Space Fractional Advection Diffusion Research Articles

Lattice Boltzmann Simulation of Spatial Fractional Convection-Diffusion Equation.

The space fractional advection-diffusion equation is a crucial type of fractional partial differential equation, widely used for its ability to more accurately describe natural phenomena. Due to the complexity of analytical approaches, this paper focuses on its numerical investigation. A lattice Boltzmann model for the spatial fractional convection-diffusion equation is developed, and an error analysis is carried out. The spatial fractional convection-diffusion equation is solved for several examples. The validity of the model is confirmed by comparing its numerical solutions with those obtained from other methods The results demonstrate that the lattice Boltzmann method is an effective tool for solving the space fractional convection-diffusion equation.

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.

Physics-informed neural network algorithm for solving forward and inverse problems of variable-order space-fractional advection–diffusion equations

A new physics-informed neural network (PINN) algorithm is proposed to solve variable-order space-fractional partial differential equations (PDEs). For the forward problem, PINN algorithm based on a power series expansion is established to solve the space-fractional advection–diffusion equations with variable coefficients in one and two dimensions. The loss function is constructed by taking the coefficients in the power series expansion as the output of the network. The learning rate range is also analyzed to ensure that the error is reduced with respect to the training time. For the inverse problem, a novel dual-network architecture based on neural network parallelism is designed to estimate the order of variable fractional operator. One of the networks outputs the coefficients of the power series expansion, which is a function that depends only on time variables, and the other is fitted to the order of variable fractional operator, which is a function that is related to both space and time variables. Compared with the forward problem, we make more comparisons between the approximated solution and the exact solution at some interior points when constructing the loss function to improve the fitting ability of the inverse problem. The exact solutions are easy to find and can be readily obtained with a fine mesh. Several numerical examples are illustrated with graphs. These results confirm the effectiveness of our method in solving variable-order space-fractional partial differential equations.

An efficient conservative splitting characteristic difference method for solving 2-d space-fractional advection–diffusion equations

An efficient conservative splitting characteristic difference method for solving 2-d space-fractional advection–diffusion equations

A unified model of suspension concentration distribution in sediment mixed turbulent flows using generalized fractional advection-diffusion equation

The fractional operator in a space fractional advection-diffusion equation (FADE) plays a significant role in the mixing and vertical movement of sediment particles in a sediment-laden turbulent flow under non-local effects. Turbulent flow exhibits non-local mixing properties, which leads to the non-Fickian diffusion process that cannot be captured by the traditional diffusion equation. In this work, we present a generalized FADE that includes the generalized fractional differential operator in the Caputo sense. The full analytical solution is proposed utilizing the general Laplace transformation method. This generalized solution contains weight and scale functions and includes the effects of non-locality. It has been shown that several existing famous models of suspension concentration distribution for sediment particles (including both type-I and type-II distributions) in turbulent flows can be obtained from the proposed generalized solution with proper choices of the scale and weight functions in particular. Here a total of fourteen different types of concentration distribution equations including type-I and type-II profiles are derived from the general solution. Further possible generalizations of the model are also discussed which are more useful for practical applications. It is found that the several existing sediment distribution models are equivalent up to choices of weight and scale functions. Further, we found that the scale function could be physically related to the characteristic Lagrangian length of sediment mixing. The choice of the scale and weight function for both the type-I and type-II profiles are discussed and analyzed. Finally, the model is validated with experimental data as well as field data from the Missouri River, Mississippi River, and Rio Grande conveyance channels, and in each case, satisfactory agreements are obtained. These suggest the broader applicability of the present study.

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

Stability and convergence of 3-point WSGD schemes for two-sided space fractional advection-diffusion equations with variable coefficients

In this paper, we consider high order numerical methods for the solution of the initial-boundary value problem of two-sided space fractional advection-diffusion equations (SFADEs). We use the Crank-Nicolson (CN) technique to discretize the temporal derivative, and apply 3-point weighted and shifted Grünwald difference (WSGD) operators to discretize the fractional derivatives of order α∈(1,2) and fractional derivatives of order β∈(0,1), respectively. As a result, a new family of CN-WSGD schemes for SFADEs with temporally 2nd order and spatially jth order (j≥2) accuracy are obtained. We then analyse the stability and the convergence of the numerical schemes. The extrapolated WSGD (EWSGD) operators are also discussed. Numerical examples are implemented to verify the theoretical results.

A non‐standard finite difference method for space fractional advection–diffusion equation

AbstractIn this paper, a non‐standard finite difference scheme is developed to solve the space fractional advection–diffusion equation. By using Fourier–Von Neumann method, we prove that non‐standard finite difference scheme is unconditionally stable. We further discuss the convergence of numerical method and give the order of convergence. The numerical examples show that the non‐standard finite difference method can effectively reduce the maximum error and improve the accuracy of numerical solution in contrast to classical numerical methods. Moreover, we find that our numerical scheme is very flexible, when we optimize the denominator function of time and space simultaneously, the performance is the best. These studies show that the non‐standard finite difference scheme is feasible and efficient for solving fractional partial differential equations.

Implicit radial point interpolation method for nonlinear space fractional advection-diffusion equations

This paper aims to employ the implicit radial point interpolation method, which is intrinsically meshless, for the numerical simulation of nonlinear space fractional advection-diffusion equations. The space fractional derivative is defined in the Caputo sense and calculated by the Gauss’s Jacobi quadrature formula. The accuracy and convergency of the proposed meshless method are demonstrated by several numerical examples with different regions and different nodal distributions. It is proved that the presented method is computational efficiency for modeling and simulation of nonlinear SFADEs.

The Impact of Chebyshev Collocation Method on Solutions of fractional Advection–Diffusion Equation

The present paper is primarily aimed at obtaining the numerical solution of space fractional advection–diffusion equation including two fractional space derivatives of order. At the first stage, a difference approach with the second-order accuracy is formulated to obtain a semi-discrete scheme. Unconditional stability and convergence analysis has been analyzed. At the second stage, the spatial discretization is accomplished by means of the second kind shifted Chebyshev polynomials. Two examples are investigated, and numerical results are reported to confirm the theoretical results.

Fourth order compact scheme for space fractional advection–diffusion reaction equations with variable coefficients

In this work, a new fourth order compact approximation is derived for Riemann–Liouville space fractional derivatives. Modified wave numbers are obtained for various approximations of fractional derivatives. Considering these modified wave numbers, Fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations. A novel fourth order compact scheme is developed for space fractional advection–diffusion reaction equations with variable coefficients in one and two dimensions. In the proposed compact scheme, the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations. This splitting of second derivative approximations allows us to obtain a similar system of linear equations as in Zhao and Tian (2017) which presents only second order accurate finite difference method for advection–diffusion reaction equations in one-dimension. The stability of the proposed compact scheme is demonstrated numerically. Furthermore, the proposed compact scheme is employed to space fractional Black–Scholes equation for pricing European options as an application of fractional derivatives in mathematical finance. Numerical illustrations are presented to validate the theoretical claims.

Compact finite difference method to numerically solving a stochastic fractional advection-diffusion equation

In this paper, a stochastic space fractional advection diffusion equation of Itô type with one-dimensional white noise process is presented. The fractional derivative is defined in the sense of Caputo. A stochastic compact finite difference method is used to study the proposed model numerically. Stability analysis and consistency for the stochastic compact finite difference scheme are proved. Two test examples are given to test the performance of the proposed method. Numerical simulations show that the results obtained are compatible with the exact solutions and with the solutions derived in the literature.

Necessity of introducing non-integer shifted parameters by constructing high accuracy finite difference algorithms for a two-sided space-fractional advection–diffusion model

In this study, we develop a second-order finite difference scheme based on the shifted convolution quadrature (SCQ) framework that approximates the space-fractional derivatives at a shifted node xn−θ where θ may not necessarily be an integer. By applying the proposed method for a space-fractional advection–diffusion equation in the spacial direction and the Crank–Nicolson scheme for the time variable discretization, we analyze the von Neumann stability for the fully discrete scheme. Further, we explore the impact of different θ on the robustness of our scheme for weak regular solutions and compare that with the shifted Grünwald–Letnikov formula. The results confirm the necessity of introducing non-integer shifted parameters θ.

Discussion of “Two-Parameter Mittag-Leffler Solution of Space Fractional Advection-Diffusion Equation for Sediment Suspension in Turbulent Flows” by Snehasis Kundu

Discussion of “Two-Parameter Mittag-Leffler Solution of Space Fractional Advection-Diffusion Equation for Sediment Suspension in Turbulent Flows” by Snehasis Kundu

Analytical Solutions of One-Dimensional Space-Fractional Advection–Diffusion Equation for Sediment Suspension Using Homotopy Analysis Method

AbstractIn this study, the full analytical solution of one-dimensional space-fractional advection–diffusion equation (fADE) is derived using the homotopy analysis method (HAM). Apart from the sever...

Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation

In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection–diffusion–reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection–diffusion–reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.

Multiple-Relaxation-Time Lattice Boltzmann scheme for fractional advection–diffusion equation

Partial differential equations (p.d.e) equipped with spatial derivatives of fractional order capture anomalous transport behaviors observed in diverse fields of Science. A number of numerical methods approximate their solutions in dimension one. Focusing our effort on such p.d.e. in higher dimension with Dirichlet boundary conditions, we present an approximation based on Lattice Boltzmann Method with Bhatnagar–Gross–Krook(BGK) or Multiple-Relaxation-Time (MRT) collision operators. First, an equilibrium distribution function is defined for simulating space-fractional diffusion equations in dimensions 2 and 3. Then, we check the accuracy of the solutions by comparing with (i) random walks derived from stable Lévy motion, and (ii) exact solutions. Because of its additional freedom degrees, the MRT collision operator provides accurate approximations to space-fractional advection–diffusion equations, even in the cases which the BGK fails to represent because of anisotropic diffusion tensor or of flow rate destabilizing the BGK LBM scheme.

Two-Parameter Mittag-Leffler Solution of Space Fractional Advection-Diffusion Equation for Sediment Suspension in Turbulent Flows

AbstractIn this paper, a simple derivation of the space fractional advection-diffusion equation (fADE) under the steady condition is proposed from a generalized mixing length concept. The obtained ...

Numerical algorithm for three-dimensional space fractional advection diffusion equation

Space fractional advection diffusion equations are better to describe anomalous diffusion phenomena because of non-locality of fractional derivatives, which causes people to confront great trouble in problem solving while enjoying the convenience from mathematical modelling, especially in high dimensional cases. In this paper, we solve the three-dimensional problem by the process of dimension by dimension, which can be achieved through a predictor-corrector algorithm. In time discretization, Crank-Nicolson scheme is adopted to match second-order difference operator of the space direction. Then, the efficiency of this method is demonstrated by some numerical examples finally.

Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diffusion Equation

The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.