Neumann problem for a stochastic Benjamin-Bona-Mahony equation with Riesz fractional derivative

Image denoising is an important component of image processing. The interest in the use of Riesz fractional order derivative has been rapidly growing for image processing recently. This paper mainly introduces the concept of fractional calculus and proposes a new mathematical model in using the convolution of fractional Tsallis entropy with the Riesz fractional derivative for image denoising. The structures of n × n fractional mask windows in the x and y directions of this algorithm are constructed. The image denoising performance is assessed using the visual perception, and the objective image quality metrics, such as peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM). The proposed algorithm achieved average PSNR of 28.92 dB and SSIM of 0.8041. The experimental results prove that the improvements achieved are compatible with other standard image smoothing filters (Gaussian, Kuan, and Homomorphic Wiener).

Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

This paper aims to approximate the fractional diffusion equations with the Riesz fractional derivative in a finite domain utilizing the McCormack numerical method with a second order accuracy. To approximate the Riesz fractional derivative, the fractional central difference is used. The fractional derivative error is obtained for the central fractional difference and the stability of the McCormack numerical method is studied. Some numerical examples are given to evidence the maximum error obtained when the McCormack method is utilized for the solution of the fractional diffusion equations using the fractional central difference.

In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.

This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results.

Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain

In this paper, a fractional diffusion equation by using the explicit numerical method in a finite domain with second-order accuracy which includes the Riesz fractional derivative approximation is studied. For the Riesz fractional derivative approximation, ''fractional centered derivative'' approach is used. The error of the Riesz fractional derivative to the fractional centered difference is calculated. We used the implicit numerical method to solve the fractional diffusion equation and also investigated the stability of explicit and implicit methods. The maximum error of the implicit method for fractional diffusion equation with using fractional centered difference approach is shown by using the numerical results.

In this paper, a digital image sharpening method using Riesz fractional order derivative (RFOD) and discrete Hartley transform (DHT) is presented. First, the definition of Riesz fractional order derivative is reviewed briefly. Then, the DHT interpolation method is described. Next, the RFOD, DHT interpolation and Mach band effect are used to construct a digital image sharpening algorithm. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed digital image sharpening approach.

The present work deals with the solutions of the Gerdjikov–Ivanov(G–I) equation with the Riesz fractional derivative by means of the time-splitting spectral approach. In this approach, the G–I equation is split into two equations and the proposed technique viz. time-splitting spectral method is employed for discretizing the equation in space and then subsequently integrating in time exactly. Furthermore, an implicit finite difference method (IMFD) is utilized here to compare the results with the above-mentioned seminumerical method viz. time-splitting spectral technique. Moreover, it has been established that the proposed method is unconditionally stable. In addition to these, the error norms have been also presented here.

Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator.Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.

Riesz fractional derivative Elite-guided sine cosine algorithm

This chapter presents basic definitions and characteristics of fractional calculus. Definitions and characteristics of the Riemann–Liouville fractional integral and derivative, Caputo fractional derivative, Grünwald–Letnikov fractional derivative, Riesz fractional derivative, modified Riemann–Liouville derivative, and local fractional derivative have been primarily explored here. However, the Riemann–Liouville definitions of fractional differentiation play a significant role in developing fractional calculus. Because the derivative of constant is not zero, the Riemann–Liouville definition is not widely applicable. In such scenarios, the Caputo fractional derivative is more suited than the standard Riemann–Liouville derivative for applications in various engineering problems. It may be noted that the Riemann–Liouville technique is challenging to understand physically, while the Caputo approach employs integer-order initial conditions, which are easier to apply in real-world applications. This chapter also covers the Euler psi function, incomplete gamma function, beta function, incomplete beta function, Wright function, Mellin-Ross function, error function, hypergeometric functions such as Gauss, Kummer, and extended hypergeometric functions, and the H -function.

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

Gaussian Lipschitz spaces Lip α(γ d ) and the boundedness properties of Riesz potentials, Bessel potentials and fractional derivatives there were studied in detail in Gatto and Urbina (On Gaussian Lipschitz Spaces and the Boundedness of Fractional Integrals and Fractional Derivatives on them, 2009. Preprint. arXiv:0911.3962v2). In this chapter we will study the boundedness of those operators on Gaussian Besov-Lipschitz spaces B p, q α(γ d ). Also, these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.KeywordsFractional DerivativeGaussian MeasureFractional IntegralHermite PolynomialBoundedness PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Riesz potentials, Bessel potentials and fractional derivatives on Triebel–Lizorkin spaces for the Gaussian measure