Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling

Abstract

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.

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Acknowledgments

The author thanks the reviewers for their useful comments and suggestions that improved the paper.

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Correspondence to Mohammad Ilati.

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Ilati, M. Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. Numer Algor 85, 485–502 (2020). https://doi.org/10.1007/s11075-019-00823-6

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Keywords

  • Extended Fisher–Kolmogorov equation
  • Interpolating element-free Galerkin method
  • Interpolating moving least squares method
  • Error estimate
  • Brain tumor dynamics modeling

Mathematics Subject Classification (2010)

  • 41A30; 65M99; 65N99