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An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

  • Ömer OruçEmail author
Original Article
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Abstract

In this paper, we consider two-space dimensional nonlinear Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation and two-space dimensional nonlinear fourth-order extended Fisher–Kolmogorov (EFK) equation which have a lot of applications in different branches of science, especially in mathematical biology. We present a wavelet collocation method based on Chebyshev wavelets combined with two different time discretization schemes. To generate more accurate results for Fisher–KPP equation, in time variable discretization, forward Euler timestepping scheme along with Taylor series expansion is used. Resultant time-discretized scheme has second-order accuracy and includes second-order time derivative which is evaluated from governing equation. On the other hand for EFK equation forward Euler timestepping scheme with first-order accuracy is used. Then space variables situated in the semi-discrete schemes are discretized with Chebyshev wavelet series expansion. In this way a full discrete scheme is obtained. By this approach obtainment of numerical solution of considered partial differential equations is turned into an operation of finding solution of an algebraic system of equations. Actually the solution of the algebraic system of equations is wavelet coefficients in wavelet series expansion. Putting these coefficients into Chebyshev wavelet series expansion the numerical solution of considered partial differential equations can be obtained successively. The main objective of this paper is to demonstrate that Chebyshev wavelet-based method is accurate, efficient, and reliable for nonlinear two-space dimensional high-order partial differential equations. Six test problems are considered and \(L_2\), \(L_{\infty }\) error norms are calculated for comparison of our numerical results with exact results whenever they are available. Also numerical results are plotted for comparison with reference solutions. The obtained results certify the applicability and efficiency of the suggested method for Fisher–KPP and EFK equations.

Keywords

Two-dimensional Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation Two-dimensional extended Fisher–Kolmogorov (EFK) equation Chebyshev wavelet method Wavelet–Taylor Numerical solution 

Mathematics Subject Classification

65T60 65M70 

Notes

Acknowledgements

The author would like to thank to the reviewers for their useful comments and suggestions towards improvement of the paper.

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Eğil Vocational and Technical Anatolian High SchoolDiyarbakırTurkey

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