Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Differential Equations: Problem Solving01:21

Differential Equations: Problem Solving

106
When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
106
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

4.5K
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.
4.5K
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

383
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...
383
Newton’s Method01:30

Newton’s Method

109
Newton’s Method is a powerful iterative technique for approximating the roots of real-valued, differentiable functions, particularly when analytical solutions are impractical. This approach is widely used in scientific computing, engineering, and finance, where equations may be too complex for traditional algebraic methods to handle. The method relies on an iterative process that refines an initial estimate using the function’s derivative to approach the true solution progressively.
109
Estimation of the Physical Quantities01:05

Estimation of the Physical Quantities

8.5K
On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound...
8.5K
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

391
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,...
391

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

In-silico combinatorial design and pharmacophore modeling of potent antimalarial 4-anilinoquinolines utilizing QSAR and computed descriptors.

SpringerPlus·2017
Same journal

Erratum to: Implication of Paris Agreement in the context of long-term climate mitigation goals.

SpringerPlus·2017
Same journal

Erratum to: Associations between adherence, depressive symptoms and health-related quality of life in young adults with cystic fibrosis.

SpringerPlus·2017
Same journal

Erratum to: Numerical method to compute acoustic scattering effect of a moving source.

SpringerPlus·2017
Same journal

Identifying appropriate protected areas for endangered fern species under climate change.

SpringerPlus·2017
Same journal

An Algorithm to detect balancing of iterated line sigraph.

SpringerPlus·2017

Related Experiment Video

Updated: Mar 15, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.5K

Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via

Ayşe Kurt Bahşı1, Salih Yalçınbaş1

  • 1Department of Mathematics, Manisa Celal Bayar University, Manisa, Turkey.

Springerplus
|September 10, 2016
PubMed
Summary

This study introduces the Fibonacci collocation method to solve fractional diffusion equations. The method offers an efficient way to find approximate solutions and estimate errors, even without knowing the exact solution.

Keywords:
Collocation methodError estimation algorithmFibonacci polynomialsFractional diffusion equationsMatrix relations

More Related Videos

A Method for Determination and Simulation of Permeability and Diffusion in a 3D Tissue Model in a Membrane Insert System for Multi-well Plates
10:33

A Method for Determination and Simulation of Permeability and Diffusion in a 3D Tissue Model in a Membrane Insert System for Multi-well Plates

Published on: February 23, 2018

26.2K
Controlled Synthesis and Fluorescence Tracking of Highly Uniform PolyN-isopropylacrylamide Microgels
11:34

Controlled Synthesis and Fluorescence Tracking of Highly Uniform PolyN-isopropylacrylamide Microgels

Published on: September 8, 2016

10.8K

Related Experiment Videos

Last Updated: Mar 15, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.5K
A Method for Determination and Simulation of Permeability and Diffusion in a 3D Tissue Model in a Membrane Insert System for Multi-well Plates
10:33

A Method for Determination and Simulation of Permeability and Diffusion in a 3D Tissue Model in a Membrane Insert System for Multi-well Plates

Published on: February 23, 2018

26.2K
Controlled Synthesis and Fluorescence Tracking of Highly Uniform PolyN-isopropylacrylamide Microgels
11:34

Controlled Synthesis and Fluorescence Tracking of Highly Uniform PolyN-isopropylacrylamide Microgels

Published on: September 8, 2016

10.8K

Area of Science:

  • Numerical analysis
  • Applied mathematics
  • Computational science

Background:

  • Fractional diffusion equations with variable coefficients present significant challenges in modeling complex phenomena.
  • Existing numerical methods may lack efficiency or accuracy for these types of equations.
  • The development of robust and accurate solution techniques is crucial for advancing scientific understanding.

Purpose of the Study:

  • To present a novel numerical method, the Fibonacci collocation method, for solving fractional diffusion equations with variable coefficients.
  • To introduce an error estimation algorithm based on residual functions to improve solution accuracy.
  • To demonstrate the efficiency and applicability of the proposed method through numerical examples.

Main Methods:

  • The Fibonacci collocation method is employed, utilizing Fibonacci polynomials for approximating the solution.
  • Fractional derivatives are handled in the Caputo sense.
  • The fractional diffusion equation is transformed into a system of linear algebraic equations.

Main Results:

  • The Fibonacci collocation method effectively reduces the fractional diffusion equation to a system of linear algebraic equations.
  • An error estimation algorithm is developed, enabling the approximation of absolute errors.
  • The proposed error estimation allows for the improvement of approximate solutions, yielding results more efficient than direct numerical solutions.

Conclusions:

  • The Fibonacci collocation method provides an efficient and usable approach for solving fractional diffusion equations with variable coefficients.
  • The integrated error estimation algorithm enhances the reliability and accuracy of the numerical solutions.
  • The method demonstrates significant potential for applications where precise solutions to fractional differential equations are required.