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Nonstandard finite difference method for solving complex-order fractional Burgers' equations.

N H Sweilam1, S M Al-Mekhlafi2, D Baleanu3,4

  • 1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

Journal of Advanced Research
|September 14, 2020
PubMed
Summary

This study introduces a new numerical method for complex fractional Burgers' equations. The unconditionally stable finite-difference scheme accurately models fractional structures in nonlinear problems.

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Area of Science:

  • Numerical analysis
  • Applied mathematics
  • Nonlinear dynamics

Background:

  • Fractional calculus extends classical calculus to non-integer orders, enabling modeling of complex systems.
  • Burgers' equation is a fundamental model in fluid dynamics and nonlinear wave phenomena.
  • Existing numerical methods often struggle with the complexities of fractional nonlinear differential equations.

Purpose of the Study:

  • To develop and analyze a numerical treatment for a one-dimensional, nonlinear, complex order fractional Burgers' equation.
  • To introduce a parameter characterizing fractional structures and derive its relation to the complex order of the time derivative.
  • To establish an unconditionally stable numerical scheme for this class of problems.

Main Methods:

  • A novel parameter was introduced to represent fractional structures within the Burgers' equation.
  • A relation between this parameter and the complex order of the time derivative was derived.
  • An unconditionally stable numerical scheme based on weighted average nonstandard finite-difference discretization was developed and analyzed.

Main Results:

  • The proposed numerical scheme demonstrates unconditional stability.
  • Numerical simulations confirm the reliability and accuracy of the method for the complex fractional Burgers' equation.
  • The introduced parameter provides a consistent link to the physical characteristics of fractional structures.

Conclusions:

  • The presented numerical method offers a robust and stable approach for solving complex order fractional nonlinear Burgers' equations.
  • The work contributes to the understanding and numerical treatment of fractional differential equations with applications in various scientific fields.
  • The findings validate the effectiveness of the proposed finite-difference scheme for problems exhibiting fractional dynamics.