Skip to main content

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.

Corresponding author
*Corresponding author. Email address: (T.-J. Wang)
Hide All
[1] Boyd J.P., The rate of convergence of Hermite function series, Math. Comp. 35, 13091316 (1980).
[2] Boyd J.P., The asymptotic coefficients of Hermite series, J. Comput. Phys. 54, 382410 (1984).
[3] Olmos D. and Shizgal B.D., A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math. 193, 219242 (2006).
[4] Fok J.C.M., Guo B.Y. and Tang T., Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp. 71, 14971528 (2001).
[5] Fisher R.A., The wave of advance of advantageous genes, Ann. Eugenics 7, 355369 (1937).
[6] Logan J. David, An Introduction to Nonlinear Differential Equations, 2nd edition, Wiley-Interscience, New York (2008).
[7] Li X.Z., Zhang W.G. and Yuan S.L., LS method and qualitative analysis of traveling wave solutions of Fisher equation, Acta Phys. Sin. 59, 744749 (2010).
[8] Guo B.Y. and Xu C.L., Hermite pseudospectral method for nonlinear partial differential equations, Math. Model. Numer. Anal. 34, 859872 (2000).
[9] Guo B.Y. and Wang T.J., Mixed Legendre-Hermite spectral method for heat transfer in an infinite plate, Comput. Math. Appl. 51, 751768 (2006).
[10] Funaro D. and Kavian O., Approximation of some diffusion evolution equation in unbounded domains by Hermite function, Math. Comp. 57, 597619 (1999).
[11] Guo B.Y., Spectral Methods and Their Applications, World Scientific, Singapore (1998).
[12] Guo B.Y., Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp. 68, 10691078 (1999).
[13] Guo B.Y., Shen J. and Xu C.L., Spectral and pseudospectral approximation using Hermite functions: Application to the Dirac equation, Adv. Comput. Math. 19, 3555 (2003).
[14] Guo B.Y. and Zhang C., Generalised Hermite spectral method: Matching different algebraic decay at infinities, J. Sci. Comput. 65, 648671 (2015).
[15] Guo B.Y. and Yi Y.G., Generalised Jacobi Rational Spectral Method and Its Applications, J. Sci. Comput. 43, 201238 (2010).
[16] Mittal R.C. and Jiwari R., Numerical study of Fisher's equation by using differential quadrature method, Int. J. Inform. Sys. Sci. 5, 143160 (2009).
[17] Mehdi B. and Davod K.S., A highly accurate method to solve Fisher's equation, Pramana-J. Phys. 78, 335346 (2012).
[18] Ma H.P., Sun W.W. and Tang T., Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains, SIAM J. Numer. Anal. 43, 5875 (2005).
[19] Ma H.P. and Zhao T.G., A stabilised Hermite spectral method for second-order differential equations in unbounded domain, Numer. Meth. Part. D. E. 23, 968983 (2007).
[20] Shen J. and Wang L.L., Some recent advances in spectral methods for unbounded domains, Comm. Comput. Phys. 5, 195241 (2009).
[21] Shen J., Tang T. and Wang L.L., Spectral Method: Algorithms, Analysis and Applications, Springer Verlag, Berlin (2011).
[22] Weideman J.A.C., The eigenvalues of Hermite and rational differentiation matrices, Numer. Math. 61, 409431 (1992).
[23] Wang T.J. and Guo B.Y., Mixed Legendre-Hermite pseudospectral method for heat transfer in an infinite plate, J. Comput. Math. 23, 587602 (2005).
[24] Verwer J.G., Hundsdorfer W.H. and Sommeijer B.P., Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math. 57, 157178 (1990).
[25] Wang T.J., Generalised Laguerre spectral method for Fisher's equation on a semi-infinite interval, Int. J. Comput. Math. 92, 10391052 (2015).
[26] Wang T.J., Mixed spectral method for heat transfer with inhomogeneous Neumann boundary condition in an infinite strip, Appl. Numer. Math. 92, 8297 (2015).
[27] Yi Y.G. and Guo B.Y., Generalised Jacobi rational spectral method on the half line, Adv. Math. Comput. 37, 137 (2012).
[28] Zhang C. and Guo B.Y., Generalised Hermite spectral method matching asymptotic behaviors, J. Comput. Appl. Math. 255, 616634 (2014).
[29] Xiang X.M. and Wang Z.Q., Generalised Hermite spectralmethod and its applications to problems in unbounded domains, SIAM J. Numer. Anal. 48, 12311253 (2010).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 43 *
Loading metrics...

Abstract views

Total abstract views: 187 *
Loading metrics...

* Views captured on Cambridge Core between 19th October 2016 - 19th January 2018. This data will be updated every 24 hours.