top
Articles
  • OpenAccess
  • On Two Birkhoff-Type Interpolations with First- and Second-Order Derivative  [ICPDE 2016]
  • DOI: 10.4236/jamp.2016.47133   PP.1269 - 1274
  • Author(s)
  • Tinggang Zhao, Yongjun Li
  • ABSTRACT
  • In this paper, we consider two interpolations of Birkhoff-type with integer-order derivative. The Birkhoff interpolation is related with collocation method for the corresponding initial or boundary value problems of differential equations. The solvability of the interpolation problems is proved. For Gauss-type interpolating points, error of interpolation approximation is deduced. Also, we give efficient algorithms to implement the concerned interpolations.
  • KEYWORDS
  • Birkhoff Interpolation, Collocation Method, Gauss-Type Interpolating Point, Error Estimation
  • References
  • [1]
    [1] Birkhoff, G.D. (1906) General Mean Value and Remainder Theorems with Applications to Mechanical Differentiation and Quadrature. Transactions of the American Mathematical Society, 7, 107-136.
    http://dx.doi.org/10.2307/1986339
    [2]
    Lorentz, G.G. and Zeller, K.L. (1971) Birkhoff Interpolation. SIAM Journal on Numerical Analysis, 8, 43-48.
    http://dx.doi.org/10.1137/0708006
    [3]
    Lorentz, G.G., Jetter, K. and Riemenschneider, S.D. (1983) Birkhoff Interpo-lation. Addison-Wesley Publ. Comp.
    [4]
    Pólya, G. (1931) Bemerkung zur interpolation und zur N?herungstheorie der dalkenbiegung. Zeitschrift für Angewandte Mathematik und Mechanik, 11, 445-449.
    http://dx.doi.org/10.1002/zamm.19310110620
    [5]
    Karlin, S. and Karon, J.M. (1972) On Hermite-Birkhoff Interpolation. Journal of Approximation Theory, 6, 90-114.
    http://dx.doi.org/10.1016/0021-9045(72)90085-8
    [6]
    Sharma, A. (1972) Some Poised and Unpoised Problems of Interpolation. SIAM Review, 14, 129-151.
    http://dx.doi.org/10.1137/1014004
    [7]
    Shi, Y.G. (2003) Theory of Birkhoff Interpolation. Nova Science Publishers, New York.
    [8]
    Ferguson, D.R. (1969) The Question of Uniqueness for G. D. Birkhoff Interpolation Problems. Jour-nal of Approximation Theory, 2, 1-28.
    http://dx.doi.org/10.1016/0021-9045(69)90028-8
    [9]
    Palacios, F. and Rubió, P. (2003) Generalised Pólya Condition for Birkhoff Interpolation with Lacunary Polynomial. Applied Mathematics E-Notes, 3, 124-129.
    [10]
    Mühlbach, G. (1981) An Algorithm Approach to Hermite-Birkhoff Interpolation. Numerische Mathematik, 37, 339- 347.
    http://dx.doi.org/10.1007/BF01400313
    [11]
    Szili, L. (1996) A Survey on (0,2) Interpolation. Annales Universitatis Scientiarum Budapestinensis de Rolando E?tv?s Nominatae, Sectio Computatorica, 16, 377-390.
    [12]
    Wang, L.L., Samson, M.D. and Zhao, X.D. (2014) A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix. SIAM: SIAM Journal on Scientific Computing, 36, A907-A929.
    http://dx.doi.org/10.1137/130922409
    [13]
    Shen, J., Tang, T. and Wang, L.L. (2011) Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin.
    http://dx.doi.org/10.1007/978-3-540-71041-7

Engineering Information Institute is the member of/source content provider to

http://www.scirp.org http://www.hanspub.org/ http://www.crossref.org/index.html http://www.oalib.com/ http://www.ebscohost.com/ http://www.proquest.co.uk/en-UK/aboutus/default.shtml http://ip-science.thomsonreuters.com/cgi-bin/jrnlst/jlresults.cgi?PC=MASTER&Full=journal%20of%20Bioequivalence%20%26%20Bioavailability http://publishers.indexcopernicus.com/index.php