### An Algorithm for Computing Spline Function

Faraidun K.H., Karwan H.F Jwamer, and Sabah Ali Mohammed

College of Science Education, Department of Mathematics, University of Sulaimani, Kurdistan Region, Iraq.
College of Science, Department of Mathematics, Kurdistan Region, Sulaimani, Iraq.
College of Engineering, Komar University of Science and Technology, Sulaimani, Kurdistan Region, Iraq.
Education College, Department of Computers, University of Salahadden, Kurdistan Region, Erbil, Iraq.

DOI: https://doi.org/10.17656/jzs.10554

Abstract

A numerical algorithm is constructed to develop numerical solution to the spline function based belonging to theC6-class. The presented method showed that the approximate solution for boundary value problems obtain by the numerical algorithm which are applied sixticspline function is effective. Convergence analysis of the proposed method and error estimates are obtained. Numerical results illustrate by two examples are given the practical usefulness and efficiency of the algorithm.

Key Words:  Algorithm spline interpolation, error optimality, approximate solution, convergence analysis.

References

[1] Abbas Y. A, Rostam K.S, and Faraidun K. H.The existence, uniqueness and error bounds of approximation splines interpolation for solving second order initial value problem; Journal of Mathematics and Statistics Vol.5, No.2, pp 123-129, (2009).

[2] MiculaGh., Revnic A., An implicit numerical spline method for systems for ODEs, Applied Mathematics and Computation-Elsevier, Vol.111, pp 121-132, (2000).

[3] Srivastava P. K., Kumar M., Numerical Algorithm Based on quinticNonpolynomial Spline for Solving Third-Order Boundary Value Problems Associated with Draining and Coating Flows, Chinese Annals of Mathematics, Series B, Vol.33 No.B(6), pp 831–840, (2012).

[4] Faraidun K. H., Fractional Polynomial Spline for solving Differential Equations of Fractional Order, Math. Sci. Lett. Vol.4, No. 3, pp 291-296. www.naturalspublishing.com/Journals.asp, (2015).

[5] Faraidun K. H. &Pshtiwan O. M., Generalized Quartic Fractional Spline Interpolation with Applications, Int. J. Open Problems Compt. Math., Vol. 8, No. 1, March 2015, ISSN 1998-6262; Copyright © ICSRS Publication, www.i-csrs.org, (2015).

[6] Meinardus G., Nürnberger G., Sommer M. and Strauss H., Algorithms for piecewise polynomials and splines with free knots, Mathematics of Computation, Vol. 53, pp 235-247, (1989).

[7] Sharon A. Johnson, Jery R. Stedinger, Christine A. Shoemaker, Ying Li and José Alberto Tejada-Guibert, Numerical Solution of Continuous-State Dynamic Programs Using Linear and Spline Interpolation, Operations Research, Vol.41, No.3, pp 484-500, (1993).

[8] Fawzy T., Notes on lacunary interpolation by splines,II, Annales Univ. Sci. Budapest., SectioCompuatorica, Vol.6, pp 117-123,(1985).

[9] Srivastava P. K., Kumar M. and Mohapatra R. N., Solution of Fourth Order Boundary Value Problems by Numerical Algorithms Based on NonpolynomialQuintic Splines, Journal of Numerical Mathematics and Stochastic, Vol.4, No.1, pp 13-25, (2012).

[10] Pittaluga G., Sacripante L. and Venturino E., Lacunary Interpolation with Arbitrary Data of High Order, Annals Univ. Sci. Budapest., Sect. Comp. Vol.20, No., pp 83-96, (2001).

[11] Rentrop P., An algorithm for the computation of the exponential spline, NumerischeMathematik, Vol. 35, Issue 1, pp 81-93, (1980).

[12] Tzimbalario J., On a Class of lnterpolatory Splines, Journal of Approximation Theory Vol.23, pp. 142- 145, (1978).

[13] Reinhold Klass, An offset spline approximation for plane cubic splines, Computer-Aided Design, Vol. 15, Issue 5, pp 297-299, September, (1983).

[14] Charles A. Hall & W. Weston Meyer, Optimal error bounds for cubic spline interpolation, Journal of Approximation Theory, Vol. 16, Issue 2, pp 105-122, February, (1976).

[15] Arvet P., Enn T., On the convergence of spline collocation methods for solving fractional differential equations, Journal of Computational and Applied Mathematics Vol.235, pp 3502–3514, (2011).

[16] Noor M. A. and Mohyud S. T. -Din, An efficient algorithm for solving fifth-order boundary value problems, Mathematical and Computer Modeling, Vol. 45, No. 7-8, pp 954-964, (2007).