Spatial Fractional Derivatives Research Articles

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

Numerical study of distributed-order Bessel fractional derivative with application to Euler–Poisson–Darboux equation

The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts.

Solutions of Time-Space Fractional Partial Differential Equations Using Picard's Iterative Method

Abstract In this paper, we present Picard's iterative method (PIM) for solving time–space fractional partial differential equations, where the derivatives are considered in the Caputo sense. We prove the existence and uniqueness of solutions. Additionally, we demonstrate the versatility of our proposed approach by obtaining exact solutions for a diverse set of equations. This method is user-friendly and directly applicable to any computer algebra system. The proposed method avoids intricate computations associated with the Adomian decomposition method, such as calculating Adomian polynomials, or the requirements of other methods like choosing a homotopy in the homotopy perturbation method, identification and manipulation of the invariant subspace in invariant subspace method or constructing a variational function in the variational iteration method. Thus, the proposed method is a versatile and efficient tool for exploring systems that involve both temporal and spatial fractional derivatives.

The [formula omitted]-asymptotic behavior of strong solutions to the incompressible magneto-hydrodynamic equations in half-spaces

We give the L1-decay estimates on the fractional spatial derivatives of the strong solution to the incompressible magneto-hydrodynamic (MHD) equations in a half-space. Moreover, by using the Stokes solution formula and the decomposition of the nonlinear terms in MHD equations, we derive asymptotic expansion properties of the strong solution in L1(R+n).

Seismic reverse time migration based on biaxial wavefield decomposition

Seismic images generated by reverse time migration (RTM) are often contaminated by low-wavenumber artifacts. Decomposition of up- and downgoing waves during propagation can effectively reduce these artifacts. However, decomposition along the vertical axis is not sufficient to eliminate the migration artifacts. We introduce biaxial wavefield decomposition to decompose seismic wavefields in the lateral and vertical directions. After decomposition, there are 16 crosscorrelation terms between different components, and there are only four crosscorrelation terms that can be used separately to image flat layers and tilted interfaces. Within these four terms, the remaining artifacts can be easily identified and reduced by other methods, and the resolution can be further improved by adjusting the weights of the four terms. The decomposition method based on the analytic wavefield via the Hilbert transform has been shown to agree with the viscoacoustic wave equation with fractional spatial derivatives. Therefore, this biaxial wavefield decomposition method is suitable for decomposing the wavefields for the attenuation-compensated RTM. The decomposed and recombined migration results can better image a complex subsurface structure with high resolution.

Numerical Schemes and Stability Analysis for Fractional Models of Transient Uni- and Bidimensional flow in Petroleum Reservoirs

Objective: To propose a modeling for the flow in oil reservoirs using Caputo’s definition of fractional derivative applied to the spatial coordinate of the problem. Furthermore, delimit a region of numerical stability for the explicit method of solving the model equations. Methodology: Starting from the material balance in a differential control volume and the generalized Darcy's law, and coupling it with an equation of state, models describing flow in porous media were derived. A numerical solution scheme using an explicit discretization was proposed. For this purpose, the L1 method for approximating spatial fractional derivatives and finite difference approximation for temporal derivatives were employed.. Results and Discussion: Through the Von Neumann stability analysis, it was observed that the explicit solution schemes for the models are conditionally stable and dependent on the order of the fractional derivative. Additionally, numerical stability regions were constructed for one and two-dimensional transient schemes, revealing that the one-dimensional scheme is stable for , and the two-dimensional scheme is stable for for all derivative orders between 0 and 1. Implications of the research: Historically, flow models in petroleum reservoirs have been based on the application of Darcy's law; however, its use is limited due to some constraints. Recently, fractional calculus has played a significant role in generalizing and obtaining more accurate models in various application areas. In this context, Darcy's law has been generalized, allowing for the derivation of fractional models that are more suitable for describing flow in porous media. These models, in general, are solved using a numerical method, and it is crucial to understand the stability region of the method for their resolution.

An accelerated novel meshless coupled algorithm for Non-local nonlinear behavior in 2D/3D space-fractional GPEs

In this work, an accelerated novel meshless coupled algorithm is first presented for the numerical study of high-dimensional space-fractional nonlinear Schrödinger or Gross-Pitaevskii equations (SFNLS/GPEs). The proposed meshless method is motivated by the novel coupled concepts: (a) a time splitting-step is adopted to decompose the NLSE; (b) the spatial Riesz fractional derivative under the Riemann-Liouville operator is discretized by the combination of numerical integration and meshless weighted-least-square schemes; (c) an accurate absorbing boundary condition is employed to treat the unbounded domain. Meanwhile, a parallel algorithm based on the above approximated schemes is designed to reduce the computing cost on a CUDA-program-based GPU card. Subsequently, the validation and numerical convergence rate of the proposed method for SFNLSE are discussed and illustrated by solving two 2D/3D examples. Furthermore, the proposed parallel meshless algorithm is extended to predict the nonlinear dynamic behaviors dominated by the space-fractional Gross-Pitaevskii equations (SFGPEs), and the influence of nonlocal effect on the nonlinear phenomena is also discussed. The presented numerical results illustrate that the proposed algorithm for multi-dimensional SFNLSEs is reliability and high-efficient.

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

A fast preconditioning iterative method for solving the discretized second-order space-fractional advection–diffusion equations

Space-fractional advection–diffusion equations (SFADEs) arise in many application areas. In this paper, we propose fast second-order numerical methods for solving the one- and two-dimensional SFADEs defined on a finite domain. The Crank–Nicolson difference scheme is utilized to discretize the temporal derivatives, the weighted and shifted Grünwald difference operators are employed to discretize the spatial fractional derivatives in SFADEs. We analyze the stability and convergence of the difference schemes by using the matrix analysis method. The coefficient matrix of the discretized system of linear equations has the structure of the sum of an identity matrix and non-symmetric Toeplitz matrices. New τ-matrix approximate preconditioners are proposed for the discretized system of linear equations for both one- and two-dimensional SFADEs, respectively. The generalized minimal residual (GMRES) methods combined with the proposed preconditioners are applied to solve the linear systems. We analyze the convergence rate of the GMRES method for solving the preconditioned linear systems. Numerical results demonstrate the effectiveness of the proposed methods.

Novel analytical technique to find diversity of solitary wave solutions for Wazwaz-Benjamin-Bona Mahony equations of fractional order

The newly modified Benjamin-Bona-Mahony equations in three dimensions are examined in the current work through the use of conformable fractional derivatives to incorporate spatial and temporal fractional order derivatives. With the aid of the Wolfram Mathematica software, a variety of solutions, including hyperbolic and periodic function solutions, are created. The information was gathered to evaluate how well the proposed novel (G’/G2)-expansion method could compute exact solutions of the WBBM equations that could be applied to the implementation of the nonlinear water model in the ocean and coastal engineering. Table 1 shows a comparison between the recently discovered solutions and those that have previously been reported in the literature. There are some visual representations of the solutions.

Dynamics of solitons in Lugiato–Lefever cavities with fractional diffraction

We study soliton dynamics in a passive optical system governed by the Lugiato–Lefever equation with effective diffraction represented by a fractional spatial derivative. Two models are considered, with the spatially uniform or strongly localized pump. Stable (quasi-) solitons are constructed, and their stability regions in the systems’ parameter spaced are identified primarily, in a numerical form. The stability is strongly affected by the Lévy index (LI) of the fractional diffraction. The dependence of the stability on the loss coefficient and pump strength, as well as the localization size in the case of the confined pump, are studied too. In the latter case, the stability is enhanced by the narrow localization. Unstable solitons spontaneously develop intrinsic oscillations. In the case of the localized pump, some unstable solitons escape from the position pinned to the pumped region.

K-space dispersion error compensators for the fractional spatial derivatives based constant-Q viscoelastic wave equation modeling

Unlike the fractional temporal derivatives based constant-Q viscoelastic wave equation, the fractional spatial derivatives based one is preferred for the wavefield simulation due to its tractable memory consumption. This viscoelastic wave equation is often solved by the staggered-grid pseudospectral method, which has the spatial spectral accuracy but temporal low order accuracy. The k-space method is one of the potential remedies for the temporal discretization errors. However, it is difficult to fully remove the temporal dispersion errors by directly applying the elastic k-space error compensators to the viscoelastic wave equation, which contains additional temporal derivatives associated with the amplitude decay terms of the P- and S- waves. To tackle the issue, we have derived a new wavenumber-time domain viscoelastic k-space equation by the eigenvalue decomposition method and by the solution to the matrix differential equations. The error compensation terms inside our k-space equation can simultaneously correct the discretization errors of different types of temporal derivatives and are wave mode adaptive. The accuracy of the k-space wave equation is verified by the viscoelastic Green's function's solution. To further boost the simulation efficiency of the k-space method in heterogeneous media, we have proposed an improved k-space method, which decreases the number of the costly mixed-domain operators. The simulation results from the simple and complex models demonstrate that our two k-space methods are superior to the conventional staggered-grid pseudospectral method for the viscoelastic wave modeling.

Fractional Schrödinger equation and time dependent potentials

We investigate the solutions for a time-dependent potential by considering two scenarios for the fractional Schrödinger equation. The first scenario analyzes the influence of the time-dependent potential in the absence of the kinetic term. We obtain analytical and numerical solutions for this case by considering the Caputo fractional time derivative, which extends Rabi’s model. In the second scenario, we incorporate the kinetic term in the Schrödinger equation and consider fractional spatial derivatives. For this case, we analyze the spreading of the Gaussian wave package under the action of the time and spatial fractional differential operators.

Tunable spatial fractional derivatives with graphene-based transmit arrays.

The optical implementation of mathematical spatial operators is a critical step toward achieving practical high-speed, low-energy analog optical processors. In recent years, it has been shown that using fractional derivatives in many engineering and science applications leads to more accurate results. In the case of optical spatial mathematical operators, the derivatives of the first and second orders have been investigated. But no research has been performed on fractional derivatives. On the other hand, in previous studies, each structure is dedicated to a single integer order derivative. This paper proposes a tunable structure made of graphene arrays on silica to implement fractional derivative orders smaller than two, as well as first and second orders. The approach used for derivatives implementation is based on the Fourier transform with two graded index lenses positioned at the structure's sides and three stacked periodic graphene-based transmit arrays in middle. The distance between the graded index lenses and the nearest graphene array is different for the derivatives of order smaller than one and between one and two. In fact, to implement all derivatives, we need two devices with the same structure having a slight difference in parameters. Simulation results based on the finite element method closely match the desired values. Given the tunability of the transmission coefficient of the proposed structure in the approximate amplitude range of [0,1] and phase range of [-180, 180], on top of the acceptable implementation of the derivative operator, this structure allows obtaining other spatial multi-purpose operators, which are a prelude to achieving analog optical processors and even improving the optical studies performed in image processing.

The non-uniform L1-type scheme coupling the finite volume method for the time–space fractional diffusion equation with variable coefficients

In this paper, we consider the time–space fractional diffusion equation on a bounded interval with the Caputo time fractional derivative and the two-sided Riemann–Liouville spatial fractional derivative. |The diffusion coefficients are variable with respect to x and t. The temporal non-uniform L1-type discretization and the finite volume method based on the nodal basis functions are adopted to discrete this fractional partial derivative equation. We strictly prove that our scheme is conditionally stable with convergence rate Oh2+N−min{2−α,rα}, where h is the spatial step and N,α,r are the temporal nodal numbers, the fractional parameter and the graded parameter respectively. Finally, some numerical results are presented to validate the theoretical analysis.

A Generalized Diffusion Equation: Solutions and Anomalous Diffusion

We investigate the solutions of a generalized diffusion-like equation by considering a spatial and time fractional derivative and the presence of non-local terms, which can be related to reaction or adsorption–desorption processes. We use the Green function approach to obtain solutions and evaluate the spreading of the system to show a rich class of behaviors. We also connect the results obtained with the anomalous diffusion processes.

An [formula omitted]-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution

A fast algorithm is proposed for solving two-dimensional distributed-order time–space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss–Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be α-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz–Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.

High order approximation scheme for a fractional order coupled system describing the dynamics of rotating two-component Bose-Einstein condensates

<abstract><p>A coupled system of fractional order Gross-Pitaevskii equations is under consideration in which the time-fractional derivative is given in Caputo sense and the spatial fractional order derivative is of Riesz type. This kind of model may shed light on some time-evolution properties of the rotating two-component Bose¢ Einstein condensates. An unconditional convergent high-order scheme is proposed based on L2-$ 1_{\sigma} $ finite difference approximation in the time direction and Galerkin Legendre spectral approximation in the space direction. This combined scheme is designed in an easy algorithmic style. Based on ideas of discrete fractional Grönwall inequalities, we can prove the convergence theory of the scheme. Accordingly, a second order of convergence and a spectral convergence order in time and space, respectively, without any constraints on temporal meshes and the specified degree of Legendre polynomials $ N $. Some numerical experiments are proposed to support the theoretical results.</p></abstract>

Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

<abstract><p>In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, <italic>Int. J. Model. Simul. Sci. Comput.</italic>, <bold>6</bold> (2015), 1540001].</p></abstract>

New Analytical Solutions for Time-Fractional Stochastic (3+1)-Dimensional Equations for Fluids with Gas Bubbles and Hydrodynamics

This paper explores the effects of spatial fractional derivatives and the multiplicative Wiener process on the analytical solutions for (3+1)-dimensional fractional stochastic equations for fluids with gas bubbles. We study the bifurcation of the analytical solutions and introduce new fractional stochastic solutions. We also discuss how the solutions differ depending on the initial conditions. The new solutions are notably more beneficial and impactful for understanding various, significant, and incredibly hard physical phenomena due to the significance of the modified fractional stochastic (3+1)-dimensional equations for fluids with gas bubbles and hydrodynamics. We also discuss the effects of the fractional order and the Wiener process on the obtained analytical solutions.