American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: Editor-in-chief: Emanuele Galligani
Open Access
Journal Browser
American Journal of Numerical Analysis. 2015, 3(1), 30-35
DOI: 10.12691/ajna-3-1-5
Open AccessArticle

Analysis of Fractional Splines Interpolation and Optimal Error Bounds

Faraidun K. Hamasalh1, and Pshtiwan O. Muhammad1

1Faculty of Science and Science Education, School of Science Education, Sulaimani Univ., Sulaimani, Iraq

Pub. Date: March 29, 2015

Cite this paper:
Faraidun K. Hamasalh and Pshtiwan O. Muhammad. Analysis of Fractional Splines Interpolation and Optimal Error Bounds. American Journal of Numerical Analysis. 2015; 3(1):30-35. doi: 10.12691/ajna-3-1-5


This paper presents a formulation and a study of three interpolatory fractional splines these are in the class of mα, m = 2, 4, 6, α = 0:5. We extend fractional splines function with uniform knots to approximate the solution of fractional equations. The developed of spline method is to analysis convergence fractional order derivatives and estimating error bounds. We propose spline fractional method to solve fractional differentiation equations. Numerical example is given to illustrate the applicability and accuracy of the methods.

fractional integral and derivative caputo derivative error bound

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


[1]  S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science, Amsterdam, 1993.
[2]  I. Podlubny, Fractional Differentional Equations, Academic Press, San Diego, 1999.
[3]  J. E. Phythian and R. Williams, Direct cubic spline approximation to integrals, Int. j. numer. methodseng., 23 (1986), 305-315.
[4]  P. G. Clarleft, M. H. Schultz and R. S. Varga, Numerical methods of high order accuracy, Numer. Math., 9 (1967), 394-430.
[5]  Zahra W. K. and Elkholy S. M., Quadratic spline solution for boundary value problem of fractional order, Numer Algor, 59 (2012), 373-391.
[6]  G. Micula, T. Fawzy, and Z. Ramadan, A polynomial spline approximation method for solving system of ordinary differential equations, Babes-Bolyai Cluj-Napoca. Mathematica, vol. 32, no. 4 (1987) 55-60.
[7]  M. A. Ramadan, Spline solutions of first order delay differential equations, Journal of the Egyptian Mathematical Society, vol. 13, no. 1 (2005) 7-18.
[8]  G. Birkhoff and A. Priver, Hermite interpolation errors for derivatives, J. Math. Phys., 46 (1967) 440-447.
[9]  A. K. Varma and G. Howell, Best error bounds for derivatives in two point Birkhoff interpolation problems, J. Approx. Theory, 38 (1983) 258-268.
[10]  Richard Herrmann, Fractional calculus: an introduction for physicists, Giga Hedron, Germany, 2nd edition, 2014.
[11]  M. Ishteva, Properties and applications of the Caputo fractional operator, Msc. Thesis, Dept. of Math., Universit├Ąt Karlsruhe (TH), Sofia, Bulgaria, 2005.
[12]  A. K. Varma and G. Howell, Cantor-type cylindrical coordinate method for differential equations with local fractional derivatives, Physics Letters A, 377 (28) (2013) 1696-1700.
[13]  A. K. Varma and G. Howell, Thermal impedance estimations by semi-derivatives and semi-integrals: 1-D semiinfinite cases, Thermal Science, 17 (2) (2013) 581-589.
[14]  Faraidun K. Hamasalh and Pshtiwan O. Muhammad, Generalized Quartic Fractional Spline Interpolation with Applications, Int. J. Open Problems Compt. Math, Vol. 8, No. 1 (2015)., 67-80.