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A Fully Discrete Spectral Method for Fisher’s Equation on the Whole Line

  • Yu-Jian Jiao (a1), Tian-Jun Wang (a2) and Qiong Zhang (a2)

A generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

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*Corresponding author. Email address: (T.-J. Wang)
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[1] J.P. Boyd , The rate of convergence of Hermite function series, Math. Comp. 35, 13091316 (1980).

[2] J.P. Boyd , The asymptotic coefficients of Hermite series, J. Comput. Phys. 54, 382410 (1984).

[3] D. Olmos and B.D. Shizgal , A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math. 193, 219242 (2006).

[4] J.C.M. Fok , B.Y. Guo and T. Tang , Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp. 71, 14971528 (2001).

[5] R.A. Fisher , The wave of advance of advantageous genes, Ann. Eugenics 7, 355369 (1937).

[8] B.Y. Guo and C.L. Xu , Hermite pseudospectral method for nonlinear partial differential equations, Math. Model. Numer. Anal. 34, 859872 (2000).

[9] B.Y. Guo and T.J. Wang , Mixed Legendre-Hermite spectral method for heat transfer in an infinite plate, Comput. Math. Appl. 51, 751768 (2006).

[10] D. Funaro and O. Kavian , Approximation of some diffusion evolution equation in unbounded domains by Hermite function, Math. Comp. 57, 597619 (1999).

[11] B.Y. Guo , Spectral Methods and Their Applications, World Scientific, Singapore (1998).

[12] B.Y. Guo , Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp. 68, 10691078 (1999).

[14] B.Y. Guo and C. Zhang , Generalised Hermite spectral method: Matching different algebraic decay at infinities, J. Sci. Comput. 65, 648671 (2015).

[15] B.Y. Guo and Y.G. Yi , Generalised Jacobi Rational Spectral Method and Its Applications, J. Sci. Comput. 43, 201238 (2010).

[18] H.P. Ma , W.W. Sun and T. Tang , Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains, SIAM J. Numer. Anal. 43, 5875 (2005).

[19] H.P. Ma and T.G. Zhao , A stabilised Hermite spectral method for second-order differential equations in unbounded domain, Numer. Meth. Part. D. E. 23, 968983 (2007).

[21] J. Shen , T. Tang and L.L. Wang , Spectral Method: Algorithms, Analysis and Applications, Springer Verlag, Berlin (2011).

[22] J.A.C. Weideman , The eigenvalues of Hermite and rational differentiation matrices, Numer. Math. 61, 409431 (1992).

[24] J.G. Verwer , W.H. Hundsdorfer and B.P. Sommeijer , Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math. 57, 157178 (1990).

[25] T.J. Wang , Generalised Laguerre spectral method for Fisher's equation on a semi-infinite interval, Int. J. Comput. Math. 92, 10391052 (2015).

[26] T.J. Wang , Mixed spectral method for heat transfer with inhomogeneous Neumann boundary condition in an infinite strip, Appl. Numer. Math. 92, 8297 (2015).

[27] Y.G. Yi and B.Y. Guo , Generalised Jacobi rational spectral method on the half line, Adv. Math. Comput. 37, 137 (2012).

[28] C. Zhang and B.Y. Guo , Generalised Hermite spectral method matching asymptotic behaviors, J. Comput. Appl. Math. 255, 616634 (2014).

[29] X.M. Xiang and Z.Q. Wang , Generalised Hermite spectralmethod and its applications to problems in unbounded domains, SIAM J. Numer. Anal. 48, 12311253 (2010).

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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