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The fractional calculus (FC) has been extensively studied by researchers due to its vast applications in sciences in the last few years. In fractional calculus, multivariate Mittag–Leffler functions are considered the powerful extension of the classical Mittag–Leffler functions. This paper defines the generalized fractional integral operator with multivariate Mittag–Leffler (M-L) function. We prove certain basic properties of the proposed operators, such as an expansion of an infinite series of Riemann–Liouville integrals, Laplace transform (LT), semigroup property, composition with Riemann–Liouville integrals. Also, we present the fractional differential operators and their properties. The application of the proposed operators like the fractional kinetic differential and the time-fractional heat equation are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in L 2 -norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete time WG-FEM. The numerical example is to illustrate the theoretical analysis with polynomial mixture { P k (K) , P k − 1 (∂ K) , P k − 1 (K) 2 }. [ABSTRACT FROM AUTHOR]

Geiger-mode avalanche photodiode (GM-APD) light detection and ranging (LiDAR) target echo signals are susceptible to interference from atmospheric backscatter, daylight, and other noise, and extracted target depth images contain a large amount of anomalous noise. Accordingly, this paper devises a method based on a spatially correlated fractional integral for denoising GM-APD LiDAR depth images. First, a multi-scale superpixel fusion algorithm is employed to perform null-pixel complementation on a target depth image extracted using histogram statistics. Next, a pixel neighborhood spatial correlation kernel function is used to optimize the Grünwald–Letnikov fractional integral operator to correct the large amount of anomalous noise and thus achieve GM-APD depth image denoising. Simulation and experimental results demonstrate that the denoising performance of the algorithm on GM-APD LiDAR depth images is significantly better than that of median filtering and bilateral filtering, with at least 22.8 % and 4.8 % improvements in the target reduction degree (K) and peak signal-to-noise ratio (PSNR), respectively. In addition, the K and PSNR reach 91.36 % and 18.0241 dB, respectively, with 25 statistical frames in an outdoor imaging experiment. This demonstrates that the algorithm can effectively denoise GM-APD LiDAR depth images with few statistical frames and thus improve the frame rate of GM-APD LiDAR imaging. [ABSTRACT FROM AUTHOR]

We prove the boundedness of the fractional integration operator of variable order α (x) in the limiting Sobolev case α (x) p (x) = n − λ (x) from variable exponent Morrey spaces L p ⋅ , λ ⋅ Ω into BMO (Ω), where Ω is a bounded open set. In the case α (x) ≡ const, we also show the boundedness from variable exponent vanishing Morrey spaces V L p ⋅ , λ ⋅ Ω into VMO (Ω). The results seem to be new even when p and λ are constant. [ABSTRACT FROM AUTHOR]

Predicting the horizontal groundwater flow in unsaturated porous media is a challenge in many areas of science and engineering. The governing equation associated with this phenomenon is a nonlinear partial differential equation known as the Richards equation. However, the numerical results obtained using this equation can differ substantially from the experimental results. In order to overcome this difficulty, a new version of the Richards equation was proposed recently that considers a time derivative of fractional order. In this study, we present a numerical method for solving this fractional Richards equation. Our method comprises an adaptive time marching scheme that uses Picard iterations to solve the corresponding nonlinear equations. A computational code was implemented for the proposed method using the Scilab programming language. We performed numerical simulations of the anomalous diffusion of water in a white siliceous brick and showed that the numerical results were consistent with the available experimental data. [ABSTRACT FROM AUTHOR]

Let X be a ball quasi-Banach function space on R n satisfying some mild assumptions and let α ∈ (0 , n) and β ∈ (1 , ∞). In this article, when α ∈ (0 , 1) , the authors first find a reasonable version I ˜ α of the fractional integral I α on the ball Campanato-type function space L X , q , s , d (R n) with q ∈ [ 1 , ∞) , s ∈ Z + n , and d ∈ (0 , ∞). Then the authors prove that I ˜ α is bounded from L X β , q , s , d (R n) to L X , q , s , d (R n) if and only if there exists a positive constant C such that, for any ball B ⊂ R n , | B | α n ≤ C ‖ 1 B ‖ X β − 1 β , where X β denotes the β -convexification of X. Furthermore, the authors extend the range α ∈ (0 , 1) in I ˜ α to the range α ∈ (0 , n) and also obtain the corresponding boundedness in this case. Moreover, I ˜ α is proved to be the adjoint operator of I α. All these results have a wide range of applications. Particularly, even when they are applied, respectively, to mixed-norm Lebesgue spaces, Morrey spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the dual theorem on L X , q , s , d (R n) and also on the special atomic decomposition of molecules of H X (R n) (the Hardy-type space associated with X) which proves the predual space of L X , q , s , d (R n). [ABSTRACT FROM AUTHOR]

In this paper, we establish some new refinements and similar extensions of Hermite–Hadamard inequality for fractional integrals and present several applications in the Beta function. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a systematic study of some extended special functions like B b ; ρ , λ α , β x , y , 2 F 1 α , β ; ρ , λ z ; b and p F q α , β ; ρ , λ z ; b . We obtain various properties of these extended functions and establish their some connections with the Laguerre polynomial and Fox’s H -function. Furthermore, we also establish the extended Riemann–Liouville type fractional integral operator and extended Kober type fractional integral operators. [ABSTRACT FROM AUTHOR]