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The document summarizes research on constructing twice continuously differentiable approximations using fifth-order integro-differential splines of the first level. It describes building basis splines on a given interval by choosing grid nodes and constructing a system of linear algebraic equations to solve for the basis spline coefficients. It proves estimates for the approximation error and derivatives of the approximated function in terms of the maximum of the fifth derivative on the interval and the grid spacing.

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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 239 CONSTRUCTION OF TWICE CONTINUOUSLY DIFFERENTIABLE APPROXIMATIONS BY INTEGRO-DIFFERENTIAL SPLINES OF FIFTH ORDER AND FIRST LEVEL I.G. Burova1 , S.V. Poluyanov2 1 Professor, Saint-Petersburg State University 2 Graduate Student, Saint-Petersburg State University ABSTRACT The construction of twice continuously differentiable approximation using basis integro-differential splines of fifth order and first level is considered. The theory of minimal splines of zero or non-zero level is worked out in details in [1]. Integro-differential approximations are distinguished by the use of the integral of the function to be approximated by one or several adjacent intervals. Polynomial smooth integro-differential splines are proposed in [2]. In this paper, we consider approximations of functions and their derivatives by continuously differentiable polynomial integro-differential splines of fifth-order (see [3]). 1. BASIS SPLINES CONSTRUCTION We consider an interval ሾܽ, ܾሿ, where ܽ, ܾ — are real numbers, choose a natural number ݊ ൒ 1 and construct a grid node ሼ‫ݔ‬௝ሽ with increment ݄ ൌ ሺ௕ି௔ሻ ௡ : ܽ ൌ ‫ݔ‬଴ ൏. . . ൏ ‫ݔ‬௝ିଵ ൏ ‫ݔ‬௝ ൏ ‫ݔ‬௝ାଵ ൏. . . ൏ ‫ݔ‬௡ ൌ ܾ. Let function ‫ݑ‬ሺ‫ݔ‬ሻ be such that ‫ݑ‬ ‫א‬ ‫ܥ‬ହ ሾܽ, ܾሿ. We have the values ‫ݑ‬ሺ‫ݔ‬௞ሻ, ‫ݑ‬′ ሺ‫ݔ‬௞ሻ, ݇ ൌ 0,1, … , ݊, ‫׬‬ ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫,ݐ‬ ݇ ൌ 0,1, … , ݊ െ 1. The approximation of ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, we take in the form INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2014): 7.8273 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 240 ‫ݑ‬෤௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬′ሺ‫ݔ‬௞ሻ߱௞,ଵሺ‫ݔ‬ሻ ൅ ൅‫ݑ‬′ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (1) where߱௞,଴ሺ‫ݔ‬ሻ, ߱௞ାଵ,଴ሺ‫ݔ‬ሻ, ߱௞,ଵሺ‫ݔ‬ሻ, ߱௞ାଵ,ଵሺ‫ݔ‬ሻ, ߱௞ ழଵவ ሺ‫ݔ‬ሻ are determined from conditions ‫ݑ‬෤௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬ሻ for ‫ݑ‬ሺ‫ݔ‬ሻ ൌ ‫ݔ‬௜ , ݅ ൌ 0,1,2,3,4. (2) We take ‫݌݌ݑݏ‬Ԝ߱௞,ఈ ൌ ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ାଵሿ, ߙ ൌ 0,1, ‫݌݌ݑݏ‬Ԝ߱௞ ழଵவ ൌ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ. Conditions (2) lead to a system of linear algebraic equations with respect to ߱௦,ఈ, ‫ݏ‬ ൌ ݇, ݇ ൅ 1, ߙ ൌ 0,1, ߱௞ ழଵவ : ߱௞,଴ሺ‫ݔ‬ሻ ൅ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ 1, (3) ‫ݔ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ߱௞,ଵሺ‫ݔ‬ሻ ൅ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ଶ /2 െ ‫ݔ‬௞ ଶ /2ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫,ݔ‬ (4) ‫ݔ‬௞ ଶ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ଶ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 2‫ݔ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ 2‫ݔ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ଷ /3 െ ‫ݔ‬௞ ଷ /3ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ଶ , (5) ‫ݔ‬௞ ଷ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ଷ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 3‫ݔ‬௞ ଶ ߱௞,ଵሺ‫ݔ‬ሻ ൅ 3‫ݔ‬௞ାଵ ଶ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ସ /4 െ ‫ݔ‬௞ ସ /4ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ସ , (6) ‫ݔ‬௞ ସ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ସ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 4‫ݔ‬௞ ଷ ߱௞,ଵሺ‫ݔ‬ሻ ൅ 4‫ݔ‬௞ାଵ ଷ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ହ /5 െ ‫ݔ‬௞ ହ /5ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ହ , (7) It’s easy to see that ߱௞,଴, ߱௞,ଵ, ߱௞ ழଵவ ‫א‬ ‫ܥ‬ଵ ሺܴଵ ሻ.Let ‫צ‬ ݂ ‫צ‬ሾ௔,௕ሿൌ maxሾ௔,௕ሿ | ݂ሺ‫ݔ‬ሻ|, ‫צ‬ ݂ ‫צ‬ሾ௔,௕ሻൌ supሾ௔,௕ሻ | ݂ሺ‫ݔ‬ሻ|. Lemma 1.Let function ‫ݑ‬ ‫א‬ ‫ܥ‬ሺହሻ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ‫ݑ‬௞ defines by (1). Nextrelationsaretrue: |‫ݑ‬ሺ‫ݔ‬ሻ െ ‫ݑ‬෤௞ሺ‫ݔ‬ሻ| ൑ ݄ହ ‫ܭ‬଴ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ‫ܭ‬଴ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0, (8) |‫ݑ‬′ ሺ‫ݔ‬ሻ െ ‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ| ൑ ݄ସ ‫ܭ‬ଵ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, ‫ܭ‬ଵ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0, (9) |‫ݑ‬′′ ሺ‫ݔ‬ሻ െ ‫ݑ‬෤′′ ௞ሺ‫ݔ‬ሻ| ൑ ݄ଷ ‫ܭ‬ଶ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻ, , ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ, ‫ܭ‬ଶ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0. (10) Proof. Indeed, if ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿrepresenting ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ and ‫ݑ‬′ ሺ‫ݔ‬௞ାଵሻ using Taylor’s formula, with help (3)-(7) we obtain ‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ ൌ ܴ, where ܴ ൌ 1 5! ‫ݑ‬ሺହሻ ሺ߬ଶሻሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻହ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 1 4! ‫ݑ‬ሺହሻ ሺ߬ଷሻሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻସ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ൅ 1 5! න ‫ݑ‬ሺହሻ ௫ೖశభ ௫ೖ ሺ߬ଵሻሺ‫ݐ‬ െ ‫ݔ‬௞ሻହ ݀‫߱ݐ‬௞ ழଵவ ሺ‫ݔ‬ሻ െ 1 5! ‫ݑ‬ሺହሻ ሺ߬ସሻሺ‫ݔ‬ െ ‫ݔ‬௞ሻହ .
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 241 Here ߬௜ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ݅ ൌ 1,2,3,4. The determinant of the system of equations (3)–(7) is equal to െ ଵ ଷ଴ ሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻଽ . Solving the system of equations (3)–(7), we obtain the following formulas for basis splines if ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ: ߱௞,଴ሺ‫ݔ‬ሻ ൌ ሺ5‫ݔ‬ ൅ ݄ െ 5‫ݔ‬௞ሻሺെ3‫ݔ‬ ൅ ݄ ൅ 3‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /݄ସ , (11) ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൌ െሺെ‫ݔ‬௞ ൅ ‫ݔ‬ሻଶ ሺെ3‫ݔ‬ ൅ 3‫ݔ‬௞ ൅ 2݄ሻሺെ5‫ݔ‬ ൅ 5‫ݔ‬௞ ൅ 6݄ሻ/݄ସ , (12) ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ 30ሺ‫ݔ‬ െ ‫ݔ‬௞ሻଶ ሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /݄ହ , (13) ߱௞,ଵሺ‫ݔ‬ሻ ൌ ሺ‫ݔ‬ െ ‫ݔ‬௞ሻሺ2݄ െ 5‫ݔ‬ ൅ 5‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /ሺ2݄ଷ ሻ, (14) ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൌ ሺ‫ݔ‬ െ ‫ݔ‬௞ሻଶ ሺെ5‫ݔ‬ ൅ 3݄ ൅ 5‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻ/ሺ2݄ଷ ሻ. (15) It is easy to see that for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ the following relations are valid: |߱௞,଴ሺ‫ݔ‬ሻ| ൑ 1, |߱௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 1 , |߱௞,ଵሺ‫ݔ‬ሻ| ൑ ‫ܥ‬ଵ݄, ‫ܥ‬ଵ ൎ 0.06778775 , |߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ ‫ܥ‬ଶ݄, ‫ܥ‬ଶ ൎ 0.0225959 . |߱௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 1.875/݄. Now, using the mean value theorem, we obtain for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ |ܴ| ൌ |‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ ሺ ݄ହ 5! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ାଵ,଴ሺ‫ݔ‬ሻ| ൅ ݄ସ 4! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൅ ൅ ݄଺ 6! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ ழଵவ ሺ‫ݔ‬ሻ| ൅ ݄ହ 5! ሻ, hence|‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑ 0.02 ݄ହ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ. Similarly, |ܴଵ| ൌ |‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′ ሺ‫ݔ‬ሻ| ൑‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ ሺ ݄ହ 5! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൅ ݄ସ 4! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൅ ൅ ݄଺ 6! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൅ ݄ହ 5! ሻ. Takingintoaccount |߱′ ௞,଴ሺ‫ݔ‬ሻ| ൑ 3.94023/݄, |߱′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 3.94023/݄, |߱′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 5.77350/݄ଶ , |߱′ ௞,ଵሺ‫ݔ‬ሻ| ൑ 1, |߱′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ 1, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, weobtain ห‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′ሺ‫ݔ‬ሻห ൑ ൬ 1 5! 3.94023 ൅ 1 4! ൅ 1 6! 5.77350 ൅ 1 4! ൰ ݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿൌ 0.12419݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 242 For ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ we have |߱′′ ௞,଴ሺ‫ݔ‬ሻ| ൑ 15.20000001/݄ଶ , |߱′′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 36/݄ଶ , |߱′′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 60/݄ଷ , |߱′′ ௞,ଵሺ‫ݔ‬ሻ| ൑ 9/݄, |߱′′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ 9/݄, then for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ ܴଶ ൌ ห‫ݑ‬෤′′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′′ሺ‫ݔ‬ሻห ൑ ቆ 1 5! 36 ൅ 1 4! 9 ൅ ݄଺ 6! 60 ൅ 1 4! ቇ ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻൌ ሺ4/5ሻ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻ. Remark 1. We can take ‫ܭ‬଴ ൌ 0.02 in (8), ‫ܭ‬ଵ ൑ 0.125 in (9), ‫ܭ‬ଶ ൑ 4/5 in (10). Remark 2. If we put ‫ݔ‬ ൌ ‫ݔ‬௞ ൅ ‫,݄ݐ‬ ‫ݐ‬ ‫א‬ ሾ0,1ሿ, then we get from (11)–(15) the following formulas of basis splines: ߱௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ቐ െሺ5‫ݐ‬ ൅ 1ሻሺ3‫ݐ‬ െ 1ሻሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, െሺ3‫ݐ‬ ൅ 1ሻሺ5‫ݐ‬ െ 1ሻሺ1 ൅ ‫ݐ‬ሻଶ , ‫ݐ‬ ‫א‬ ሾെ1,0ሿ, 0, ‫ݐ‬ ‫ב‬ ሾെ1,1ሿ, (16) ߱௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬െ 1 2 ‫݄ݐ‬ሺ5‫ݐ‬ െ 2ሻሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, 1 2 ‫݄ݐ‬ሺ2 ൅ 5‫ݐ‬ሻሺ1 ൅ ‫ݐ‬ሻଶ , ‫ݐ‬ ‫א‬ ሾെ1,0ሿ, 0, ‫ݐ‬ ‫ב‬ ሾെ1,1ሿ, (17) ߱௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ቐ 30‫ݐ‬ଶ ݄ ሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, 0, ‫ݐ‬ ‫ב‬ ሾ0,1ሿ, (18) and also for ‫ݐ‬ ‫א‬ ሾ0,1ሿ we have ߱௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െሺ5‫ݐ‬ െ 6ሻሺ3‫ݐ‬ െ 2ሻ‫ݐ‬ଶ , ߱௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 1 2 ݄ሺ5‫ݐ‬ െ 3ሻሺ‫ݐ‬ െ 1ሻ‫ݐ‬ଶ , (19) ߱′ ௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12‫ݐ‬ሺ5‫ݐ‬ െ 3ሻሺ‫ݐ‬ െ 1ሻ/݄, ߱′ ௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12‫ݐ‬ሺ5‫ݐ‬ଶ െ 7‫ݐ‬ ൅ 2ሻ/݄, (20) ߱′ ௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 60‫ݐ‬ሺ2‫ݐ‬ െ 1ሻሺ‫ݐ‬ െ 1ሻ/݄ଶ , (21) ߱′ ௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െሺ‫ݐ‬ െ 1ሻሺ10‫ݐ‬ଶ െ 8‫ݐ‬ ൅ 1ሻ, ߱′ ௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ‫ݐ‬ሺ3 െ 12‫ݐ‬ ൅ 10‫ݐ‬ଶ ሻ. (22) ߱′′ ௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12ሺ3 െ 16‫ݐ‬ ൅ 15‫ݐ‬ଶ ሻ/݄ଶ , ߱′′ ௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12ሺ15‫ݐ‬ଶ െ 14‫ݐ‬ ൅ 2ሻ/݄ଶ , (23) ߱′′ ௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 60ሺ1 െ 6‫ݐ‬ ൅ 6‫ݐ‬ଶ ሻ/݄ଷ , (24)
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 243 ߱′′ ௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ3ሺ3 െ 12‫ݐ‬ ൅ 10‫ݐ‬ଶ ሻ/݄, ߱′′ ௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 3ሺെ8‫ݐ‬ ൅ 10‫ݐ‬ଶ ൅ 1ሻ/݄. (25) Remark 3. We obtain approximation for ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ, ߙ ൌ 1,2, using the formulas: ‫ݑ‬෤௞ ሺఈሻ ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ൅‫ݑ‬′ ሺ‫ݔ‬௞ሻ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬′ ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ሺ߱௞ ழଵவ ሻሺఈሻ ሺ‫ݔ‬ሻ, where ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ, ሺ߱௞ ழଵவ ሻሺఈሻ ሺ‫ݔ‬ሻ are determined by the formulas (20)–(25). We introduce the function ܷ෩ሺ‫ݔ‬ሻ , ‫ݔ‬ ‫א‬ ሾܽ, ܾሿ , associated with ‫ݑ‬෤௞ሺ‫ݔ‬ሻ by the relation ܷ෩ሺ‫ݔ‬ሻ ൌ ‫ݑ‬෤௞ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Now the inequalities are obvious: ‫צ‬ ܷ෩ െ ‫ݑ‬ ‫צ‬ሾ௔,௕ሿ൑ ‫ܭ‬଴݄ହ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ‫צ‬ ܷ෩′ െ ‫ݑ‬′ ‫צ‬ሾ௔,௕ሿ൑ ‫ܭ‬ଵ݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ‫צ‬ ܷ෩′′ െ ‫ݑ‬′′ ‫צ‬ሾ௔,௕ሻ൑ ‫ܭ‬ଶ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሻ. In tables 1 and 2 for ݄ ൌ 0.1, ݄ ൌ 0.01, respectively, the errors of approximations of some functions and their first derivatives of the integro-differential fifth-order splines are presented. Here we use the following designations: for the actual error function and its first derivative — ܴ෨ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬ െ ܷ෩|, ܴଵ ෪ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬′ െ ܷ′෩ |, for theoretical error function and its first derivative — ܴ and ܴଵ respectively. Table 1 No u(x) ܴ෨ ܴ෨ଵ R0 R1 1 sin(3x)cos(5x) 0.12⋅10−4 0.81⋅10−3 0.33⋅10−3 0.20 2 tg(x) 0.16⋅10−5 0.11⋅10−3 0.69⋅10−4 0.43⋅10−1 3 cos(2x) 0.24⋅10−7 0.17⋅10−5 0.64⋅10−6 0.40⋅10−3 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.21⋅10−3 0.14⋅10−1 0.64⋅10−6 0.40⋅10−3 Table 2 No u(x) ܴ෨ ܴ෨ଵ R0 R1 1 sin(3x)cos(5x) 0.12⋅10−9 0.85⋅10−7 0.33⋅10−8 0.20⋅10−4 2 tg(x) 0.24⋅10−10 0.17⋅10−7 0.69⋅10−9 0.43⋅10−5 3 cos(2x) 0.24⋅10−12 0.17⋅10−9 0.64⋅10−11 0.40⋅10−7 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.23⋅10−8 0.16⋅10−5 0.63⋅10−7 0.39⋅10−3
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 244 2. THE CONSTRUCTION OF TWICE CONTINUOUSLY DIFFERENTIABLE APPROXIMATIONS Let mesh nodes ሼ‫ݔ‬௝ሽ be uniform. With each ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ consider an approximation for ‫ݑ‬ሺ‫ݔ‬ሻ in the form ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (26) where the real numbers ‫ܥ‬௞ we find later. We denote ‫ݑ‬′ ௞ ൌ ‫ݑ‬′ ሺ‫ݔ‬௞ሻ and take on the interval ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬௞ାଵ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ න ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (27) and on the adjacent interval ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ሻ the relation ‫ݑ‬෤෨௞ିଵሺ‫ݔ‬ሻ ൌ ‫ݑ‬௞ିଵ߱௞ିଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ିଵ߱௞ିଵ,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ න ‫ݑ‬ ௫ೖ ௫ೖషభ ሺ‫ݐ‬ሻ݀‫߱ݐ‬௞ିଵ ழଵவ ሺ‫ݔ‬ሻ. (28) Twice differentiating the relations (27)–(28) from the condition ‫ݑ‬෤෨௞ ′′ ሺ‫ݔ‬௞൅ሻ ൌ ‫ݑ‬෤෨௞ିଵ ′′ ሺ‫ݔ‬௞െሻ we obtain the system of equations ‫ܥ‬௞ିଵ െ 6‫ܥ‬௞ ൅ ‫ܥ‬௞ାଵ ൌ ݂௞, (29) where ݂௞ ൌ 8ሺ‫ݑ‬௞ାଵ െ ‫ݑ‬௞ିଵሻ/݄ ൅ 20 ቆන ‫ݑ‬ ௫ೖ ௫ೖషభ ሺ‫ݐ‬ሻ݀‫ݐ‬ െ න ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ቇ /݄ଶ , ݇ ൌ 1, … , ݊ െ 1. Now relations for twice continuously differentiable splines ܷሺ‫ݔ‬ሻ, on each interval ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ, ݇ ൌ 1,2, … , ݊ െ 1, are defined by (26) with coefficients ‫ܥ‬௞, which are the solution of the system (29). Lemma 2. |‫ܥ‬௞ െ ‫ݑ‬′ ሺ‫ݔ‬௞ሻ| ൑ ‫,ݍ‬ ‫ݍ‬ ൌ ‫ܭ‬෩݄ସ , ‫ܭ‬෩ ൐ 0. (30) Proof. In the system of equations (29) we make the change of variables. We put ܵ௞ ൌ ‫ܥ‬௞ െ ‫ݑ‬′ ௞. Now we have the system of equations ܵ௞ିଵ െ 6ܵ௞ ൅ ܵ௞ାଵ ൌ ‫ܨ‬௞, where ‫ܨ‬௞ ൌ ݂௞ െ ሺ‫ݑ‬′ ௞ିଵ െ 6‫ݑ‬′ ௞ ൅ ‫ݑ‬′ ௞ାଵሻ, ݇ ൌ 1, … , ݊ െ 1. We apply Taylor’s formula for representation ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݑ‬′ ௞ିଵ, ‫ݑ‬′ ௞ାଵ in the neighborhood ‫ݔ‬௞, and after reduction of similar terms we obtain ‫ܨ‬௞ ൌ ݄ସ ሺ 8 5! ሺ‫ݑ‬ሺହሻ ሺ߬ଵሻ ൅ ‫ݑ‬ሺହሻ ሺ߬ଶሻሻ െ 20 6! ሺ‫ݑ‬ሺହሻ ሺ߬ଷሻ ൅ ‫ݑ‬ሺହሻ ሺ߬ସሻሻ െ 1 4! ሺ‫ݑ‬ሺହሻ ሺ߬ହሻ ൅ ‫ݑ‬ሺହሻ ሺ߬଺ሻሻሻ ߬ଵ, ߬ସ, ߬ହ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ,߬ଶ, ߬ଷ, ߬଺ ‫א‬ ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ሿ.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 245 Hence |‫ܨ‬௞| ൑ ݄ସ ‫ܭ‬෩ max ሾ௫ೖషభ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, where ‫ܭ‬෩ ൌ 2 ଼ ହ! ൅ 2 ଶ଴ ଺! ൅ 2 ଵ ସ! ൌ 49/180 ൎ 0.2722. As known (see [4]) |ܵ௞| ൑ ‫ݍ‬ ൌ max ௜ | ‫ܨ‬௜|. Thus, the inequality (30) is proved with ‫ܭ‬෩ ൌ 49/180 ൎ 0.2722. We introduce the function ܷ෩෩ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾܽ, ܾሿ, associated with ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ by relation ܷ෩෩ሺ‫ݔ‬ሻ ൌ ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Theorem.Let ‫ݑ‬ ‫א‬ ‫ܥ‬ହ ሾܽ, ܾሿ , ‫ݑ‬෤෨௞ — twice continuously differentiable approximation (26) constructed using polynomial basis splines (11)-(15), then ‫צ‬ ܷ෩෩ሺఈሻ െ ‫ݑ‬ሺఈሻ ‫צ‬ሾ௔,௕ሻ൑ ‫ܭ‬෩ఈ݄ହିఈ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ߙ ൌ 0,1,2, (31) where‫ܭ‬෩଴ ൌ 0.5464‫ܭ‬෩ଵ ൌ 2.2692, ‫ܭ‬෩ଶ ൌ 5.6996. Proof. We have ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. in view of Lemma 1, Lemma 2 and relations (11)–(13), we obtain |்ܴ | ൌ |‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑ |‫ݑ‬෤௞ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൅ ൅|ሺ‫ܥ‬௞ െ ‫ݑ‬′ ௞ሻ߱௞,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ܥ‬௞ାଵ െ ‫ݑ‬′ ௞ାଵሻ߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ ݄ସ max ሾ௫ೖ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Hence we obtain (31) with ‫ܭ‬෩଴ ൌ 0.5464. Similarly, forthederivatives ‫ݑ‬෤෨ ௞ ሺఈሻ ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவሺఈሻ ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. withhelpof (20), (21), (23), (24) wehave |ܴఈ ் | ൌ |‫ݑ‬෤෨ ௞ ሺఈሻ ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ| ൑ |‫ݑ‬෤௞ ሺఈሻ ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ| ൅ ൅|ሺ‫ܥ‬௞ െ ‫ݑ‬′ ௞ሻ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺ‫ܥ‬௞ାଵ െ ‫ݑ‬′ ௞ାଵሻ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ| ൑ ൑ ݄ହିఈ max ሾ௫ೖ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, ߙ ൌ 1,2, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Now we obtain (31) with ‫ܭ‬෩ଵ ൌ 2.2692, ‫ܭ‬෩ଶ ൌ 5.6996. Tables 3 and 4 shows actual ܴ஺ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬ െ ܷ෩෩| , ܴଵ ஺ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬′ െ ܷ෩෩′ | and theoretical ்ܴ , ܴଵ ் errors of function and its first derivative approximations with ݄ ൌ 0.1and݄ ൌ 0.01.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 246 Table 3 No u(x) RA RT RA RT h=0.1 h=0.1 h=0.01 h=0.01 1 sin(3x)cos(5x) 0.51⋅10−3 0.89⋅10−1 0.106⋅10−7 0.89⋅10−6 2 tg(x) 0.45⋅10−4 0.19⋅10−1 0.15⋅10−8 0.19⋅10−6 3 cos(2x) 0.19⋅10−5 0.17⋅10−3 0.21⋅10−10 0.17⋅10−8 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.270⋅10−2 1.72 0.183⋅10−6 0.17⋅10−4 Table 4 No u(x) ܴଵ ி ܴଵ ் ܴଵ ஺ ܴଵ ் h=0.1 h=0.1 h=0.01 h=0.01 1 sin(3x)cos(5x) 0.18⋅10−1 0.20 0.36⋅10−5 0.20⋅10−4 2 tg(x) 0.14⋅10−2 0.79 0.50⋅10−6 0.79⋅10−4 3 cos(2x) 0.61⋅10−4 0.73⋅10−2 0.71⋅10−8 0.73⋅10−6 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.89⋅10−1 71.48 0.60⋅10−4 0.71⋅10−2 In Table 5, when ݄ ൌ 0.01 absolute values are presented for actual — ܴ෨ଶ, ܴଶ ஺ and theoretical — ܴଶ, ܴଶ ் errors defined by relations (1), (23)–(25) and (26), (23)–(25), second derivatives of ‫.ݑ‬ Table 5 No u(x) ܴ෨ଶ R2 ܴଶ ஺ ܴଶ ் 1 sin(3x)cos(5x) 0.11⋅10−2 0.13⋅10−1 0.10⋅10−2 0.93⋅10−1 2 tg(x) 0.22⋅10−3 0.28⋅10−2 0.29⋅10−3 0.20⋅10−1 3 cos(2x) 0.22⋅10−5 0.26⋅10−4 0.28⋅10−5 0.18⋅10−3 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.21⋅10−1 0.25 0.18⋅10−1 1.80 REFERENCES [1] Burova I.G., Dem’yanovich Y.K. Minimal splines and its applications. Theory of minimal splines.Publishing House of St. Petersburg State University, 2010. 364p. [2] Kireev V.I. Panteleev A.V. Numerical methods in examples and tasks. Moscow. 2008. 480p. [3] Burova I.G. Approximation of real and complex minimal splines. St. Petersburg. Publishing House of St. Petersburg State University, 2013. 142p. [4] Zav’yalov Y.S., Kvasov B.I., Miroshnichenko V.L. Methods of spline functions. Moscow. 1980. 353p. [5] Mehdi Zamani, “An Applied Two-Dimensional B-Spline Model for Interpolation of Data”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 322 - 336, ISSN Print: 0976-6480, ISSN Online: 0976-6499.

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20120140504026

  • 1.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 239 CONSTRUCTION OF TWICE CONTINUOUSLY DIFFERENTIABLE APPROXIMATIONS BY INTEGRO-DIFFERENTIAL SPLINES OF FIFTH ORDER AND FIRST LEVEL I.G. Burova1 , S.V. Poluyanov2 1 Professor, Saint-Petersburg State University 2 Graduate Student, Saint-Petersburg State University ABSTRACT The construction of twice continuously differentiable approximation using basis integro-differential splines of fifth order and first level is considered. The theory of minimal splines of zero or non-zero level is worked out in details in [1]. Integro-differential approximations are distinguished by the use of the integral of the function to be approximated by one or several adjacent intervals. Polynomial smooth integro-differential splines are proposed in [2]. In this paper, we consider approximations of functions and their derivatives by continuously differentiable polynomial integro-differential splines of fifth-order (see [3]). 1. BASIS SPLINES CONSTRUCTION We consider an interval ሾܽ, ܾሿ, where ܽ, ܾ — are real numbers, choose a natural number ݊ ൒ 1 and construct a grid node ሼ‫ݔ‬௝ሽ with increment ݄ ൌ ሺ௕ି௔ሻ ௡ : ܽ ൌ ‫ݔ‬଴ ൏. . . ൏ ‫ݔ‬௝ିଵ ൏ ‫ݔ‬௝ ൏ ‫ݔ‬௝ାଵ ൏. . . ൏ ‫ݔ‬௡ ൌ ܾ. Let function ‫ݑ‬ሺ‫ݔ‬ሻ be such that ‫ݑ‬ ‫א‬ ‫ܥ‬ହ ሾܽ, ܾሿ. We have the values ‫ݑ‬ሺ‫ݔ‬௞ሻ, ‫ݑ‬′ ሺ‫ݔ‬௞ሻ, ݇ ൌ 0,1, … , ݊, ‫׬‬ ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫,ݐ‬ ݇ ൌ 0,1, … , ݊ െ 1. The approximation of ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, we take in the form INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2014): 7.8273 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
  • 2.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 240 ‫ݑ‬෤௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬′ሺ‫ݔ‬௞ሻ߱௞,ଵሺ‫ݔ‬ሻ ൅ ൅‫ݑ‬′ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (1) where߱௞,଴ሺ‫ݔ‬ሻ, ߱௞ାଵ,଴ሺ‫ݔ‬ሻ, ߱௞,ଵሺ‫ݔ‬ሻ, ߱௞ାଵ,ଵሺ‫ݔ‬ሻ, ߱௞ ழଵவ ሺ‫ݔ‬ሻ are determined from conditions ‫ݑ‬෤௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬ሻ for ‫ݑ‬ሺ‫ݔ‬ሻ ൌ ‫ݔ‬௜ , ݅ ൌ 0,1,2,3,4. (2) We take ‫݌݌ݑݏ‬Ԝ߱௞,ఈ ൌ ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ାଵሿ, ߙ ൌ 0,1, ‫݌݌ݑݏ‬Ԝ߱௞ ழଵவ ൌ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ. Conditions (2) lead to a system of linear algebraic equations with respect to ߱௦,ఈ, ‫ݏ‬ ൌ ݇, ݇ ൅ 1, ߙ ൌ 0,1, ߱௞ ழଵவ : ߱௞,଴ሺ‫ݔ‬ሻ ൅ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ 1, (3) ‫ݔ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ߱௞,ଵሺ‫ݔ‬ሻ ൅ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ଶ /2 െ ‫ݔ‬௞ ଶ /2ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫,ݔ‬ (4) ‫ݔ‬௞ ଶ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ଶ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 2‫ݔ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ 2‫ݔ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ଷ /3 െ ‫ݔ‬௞ ଷ /3ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ଶ , (5) ‫ݔ‬௞ ଷ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ଷ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 3‫ݔ‬௞ ଶ ߱௞,ଵሺ‫ݔ‬ሻ ൅ 3‫ݔ‬௞ାଵ ଶ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ସ /4 െ ‫ݔ‬௞ ସ /4ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ସ , (6) ‫ݔ‬௞ ସ ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݔ‬௞ାଵ ସ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 4‫ݔ‬௞ ଷ ߱௞,ଵሺ‫ݔ‬ሻ ൅ 4‫ݔ‬௞ାଵ ଷ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ݔ‬௞ାଵ ହ /5 െ ‫ݔ‬௞ ହ /5ሻ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ହ , (7) It’s easy to see that ߱௞,଴, ߱௞,ଵ, ߱௞ ழଵவ ‫א‬ ‫ܥ‬ଵ ሺܴଵ ሻ.Let ‫צ‬ ݂ ‫צ‬ሾ௔,௕ሿൌ maxሾ௔,௕ሿ | ݂ሺ‫ݔ‬ሻ|, ‫צ‬ ݂ ‫צ‬ሾ௔,௕ሻൌ supሾ௔,௕ሻ | ݂ሺ‫ݔ‬ሻ|. Lemma 1.Let function ‫ݑ‬ ‫א‬ ‫ܥ‬ሺହሻ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ‫ݑ‬௞ defines by (1). Nextrelationsaretrue: |‫ݑ‬ሺ‫ݔ‬ሻ െ ‫ݑ‬෤௞ሺ‫ݔ‬ሻ| ൑ ݄ହ ‫ܭ‬଴ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ‫ܭ‬଴ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0, (8) |‫ݑ‬′ ሺ‫ݔ‬ሻ െ ‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ| ൑ ݄ସ ‫ܭ‬ଵ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, ‫ܭ‬ଵ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0, (9) |‫ݑ‬′′ ሺ‫ݔ‬ሻ െ ‫ݑ‬෤′′ ௞ሺ‫ݔ‬ሻ| ൑ ݄ଷ ‫ܭ‬ଶ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻ, , ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ, ‫ܭ‬ଶ ൌ ܿ‫ݐݏ݊݋‬ ൐ 0. (10) Proof. Indeed, if ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿrepresenting ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ and ‫ݑ‬′ ሺ‫ݔ‬௞ାଵሻ using Taylor’s formula, with help (3)-(7) we obtain ‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ ൌ ܴ, where ܴ ൌ 1 5! ‫ݑ‬ሺହሻ ሺ߬ଶሻሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻହ ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ 1 4! ‫ݑ‬ሺହሻ ሺ߬ଷሻሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻସ ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ൅ 1 5! න ‫ݑ‬ሺହሻ ௫ೖశభ ௫ೖ ሺ߬ଵሻሺ‫ݐ‬ െ ‫ݔ‬௞ሻହ ݀‫߱ݐ‬௞ ழଵவ ሺ‫ݔ‬ሻ െ 1 5! ‫ݑ‬ሺହሻ ሺ߬ସሻሺ‫ݔ‬ െ ‫ݔ‬௞ሻହ .
  • 3.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 241 Here ߬௜ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ, ݅ ൌ 1,2,3,4. The determinant of the system of equations (3)–(7) is equal to െ ଵ ଷ଴ ሺ‫ݔ‬௞ାଵ െ ‫ݔ‬௞ሻଽ . Solving the system of equations (3)–(7), we obtain the following formulas for basis splines if ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ: ߱௞,଴ሺ‫ݔ‬ሻ ൌ ሺ5‫ݔ‬ ൅ ݄ െ 5‫ݔ‬௞ሻሺെ3‫ݔ‬ ൅ ݄ ൅ 3‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /݄ସ , (11) ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൌ െሺെ‫ݔ‬௞ ൅ ‫ݔ‬ሻଶ ሺെ3‫ݔ‬ ൅ 3‫ݔ‬௞ ൅ 2݄ሻሺെ5‫ݔ‬ ൅ 5‫ݔ‬௞ ൅ 6݄ሻ/݄ସ , (12) ߱௞ ழଵவ ሺ‫ݔ‬ሻ ൌ 30ሺ‫ݔ‬ െ ‫ݔ‬௞ሻଶ ሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /݄ହ , (13) ߱௞,ଵሺ‫ݔ‬ሻ ൌ ሺ‫ݔ‬ െ ‫ݔ‬௞ሻሺ2݄ െ 5‫ݔ‬ ൅ 5‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻଶ /ሺ2݄ଷ ሻ, (14) ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൌ ሺ‫ݔ‬ െ ‫ݔ‬௞ሻଶ ሺെ5‫ݔ‬ ൅ 3݄ ൅ 5‫ݔ‬௞ሻሺ‫ݔ‬௞ ൅ ݄ െ ‫ݔ‬ሻ/ሺ2݄ଷ ሻ. (15) It is easy to see that for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ the following relations are valid: |߱௞,଴ሺ‫ݔ‬ሻ| ൑ 1, |߱௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 1 , |߱௞,ଵሺ‫ݔ‬ሻ| ൑ ‫ܥ‬ଵ݄, ‫ܥ‬ଵ ൎ 0.06778775 , |߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ ‫ܥ‬ଶ݄, ‫ܥ‬ଶ ൎ 0.0225959 . |߱௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 1.875/݄. Now, using the mean value theorem, we obtain for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ |ܴ| ൌ |‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ ሺ ݄ହ 5! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ାଵ,଴ሺ‫ݔ‬ሻ| ൅ ݄ସ 4! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൅ ൅ ݄଺ 6! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱௞ ழଵவ ሺ‫ݔ‬ሻ| ൅ ݄ହ 5! ሻ, hence|‫ݑ‬෤௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑ 0.02 ݄ହ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ. Similarly, |ܴଵ| ൌ |‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′ ሺ‫ݔ‬ሻ| ൑‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ ሺ ݄ହ 5! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൅ ݄ସ 4! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൅ ൅ ݄଺ 6! max ௫‫א‬ሾ௫ೖ,௫ೖశభሿ | ߱′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൅ ݄ହ 5! ሻ. Takingintoaccount |߱′ ௞,଴ሺ‫ݔ‬ሻ| ൑ 3.94023/݄, |߱′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 3.94023/݄, |߱′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 5.77350/݄ଶ , |߱′ ௞,ଵሺ‫ݔ‬ሻ| ൑ 1, |߱′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ 1, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞,‫ݔ‬௞ାଵሿ, weobtain ห‫ݑ‬෤′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′ሺ‫ݔ‬ሻห ൑ ൬ 1 5! 3.94023 ൅ 1 4! ൅ 1 6! 5.77350 ൅ 1 4! ൰ ݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿൌ 0.12419݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሿ.
  • 4.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 242 For ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ we have |߱′′ ௞,଴ሺ‫ݔ‬ሻ| ൑ 15.20000001/݄ଶ , |߱′′ ௞ାଵ,଴ሺ‫ݔ‬ሻ| ൑ 36/݄ଶ , |߱′′ ௞ ழଵவ ሺ‫ݔ‬ሻ| ൑ 60/݄ଷ , |߱′′ ௞,ଵሺ‫ݔ‬ሻ| ൑ 9/݄, |߱′′ ௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ 9/݄, then for ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ ܴଶ ൌ ห‫ݑ‬෤′′ ௞ሺ‫ݔ‬ሻ െ ‫ݑ‬′′ሺ‫ݔ‬ሻห ൑ ቆ 1 5! 36 ൅ 1 4! 9 ൅ ݄଺ 6! 60 ൅ 1 4! ቇ ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻൌ ሺ4/5ሻ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௫ೖ,௫ೖశభሻ. Remark 1. We can take ‫ܭ‬଴ ൌ 0.02 in (8), ‫ܭ‬ଵ ൑ 0.125 in (9), ‫ܭ‬ଶ ൑ 4/5 in (10). Remark 2. If we put ‫ݔ‬ ൌ ‫ݔ‬௞ ൅ ‫,݄ݐ‬ ‫ݐ‬ ‫א‬ ሾ0,1ሿ, then we get from (11)–(15) the following formulas of basis splines: ߱௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ቐ െሺ5‫ݐ‬ ൅ 1ሻሺ3‫ݐ‬ െ 1ሻሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, െሺ3‫ݐ‬ ൅ 1ሻሺ5‫ݐ‬ െ 1ሻሺ1 ൅ ‫ݐ‬ሻଶ , ‫ݐ‬ ‫א‬ ሾെ1,0ሿ, 0, ‫ݐ‬ ‫ב‬ ሾെ1,1ሿ, (16) ߱௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ‫ە‬ ۖ ‫۔‬ ۖ ‫ۓ‬െ 1 2 ‫݄ݐ‬ሺ5‫ݐ‬ െ 2ሻሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, 1 2 ‫݄ݐ‬ሺ2 ൅ 5‫ݐ‬ሻሺ1 ൅ ‫ݐ‬ሻଶ , ‫ݐ‬ ‫א‬ ሾെ1,0ሿ, 0, ‫ݐ‬ ‫ב‬ ሾെ1,1ሿ, (17) ߱௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ቐ 30‫ݐ‬ଶ ݄ ሺ‫ݐ‬ െ 1ሻଶ , ‫ݐ‬ ‫א‬ ሾ0,1ሿ, 0, ‫ݐ‬ ‫ב‬ ሾ0,1ሿ, (18) and also for ‫ݐ‬ ‫א‬ ሾ0,1ሿ we have ߱௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െሺ5‫ݐ‬ െ 6ሻሺ3‫ݐ‬ െ 2ሻ‫ݐ‬ଶ , ߱௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 1 2 ݄ሺ5‫ݐ‬ െ 3ሻሺ‫ݐ‬ െ 1ሻ‫ݐ‬ଶ , (19) ߱′ ௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12‫ݐ‬ሺ5‫ݐ‬ െ 3ሻሺ‫ݐ‬ െ 1ሻ/݄, ߱′ ௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12‫ݐ‬ሺ5‫ݐ‬ଶ െ 7‫ݐ‬ ൅ 2ሻ/݄, (20) ߱′ ௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 60‫ݐ‬ሺ2‫ݐ‬ െ 1ሻሺ‫ݐ‬ െ 1ሻ/݄ଶ , (21) ߱′ ௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െሺ‫ݐ‬ െ 1ሻሺ10‫ݐ‬ଶ െ 8‫ݐ‬ ൅ 1ሻ, ߱′ ௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ ‫ݐ‬ሺ3 െ 12‫ݐ‬ ൅ 10‫ݐ‬ଶ ሻ. (22) ߱′′ ௞,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12ሺ3 െ 16‫ݐ‬ ൅ 15‫ݐ‬ଶ ሻ/݄ଶ , ߱′′ ௞ାଵ,଴ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ12ሺ15‫ݐ‬ଶ െ 14‫ݐ‬ ൅ 2ሻ/݄ଶ , (23) ߱′′ ௞ ழଵவ ሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 60ሺ1 െ 6‫ݐ‬ ൅ 6‫ݐ‬ଶ ሻ/݄ଷ , (24)
  • 5.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 243 ߱′′ ௞,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ െ3ሺ3 െ 12‫ݐ‬ ൅ 10‫ݐ‬ଶ ሻ/݄, ߱′′ ௞ାଵ,ଵሺ‫ݔ‬௞ ൅ ‫݄ݐ‬ሻ ൌ 3ሺെ8‫ݐ‬ ൅ 10‫ݐ‬ଶ ൅ 1ሻ/݄. (25) Remark 3. We obtain approximation for ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ, ߙ ൌ 1,2, using the formulas: ‫ݑ‬෤௞ ሺఈሻ ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ൅‫ݑ‬′ ሺ‫ݔ‬௞ሻ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬′ ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ሺ߱௞ ழଵவ ሻሺఈሻ ሺ‫ݔ‬ሻ, where ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ, ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ, ሺ߱௞ ழଵவ ሻሺఈሻ ሺ‫ݔ‬ሻ are determined by the formulas (20)–(25). We introduce the function ܷ෩ሺ‫ݔ‬ሻ , ‫ݔ‬ ‫א‬ ሾܽ, ܾሿ , associated with ‫ݑ‬෤௞ሺ‫ݔ‬ሻ by the relation ܷ෩ሺ‫ݔ‬ሻ ൌ ‫ݑ‬෤௞ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Now the inequalities are obvious: ‫צ‬ ܷ෩ െ ‫ݑ‬ ‫צ‬ሾ௔,௕ሿ൑ ‫ܭ‬଴݄ହ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ‫צ‬ ܷ෩′ െ ‫ݑ‬′ ‫צ‬ሾ௔,௕ሿ൑ ‫ܭ‬ଵ݄ସ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ‫צ‬ ܷ෩′′ െ ‫ݑ‬′′ ‫צ‬ሾ௔,௕ሻ൑ ‫ܭ‬ଶ݄ଷ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሻ. In tables 1 and 2 for ݄ ൌ 0.1, ݄ ൌ 0.01, respectively, the errors of approximations of some functions and their first derivatives of the integro-differential fifth-order splines are presented. Here we use the following designations: for the actual error function and its first derivative — ܴ෨ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬ െ ܷ෩|, ܴଵ ෪ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬′ െ ܷ′෩ |, for theoretical error function and its first derivative — ܴ and ܴଵ respectively. Table 1 No u(x) ܴ෨ ܴ෨ଵ R0 R1 1 sin(3x)cos(5x) 0.12⋅10−4 0.81⋅10−3 0.33⋅10−3 0.20 2 tg(x) 0.16⋅10−5 0.11⋅10−3 0.69⋅10−4 0.43⋅10−1 3 cos(2x) 0.24⋅10−7 0.17⋅10−5 0.64⋅10−6 0.40⋅10−3 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.21⋅10−3 0.14⋅10−1 0.64⋅10−6 0.40⋅10−3 Table 2 No u(x) ܴ෨ ܴ෨ଵ R0 R1 1 sin(3x)cos(5x) 0.12⋅10−9 0.85⋅10−7 0.33⋅10−8 0.20⋅10−4 2 tg(x) 0.24⋅10−10 0.17⋅10−7 0.69⋅10−9 0.43⋅10−5 3 cos(2x) 0.24⋅10−12 0.17⋅10−9 0.64⋅10−11 0.40⋅10−7 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.23⋅10−8 0.16⋅10−5 0.63⋅10−7 0.39⋅10−3
  • 6.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 244 2. THE CONSTRUCTION OF TWICE CONTINUOUSLY DIFFERENTIABLE APPROXIMATIONS Let mesh nodes ሼ‫ݔ‬௝ሽ be uniform. With each ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ consider an approximation for ‫ݑ‬ሺ‫ݔ‬ሻ in the form ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (26) where the real numbers ‫ܥ‬௞ we find later. We denote ‫ݑ‬′ ௞ ൌ ‫ݑ‬′ ሺ‫ݔ‬௞ሻ and take on the interval ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬௞ାଵ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ න ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, (27) and on the adjacent interval ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ሻ the relation ‫ݑ‬෤෨௞ିଵሺ‫ݔ‬ሻ ൌ ‫ݑ‬௞ିଵ߱௞ିଵ,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬௞߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ିଵ߱௞ିଵ,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ න ‫ݑ‬ ௫ೖ ௫ೖషభ ሺ‫ݐ‬ሻ݀‫߱ݐ‬௞ିଵ ழଵவ ሺ‫ݔ‬ሻ. (28) Twice differentiating the relations (27)–(28) from the condition ‫ݑ‬෤෨௞ ′′ ሺ‫ݔ‬௞൅ሻ ൌ ‫ݑ‬෤෨௞ିଵ ′′ ሺ‫ݔ‬௞െሻ we obtain the system of equations ‫ܥ‬௞ିଵ െ 6‫ܥ‬௞ ൅ ‫ܥ‬௞ାଵ ൌ ݂௞, (29) where ݂௞ ൌ 8ሺ‫ݑ‬௞ାଵ െ ‫ݑ‬௞ିଵሻ/݄ ൅ 20 ቆන ‫ݑ‬ ௫ೖ ௫ೖషభ ሺ‫ݐ‬ሻ݀‫ݐ‬ െ න ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ቇ /݄ଶ , ݇ ൌ 1, … , ݊ െ 1. Now relations for twice continuously differentiable splines ܷሺ‫ݔ‬ሻ, on each interval ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ, ݇ ൌ 1,2, … , ݊ െ 1, are defined by (26) with coefficients ‫ܥ‬௞, which are the solution of the system (29). Lemma 2. |‫ܥ‬௞ െ ‫ݑ‬′ ሺ‫ݔ‬௞ሻ| ൑ ‫,ݍ‬ ‫ݍ‬ ൌ ‫ܭ‬෩݄ସ , ‫ܭ‬෩ ൐ 0. (30) Proof. In the system of equations (29) we make the change of variables. We put ܵ௞ ൌ ‫ܥ‬௞ െ ‫ݑ‬′ ௞. Now we have the system of equations ܵ௞ିଵ െ 6ܵ௞ ൅ ܵ௞ାଵ ൌ ‫ܨ‬௞, where ‫ܨ‬௞ ൌ ݂௞ െ ሺ‫ݑ‬′ ௞ିଵ െ 6‫ݑ‬′ ௞ ൅ ‫ݑ‬′ ௞ାଵሻ, ݇ ൌ 1, … , ݊ െ 1. We apply Taylor’s formula for representation ‫ݑ‬ሺ‫ݔ‬ሻ, ‫ݑ‬′ ௞ିଵ, ‫ݑ‬′ ௞ାଵ in the neighborhood ‫ݔ‬௞, and after reduction of similar terms we obtain ‫ܨ‬௞ ൌ ݄ସ ሺ 8 5! ሺ‫ݑ‬ሺହሻ ሺ߬ଵሻ ൅ ‫ݑ‬ሺହሻ ሺ߬ଶሻሻ െ 20 6! ሺ‫ݑ‬ሺହሻ ሺ߬ଷሻ ൅ ‫ݑ‬ሺହሻ ሺ߬ସሻሻ െ 1 4! ሺ‫ݑ‬ሺହሻ ሺ߬ହሻ ൅ ‫ݑ‬ሺହሻ ሺ߬଺ሻሻሻ ߬ଵ, ߬ସ, ߬ହ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሿ,߬ଶ, ߬ଷ, ߬଺ ‫א‬ ሾ‫ݔ‬௞ିଵ, ‫ݔ‬௞ሿ.
  • 7.
    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 245 Hence |‫ܨ‬௞| ൑ ݄ସ ‫ܭ‬෩ max ሾ௫ೖషభ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, where ‫ܭ‬෩ ൌ 2 ଼ ହ! ൅ 2 ଶ଴ ଺! ൅ 2 ଵ ସ! ൌ 49/180 ൎ 0.2722. As known (see [4]) |ܵ௞| ൑ ‫ݍ‬ ൌ max ௜ | ‫ܨ‬௜|. Thus, the inequality (30) is proved with ‫ܭ‬෩ ൌ 49/180 ൎ 0.2722. We introduce the function ܷ෩෩ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾܽ, ܾሿ, associated with ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ by relation ܷ෩෩ሺ‫ݔ‬ሻ ൌ ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Theorem.Let ‫ݑ‬ ‫א‬ ‫ܥ‬ହ ሾܽ, ܾሿ , ‫ݑ‬෤෨௞ — twice continuously differentiable approximation (26) constructed using polynomial basis splines (11)-(15), then ‫צ‬ ܷ෩෩ሺఈሻ െ ‫ݑ‬ሺఈሻ ‫צ‬ሾ௔,௕ሻ൑ ‫ܭ‬෩ఈ݄ହିఈ ‫צ‬ ‫ݑ‬ሺହሻ ‫צ‬ሾ௔,௕ሿ, ߙ ൌ 0,1,2, (31) where‫ܭ‬෩଴ ൌ 0.5464‫ܭ‬෩ଵ ൌ 2.2692, ‫ܭ‬෩ଶ ൌ 5.6996. Proof. We have ‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவ ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. in view of Lemma 1, Lemma 2 and relations (11)–(13), we obtain |்ܴ | ൌ |‫ݑ‬෤෨௞ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൑ |‫ݑ‬෤௞ െ ‫ݑ‬ሺ‫ݔ‬ሻ| ൅ ൅|ሺ‫ܥ‬௞ െ ‫ݑ‬′ ௞ሻ߱௞,ଵሺ‫ݔ‬ሻ ൅ ሺ‫ܥ‬௞ାଵ െ ‫ݑ‬′ ௞ାଵሻ߱௞ାଵ,ଵሺ‫ݔ‬ሻ| ൑ ݄ସ max ሾ௫ೖ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Hence we obtain (31) with ‫ܭ‬෩଴ ൌ 0.5464. Similarly, forthederivatives ‫ݑ‬෤෨ ௞ ሺఈሻ ሺ‫ݔ‬ሻ ൌ ‫ݑ‬ሺ‫ݔ‬௞ሻ߱௞,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ݑ‬ሺ‫ݔ‬௞ାଵሻ߱௞ାଵ,଴ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ൅‫ܥ‬௞߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ‫ܥ‬௞ାଵ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺන ‫ݑ‬ ௫ೖశభ ௫ೖ ሺ‫ݐ‬ሻ݀‫ݐ‬ሻ ߱௞ ழଵவሺఈሻ ሺ‫ݔ‬ሻ, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. withhelpof (20), (21), (23), (24) wehave |ܴఈ ் | ൌ |‫ݑ‬෤෨ ௞ ሺఈሻ ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ| ൑ |‫ݑ‬෤௞ ሺఈሻ ሺ‫ݔ‬ሻ െ ‫ݑ‬ሺఈሻ ሺ‫ݔ‬ሻ| ൅ ൅|ሺ‫ܥ‬௞ െ ‫ݑ‬′ ௞ሻ߱௞,ଵ ሺఈሻ ሺ‫ݔ‬ሻ ൅ ሺ‫ܥ‬௞ାଵ െ ‫ݑ‬′ ௞ାଵሻ߱௞ାଵ,ଵ ሺఈሻ ሺ‫ݔ‬ሻ| ൑ ൑ ݄ହିఈ max ሾ௫ೖ,௫ೖశభሿ | ‫ݑ‬ሺହሻ ሺ‫ݔ‬ሻ|, ߙ ൌ 1,2, ‫ݔ‬ ‫א‬ ሾ‫ݔ‬௞, ‫ݔ‬௞ାଵሻ. Now we obtain (31) with ‫ܭ‬෩ଵ ൌ 2.2692, ‫ܭ‬෩ଶ ൌ 5.6996. Tables 3 and 4 shows actual ܴ஺ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬ െ ܷ෩෩| , ܴଵ ஺ ൌ maxሾିଵ,ଵሿ | ‫ݑ‬′ െ ܷ෩෩′ | and theoretical ்ܴ , ܴଵ ் errors of function and its first derivative approximations with ݄ ൌ 0.1and݄ ൌ 0.01.
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    International Journal ofAdvanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 4, April (2014), pp. 239-246 © IAEME 246 Table 3 No u(x) RA RT RA RT h=0.1 h=0.1 h=0.01 h=0.01 1 sin(3x)cos(5x) 0.51⋅10−3 0.89⋅10−1 0.106⋅10−7 0.89⋅10−6 2 tg(x) 0.45⋅10−4 0.19⋅10−1 0.15⋅10−8 0.19⋅10−6 3 cos(2x) 0.19⋅10−5 0.17⋅10−3 0.21⋅10−10 0.17⋅10−8 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.270⋅10−2 1.72 0.183⋅10−6 0.17⋅10−4 Table 4 No u(x) ܴଵ ி ܴଵ ் ܴଵ ஺ ܴଵ ் h=0.1 h=0.1 h=0.01 h=0.01 1 sin(3x)cos(5x) 0.18⋅10−1 0.20 0.36⋅10−5 0.20⋅10−4 2 tg(x) 0.14⋅10−2 0.79 0.50⋅10−6 0.79⋅10−4 3 cos(2x) 0.61⋅10−4 0.73⋅10−2 0.71⋅10−8 0.73⋅10−6 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.89⋅10−1 71.48 0.60⋅10−4 0.71⋅10−2 In Table 5, when ݄ ൌ 0.01 absolute values are presented for actual — ܴ෨ଶ, ܴଶ ஺ and theoretical — ܴଶ, ܴଶ ் errors defined by relations (1), (23)–(25) and (26), (23)–(25), second derivatives of ‫.ݑ‬ Table 5 No u(x) ܴ෨ଶ R2 ܴଶ ஺ ܴଶ ் 1 sin(3x)cos(5x) 0.11⋅10−2 0.13⋅10−1 0.10⋅10−2 0.93⋅10−1 2 tg(x) 0.22⋅10−3 0.28⋅10−2 0.29⋅10−3 0.20⋅10−1 3 cos(2x) 0.22⋅10−5 0.26⋅10−4 0.28⋅10−5 0.18⋅10−3 4 1 ሺ1 ൅ 25‫ݔ‬ଶሻ 0.21⋅10−1 0.25 0.18⋅10−1 1.80 REFERENCES [1] Burova I.G., Dem’yanovich Y.K. Minimal splines and its applications. Theory of minimal splines.Publishing House of St. Petersburg State University, 2010. 364p. [2] Kireev V.I. Panteleev A.V. Numerical methods in examples and tasks. Moscow. 2008. 480p. [3] Burova I.G. Approximation of real and complex minimal splines. St. Petersburg. Publishing House of St. Petersburg State University, 2013. 142p. [4] Zav’yalov Y.S., Kvasov B.I., Miroshnichenko V.L. Methods of spline functions. Moscow. 1980. 353p. [5] Mehdi Zamani, “An Applied Two-Dimensional B-Spline Model for Interpolation of Data”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 322 - 336, ISSN Print: 0976-6480, ISSN Online: 0976-6499.