Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T19:28:28.472Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: General theory in ordinary differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems Boundary value problems Approximations and expansions Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  19 October 2016

Yu-Jian Jiao ,
Tian-Jun Wang  and
Qiong Zhang
Yu-Jian Jiao
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P. R. China; Scientific Computing Key Laboratory of Shanghai Universities
Tian-Jun Wang*
Affiliation:
Henan University of Science and Technology, Luoyang, 471003, P. R. China
Qiong Zhang
Affiliation:
Henan University of Science and Technology, Luoyang, 471003, P. R. China
*
*Corresponding author. Email address:[email protected] (T.-J. Wang)
Get access

Abstract

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.

Keywords

Fisher's equationspectral methodgeneralised Hermite functionproblem on the whole line

MSC classification

Secondary: 34A34: Nonlinear equations and systems, general 34B40: Boundary value problems on infinite intervals 41A30: Approximation by other special function classes 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 65M70: Spectral, collocation and related methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 400 - 415
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Boyd, J.P., The rate of convergence of Hermite function series, Math. Comp. 35, 1309–1316 (1980).Google Scholar
[2] Boyd, J.P., The asymptotic coefficients of Hermite series, J. Comput. Phys. 54, 382–410 (1984).CrossRefGoogle Scholar
[3] Olmos, D. and Shizgal, B.D., A pseudospectral method of solution of Fisher's equation, J. Comput. Appl. Math. 193, 219–242 (2006).Google Scholar
[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, 1497–1528 (2001).Google Scholar
[5] Fisher, R.A., The wave of advance of advantageous genes, Ann. Eugenics 7, 355–369 (1937).Google Scholar
[6] Logan, J. David, An Introduction to Nonlinear Differential Equations, 2nd edition, Wiley-Interscience, New York (2008).Google Scholar
[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, 744–749 (2010).Google Scholar
[8] Guo, B.Y. and Xu, C.L., Hermite pseudospectral method for nonlinear partial differential equations, Math. Model. Numer. Anal. 34, 859–872 (2000).CrossRefGoogle Scholar
[9] Guo, B.Y. and Wang, T.J., Mixed Legendre-Hermite spectral method for heat transfer in an infinite plate, Comput. Math. Appl. 51, 751–768 (2006).Google Scholar
[10] Funaro, D. and Kavian, O., Approximation of some diffusion evolution equation in unbounded domains by Hermite function, Math. Comp. 57, 597–619 (1999).Google Scholar
[11] Guo, B.Y., Spectral Methods and Their Applications, World Scientific, Singapore (1998).Google Scholar
[12] Guo, B.Y., Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp. 68, 1069–1078 (1999).Google Scholar
[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, 35–55 (2003).CrossRefGoogle Scholar
[14] Guo, B.Y. and Zhang, C., Generalised Hermite spectral method: Matching different algebraic decay at infinities, J. Sci. Comput. 65, 648–671 (2015).Google Scholar
[15] Guo, B.Y. and Yi, Y.G., Generalised Jacobi Rational Spectral Method and Its Applications, J. Sci. Comput. 43, 201–238 (2010).Google Scholar
[16] Mittal, R.C. and Jiwari, R., Numerical study of Fisher's equation by using differential quadrature method, Int. J. Inform. Sys. Sci. 5, 143–160 (2009).Google Scholar
[17] Mehdi, B. and Davod, K.S., A highly accurate method to solve Fisher's equation, Pramana-J. Phys. 78, 335–346 (2012).Google Scholar
[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, 58–75 (2005).CrossRefGoogle Scholar
[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, 968–983 (2007).CrossRefGoogle Scholar
[20] Shen, J. and Wang, L.L., Some recent advances in spectral methods for unbounded domains, Comm. Comput. Phys. 5, 195–241 (2009).Google Scholar
[21] Shen, J., Tang, T. and Wang, L.L., Spectral Method: Algorithms, Analysis and Applications, Springer Verlag, Berlin (2011).Google Scholar
[22] Weideman, J.A.C., The eigenvalues of Hermite and rational differentiation matrices, Numer. Math. 61, 409–431 (1992).Google Scholar
[23] Wang, T.J. and Guo, B.Y., Mixed Legendre-Hermite pseudospectral method for heat transfer in an infinite plate, J. Comput. Math. 23, 587–602 (2005).Google Scholar
[24] Verwer, J.G., Hundsdorfer, W.H. and Sommeijer, B.P., Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math. 57, 157–178 (1990).Google Scholar
[25] Wang, T.J., Generalised Laguerre spectral method for Fisher's equation on a semi-infinite interval, Int. J. Comput. Math. 92, 1039–1052 (2015).Google Scholar
[26] Wang, T.J., Mixed spectral method for heat transfer with inhomogeneous Neumann boundary condition in an infinite strip, Appl. Numer. Math. 92, 82–97 (2015).CrossRefGoogle Scholar
[27] Yi, Y.G. and Guo, B.Y., Generalised Jacobi rational spectral method on the half line, Adv. Math. Comput. 37, 1–37 (2012).Google Scholar
[28] Zhang, C. and Guo, B.Y., Generalised Hermite spectral method matching asymptotic behaviors, J. Comput. Appl. Math. 255, 616–634 (2014).CrossRefGoogle Scholar
[29] Xiang, X.M. and Wang, Z.Q., Generalised Hermite spectralmethod and its applications to problems in unbounded domains, SIAM J. Numer. Anal. 48, 1231–1253 (2010).Google Scholar