## Integro-Differential Splines and Quadratic Formulae

 AUTHOR(S):  I. G. Burova, O. V. Rodnikova TITLE Integro-Differential Splines and Quadratic Formulae PDF ABSTRACT This work is devoted to the further investigation of splines of the fifth order approximation. Here we present some new formulae which are useful for the approximation of functions with one or two variables. For each grid interval (or elementary rectangular) we construct the approximation separately. Here we construct the basic one-dimensional polynomial splines of the fifth order approximation when the values of the function and the values of its first derivative are known in each point of interpolation. Sometimes it is important that the integrals of the function over the intervals are equal to the integrals of the approximation of the function over the intervals. In that case the approximation has some physical parallel. For this aim we use quadratic formulae here with the sixth order of approximation instead of the value of integral. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented. KEYWORDS Polynomial splines, Integro-Differential Splines, Interpolation REFERENCES  Safak, S., On the trivariate polynomial interpolation, WSEAS Transactions on Mathematics. Vol. 11, Iss. 8, 2012, pp. 738–746.  Safak, S., On the trivariate polynomial interpolation, WSEAS Transactions on Mathematics. Vol. 11, Iss. 8, 2012, pp. 738–746.  Skala, V., Fast interpolation and approximation of scattered multidimensional and dynamic data using radial basis functions, WSEAS Transactions on Mathematics. Vol. 12, Iss. 5, 2013, pp. 501–511.  Sarfraz, M., Al-Dabbous, N., Curve representation for outlines of planar images using multilevel coordinate search, WSEAS Transactions on Computers. Vol. 12, Iss. 2, 2013, pp. 62–73.  Sarfraz, M., Generating outlines of generic shapes by mining feature points, WSEAS Transactions on Systems, Vol. 13, 2014, pp. 584–595.  Zamani, M., A new, robust and applied model for approximation of huge data, WSEAS Transactions on Mathematics, Vol. 12, Iss. 6, 2013, pp. 727–735.  C. K. Chui. Multivariate Splines. Society for Industrial and Applied Mathematics (SIAM), Pensylvania, USA, 1988.  Kuragano, T., Quintic B-spline curve generation using given points and gradients and modification based on specified radius of curvature, WSEAS Transactions on Mathematics, Vol. 9, Iss. 2, 2010, pp. 79–89.  Fengmin Chen, Patricia J.Y.Wong, On periodic discrete spline interpolation: Quintic and biquintic case, Journal of Computational and Applied Mathematics, 255, 2014, pp. 282-296.  Abbas, M., Majid, A.A., Awang, M.N.H., Ali, J.Md., Shape-preserving rational bi-cubic spline for monotone surface data, WSEAS Transactions on Mathematics, Vol. 11, Issue 7, July 2012, pp. 660-673.  Xiaodong Zhuang, N. E. Mastorakis., A Model of Virtual Carrier Immigration in Digital Images for Region Segmentation, Wseas Transactions On Computers, Vol. 14, 2015, pp.708–718.  C. de Boor, Efficient computer manipulation of tensor products, ACM Trans. Math. Software 5, 1979, pp. 173-182.  C. de Boor. C. de Boor, A Practical Guide to Splines, Springer, New York, NY, USA, 1978.  Eric Grosse. Tensor spline approximation, Linear Algebra and its Applications, Vol. 34, December 1980, pp. 29-41.  Burova Irina. On Integro- Differential Splines Construction. Advances in Applied and Pure Mathematics. Proceedinngs of the 7-th International Conference on Finite Differences, Finite Elements, Finite Volumes, Boundary Elements (F-and-B’14). Gdansk. Poland. May 15– 17, 2014, pp.57–61.  Burova Irina, On Integro-Differential Splines and Solution of Cauchy Problem. Mathematical Methods and Systems in Science and Engineering, Proc. of the 17th International Conf. on Mathematical Methods, Computational Techniques and Intelligent Systems (MAMECTIS’15), Tenerife, Canary Islands, Spain, January 10-12, 2015, pp.48–52.  Burova Irina, Evdokimova Tatjana, On Splines of the Fifth Order, Recent Advances in Mathematical and Computational Methods, Proc. of the 17th International Conf. on Mathematical and Computational Methods in Science and Engineering (MACMESE’15), Kuala Lumpur, Malaysia, April 23-25, 2015, pp.60–65. Cite this paper I. G. Burova, O. V. Rodnikova. (2016) Integro-Differential Splines and Quadratic Formulae. International Journal of Mathematical and Computational Methods, 1, 384-388 Copyright © 2016 Author(s) retain the copyright of this article.This article is published under the terms of the Creative Commons Attribution License 4.0