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Finite Difference Method for Time-Space Fractional Advection-Diffusion Equations with Riesz Derivative.

Sadia Arshad1,2, Dumitru Baleanu3,4, Jianfei Huang5

  • 1The State Key Laboratory of Scientific and Engineering Computing (LSEC), The Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a novel numerical method for solving fractional advection-diffusion equations. The scheme achieves second-order accuracy in both time and space, validated by numerical experiments.

Keywords:
caputo derivativefractional advection dispersion equationriesz derivativetrapezoidal formula

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Partial differential equations

Background:

  • Fractional calculus extends traditional calculus to non-integer orders, enabling modeling of complex phenomena.
  • Advection-diffusion equations describe transport processes in various scientific fields.
  • Time-space fractional differential equations capture anomalous diffusion and complex transport behaviors.

Purpose of the Study:

  • To develop and analyze a numerical scheme for the time-space fractional advection-diffusion equation.
  • To approximate Riesz spatial derivatives and Caputo temporal derivatives.
  • To rigorously investigate the stability and convergence properties of the proposed numerical method.

Main Methods:

  • Approximation of the Riesz space derivative using a second-order fractional weighted and shifted Grünwald-Letnikov formula.
  • Transformation of the fractional differential equation into an equivalent integral equation.
  • Approximation of the resulting integral equation using the trapezoidal formula.

Main Results:

  • A numerical scheme is formulated and analyzed for the time-space fractional advection-diffusion equation.
  • The scheme is proven to have second-order accuracy in both temporal and spatial directions.
  • Stability and convergence analyses are rigorously performed.

Conclusions:

  • The developed numerical scheme accurately solves the time-space fractional advection-diffusion equation.
  • The theoretical findings on accuracy, stability, and convergence are supported by numerical experiments.
  • This work provides an efficient and reliable tool for simulating fractional advection-diffusion processes.