This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem ...This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem for general boundary conditions is solved in detail, on the basis of which Neumann and Robin conditions are easily handled. The solution to the simpler problem in cylindrical coordinates is also provided. Ways to efficiently implement the formulae are explained. Minor adjustments result in solutions to the wave equation and to the heat equation on the same domain as well, since the latter are particular cases of the more general telegrapher’s equation.展开更多
对于固定收益产品定价这个问题已经有很多种方法,将从另外一种角度来考虑这个问题,先通过T ay lor展开得到一个双曲型的偏微分方程,利用这个方程可以求出未定权益组合的最好最坏情景下的价格,然后再利用市场上已有的产品对此未定权益静...对于固定收益产品定价这个问题已经有很多种方法,将从另外一种角度来考虑这个问题,先通过T ay lor展开得到一个双曲型的偏微分方程,利用这个方程可以求出未定权益组合的最好最坏情景下的价格,然后再利用市场上已有的产品对此未定权益静态对冲,将会得到一个收益率曲线包络.展开更多
When a valve is suddenly closed in fluid transport pipelines,a pressure surge or shock is created along the pipeline due to the momentum change.This phenomenon,called hydraulic shock,can cause major damage to the pipe...When a valve is suddenly closed in fluid transport pipelines,a pressure surge or shock is created along the pipeline due to the momentum change.This phenomenon,called hydraulic shock,can cause major damage to the pipelines.In this paper,we introduce a hyperbolic partial differential equation(PDE)system to describe the fluid flow in the pipeline and propose an optimal boundary control problem for pressure suppression during the valve closure.The boundary control in this system is related to the valve actuation located at the pipeline terminus through a valve closing model.To solve this optimal boundary control problem,we use the method of lines and orthogonal collocation to obtain a spatial-temporal discretization model based on the original pipeline transmission PDE system.Then,the optimal boundary control problem is reduced to a nonlinear programming(NLP)problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming(SQP).Finally,we conclude the paper with simulation results demonstrating that the full parameterization(FP)method eliminates pressure shock effectively and costs less computation time compared with the control vector parameterization(CVP)method.展开更多
文摘This article provides a closed form solution to the telegrapher’s equation with three space variables defined on a subset of a sphere within two radii, two azimuthal angles and one polar angle. The Dirichlet problem for general boundary conditions is solved in detail, on the basis of which Neumann and Robin conditions are easily handled. The solution to the simpler problem in cylindrical coordinates is also provided. Ways to efficiently implement the formulae are explained. Minor adjustments result in solutions to the wave equation and to the heat equation on the same domain as well, since the latter are particular cases of the more general telegrapher’s equation.
基金partially supported by the National Natural Science Foundation of China(61703217,61703114)the K.C.Wong Magna Fund in Ningbo University,the Open Project of Key Laboratory of Industrial Internet of Things and Networked Control(2018FF02)the Open Research Project of the State Key Laboratory of Industrial Control Technology,Zhejiang University,China(ICT1900313)
文摘When a valve is suddenly closed in fluid transport pipelines,a pressure surge or shock is created along the pipeline due to the momentum change.This phenomenon,called hydraulic shock,can cause major damage to the pipelines.In this paper,we introduce a hyperbolic partial differential equation(PDE)system to describe the fluid flow in the pipeline and propose an optimal boundary control problem for pressure suppression during the valve closure.The boundary control in this system is related to the valve actuation located at the pipeline terminus through a valve closing model.To solve this optimal boundary control problem,we use the method of lines and orthogonal collocation to obtain a spatial-temporal discretization model based on the original pipeline transmission PDE system.Then,the optimal boundary control problem is reduced to a nonlinear programming(NLP)problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming(SQP).Finally,we conclude the paper with simulation results demonstrating that the full parameterization(FP)method eliminates pressure shock effectively and costs less computation time compared with the control vector parameterization(CVP)method.
