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Related Concept Videos

State Space Representation01:27

State Space Representation

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Difference Equation Solution using z-Transform01:24

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The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
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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.
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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
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Transfer Function to State Space01:23

Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
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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.
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Explicit scheme for solving variable-order time-fractional initial boundary value problems.

Asia Kanwal1, Salah Boulaaras2, Ramsha Shafqat3

  • 1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People's Republic of China.

Scientific Reports
|March 5, 2024
PubMed
Summary

This study introduces an explicit finite difference scheme for solving fractional differential equations with variable-order temporal fractional derivatives. The method utilizes the Caputo derivative and is proven conditionally stable via Fourier analysis.

Keywords:
Caputo derivativeExplicit schemeFractional derivativesFractional diffusion equationsInitial boundary value problemStability analysis

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Area of Science:

  • Numerical Analysis
  • Fractional Calculus
  • Applied Mathematics

Background:

  • Fractional calculus is crucial for modeling systems with memory and hereditary properties.
  • Variable-order temporal fractional derivatives offer a more flexible approach to modeling complex phenomena.
  • Initial boundary value problems (IBVPs) are fundamental in many scientific and engineering disciplines.

Purpose of the Study:

  • To develop an explicit finite difference scheme for linear and semi-linear IBVPs with variable-order temporal fractional derivatives.
  • To establish the stability of the proposed numerical scheme.
  • To demonstrate the scheme's efficacy through numerical examples.

Main Methods:

  • Development of an explicit finite difference scheme.
  • Utilization of the Caputo fractional derivative.
  • Fourier stability analysis of the numerical scheme.
  • Numerical simulations using MATLAB.

Main Results:

  • The explicit finite difference scheme effectively resolves the targeted fractional IBVPs.
  • Fourier analysis confirms the conditional stability of the scheme.
  • Numerical examples validate the accuracy and applicability of the method.

Conclusions:

  • The proposed explicit finite difference scheme is a viable tool for solving variable-order fractional differential equations.
  • The Caputo derivative is well-suited for capturing memory effects in these problems.
  • The study provides a foundation for further research in numerical methods for fractional calculus.