Abstract
In this paper, we propose a new generalized Hermite spectral method. We introduce an orthogonal family of new generalized Hermite functions, with the weight function \((1+\frac{2}{\pi } \arctan x)^{\alpha }(1-\frac{2}{\pi }\arctan x)^{\gamma }\), \(\alpha \) and \(\gamma \) being arbitrary real numbers. The basic results on the corresponding orthogonal approximation and interpolation are established. As examples of applications, we provide the spectral schemes for a linear problem and the Fisher equation, which possess the spectral accuracy in space and match the different algebraic decay at infinities reasonably. Numerical results demonstrate their high efficiency and coincide well with the analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boyd, J.P.: The rate of convergence of Hermite function series. Math. Comput. 35, 1309–1316 (1980)
Boyd, J.P.: The asymptotic coefficients of Hermite series. J. Comput. Phys. 54, 382–410 (1984)
Fok, J.C.M., Guo, B., Tang, T.: Hermite spectral-finite difference method for the Fokker–Planck equation. Math. Comput. 71, 1497–1528 (2002)
Funaro, D., Kavian, O.: Approximation of some diffusion evolution equations in unbounded domains by Hermite function. Math. Comput. 57, 597–619 (1999)
Guo, B., Chen, Z.: Analytic solutions of the Fisher equation. J. Phys. A Math. Gen. 24, 645–650 (1991)
Guo, B.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)
Guo, B.: Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68, 1067–1078 (1999)
Guo, B., Xu, C.: Hermite pseudospectral method for nonlinear partial differential equations. Math. Model. Numer. Anal. 34, 859–872 (2000)
Guo, B., Shen, J., Xu, C.: Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation. Adv. Comput. Math. 19, 35–55 (2003)
Guo, B., Shen, J., Wang, L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)
Guo, B., Shen, J., Wang, L.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59, 1011–1028 (2009)
Guo, B., Yi, Y.: Generalized Jacobi rational spectral method and its applications. J. Sci. Comput. 43, 201–238 (2010)
Guo, B., Zhang, C.: Spectral method for high order problems with proper simulations of asymptotic behaviors at infinity. J. Comput. Appl. Math. 237, 269–294 (2013)
Guo, B.: Some progress in spectral methods. Sci. China Math. 56, 2411–2438 (2013)
Ma, H., Sun, W., Tang, T.: Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains. SIAM J. Numer. Anal. 43, 58–75 (2005)
Ma, H., Zhao, T.: A stabilized Hermite spectral method for second-order differential equations in unbounded domain. Numer. Methods PDEs. 23, 968–983 (2007)
Shen, J., Wang, L.: Some recent advances in spectral methods for unbounded domains. Comm. Comput. Phys. 5, 195–241 (2009)
Sun, T., Guo, B.: Generalized Jacobi approximation in multiple dimensions and its applications. J. Sci. Comput. 55, 327–350 (2013)
Weideman, J.A.C.: The eigenvalues of Hermite and rational differentiation matrices. Numer. Math. 61, 409–431 (1992)
Xiang, X., Wang, Z.: Generalized Hermite spectral method and its applications to problems in unbounded domains. SIAM J. Numer. Anal. 48, 1231–1253 (2010)
Yi, Y., Guo, B.: Generalized Jacobi rational spectral method on the half line. Adv. Math. Comput. 37, 1–37 (2012)
Zhang, C., Guo, B.: Generalized Hermite spectral method matching asymptotic behaviors. J. Comput. Appl. Math. 255, 616–634 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of Chao Zhang is supported in part by NSF of China N.11171227 and N.11371123, Research Fund for Young Teachers of Jiangsu Normal University N.11XLR27, and Priority Academic Program Development of Jiangsu Higher Education Institutions.
The work of Ben-yu Guo is supported in part by NSF of China N.11171227, Fund for Doctoral Authority of China N.20123127110001, Fund for E-institute of Shanghai Universities N.E03004, and Leading Academic Discipline Project of Shanghai Municipal Education Commission N.J50101.
Rights and permissions
About this article
Cite this article
Guo, By., Zhang, C. Generalized Hermite Spectral Method Matching Different Algebraic Decay at infinities. J Sci Comput 65, 648–671 (2015). https://doi.org/10.1007/s10915-014-9981-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9981-3
Keywords
- New generalized Hermite orthogonal approximation and interpolation
- Spectral method on the whole line
- Matching different algebraic decay at infinities