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

Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

3.1K
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.
3.1K
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

1.3K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
1.3K
Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

350
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...
350
Differential Form of Maxwell's Equations01:17

Differential Form of Maxwell's Equations

529
James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
529
RLC Circuit as a Damped Oscillator01:30

RLC Circuit as a Damped Oscillator

1.1K
An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...
1.1K
RLC Series Circuits01:30

RLC Series Circuits

3.0K
An RLC series circuit comprises an inductor, a resistor, and a charged capacitor connected in series. When the circuit is closed, the capacitor begins to discharge through the resistor and inductor by transferring energy from the electric field to the magnetic field. Here, the resistor connected to the circuit causes energy losses; therefore, on the complete discharge of the capacitor, the magnetic field energy acquired by the inductor is less than the original electric field energy of the...
3.0K

You might also read

Related Articles

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

Sort by
Same journal

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order.

Communications on applied mathematics and computation·2025
Same journal

A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity.

Communications on applied mathematics and computation·2024
Same journal

Piecewise Acoustic Source Imaging with Unknown Speed of Sound Using a Level-Set Method.

Communications on applied mathematics and computation·2024
Same journal

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic.

Communications on applied mathematics and computation·2024
Same journal

A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations.

Communications on applied mathematics and computation·2023
Same journal

Von Neumann Stability Analysis of DG-Like and P<i>N</i>P<i>M</i>-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields.

Communications on applied mathematics and computation·2022

Related Experiment Video

Updated: Jul 25, 2025

The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

The Diffusion of Passive Tracers in Laminar Shear Flow

Published on: May 1, 2018

8.6K

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

Zhen Wang1

  • 1School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013 Jiangsu China.

Communications on Applied Mathematics and Computation
|June 26, 2023
PubMed
Summary

This study introduces novel discrete Gronwall inequalities to analyze L1/local discontinuous Galerkin (LDG) methods for fractional diffusion equations. These inequalities ensure the numerical methods are robust, even for small time steps.

Keywords:
Caputo-Hadamard derivativeDiscrete Gronwall inequalityError estimateL1 formulaLocal discontinuous Galerkin (LDG) method

More Related Videos

Spot Variation Fluorescence Correlation Spectroscopy for Analysis of Molecular Diffusion at the Plasma Membrane of Living Cells
05:56

Spot Variation Fluorescence Correlation Spectroscopy for Analysis of Molecular Diffusion at the Plasma Membrane of Living Cells

Published on: November 12, 2020

2.8K
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

7.9K

Related Experiment Videos

Last Updated: Jul 25, 2025

The Diffusion of Passive Tracers in Laminar Shear Flow
08:01

The Diffusion of Passive Tracers in Laminar Shear Flow

Published on: May 1, 2018

8.6K
Spot Variation Fluorescence Correlation Spectroscopy for Analysis of Molecular Diffusion at the Plasma Membrane of Living Cells
05:56

Spot Variation Fluorescence Correlation Spectroscopy for Analysis of Molecular Diffusion at the Plasma Membrane of Living Cells

Published on: November 12, 2020

2.8K
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

7.9K

Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Fractional diffusion equations model complex physical phenomena.
  • Existing numerical methods may face challenges with robustness, especially for small time steps.
  • The Caputo-Hadamard fractional derivative is a key component in certain diffusion models.

Purpose of the Study:

  • To propose a new class of discrete Gronwall inequalities.
  • To apply these inequalities to analyze L1/local discontinuous Galerkin (LDG) finite element methods.
  • To demonstrate the -robustness of the numerical methods for the Caputo-Hadamard time fractional diffusion equation.

Main Methods:

  • Development of discrete Gronwall inequalities.
  • Analysis of L1/local discontinuous Galerkin (LDG) finite element methods.
  • Theoretical analysis of numerical scheme stability and convergence.

Main Results:

  • A novel class of discrete Gronwall inequalities was established.
  • The L1/LDG finite element methods were analyzed using the new inequalities.
  • The numerical methods were proven to be -robust, maintaining validity for small values.

Conclusions:

  • The proposed discrete Gronwall inequalities are effective tools for analyzing numerical methods for fractional diffusion equations.
  • The L1/LDG methods are robust and suitable for solving the Caputo-Hadamard time fractional diffusion equation.
  • Theoretical findings are supported by numerical experiments.