Space Fractional Diffusion Equation Research Articles

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Midpoint splitting methods for nonlinear space fractional diffusion equations

We propose a midpoint splitting method to solve nonlinear space fractional diffusion equations. We first discretize the equation in space using the weighted and shifted Grünwald difference (WSGD) operators. Then, we obtain the fully discrete scheme of the equation using the midpoint splitting method, which can alleviate computational costs and mitigate instability issues that may arise. We establish the stability result of the fully discrete scheme under certain conditions. Finally, we demonstrate its feasibility and efficiency through numerical examples.

A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation

As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method.

A new diagonal and Toeplitz splitting preconditioning method for solving time-dependent Riesz space-fractional diffusion equations

The initial–boundary value problem of the Riesz space-fractional diffusions equation is an important class of equations arising in many application fields. In this paper, we apply the Grünwald–Letnikov type formulas to discretize the time-dependent Riesz space-fractional diffusion equations, and obtain a system of linear equations from the discretization results. A new diagonal and Toeplitz splitting (NDTS) iteration method is constructed for this linear system. Based on the NDTS iteration method, an NDTSτ preconditioner is proposed and the generalized minimal residual (GMRES) method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the NDTSτ preconditioned matrix are clustered. Numerical experiments illustrate the efficiency of the proposed method.

Richardson extrapolation method for solving the Riesz space fractional diffusion problem

AbstractThe Richardson extrapolation is widely utilized in numerically solving the differential equations since it enjoys both high accuracy and ease of implementation. For the Riesz space fractional diffusion equation, the fractional centered difference operator is employed to approximate the fractional derivative, then the Richardson extrapolation methods for two difference schemes are constructed with the help of the asymptotic expansions of the discrete solutions. Specifically, for the Crank–Nicolson difference scheme, the extrapolation method contains two extrapolation formulae that achieve the fourth order and the sixth order both in the temporal and spatial directions, respectively. The extrapolation method for the compact difference scheme involves one extrapolation formula by which the sixth order can be obtained when the time step size is proportional to the squares of the space step size. The maximum norm error estimates of the extrapolation solutions are proved by the discrete fractional Sobolev embedding inequalities. The extension to the high dimensional and nonlinear cases is also demonstrated. Numerical results verify the theoretical convergence orders and efficiency of our methods.

A nonstationary iterated quasi-boundary value method for reconstructing the source term in a time–space fractional diffusion equation

An inverse source problem of a time–space fractional diffusion equation is considered in this paper. Due to the ill-posedness of the inverse problem, we propose a novel nonstationary iterated quasi-boundary value regularization method for reconstructing the source function, and show that the regularization problem is well-posed. The convergence rates are established under a priori and a posterior choice rules of regularization parameters, respectively. A numerical scheme for solving the regularization problem in one-dimensional case is derived from a finite difference method. Moreover, various of numerical examples are performed to test the efficiency of our method.

A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha }$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

Numerical Method for Solving the Nonlinear Superdiffusion Equation with Functional Delay

For a space-fractional diffusion equation with a nonlinear superdiffusion coefficient and with the presence of a delay effect, the grid numerical method is constructed. Interpolation and extrapolation procedures are used to account for the functional delay. At each time step, the algorithm reduces to solving a linear system with a main matrix that has diagonal dominance. The convergence of the method in the maximum norm is proved. The results of numerical experiments with constant and variable delays are presented.

On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Continuation of Radial Positive Definite Functions and Their Characterization

This paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann–Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Approximate solution of space fractional order diffusion equations by Gegenbauer collocation and compact finite difference scheme

In this paper, approximation of space fractional order diffusion equation are considered using compact finite difference technique to discretize the time derivative, which was then approximated via shifted Gegenbauer polynomials using zeros of (N - 1) degree shifted Gegenbauer polynomial as collocation points. The important feature in this approach is that it reduces the problems to algebraic linear system of equations together with the boundary conditions gives (N + 1) linear equations. Some theorems are given to establish the convergence and the stability of the proposed method. To validate the efficiency and the accuracy of the method, obtained results are compared with the existing results in the literature. The graphical representation are also displayed for various values of \beta Gegenbauer polynomials. It can be observe in the tables of the results and figures that the proposed method performs better than the existing one in the literature.

An Implicit Numerical Method for the Riemann–Liouville Distributed-Order Space Fractional Diffusion Equation

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O(σ2+ρ2+τ+hx+hy). After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus.

A Second-Order Accurate Numerical Approximation for a Two-Sided Space-Fractional Diffusion Equation

In this paper, we investigate a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain, which is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation. Second-order exact numerical estimates in time and space are obtained. The unconditional stability and convergence of the method are tested. Two numerical examples are also presented and compared with the exact solution.

The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time–Space Fractional Diffusion Equation

In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time–space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.

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.

On the convergence of piecewise polynomial collocation methods for variable-order space-fractional diffusion equations

In this study, we present a piecewise polynomial collocation method for the boundary-value problem of variable-order linear space-fractional diffusion equations. The proposed model is transformed to a weakly singular Volterra integral equation (VIE) of the second kind by an auxiliary variable, then a collocation method is constructed and analyzed for the obtained VIE. We demonstrate the existence and uniqueness of the collocation solution, as well as the optimal convergence order of the collocation method. Some numerical experiments are given to illustrate the theoretical results.

A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

In this paper, we introduce an efficient stochastic method for solving space-fractional diffusion equations in high dimensions based on the Feynman–Kac formula. The key idea is to approximate the trajectory of the process by using a series of balls. As an extension of our first work Sheng et al. (2022) for the Poisson equation, the new algorithm finds remarkably efficient in solving time-dependent linear problems with integral fractional Laplacian on the bounded and unbounded domains. We present some numerical examples to validate the robustness and efficiency of our methods.

Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients

In this paper, efficient preconditioned iterative methods for solving the Riesz space fractional diffusion equations with variable diffusion coefficients are considered. Crank–Nicolson and weighted and shifted Grünwald difference scheme are applied to discretize the problem. For one-dimensional problems, we transform the asymmetric discretized linear system into a symmetric one and solve the resulted linear system by preconditioned conjugate gradient method. The preconditioned conjugate gradient method with a symmetric Toeplitz preconditioner and a sine transform based preconditioner are employed to solve the resulting linear system. Theoretically, we respectively prove that the condition numbers of preconditioned matrices have uniform upper bounds, which is independent of discretization step-sizes and fractional order. For two-dimensional problems, we propose an optional and symmetric splitting iteration method, which can be embedded by conjugate gradient method although the diffusion coefficients are non-separable. We prove that the optional and symmetric splitting iteration method is unconditionally convergent. Numerical results are reported to illustrate the efficiency of the proposed methods.