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

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

Reinforcement Learning for Robot Navigation with Adaptive Forward Simulation Time (AFST) in a Semi-Markov Model

Deep reinforcement learning (DRL) algorithms have proven effective in robot navigation, especially in unknown environments, by directly mapping perception inputs into robot control commands. However, most existing methods ignore the local minimum problem in navigation and thereby cannot handle complex unknown environments. In this paper, we propose the first DRL-based navigation method modeled by a semi-Markov decision process (SMDP) with continuous action space, named Adaptive Forward Simulation Time (AFST), to overcome this problem. Specifically, we reduce the dimensions of the action space and improve the distributed proximal policy optimization (DPPO) algorithm for the specified SMDP problem by modifying its GAE to better estimate the policy gradient in SMDPs. Experiments in various unknown environments demonstrate the effectiveness of AFST

Application of a new self-regulating temperature magnetic material Fe83Zr10B7 in magnetic induction hyperthermia

AbstractIntroduction The temperature control of magnetic hyperthermia therapy mainly relies on circulating water cooling and regulating magnetic field intensity, which increases complexity in clinical applications. Using magnetic materials with appropriate Curie temperature has become an effective means to solve temperature monitoring and potentially achieve self-regulating temperature.Methods A self-temperature-regulating Fe83Zr10B7 magnetic material was prepared. Based on this material, a simplified model of magnetic hyperthermia for arm tumors was established and verified using the finite- element method. The influence of magnetic field intensity and frequency on the heating power and temperature rise rate of different-sized and shaped magnetic media was studied. Additionally, factors such as the size, quantity, and spatial arrangement of the magnetic media were analyzed for their impact on the damage to tumors with different volumes and shapes.Results Spherical shape is the most suitable for magnetic hyperthermia media, and the radius of the spherical magnetic media can be chosen according to the size of the tumor. For tumors with a radius below 10 mm, using magnetic media with a particle size of 3.5 mm is recommended. The optimal magnetic field conditions are H0 (10–12 kA/m) and f (110–120 kHz).Conclusion Based on the good magnetic properties and heating performance of the Fe83Zr10B7 magnetic material, it is feasible to use it as a magnetic medium for magnetic hyperthermia. The results of this study provide references for the selection of thermal seed size and magnetic field parameters in magnetic hyperthermia

Modulating built-in electric field via variable oxygen affinity for robust hydrogen evolution reaction in neutral media

Work function strongly impacts the surficial charge distribution, especially for metal-support electrocatalysts when a built-in electric field (BEF) is constructed. Therefore, studying the correlation between work function and BEF is crucial for understanding the intrinsic reaction mechanism. Herein, we present a Pt@CoOx electrocatalyst with a large work function difference (ΔΦ) and strong BEF, which shows outstanding hydrogen evolution activity in a neutral medium with a 4.5-fold mass activity higher than 20 % Pt/C. Both experimental and theoretical results confirm the interfacial charge redistribution induced by the strong BEF, thus subtly optimizing hydrogen and hydroxide adsorption energy. This work not only provides fresh insights into the neutral hydrogen evolution mechanism but also proposes new design principles toward efficient electrocatalysts for hydrogen production in a neutral medium.Agency for Science, Technology and Research (A*STAR)This work was financially supported by the Research Grants Council of Hong Kong (Poly U253009/18P) and the Hong Kong Polytechnic University (1-ZVGH). H.J.F. thanks the financial support from Agency for Science, Technology, and Research (A*STAR), Singapore by AME Individual Research Grants (A1983c0026)