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This study introduces a multivariable linear algebraic discretization method to solve nonlinear parabolic equations (NPEs). The approach effectively addresses challenges in reaction-diffusion convergence and two-grid algorithms.

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Area of Science:

  • Numerical analysis
  • Computational mathematics

Background:

  • Nonlinear parabolic equations (NPEs) present computational challenges due to their complex algebraic nature.
  • Existing methods like the extended mixed finite element method can introduce significant nonlinearity.
  • Solving reaction-diffusion problems and two-grid algorithms for NPEs often faces convergence issues.

Purpose of the Study:

  • To propose a novel multivariable linear algebraic discretization method for NPEs.
  • To overcome the nonlinearity and convergence problems associated with traditional methods.
  • To enhance the efficiency and adaptability of solving NPEs.

Main Methods:

  • Discretizing the NPE and transforming its algebraic form into a vector form.
  • Utilizing rough set (RS) and information entropy (IE) to determine attribute weights for variables.
  • Applying linear algebra for the discretization of multivariable equations based on calculated weights.

Main Results:

  • The proposed method effectively solves the two-grid algorithm problems for NPEs.
  • It successfully addresses the convergence problem in reaction-diffusion processes.
  • Demonstrates good adaptability and performance in the field of NPEs.

Conclusions:

  • The multivariable linear algebraic discretization method offers an effective solution for NPEs.
  • This approach improves upon existing methods by mitigating nonlinearity and enhancing convergence.
  • The technique shows promise for broader applications in solving complex differential equations.