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相关概念视频

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

85
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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Navier–Stokes Equations01:28

Navier–Stokes Equations

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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
526
Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

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Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
316
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
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Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

3.0K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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Second Order systems II01:18

Second Order systems II

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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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相关实验视频

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视觉PDE:部分微分方程的快速交互模拟.

Benjamin J Walker1,2, Adam K Townsend3, Alexander K Chudasama3

  • 1Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK.

Bulletin of mathematical biology
|October 12, 2023
PubMed
概括

VisualPDE是一个新的在线工具,通过交互式探索使复杂的非线性系统更容易理解. 这种部分微分方程的开源解决方案通过允许立即操纵模型来增强教育和研究.

关键词:
交互式数学 交互式数学空间建模 空间建模时间依赖的部分微分方程.基于网络的可视化.

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科学领域:

  • 计算科学 计算科学
  • 应用数学 应用数学 应用数学
  • 非线性动力学是一种非线性动力学.

背景情况:

  • 计算机已经改变了对复杂非线性系统的研究,为难以处理的模型和新的可视化技术提供了解决方案.
  • 通过无处不在的计算基础设施进行交互式探索,为理解复杂模型提供了一种强大的方法.
  • 现有的工具往往缺乏用于直观探索动态系统所需的即时和交互功能.

研究的目的:

  • 介绍VisualPDE,一个在线的,部分微分方程 (PDEs) 的交互式解法器.
  • 通过实践探索,促进对抽象动态系统概念的直观理解.
  • 为教育,研究和知识交流提供免费可用的,开源和可定制的工具.

主要方法:

  • 为1D和2D部分微分方程系统开发在线交互式解答器.
  • 实现允许用户探索系统对参数变化,初始条件,边界条件和时空强迫的响应的功能.
  • 设计为高可定制性和易于通过URL共享模拟.

主要成果:

  • 视觉PDE能够直观地理解诸如对称性破坏不稳定性和分叉等概念.
  • 该工具允许立即调查模型如何应对各种动态变化.
  • 作为免费可用的开源软件,VisualPDE对于各种应用程序具有高度可定制性.

结论:

  • VisualPDE提供了一个强大的交互平台,用于探索复杂的非线性系统和PDEs.
  • 它在增强数学生物学和相关领域的教学,研究和知识交流方面具有显著的潜力.
  • 该工具旨在促进更具互动性和可访问性的数学探索.