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Related Concept Videos

Second Order systems II01:18

Second Order systems II

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.
If  ζ...
Partial Fractions01:28

Partial Fractions

A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
Second Order systems I01:20

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...
The Quotient Rule01:30

The Quotient Rule

The quotient rule is a fundamental differentiation technique in calculus used to differentiate functions expressed as a ratio of two differentiable functions. Given a function of the form:Where g(x) and h(x) are both differentiable and h(x) ≠ 0, the derivative of f(x) is given by:Example:The quotient rule is beneficial when differentiating rational functions, trigonometric ratios, and exponential functions. For example, given:applying the quotient rule,This rule is essential in solving problems...
Rational Expressions01:28

Rational Expressions

Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.Determining the DomainThe domain of a rational expression includes all real...
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Integration of Rational Functions Using Partial Fractions

Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...

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Related Experiment Video

Updated: May 12, 2026

Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

Equivalent system for a multiple-rational-order fractional differential system.

Changpin Li1, Fengrong Zhang, Jürgen Kurths

  • 1Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China. lcp@shu.edu.cn

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|April 3, 2013
PubMed
Summary
This summary is machine-generated.

This study analyzes multiple-rational-order fractional differential systems using Caputo and Riemann-Liouville derivatives. It establishes equivalent systems to determine the stability of the zero solution for these complex fractional differential equations.

Related Experiment Videos

Last Updated: May 12, 2026

Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

Area of Science:

  • Mathematics
  • Applied Mathematics
  • Fractional Calculus

Background:

  • Fractional differential equations (FDEs) offer advanced modeling capabilities.
  • Multiple-rational-order (MRO) FDEs present unique analytical challenges.
  • Stability analysis is crucial for understanding FDE system behavior.

Purpose of the Study:

  • To investigate the stability of the zero solution for MRO fractional differential systems.
  • To develop equivalent systems for analyzing MRO FDEs with Caputo and Riemann-Liouville derivatives.
  • To provide a framework for stability analysis in fractional calculus.

Main Methods:

  • Transformation of MRO fractional differential systems into higher-dimensional equivalent systems.
  • Utilizing the relationship between Caputo derivatives and generalized fractional derivatives.
  • Applying properties of Riemann-Liouville derivative and fractional integral operators.
  • Analysis of equivalent systems to determine stability of the zero solution.

Main Results:

  • An equivalent system is derived for MRO fractional differential systems with Caputo derivatives, simplifying stability analysis.
  • A transformation is presented for MRO fractional differential systems with Riemann-Liouville derivatives, enabling stability studies.
  • The stability of the zero solution is rigorously analyzed through the derived equivalent systems.
  • Numerical examples validate the theoretical findings.

Conclusions:

  • The study successfully establishes methods for analyzing the stability of MRO fractional differential systems.
  • The transformation into equivalent systems provides a powerful tool for understanding fractional dynamics.
  • The findings contribute to the theoretical framework of fractional calculus and its applications.