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Related Concept Videos

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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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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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.
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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Linear Approximation in Frequency Domain01:26

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
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Difference Equation Solution using z-Transform01:24

Difference Equation Solution using z-Transform

220
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.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
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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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Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
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Parameter-uniform numerical method for a coupled system of singularly perturbed turning point problems with Robin boundary conditions.

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A new approach for solving fuzzy non-linear equations using higher order iterative method.

Srilakshmi Katuri1, Prashanth Maroju2

  • 1Department of Mathematics, School of Advanced Sciences, VIT AP University, Amaravathi, Andhra Pradesh, India.

Scientific Reports
|April 15, 2025
PubMed
Summary

We introduce a new tenth-order iterative method for solving fuzzy nonlinear equations that avoids Jacobian matrix computations. This novel approach significantly reduces computational complexity and demonstrates superior efficiency compared to existing methods.

Keywords:
Dual fuzzy non-linear equationsFuzzyIterative methodsJacobian matrix

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Area of Science:

  • Numerical Analysis
  • Fuzzy Mathematics
  • Computational Science

Background:

  • Fuzzy nonlinear equations are crucial in optimization, decision-making, control theory, and chemical engineering.
  • Solving these equations often involves computationally intensive Jacobian matrix calculations and inversions.
  • Existing methods face challenges due to high computational demands.

Purpose of the Study:

  • To present a novel multi-step, tenth-order iterative method for solving fuzzy nonlinear equations.
  • To develop an approach that eliminates the need for Jacobian matrix computations.
  • To enhance the efficiency and reduce the computational complexity of solving fuzzy nonlinear equations.

Main Methods:

  • A novel multi-step iterative scheme with tenth-order convergence is proposed.
  • The method is designed to avoid the calculation and inversion of Jacobian matrices.
  • Rigorous convergence analysis is performed to establish the method's order of convergence.

Main Results:

  • The developed iterative method achieves a tenth-order convergence rate.
  • The approach significantly reduces computational complexity by eliminating Jacobian matrix operations.
  • Numerical examples and real-life applications demonstrate the method's effectiveness and robustness.

Conclusions:

  • The proposed tenth-order iterative method offers a computationally efficient alternative for solving fuzzy nonlinear equations.
  • The elimination of Jacobian matrix computations leads to marked improvements in performance.
  • The method's effectiveness and superior efficiency are validated through comprehensive numerical studies.