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A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme. [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]

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]

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]

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics. [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]

Abstract: We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for coincides with the classical definitions on polynomials (up to a constant). Further, if , the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results. [Copyright &y& Elsevier]

This paper discusses the analytical solutions of fractional partial differential equations using Integral Transform method. The fractional derivatives are considered with reference to modified Riemann–Liouville derivatives. Fractional Fourier transform (FrFT) is applied to solve fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. The method proposed here is effective enough to work on these equations efficiently. [ABSTRACT FROM AUTHOR]