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Related Experiment Video

Updated: Sep 26, 2025

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An efficient numerical method for a time-fractional telegraph equation.

Jian Huang1, Zhongdi Cen1, Aimin Xu1

  • 1Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China.

Mathematical Biosciences and Engineering : MBE
|April 18, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces a novel numerical method for the time-fractional telegraph equation. The developed integral-difference scheme achieves second-order convergence for both time and spatial discretizations, validated by experiments.

Keywords:
caputo derivativefinite differenceintegro-differential equationsingularitytelegraph equation

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Area of Science:

  • Numerical analysis
  • Partial differential equations
  • Fractional calculus

Background:

  • The time-fractional telegraph equation models various physical phenomena.
  • Existing numerical methods may face challenges with solution singularities.
  • Efficient and accurate approximation techniques are crucial for these equations.

Purpose of the Study:

  • To develop and analyze a new numerical scheme for the time-fractional telegraph equation.
  • To address the potential singularity of the exact solution in the approximation.
  • To achieve high-order accuracy in both temporal and spatial dimensions.

Main Methods:

  • Transformation of the time-fractional telegraph equation into an integral-differential equation with a weakly singular kernel.
  • Development of an integral-difference discretization scheme.
  • Utilizing a graded mesh for improved accuracy.
  • Convergence analysis considering solution singularities.

Main Results:

  • The proposed scheme accurately approximates the integral-differential equation.
  • The scheme is proven to be second-order convergent for spatial discretization.
  • The scheme is proven to be second-order convergent for time discretization.
  • Numerical experiments corroborate the theoretical convergence rates.

Conclusions:

  • The developed integral-difference scheme provides an efficient and accurate method for solving the time-fractional telegraph equation.
  • The scheme's second-order convergence is robust even in the presence of solution singularities.
  • This work contributes a valuable tool for the numerical simulation of fractional-order models.