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Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations.

Zhi Mao1, Aiguo Xiao2, Zuguo Yu2

  • 1Hunan Key Laboratory for Computation and Simulation in Science and Engineering and Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, China ; Mathematics and Information Engineering Department, Tongren University, Tongren, Guizhou 554300, China.

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Summary

This study presents a numerical method for fractional diffusion-wave equations using shifted Chebyshev polynomials and sinc functions. The approach efficiently solves these complex equations, confirmed by a numerical example.

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Area of Science:

  • Numerical analysis
  • Partial differential equations
  • Fractional calculus

Background:

  • Fractional diffusion-wave equations model complex phenomena.
  • Variable coefficients and Caputo derivatives present numerical challenges.
  • Efficient and accurate numerical solutions are crucial.

Purpose of the Study:

  • To develop and validate a numerical method for fractional diffusion-wave equations with variable coefficients.
  • To utilize spectral methods for enhanced accuracy and efficiency.
  • To confirm the method's effectiveness through a numerical example.

Main Methods:

  • Collocation technique.
  • Shifted Chebyshev polynomials for temporal discretization.
  • Sinc functions for spatial discretization.
  • Reduction to a system of linear algebraic equations.

Main Results:

  • The proposed method effectively solves the targeted fractional diffusion-wave equations.
  • Numerical results demonstrate the accuracy and efficiency of the technique.
  • The method successfully handles variable coefficients and Caputo fractional derivatives.

Conclusions:

  • The collocation method with shifted Chebyshev polynomials and sinc functions provides a robust numerical solution.
  • This approach offers a reliable tool for analyzing fractional diffusion-wave phenomena.
  • The study confirms the efficiency and applicability of the presented numerical procedure.