Search

An offline and online bi-level structure-based dynamic path planning algorithm is proposed for an unmanned aerial vehicle (UAV) in low-altitude complex urban environment. First, an improved Hunger Games Search (HGS) algorithm is developed to generate an offline optimized path under the UAV’s performance constraints and the known static obstacles’ constraints. The individuals of the proposed algorithm will be divided into multiple groups to increase the population diversity. And then, a dynamic grouping strategy and a quantum-behaved behavior are proposed to solve the premature convergence’s problem and the imbalance problem between exploration and exploitation ability in HGS. To improve the dynamic obstacle avoidance efficiency of the algorithm, the dynamic obstacles are classified into three categories: newly added no-fly zone, known and unknown dynamic obstacles. Then, utilizing the information of the offline optimized path and the airborne sensors, three kinds of online planning strategies—an improved rapid-exploring random tree (RRT), a changing speed strategy, and a novel three-dimensional rolling windows—are introduced to dynamically update the path or speed of the UAV. Simulation results indicated that the improved HGS can enhance the performances of the traditional HGS and outperform other compared algorithms on the benchmark functions. Meanwhile, the online planning strategies can effectively achieve dynamic obstacle avoidance within the constraints of offline path. More specially, the planning time and angles of the local path to avoid the no-fly-zone’s influence are improved by 11.3% and 56.8% through utilizing the improved RRT.

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

In this paper, the exact solutions of a reduced extended $$(3+1)$$ -dimensional Jimbo–Miwa equation are investigated with the help of its bilinear representation and symbolic computation. Firstly, a kind of bright–dark lump wave solutions is directly obtained by taking the solution F in bilinear equation as a quadratic function. Furthermore, the interaction solutions between one lump wave and one stripe wave are also presented by taking F as a combination of quadratic function and exponential function. Finally, by taking F as a combination of quadratic function and hyperbolic cosine function, the rogue wave which aroused by the interaction between lump soliton and a pair of stripe solitons are obtained. The dynamic properties of the above three kinds of exact solutions are displayed vividly by figures.

As we known, reproducing kernel method (RKM) has been presented for solving differential equations for initial and boundary value problems. However, the direct application of the RKM presented in the previous works cannot produce good numerical results for Burgers’ equation. To solve this problem, this paper give a modified reproducing kernel method by piecewise technique. The exact solution is given by reproducing kernel functions in a series expansion form, the approximation solution is expressed by n-term summation of reproducing kernel functions. The three numerical experiments results show that the piecewise method is more easily implemented and effective. Some numerical results are also compared with the results obtained by other methods.

This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n-term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective.

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations (PDEs).

The aim of this work is to obtain a numerical solution of a time-fractional telegraph equation by a modified reproducing kernel method. Two numerical examples are given to show that the present method overcomes the drawback of the traditional reproducing kernel method and it is an easy and effective method.