Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach

From Springer Nature via Jisc Publications RouterHistory: received 2018-09-29, rev-recd 2018-11-09, accepted 2018-11-10, registration 2019-06-11, epub 2019-07-26, online 2019-07-26, ppub 2019-12Publication status: PublishedAbstract: We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann–Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h3-α), where h is the space step size and α∈(1, 2) is the order of Riemann–Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders O(τ+h3-α+hβ), where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald–Letnikov formula or higher order Lubich’s methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald–Letnikov formula or Lubich’s higher order approximation schemes

Deep Imaging of the HCG 95 Field.I.Ultra-diffuse Galaxies

We present a detection of 89 candidates of ultra-diffuse galaxies (UDGs) in a 4.9 degree$^2$ field centered on the Hickson Compact Group 95 (HCG 95) using deep $g$- and $r$-band images taken with the Chinese Near Object Survey Telescope. This field contains one rich galaxy cluster (Abell 2588 at $z$=0.199) and two poor clusters (Pegasus I at $z$=0.013 and Pegasus II at $z$=0.040). The 89 candidates are likely associated with the two poor clusters, giving about 50 $-$ 60 true UDGs with a half-light radius $r_{\rm e} > 1.5$ kpc and a central surface brightness $\mu(g,0) > 24.0$ mag arcsec$^{-2}$. Deep $z$'-band images are available for 84 of the 89 galaxies from the Dark Energy Camera Legacy Survey (DECaLS), confirming that these galaxies have an extremely low central surface brightness. Moreover, our UDG candidates are spread over a wide range in $g-r$ color, and $\sim$26% are as blue as normal star-forming galaxies, which is suggestive of young UDGs that are still in formation. Interestingly, we find that one UDG linked with HCG 95 is a gas-rich galaxy with H I mass $1.1 \times 10^{9} M_{\odot}$ detected by the Very Large Array, and has a stellar mass of $M_\star \sim 1.8 \times 10^{8}$ $M_{\odot}$. This indicates that UDGs at least partially overlap with the population of nearly dark galaxies found in deep H I surveys. Our results show that the high abundance of blue UDGs in the HCG 95 field is favored by the environment of poor galaxy clusters residing in H I-rich large-scale structures.Comment: Published in Ap

Numerical methods for solving space fractional partial differential equations by using Hadamard finite-part integral approach

We introduce a novel numerical method for solving two-sided space fractional partial differential equation in two dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order $O(h^{3- \alpha})$, where $h$ is the space step size and $\alpha\in (1, 2)$ is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equation. We obtained the error estimates with the convergence orders $O(\tau +h^{3-\alpha}+ h^{\beta})$, where $\tau$ is the time step size and $\beta >0$ is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equation constructed by using the standard shifted Gr\"unwald-Letnikov formula or higher order Lubich'e methods which require the solution of the equation satisfies the homogeneous Dirichlet boundary condition in order to get the first order convergence, the numerical method for solving space fractional partial differential equation constructed by using Hadamard finite-part integral approach does not require the solution of the equation satisfies the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained by using the Hadamard finite-part integral approach for solving space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained by using the numerical methods constructed with the standard shifted Gr\"unwald-Letnikov formula or Lubich's higer order approximation schemes

Indoor Scene Reconstruction with Fine-Grained Details Using Hybrid Representation and Normal Prior Enhancement

The reconstruction of indoor scenes from multi-view RGB images is challenging due to the coexistence of flat and texture-less regions alongside delicate and fine-grained regions. Recent methods leverage neural radiance fields aided by predicted surface normal priors to recover the scene geometry. These methods excel in producing complete and smooth results for floor and wall areas. However, they struggle to capture complex surfaces with high-frequency structures due to the inadequate neural representation and the inaccurately predicted normal priors. To improve the capacity of the implicit representation, we propose a hybrid architecture to represent low-frequency and high-frequency regions separately. To enhance the normal priors, we introduce a simple yet effective image sharpening and denoising technique, coupled with a network that estimates the pixel-wise uncertainty of the predicted surface normal vectors. Identifying such uncertainty can prevent our model from being misled by unreliable surface normal supervisions that hinder the accurate reconstruction of intricate geometries. Experiments on the benchmark datasets show that our method significantly outperforms existing methods in terms of reconstruction quality

Research on the Concentration Prediction of Nitrogen in Red Tide Based on an Optimal Grey Verhulst Model

In order to reduce the harm of red tide to marine ecological balance, marine fisheries, aquatic resources, and human health, an optimal Grey Verhulst model is proposed to predict the concentration of nitrogen in seawater, which is the key factor in red tide. The Grey Verhulst model is established according to the existing concentration data series of nitrogen in seawater, which is then optimized based on background value and time response formula to predict the future changes in the nitrogen concentration in seawater. Finally, the accuracy of the model is tested by the posterior test. The results show that the prediction value based on the optimal Grey Verhulst model is in good agreement with the measured nitrogen concentration in seawater, which proves the effectiveness of the optimal Grey Verhulst model in the forecast of red tide

Research on the Concentration Prediction of Nitrogen in Red Tide Based on an Optimal Grey Verhulst Model

In order to reduce the harm of red tide to marine ecological balance, marine fisheries, aquatic resources, and human health, an optimal Grey Verhulst model is proposed to predict the concentration of nitrogen in seawater, which is the key factor in red tide. The Grey Verhulst model is established according to the existing concentration data series of nitrogen in seawater, which is then optimized based on background value and time response formula to predict the future changes in the nitrogen concentration in seawater. Finally, the accuracy of the model is tested by the posterior test. The results show that the prediction value based on the optimal Grey Verhulst model is in good agreement with the measured nitrogen concentration in seawater, which proves the effectiveness of the optimal Grey Verhulst model in the forecast of red tide