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### 20120140504026

1. 1. 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
2. 2. 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 .
3. 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 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. 4. 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)
5. 5. 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
6. 6. 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.
7. 7. 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.
8. 8. 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.