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Lattice Boltzmann method for the multiterm time-fractional mixed diffusion and diffusion-wave equations.

Jia Tong1, Fangfang Wu1, Xiaoxiao Dong1

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A novel lattice Boltzmann model effectively solves complex multiterm time-fractional equations. This computational fluid dynamics approach shows excellent agreement with analytical solutions, validating its efficiency for diffusion and wave phenomena.

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Area of Science:

  • Computational physics
  • Applied mathematics
  • Numerical analysis

Background:

  • Fractional differential equations (FDEs) model complex phenomena.
  • Multiterm time-fractional equations present significant computational challenges.
  • Existing methods may lack efficiency or generality.

Purpose of the Study:

  • To develop a unified lattice Boltzmann model for multiterm time-fractional mixed diffusion and diffusion-wave equations.
  • To validate the model's accuracy and efficiency through numerical simulations.
  • To provide a robust computational tool for fractional calculus applications.

Main Methods:

  • Approximation of Caputo derivative terms using numerical differentiation and composite integration rules.
  • Selection of specific auxiliary and equilibrium distribution functions.
  • Implementation of the lattice Boltzmann method for solving FDEs.

Main Results:

  • The proposed lattice Boltzmann model successfully recovers the macroscopic equation.
  • Numerical simulations demonstrate excellent agreement between the model's results and analytical solutions.
  • The efficiency and accuracy of the unified model are validated.

Conclusions:

  • The developed lattice Boltzmann model offers an efficient and accurate approach for solving multiterm time-fractional mixed diffusion and diffusion-wave equations.
  • This method provides a reliable computational framework for problems involving fractional calculus.
  • The study validates the broad applicability of the lattice Boltzmann method in complex scientific domains.