A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Two New Approximations for Variable-Order Fractional Derivatives

We introduced a parameter σ(t) which was related to α(t); then two numerical schemes for variable-order Caputo fractional derivatives were derived; the second-order numerical approximation to variable-order fractional derivatives α(t)∈(0,1) and 3-α(t)-order approximation for α(t)∈(1,2) are established. For the given parameter σ(t), the error estimations of formulas were proven, which were higher than some recently derived schemes. Finally, some numerical examples with exact solutions were studied to demonstrate the theoretical analysis and verify the efficiency of the proposed methods

Finite difference method for time-fractional Klein-Gordon equation on an unbounded domain using artificial boundary conditions

A finite difference method for time-fractional Klein-Gordon equation with the fractional order $\alpha \in (1, 2]$ on an unbounded domain is studied. The artificial boundary conditions involving the generalized Caputo derivative are derived using the Laplace transform technique. Stability and error estimates of the proposed finite difference scheme are proved in detail by using the discrete energy method. Numerical examples show that the artificial boundary method is a robust and efficient method for solving the time-fractional Klein-Gordon equation on an unbounded domain

Error estimates of a continuous Galerkin time stepping method for subdiffusion problem

A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and L-type methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity

On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation

By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in [0,T]×R2: ∂xxu+u∂yu−∂tu=f(⋅,u), provided that T is suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time

A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

A new high-order finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k-1}) \big ), k=1, 2, \dots, N,$ with the convergence order $O(\Delta t^{4-\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4-\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Anchor Retouching via Model Interaction for Robust Object Detection in Aerial Images

Object detection has made tremendous strides in computer vision. Small object detection with appearance degradation is a prominent challenge, especially for aerial observations. To collect sufficient positive/negative samples for heuristic training, most object detectors preset region anchors in order to calculate Intersection-over-Union (IoU) against the ground-truthed data. In this case, small objects are frequently abandoned or mislabeled. In this paper, we present an effective Dynamic Enhancement Anchor (DEA) network to construct a novel training sample generator. Different from the other state-of-the-art techniques, the proposed network leverages a sample discriminator to realize interactive sample screening between an anchor-based unit and an anchor-free unit to generate eligible samples. Besides, multi-task joint training with a conservative anchor-based inference scheme enhances the performance of the proposed model while reducing computational complexity. The proposed scheme supports both oriented and horizontal object detection tasks. Extensive experiments on two challenging aerial benchmarks (i.e., DOTA and HRSC2016) indicate that our method achieves state-of-the-art performance in accuracy with moderate inference speed and computational overhead for training. On DOTA, our DEA-Net which integrated with the baseline of RoI-Transformer surpasses the advanced method by 0.40% mean-Average-Precision (mAP) for oriented object detection with a weaker backbone network (ResNet-101 vs ResNet-152) and 3.08% mean-Average-Precision (mAP) for horizontal object detection with the same backbone. Besides, our DEA-Net which integrated with the baseline of ReDet achieves the state-of-the-art performance by 80.37%. On HRSC2016, it surpasses the previous best model by 1.1% using only 3 horizontal anchors

A reconfigurable multi-terrain adaptive casualty transport aid base on Watt II six-bar linkage for industrial environment

Introduction: This paper presents the Reconfigurable Multi-Terrain Adaptive Casualty Transport Aid (RMTACTA), an innovative solution addressing the critical need for rapid and safe pre-hospital casualty transport in industrial environments. The RMTACTA, leveraging the Watt II six-bar linkage, offers enhanced adaptability through six modes of motion, overcoming the limitations of traditional stretchers and stretcher vehicles by facilitating navigation across narrow and challenging terrains.Methods: The RMTACTA's design incorporates two branching four-bar mechanisms to form a compact, reconfigurable Watt II six-bar linkage mechanism. This setup is controlled via a single remote rope, allowing for easy transition between its multiple operational modes, including stretcher, stretcher vehicle, folding, gangway-passing, obstacle-crossing, and upright modes. The mechanical design and kinematics of this innovative linkage are detailed, alongside an analysis of the optimal design and mechanical evaluation of rope control.Results: A prototype of the RMTACTA was developed, embodying the proposed mechanical and kinematic solutions. Preliminary tests were conducted to verify the prototype's feasibility and operability across different terrains, demonstrating its capability to safely and efficiently transport casualties.Discussion: The development of the proposed Reconfigurable Multi-Terrain Adaptive Casualty Transport Aid (RMTACTA) introduces a novel perspective on the design of emergency medical transport robots and the enhancement of casualty evacuation strategies. Its innovative application of the Watt II six-bar linkage mechanism not only showcases the RMTACTA's versatility across varied terrains but also illuminates its potential utility in critical scenarios such as earthquake relief, maritime rescue, and battlefield medical support

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear space-fractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results