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
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ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 4, April (2014), pp. 239-246
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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. 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. 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. 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. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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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 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. 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.