distributed Systems

The goal of this paper is to study a fractional output stabilization problem: the stabilization of the state fractional spatial derivative of complex purely imaginary order i α with α ∈]0,1[, for bilinear distributed systems. Firstly, we develop sufficient conditions for exponential, strong and weak fractional output stabilization for the considered system, also, we offer some examples illustrating the obtained results. Moreover, we characterise the stabilizing control which minimizes an appropriate cost. Finally, an illustrating example with numerical simulations is given.

Discrete monotone method for space-fractional nonlinear reaction\u2013diffusion equations

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.

On (2 + 1)-dimensional physical models endowed with decoupled spatial and temporal memory indices⋆

The current work concerns the development of an analytical scheme to handle (2 + 1) -dimensional partial differential equations endowed with decoupled spatial and temporal fractional derivatives (abbreviated by $ (\alpha,\beta)$ -models). For this purpose, a new bivariate fractional power series expansion has been integrated with the differential transform scheme. The mechanism of the submitted scheme depends mainly on converting the $ (\alpha,\beta)$ -model to a recurrence-differential equation that can be easily solved by virtue of an iterative procedure. This, in turn, reduces the computational cost of the Taylor power series method and consequently introduces a significant refinement for solving such hybrid models. To elucidate the novelty and efficiency of the proposed scheme, several $ (\alpha,\beta)$ -models are solved and the presence of remnant memory, due to the fractional derivatives, is graphically illustrated.

Some implications of position-dependent mass quantum fractional Hamiltonian in quantum mechanics

In this study, we have constructed a new fractional Schrodinger Hamiltonian for a quantum system characterized by a position-dependent mass and a modified fractional spatial derivative operator. This approach is characterized by a modified commutation relation and a modified Heisenberg’s uncertainty relation which both led to a modified Schrodinger operator typically used to describe quantum mechanics in fractional dimensions. By selecting an effective mass involved in semiconductor heterostructures, we have discussed two independent cases: the confinement of a particle in a one-dimensional infinite well and the motion of a particle in a linear potential used to describe the falling of a particle under gravity. Several features were obtained, yet the main outcome of the present manuscript concerns the fact that, for some specific fractional dimensions, a particle fallen under gravity may have a quantized energy.

A new time and spatial fractional heat conduction model for Maxwell nanofluid in porous medium

A new time and spatial fractional heat conduction model with Brownian diffusion and thermophoresis is presented to investigate the heat and mass transfer of Maxwell viscoelastic nanofluid over a moving flat plate in porous medium. New dimensionless variables are introduced to nondimensionalize the governing equations which are solved numerically by L1 algorithm and shifted Grünwald formula. The main observations of the model are the effects of embedded time and space fractional parameters on velocity, temperature and concentration profiles which are analyzed graphically. Numerical computations for a comparison between the heat conduction model with the time and spatial fractional derivatives and the model with spatial fractional derivative are made. It is found that the heat transfer is enhanced by the time and spatial fractional heat conduction model. Moreover, the larger fractional derivatives refer to the stronger memory characteristic which exhibits the physical meanings of fractional derivative.

A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations

Abstract In the present paper, the new three-dimensional modified Benjamin-Bona-Mahony equations recently introduced are analyzed with the introduction of the spatial and temporal fractional order derivatives using conformable fractional derivative. Various solutions including hyperbolic and periodic function solutions are constructed using simple ansatzs with the aid of the Wolfram Mathematica software. Some graphical representations of the solutions are depicted.

A Second‐Order Uniformly Stable Explicit Asymmetric Discretization Method for One‐Dimensional Fractional Diffusion Equations

Using the asymmetric discretization technique, an explicit finite difference scheme is constructed for one‐dimensional spatial fractional diffusion equations (FDEs). The spatial fractional derivative is approximated by the weighted and shifted Grünwald difference operator. The scheme can be solved explicitly by calculating unknowns in the different nodal‐point sequences at the odd time‐step and the even time‐step. The uniform stability is proven and the error between the discrete solution and analytical solution is theoretically estimated. Numerical examples are given to verify theoretical analysis.

Reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains

In recent years, the fractional differential equations have attracted a lot of attention due to their interested characteristics. Meshfree methods are highly accurate and have been extensively explored in engineering and mechanics fields. However, there is few research to develop the reproducing kernel particle method (RKPM), one of the widely used meshfree approach, for fractional partial differential equations. In this work, we solve time-space fractional diffusion equations in 2D regular and irregular domains. The temporal Caputo fractional derivatives are discretized by the L1 finite difference scheme and the spatial Laplacian fractional derivatives are discretized by RKPM based on the matrix transfer method. Especially, the corrected weighted shifted Grünwald–Letnikov scheme is utilized for temporally non-smooth solutions. Numerical examples in rectangular, circular, sector and human brain-like irregular domains are given to assess the efficiency and accuracy of the proposed numerical scheme. The spatial Laplacian fractional derivatives discretized by conventional finite difference method in the rectangular domain are also presented for comparison. The results indicate that RKPM is very effective for analyzing the considered fractional equations in various domains, which lays a concrete foundation for our further research of real application of human brain modeling.

A numerically efficient Hamiltonian method for fractional wave equations

In this work, we consider a partial differential equation that extends the well known wave equation. The model under consideration is a multidimensional equation which includes the presence of both a damping term and a fractional Laplacian of the Riesz type. Homogeneous Dirichlet boundary conditions on a closed and bounded spatial interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an implicit scheme which is based on the use of fractional centered differences to approximate the spatial fractional derivatives of the model. A discretized form of the Hamiltoninan is also proposed in this work, and we prove analytically that the method is capable of preserving/dissipating the discrete energy when the continuous model preserves/dissipates the energy. We establish rigorously the properties of consistency, stability and convergence of the method, and provide some a priori bounds for the numerical solutions. Moreover, we prove the existence and the uniqueness of the numerical solutions as well as the unconditional stability of the method in the linear regime. Some computer simulations that assess the capability of the method to preserve/dissipate the energy are carried out for illustration purposes.

Decay Results of the Nonstationary Navier–Stokes Flows in Half-Spaces

There is a long standing problem on the nonstationary Navier–Stokes equations which pertains to how to characterize the L1−time asymptotic expansion of the Navier–Stokes flows in the half space. Beyond a few partial results, new progress has not yet to be made on this open question. In this article, we give a confirmed answer to this problem; namely, a thorough characterization on L1−summability is revealed. In order to prove this result, we need to avoid the unboundedness of the projection operator, which is overcome by treating an elliptic Neumann problem. Finally, using the weighted estimates on the heat kernel’s convolution, we obtain the exact profile structure of the asymptotic expansion in $${L^1(\mathbb{R}^n_+)}$$ . In addition, some crucial estimates on the fractional spatial derivatives of the non-stationary Stokes and Navier–Stokes flows are established for the first time, which allows for a better understanding of the L1−decay problem.