Riesz Derivative Research Articles

The Orthogonal Riesz Fractional Derivative

The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials Cn(ν)(x), where ν>−12. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as Dαx, providing an alternative framework for fractional calculus.

An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations

This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.

Research on fractional symmetry based on Riesz derivative

Research on fractional symmetry based on Riesz derivative

Interactions between fractional solitons in bimodal fiber cavities

AbstractWe introduce a system of fractional nonlinear Schrödinger equations (FNLSEs) which model the copropagation of optical waves carried by different wavelengths or mutually orthogonal circular polarizations in fiber‐laser cavities with the effective fractional group‐velocity dispersion (FGVD), which were recently made available to the experiment. In the FNLSE system, the FGVD terms are represented by the Riesz derivative, with the respective Lévy index (LI). The FNLSEs, which include the self‐phase‐modulation (SPM) nonlinearity, are coupled by the cross‐phase‐modulation (XPM) terms, and separated by a group‐velocity (GV) mismatch (rapidity). By means of systematic simulations, we analyze collisions and bound states of solitons in the XPM‐coupled system, varying the LI and GV mismatch. Outcomes of collisions between the solitons include rebound, conversion of the colliding single‐component solitons into a pair of two‐component ones, merger of the solitons into a breather, their mutual passage leading to excitation of intrinsic vibrations, and the elastic interaction. Families of stable two‐component soliton bound states are constructed too, featuring a rapidity which is intermediate between those of the two components.

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

Superdiffusion of energetic particles at shocks: A fractional diffusion and Lévy flight model of spatial transport

The observed power laws in space and time profiles of energetic particles in the heliosphere can be the result of an underlying superdiffusive transport behavior. Such anomalous, non-Gaussian transport regimes can arise, for example, as a consequence of intermittent structures in the solar wind. Non-diffusive transport regimes may also play a critical role in other astrophysical environments such as supernova remnant shocks. To clarify the role of superdiffusion in the transport of particles near shocks, we study the solutions of a fractional diffusion-advection equation to investigate this issue. A fractional generalization of the Laplace operator, the Riesz derivative, provides a model of superdiffusive propagation. e vy motion. The expected power law profiles of particles upstream of the plasma shock, where particles are injected, can be reproduced with this approach. The method provides a full, time-dependent solution of the fractional diffusion-advection equation. The developed models enable a quantitative comparison to energetic particle properties based on a comprehensive, superdiffusive transport equation and allow for an application in a number of scenarios in astrophysics and space science.

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

Bernstein Inequality for the Riesz Derivative of Order $$0<\alpha<1$$ of Entire Functions of Exponential Type in the Uniform Norm

A finite difference/Kansa method for the two-dimensional time and space fractional Bloch-Torrey equation

In this paper, an implicit finite difference scheme combined with Kansa meshless method is proposed for the two-dimensional time and space fractional Bloch-Torrey equation. The Caputo derivative with respect to time is discretized by finite difference method based on Alikhanov's super convergent approximation while Riesz derivative with respect to spatial variables by Kansa meshless method. The convergence and stability of the time semi-discretization are proven using energy method. The convergence order in time direction is proven to be 2. The implementation of the full discretization is discussed in detail. Especially, the issue of calculation of Riesz derivative of RBF is addressed. Numerical examples are given which verify the feasibility of the proposed method.

Bernstein Inequality for the Riesz Derivative of Fractional Order Less Than Unity of Entire Functions of Exponential Type

Bernstein Inequality for the Riesz Derivative of Fractional Order Less Than Unity of Entire Functions of Exponential Type

Numerical solution of the boundary value problem for the heat equation with fractional Riesz derivative

Numerical solution of the boundary value problem for the heat equation with fractional Riesz derivative

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

BERNSTEIN INEQUALITY FOR RIESZ DERIVATIVE OF FRACTIONAL ORDER LESS THAN 1 OF ENTIRE FUNCTION OF EXPONENTIAL TYPE

We consider Bernstein inequality for the Riesz derivative of order \(0 \alpha 1\) of entire functions of exponential type in the uniform norm on the real line. The interpolation formula for this operator is obtained; this formula has non-equidistant nodes. By means of this formula, the sharp Bernstein inequality is obtained for all \(0 \alpha 1\), more precisely, the extremal entire function and the exact constant are written out.

Advancing Fractional Riesz Derivatives through Dunkl Operators

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

L1/LDG algorithm for time‐Caputo space‐Riesz fractional convection equation in two dimensions

This paper proposes an L1/LDG algorithm for a time–space fractional convection equation on a bounded rectangular domain . Here, the temporal partial derivative is in the sense of Caputo with order , and the spatial partial derivatives are of Riesz type with orders in ‐ and ‐directions, respectively. L1 approximation is utilized to evaluate the temporal Caputo derivative, while the local discontinuous Galerkin (LDG) method is adopted to deal with the spatial Riesz derivative. The rigorous numerical stability analysis and error estimate are presented which are supported by illustrative numerical examples.

An adaptive finite element method for Riesz fractional partial integro-differential equations

AbstractThe Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.

Numerical solution of multidimensional time‐space fractional differential equations of distributed order with Riesz derivative

In this article, we derive a computational scheme joining the finite difference and operational matrix methods to numerically simulate fractional differential equations of distributed order with Riesz derivatives subject to the initial and Dirichlet boundary conditions. The finite difference method is used to discretize Riesz‐space fractional derivatives, and the operational matrix method is based on fractional shifted Gegenbauer functions for the approximation of time derivatives. The derived method transforms the given equation into a system of linear algebraic equations. For the numerical solution, we derive the optimal error bound and prove the theoretical unconditional stability with respect to the ‐norm. Furthermore, we numerically verify the stability of the proposed method. We observe the accuracy of the second order of the given schemes. The accuracy and effectiveness of the method are checked to solve some problems of telegraph equations. The CPU running time for the method with a fractional shifted Gegenbauer basis is much less than for methods with other bases.

The construction of an optimal fourth-order fractional-compact-type numerical differential formula of the Riesz derivative and its application

In this paper, a novel optimal fourth-order fractional-compact-type numerical differential formula for the Riesz derivatives is derived by constructing the appropriate generating function. Meanwhile, some interesting and important properties of the proposed formula are also presented and proved. Then, we take the nonlinear space fractional Ginzburg–Landau equations as an example and establish a high-order finite difference scheme by combining the compact integral factor method with Padé approximation. Furthermore, based on the discrete energy analysis method, the proposed difference scheme is proved to be uniquely solvable and convergent with order O(τ2+h4) under different norms, where τ and h denote the temporal and spatial mesh step sizes, respectively. Finally, some numerical results are provided to illustrate the accuracy of numerical differential formula and the effectiveness of numerical method, and further demonstrate the correctness of theoretical analysis.

Mathematical Model of Heat Conduction for a Semi-Infinite Body, Taking into Account Memory Effects and Spatial Correlations

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed.

Numerical analysis of the high-order scheme of the damped nonlinear space fraction Schrödinger equation

Firstly, we construct a fourth-order numerical differential formula for approximating Riesz derivative. Substituting it into the damped nonlinear space fractional Schrödinger equation, the original equation becomes a matrix form of nonlinear ordinary differential equation system. Then, by means of implicit integrating factor method and Padé approximation, we obtain a new numerical scheme whose convergence order is higher than the existing algorithms. Finally, a numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.