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## Integro-Differential Polynomial and Trigonometrical Splines and Quadrature Formulas

PIIS004446690000308-9-1
DOI10.31857/S004446690000308-9
Publication type Article
Status Published
Authors
Affiliation: St. Petersburg State University
Affiliation: St. Petersburg State University
Affiliation: St. Petersburg State University
Journal nameZhurnal vychislitelnoi matematiki i matematicheskoi fiziki
EditionVolume 58 Issue 7
Pages1059-1072
Abstract

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

Keywordspolynomial splines, trigonometrical splines, integro-differential splines, interpolation
1. Safak S. On the trivariate polynomial interpolation // WSEAS Transactions on Math. 2012. Vol. 11. Iss. 8. P. 738–746.2. Skala V. Fast interpolation and approximation of scattered multidimensional and dynamic data using radial basis functions // WSEAS Transactions on Math. 2013. Vol. 12. Iss. 5. P. 501–511.3. Sarfraz M., Al-Dabbous N. Curve representation for outlines of planar images using multilevel coordinate search // WSEAS Transactions on Computers. 2013. Vol. 12. Iss. 2. P. 62–73.4. Sarfraz M. Generating outlines of generic shapes by mining feature points // WSEAS Transactions on Systems. 2014. Vol. 13. P. 584–595.5. Zamani M. A new, robust and applied model for approximation of huge data // WSEAS Transactions on Math. 2013. Vol. 12. Iss. 6. P. 727–735.6. Chui C.K. Multivariate splines. Society for industrial and applied mathematics (SIAM). Pensylvania, USA, 1988.7. Kuragano T. Quintic B-spline curve generation using given points and gradients and modification based on specified radius of curvature // WSEAS Transactions on Math. 2010. Vol. 9. Iss. 2. P. 79–89.8. Fengmin Chen, Wong Patricia J.Y. On periodic discrete spline interpolation: Quintic and biquintic case // J. of Comput. and Appl. Math. 2014. № 255. P. 282–296.9. Abba M., Majid A.A., Awang M.N.H., Ali J.Md. Shape-preserving rational bi-cubic spline for monotone surface data // WSEAS Transactions on Math. 2012. Vol. 11. Iss. 7. P. 660–673.10. Xiaodong Zhuang, Mastorakis N.E. A model of virtual carrier immigration in digital images for region segmentation // Wseas Transactions On Computers. 2015. Vol. 14. P.708–718.11. Simani S. Residual generator fuzzy identification for automotive diesel engine fault diagnosis // Internat. Journal of Appl. Math. and Comput. Science. 2013. Vol. 23. Iss. 2. P. 419–438.12. de Boor C. Efficient computer manipulation of tensor products // ACM Trans. Math. Software. 1979. № 5. P. 173–182.13. de Boor C. A practical guide to splines. New York: Springer, 1978.14. Grosse E. Tensor spline approximation // Linear Algebra and its Applicat. 1980. Vol. 34. P. 29–41.15. Burova I. On integro-differential splines construction // Advances in Applied and Pure Math. Proc. of the 7th Internat. Conf. Finite Differences, Finite Elements, Finite Volumes, Boundary Elements (F-and-B’14). May 15–17. Gdansk, Poland, 2014. P. 57–61.16. Burova I. On integro-differential splines and solution of cauchy problem // Math. Meth. and Systems in Sci. and Eng. Proc. of the 17th Internat. Conf. on Math. Methods, Comput. Techniques and Intelligent Systems (MAMECTIS’15), Tenerife, Canary Islands, Spain, January 10–12, 2015. P.48–52.17. Burova I.G., Rodnikova O.V. Integro-differential splines and quadratic formulae // Internat. J. Math. and Comput. Meth. 2016. Vol. 1. P. 384–388.18. Krylov V.I. Priblizhennoe vychislenie integralov. M.: 1976. 500 s.