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- 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 ሼݔሽ with increment ݄ ൌ
ሺିሻ
:
ܽ ൌ ݔ ൏. . . ൏ ݔିଵ ൏ ݔ ൏ ݔାଵ ൏. . . ൏ ݔ ൌ ܾ.
Let function ݑሺݔሻ be such that ݑ א ܥହ
ሾܽ, ܾሿ. We have the values ݑሺݔሻ, ݑ′
ሺݔሻ, ݇ ൌ
0,1, … , ݊, ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀,ݐ ݇ ൌ 0,1, … , ݊ െ 1. The approximation of ݑሺݔሻ, ݔ א ሾݔ,ݔାଵሿ, we take in
the form
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING
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ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 4, April (2014), pp. 239-246
© IAEME: www.iaeme.com/ijaret.asp
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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
ݑሺݔሻ ൌ ݑሺݔሻ߱,ሺݔሻ ݑሺݔାଵሻ߱ାଵ,ሺݔሻ ݑ′ሺݔሻ߱,ଵሺݔሻ
ݑ′ሺݔାଵሻ߱ାଵ,ଵሺݔሻ ሺන ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀ݐሻ ߱
ழଵவ
ሺݔሻ, (1)
where߱,ሺݔሻ, ߱ାଵ,ሺݔሻ, ߱,ଵሺݔሻ, ߱ାଵ,ଵሺݔሻ, ߱
ழଵவ
ሺݔሻ are determined from conditions
ݑሺݔሻ ൌ ݑሺݔሻ for ݑሺݔሻ ൌ ݔ
, ݅ ൌ 0,1,2,3,4. (2)
We take ݑݏԜ߱,ఈ ൌ ሾݔିଵ, ݔାଵሿ, ߙ ൌ 0,1, ݑݏԜ߱
ழଵவ
ൌ ሾݔ, ݔାଵሿ.
Conditions (2) lead to a system of linear algebraic equations with respect to ߱௦,ఈ, ݏ ൌ ݇, ݇
1, ߙ ൌ 0,1, ߱
ழଵவ
:
߱,ሺݔሻ ߱ାଵ,ሺݔሻ ሺݔାଵ െ ݔሻ߱
ழଵவ
ሺݔሻ ൌ 1, (3)
ݔ߱,ሺݔሻ ݔାଵ߱ାଵ,ሺݔሻ ߱,ଵሺݔሻ ߱ାଵ,ଵሺݔሻ
ሺݔାଵ
ଶ
/2 െ ݔ
ଶ
/2ሻ߱
ழଵவ
ሺݔሻ ൌ ,ݔ
(4)
ݔ
ଶ
߱,ሺݔሻ ݔାଵ
ଶ
߱ାଵ,ሺݔሻ 2ݔ߱,ଵሺݔሻ 2ݔାଵ߱ାଵ,ଵሺݔሻ
ሺݔାଵ
ଷ
/3 െ ݔ
ଷ
/3ሻ߱
ழଵவ
ሺݔሻ ൌ ݔଶ
,
(5)
ݔ
ଷ
߱,ሺݔሻ ݔାଵ
ଷ
߱ାଵ,ሺݔሻ 3ݔ
ଶ
߱,ଵሺݔሻ 3ݔାଵ
ଶ
߱ାଵ,ଵሺݔሻ
ሺݔାଵ
ସ
/4 െ ݔ
ସ
/4ሻ߱
ழଵவ
ሺݔሻ ൌ ݔସ
,
(6)
ݔ
ସ
߱,ሺݔሻ ݔାଵ
ସ
߱ାଵ,ሺݔሻ 4ݔ
ଷ
߱,ଵሺݔሻ 4ݔାଵ
ଷ
߱ାଵ,ଵሺݔሻ
ሺݔାଵ
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, ݔାଵሿ, ݑ defines by (1). Nextrelationsaretrue:
|ݑሺݔሻ െ ݑሺݔሻ| ݄ହ
ܭ צ ݑሺହሻ
צሾ௫ೖ,௫ೖశభሿ, ݔ א ሾݔ, ݔାଵሿ,
ܭ ൌ ܿݐݏ݊ 0,
(8)
|ݑ′
ሺݔሻ െ ݑ′
ሺݔሻ| ݄ସ
ܭଵ צ ݑሺହሻ
צሾ௫ೖ,௫ೖశభሿ, ݔ א ሾݔ,ݔାଵሿ,
ܭଵ ൌ ܿݐݏ݊ 0,
(9)
|ݑ′′
ሺݔሻ െ ݑ′′
ሺݔሻ| ݄ଷ
ܭଶ צ ݑሺହሻ
צሾ௫ೖ,௫ೖశభሻ, , ݔ א ሾݔ, ݔାଵሻ,
ܭଶ ൌ ܿݐݏ݊ 0.
(10)
Proof. Indeed, if ݔ א ሾݔ, ݔାଵሿrepresenting ݑሺݔሻ, ݑሺݔାଵሻ and ݑ′
ሺݔାଵሻ using Taylor’s
formula, with help (3)-(7) we obtain
ݑሺݔሻ െ ݑሺݔሻ ൌ ܴ,
where
ܴ ൌ
1
5!
ݑሺହሻ
ሺ߬ଶሻሺݔାଵ െ ݔሻହ
߱ାଵ,ሺݔሻ
1
4!
ݑሺହሻ
ሺ߬ଷሻሺݔାଵ െ ݔሻସ
߱ାଵ,ଵሺݔሻ
1
5!
න ݑሺହሻ
௫ೖశభ
௫ೖ
ሺ߬ଵሻሺݐ െ ݔሻହ
݀߱ݐ
ழଵவ
ሺݔሻ െ
1
5!
ݑሺହሻ
ሺ߬ସሻሺݔ െ ݔሻହ
.
- 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, ݔାଵሿ, ݅ ൌ 1,2,3,4.
The determinant of the system of equations (3)–(7) is equal to െ
ଵ
ଷ
ሺݔାଵ െ ݔሻଽ
.
Solving the system of equations (3)–(7), we obtain the following formulas for basis splines if
ݔ א ሾݔ, ݔାଵሿ:
߱,ሺݔሻ ൌ ሺ5ݔ ݄ െ 5ݔሻሺെ3ݔ ݄ 3ݔሻሺݔ ݄ െ ݔሻଶ
/݄ସ
, (11)
߱ାଵ,ሺݔሻ ൌ െሺെݔ ݔሻଶ
ሺെ3ݔ 3ݔ 2݄ሻሺെ5ݔ 5ݔ
6݄ሻ/݄ସ
,
(12)
߱
ழଵவ
ሺݔሻ ൌ 30ሺݔ െ ݔሻଶ
ሺݔ ݄ െ ݔሻଶ
/݄ହ
, (13)
߱,ଵሺݔሻ ൌ ሺݔ െ ݔሻሺ2݄ െ 5ݔ 5ݔሻሺݔ ݄ െ ݔሻଶ
/ሺ2݄ଷ
ሻ, (14)
߱ାଵ,ଵሺݔሻ ൌ ሺݔ െ ݔሻଶ
ሺെ5ݔ 3݄ 5ݔሻሺݔ ݄ െ ݔሻ/ሺ2݄ଷ
ሻ. (15)
It is easy to see that for ݔ א ሾݔ, ݔାଵሿ 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, ݔାଵሿ
|ܴ| ൌ |ݑሺݔሻ െ 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|ݑሺݔሻ െ 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,ݔାଵሿ,
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, ݔାଵሻ we have
|߱′′
,ሺݔሻ| 15.20000001/݄ଶ
, |߱′′
ାଵ,ሺݔሻ| 36/݄ଶ
, |߱′′
ழଵவ
ሺݔሻ| 60/݄ଷ
,
|߱′′
,ଵሺݔሻ| 9/݄, |߱′′
ାଵ,ଵሺݔሻ| 9/݄,
then for ݔ א ሾݔ, ݔାଵሻ
ܴ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 ൌ ݑሺݔሻ߱,
ሺఈሻ
ሺݔሻ ݑሺݔାଵሻ߱ାଵ,
ሺఈሻ
ሺݔሻ
ݑ′
ሺݔሻ߱,ଵ
ሺఈሻ
ሺݔሻ ݑ′
ሺݔାଵሻ߱ାଵ,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 ݑሺݔሻ by the relation
ܷ෩ሺݔሻ ൌ ݑሺݔሻ, ݔ א ሾݔ, ݔାଵሻ. Now the inequalities are obvious:
צ ܷ෩ െ ݑ צሾ,ሿ ܭ݄ହ
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 –
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 ሼݔሽ be uniform. With each ሾݔ, ݔାଵሻ consider an approximation for ݑሺݔሻ in
the form
ݑ෨ሺݔሻ ൌ ݑሺݔሻ߱,ሺݔሻ ݑሺݔାଵሻ߱ାଵ,ሺݔሻ
ܥ߱,ଵሺݔሻ ܥାଵ߱ାଵ,ଵሺݔሻ ሺන ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀ݐሻ ߱
ழଵவ
ሺݔሻ, (26)
where the real numbers ܥ we find later.
We denote ݑ′
ൌ ݑ′
ሺݔሻ and take on the interval ሾݔ, ݔାଵሻ
ݑ෨ሺݔሻ ൌ ݑ߱,ሺݔሻ ݑାଵ߱ାଵ,ሺݔሻ ܥ߱,ଵሺݔሻ ܥାଵ߱ାଵ,ଵሺݔሻ
න ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀ݐ ߱
ழଵவ
ሺݔሻ, (27)
and on the adjacent interval ሾݔିଵ, ݔሻ the relation
ݑ෨ିଵሺݔሻ ൌ ݑିଵ߱ିଵ,ሺݔሻ ݑ߱,ሺݔሻ ܥିଵ߱ିଵ,ଵሺݔሻ
ܥ߱,ଵሺݔሻ න ݑ
௫ೖ
௫ೖషభ
ሺݐሻ݀߱ݐିଵ
ழଵவ
ሺݔሻ.
(28)
Twice differentiating the relations (27)–(28) from the condition ݑ෨
′′
ሺݔሻ ൌ ݑ෨ିଵ
′′
ሺݔെሻ we obtain
the system of equations
ܥିଵ െ 6ܥ ܥାଵ ൌ ݂, (29)
where
݂ ൌ 8ሺݑାଵ െ ݑିଵሻ/݄ 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, ݔାଵሻ,
݇ ൌ 1,2, … , ݊ െ 1, are defined by (26) with coefficients ܥ, which are the solution of the system (29).
Lemma 2.
|ܥ െ ݑ′
ሺݔሻ| ,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
ሺ߬ሻሻሻ
߬ଵ, ߬ସ, ߬ହ א ሾݔ, ݔାଵሿ,߬ଶ, ߬ଷ, ߬ א ሾݔିଵ, ݔሿ.
- 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 ݑ෨ሺݔሻ by relation ܷ෩෩ሺݔሻ ൌ ݑ෨ሺݔሻ,
ݔ א ሾݔ, ݔାଵሻ.
Theorem.Let ݑ א ܥହ
ሾܽ, ܾሿ , ݑ෨ — 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
ݑ෨ሺݔሻ ൌ ݑሺݔሻ߱,ሺݔሻ ݑሺݔାଵሻ߱ାଵ,ሺݔሻ
ܥ߱,ଵሺݔሻ ܥାଵ߱ାଵ,ଵሺݔሻ ሺන ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀ݐሻ ߱
ழଵவ
ሺݔሻ, ݔ א ሾݔ, ݔାଵሻ.
in view of Lemma 1, Lemma 2 and relations (11)–(13), we obtain
|்ܴ
| ൌ |ݑ෨ሺݔሻ െ ݑሺݔሻ| |ݑ െ ݑሺݔሻ|
|ሺܥ െ ݑ′
ሻ߱,ଵሺݔሻ ሺܥାଵ െ ݑ′
ାଵሻ߱ାଵ,ଵሺݔሻ| ݄ସ
max
ሾ௫ೖ,௫ೖశభሿ
| ݑሺହሻ
ሺݔሻ|, ݔ א ሾݔ, ݔାଵሻ.
Hence we obtain (31) with ܭ෩ ൌ 0.5464.
Similarly, forthederivatives
ݑ෨
ሺఈሻ
ሺݔሻ ൌ ݑሺݔሻ߱,
ሺఈሻ
ሺݔሻ ݑሺݔାଵሻ߱ାଵ,
ሺఈሻ
ሺݔሻ
ܥ߱,ଵ
ሺఈሻ
ሺݔሻ ܥାଵ߱ାଵ,ଵ
ሺఈሻ
ሺݔሻ ሺන ݑ
௫ೖశభ
௫ೖ
ሺݐሻ݀ݐሻ ߱
ழଵவሺఈሻ
ሺݔሻ, ݔ א ሾݔ, ݔାଵሻ.
withhelpof (20), (21), (23), (24) wehave
|ܴఈ
்
| ൌ |ݑ෨
ሺఈሻ
ሺݔሻ െ ݑሺఈሻ
ሺݔሻ| |ݑ
ሺఈሻ
ሺݔሻ െ ݑሺఈሻ
ሺݔሻ|
|ሺܥ െ ݑ′
ሻ߱,ଵ
ሺఈሻ
ሺݔሻ ሺܥାଵ െ ݑ′
ାଵሻ߱ାଵ,ଵ
ሺఈሻ
ሺݔሻ|
݄ହିఈ
max
ሾ௫ೖ,௫ೖశభሿ
| ݑሺହሻ
ሺݔሻ|, ߙ ൌ 1,2, ݔ א ሾݔ, ݔାଵሻ.
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.