关于一类微积分计算方法的探究

通过对一类微积分计算方法的探究,从本质上分析所形成的思维模式,给出了一元微分以及积分区域为圆的二重积分通过平移变换简单巧妙的计算方法.由此可以类比出多元微分,以及多重积分的计算方法.     

Periodic and solitary wave solutions of the attractive and repulsive nonlinear Schrödinger equation

By means of symbolic computation, the first integral method is presented to obtain novel exact solutions of the nonlinear evolution equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed and the availability of symbolic computation is demonstrated.     

Applications of the Conjugate Gradient method in optimal surface parameterizations

This paper concerns the solutions of very large symmetric semipositive definite (singular) linear systems involved in the problem of optimal surface parameterizations using inverse curvature mapping. Two approaches are presented that transform the singular linear systems into two kinds of symmetric positive definite linear systems, so that the famous Conjugate Gradient (CG) method can be used for solving them. Numerical experiments are run on two practical large problems to illustrate that the CG algorithm works very efficiently.     

Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation X+A T X −2 A=I

Necessary and sufficient conditions for the matrix equation X+A ^T X ^?2 A=I to have a real symmetric positive definite solution X are derived. Based on these conditions, some properties of the matrix A as well as relations between the solution X and A are derived.     

Robust projection method for Fredholm integral equation of the second kind

The projection method based on the Schauder basis expansion for solving the Fredholm integral equation of the second kind is investigated. (1) It is shown that any given finite linearly independent function set in L ^p [0, 1] can be extended as a Schauder basis. The truncation of the dual Schauder bases is then used to form the projection in approximating the kernel. (2) With the finite rank approximation, the use of the Sherman–Morrison–Woodbury formula achieves efficient implementation. The computation is successive when dimension of projection space increases. Numerical experiments demonstrate the robustness of the Schauder basis expansion.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

This paper presents the mechanical quadrature methods (MQMs) for solving the boundary integral equations of steady-state anisotropic heat conduction equation on the smooth domains and polygons, respectively. The costless and high-accurate Sidi–Israeli quadrature formula are applied to deal with the integrals in which the kernels have a logarithmic singularity. Especially, the Sidi transformation is used for the polygon cases in order to obtain a rapid convergence by degrading the singularity at the corners on the boundary. The convergence and stability of the MQMs solution are proved based on Anselone's collective compact theory. In addition, asymptotic error expansion of the MQMs shows that the approximation order is of O(h³), where h is the partition size of the boundary. Finally, numerical examples are tested and results verify the theoretical analysis.     

沿虚轴无穷区问主值积分与逆傅里叶积分变换

在工程技术和科学研究的许多领域,傅里叶积分变换极为重要,但逆傅里叶积分变换手工计算比较困难,限制了傅里叶积分变换的应用范围。研究发现,逆傅里叶积分变换可以变换成沿复平面虚轴上的无穷区间主值积分,由此,导出一个逆傅里叶积分变换的计算公式,可用来快速完成逆傅里叶积分变换计算。     

一种积分型切换函数的滑模控制算法在倒立摆系统中的仿真应用

针对实际控制系统的非线性和某种程度上存在扰动的特点,采用基于积分切换函数的滑模变结构控制算法不仅能减小由于扰动带来的稳态误差,同时对抑制抖振现象有很好的效果。另一方面为了使系统能够快速到达切换面,在指数趋近律的基础上增加变速部分保证整个趋近过程用时很短。将此算法应用到一级倒立摆系统中进行仿真验证,仿真结果显示了此算法克服抖振的效果。     

基于反演法的智能车辆弯路换道轨迹跟踪控制

基于车辆的运动学模型,研究了具有非完整约束特性的智能车辆弯路换道的轨迹跟踪控制问题。结合弯曲车道的曲率半径,针对换道过程中车辆横向运动和纵向运动对换道轨迹的耦合作用,提出了一种适用于弯道的换道轨迹规划方法。根据换道轨迹,利用积分反演法,引入中间虚拟控制量,设计了轨迹跟踪控制算法,并用Lyapunov稳定性理论证明了控制系统的稳定性。仿真结果表明:该控制器可以保证车辆在弯路换道的轨迹跟踪误差一致有界收敛,且具有快速、准确跟踪和全局稳定的特性。