General Mathematics Vol. 18, No. 1 (2010), 3–17
Natural splines of Birkhoff type
approximating the solution of differential
equations 1
Ana Maria Acu
Abstract
Let us consider the following initial value problem

 y (r) = f (x, y, y 0 , . . . , y (r−1) )
(1)
 y (j) (x ) = y (j) , j = 0, r − 1
1
1
and suppose that f : D → R, D ⊂ [a, b] × Rr satisfies all the conditions assuring the existence and uniqueness of the solution y : [a, b] →
R of the problem (1).
If y is the exact solution of the problem (1), denote by
Ũy =
n
u ∈ H m,2 [a, b] | u(j) (x1 ) = y (j) (x1 ), j = 0, r − 1,
o
u(r) (xi ) = y (r) (xi ), i = 2, n ,
where x1 , x2 , . . . , xn are distinct knots in the interval [a, b], a ≤ x 1 <
x2 < · · · < xn ≤ b.
1
Received 15 September, 2009
Accepted for publication (in revised form) 14 October, 2009
3
4
A.M. Acu
We will give an algorithm to approximate the solution y of (1) by a
Birkhoff-type natural spline interpolation function, s y , corresponding
to the interpolatory set Ũy .
2000 Mathematics Subject Classification:65D07, 65D05
Key words and phrases: Natural splines of Birkhoff type, differential
equations, interpolation.
1
Introduction
Let us consider the following initial value problem

 y 0 = f (x, y)
(2)
 y(x ) = y
0
0
and suppose that f : D ⊂ R2 → R satisfies all the conditions assuring the
existence and uniqueness of the solution y : [a, b] → R of the problem (2).
In paper [2] G. Micula and P. Blaga develop an algorithm to approximate
the solution y of (2) by a spline function of even degree.
Using natural splines of Birkhoff type we will give an algorithm to
approximate the solution of the differential equation problem

 y (r) = f (x, y, y 0, . . . , y (r−1) )
 y (j) (x ) = y (j) , j = 0, r − 1
1
1
and suppose that f : D → R, D ⊂ [a, b] × Rr satisfies all the conditions
assuring the existence and uniqueness of the solution y : [a, b] → R of the
problem.
Natural splines of Birkhoff type approximating...
5
Let us consider an arbitrary finite interval [a, b], a < b, on the real line
and the Lebesque space L2 [a, b] with the corresponding norm
Z b
2
kf k2 :=
f 2 (x)dx.
a
Denote by
H m,2 [a, b] = f : [a, b] → R | f ∈ C m−1 [a, b], f (m−1) abs.cont., f (m) ∈ L2 [a, b] .
Let us take x1 , x2 , . . . , xn as distinct knots in the interval [a, b], a ≤ x1 <
x2 < · · · < xn ≤ b, the numbers αi ∈ N, where 1 ≤ αi ≤ m, i = 1, n and
the set Ii ⊆ {0, 1, . . . , αi − 1} , i = 1, n.
Denote the number of interpolation condition by N :=
n
X
card(Ii ).
i=1
Definition 1 The set
Λ := λi,νi : H m,2 [a, b] → R, λi,νi (f ) = f (νi ) (xi ), i = 1, n, νi ∈ Ii
is named a set of Birkhoff-type functionals on H m,2 [a, b].
Definition 2 For each y = (y1 , y2 , · · · , yN ) ∈ RN we define the Birkhofftype interpolatory set
Uy := u ∈ H m,2 [a, b] | u(νi ) (xi ) = yi,νi , i = 1, n, νi ∈ Ii ,
where y := (y1 , y2 , . . . , yn ) = ((y1,ν1 )ν1 ∈I1 , (y2,ν2 )ν2 ∈I2 , . . . , (yn,νn )νn ∈In ) .
Definition 3 The problem of finding function s ∈ Uy which satisfy
(3)
s(m)
2
= min u(m)
u∈Uy
2
is called Birkhoff-type natural spline interpolation problem, corresponding to
the interpolatory set Uy .
6
A.M. Acu
Lemma 1 For each y ∈ RN , if the set Uy is nonempty, then problem (3)
has unique solution sy given by
sy (x) =
m−1
X
k=0
n
ak
i
(b − x)k X X
(x − xi )2m−1−ν
+
+
bi,νi
,
k!
(2m
−
1
−
ν
)!
i
i=1 ν ∈I
i
i
where the coefficients ak , k = 0, m − 1, bi,νi , i = 1, n, νi ∈ Ii , are given by
the following linear system

m−1
k−µ1
X


µ1 (b − x1 )

ak (−1)

= y1,µ1 , µ1 ∈ I1


(k − µ1 )!

k=µ

1


j−1
m−1

k−µj
X
X
X

(xj − xi )2m−1−νi −µi

µj (b − xj )

+
bi,νi
= yj,µj ,
ak (−1)
(k
−
µ
(2m
−
1
−
ν
−
µ
)!
j )!
i
j
i=1 νi ∈Ii
k=µj




j = 2, n, µj ∈ Ij ,




n

X
X
(b − xi )l−νi



= 0, l = 0, m − 1.
b
i,ν

i

(l − νi )!
i=1 ν ∈I ,ν ≤l
i
i
i
Definition 4 For each j = 1, n and µj ∈ Ij , let yj,µj ∈ RN be defined by
yj,µj := (δij δνi ,µj )i=1,n,νi ∈Ii = (δ1,j δν1 ,µj )ν1 ∈I1 , . . . , (δn,j δνn ,µj )νn ∈In
and the corresponding interpolatory set
Uj,µj := u ∈ H m,2 [a, b] | u(νi ) (xi ) = δij δνi ,µj , i = 1, n, νi ∈ Ii .
We assume that the sets Uj,µj , j = 1, n, µj ∈ Ij are nonempty. For each
j = 1, n and µj ∈ Ij , denote by sj,µj the unique solution of problem (3)
corresponding to the interpolatory set Uj,µj . The functions sj,µj are called
the fundamental Birkhoff-type natural spline interpolation functions.
Natural splines of Birkhoff type approximating...
7
Lemma 2 For each y ∈ RN the set Uy is nonempty and the problem (3),
corresponding to Uy , has unique solution given by
sy =
n X
X
yi,νi si,νi ,
i=1 νi ∈Ii
where y := (y1 , . . . , yn ) = ((y1,ν1 )ν1 ∈I1 , . . . , (yn,νn )νn ∈In ) .
Remark 1 For each f ∈ H m,2 [a, b] can be considered
y := f (νi ) (xi )
i=1,n,νi ∈Ii
= (f (ν1 ) (x1 ))ν1 ∈I1 , . . . , (f (νn ) (xn ))νn ∈In ,
and the corresponding interpolatory set
Uf := u ∈ H m,2 [a, b] | u(νi ) (xi ) = f (νi ) (xi ), i = 1, n, νi ∈ Ii .
Since f ∈ Uf , the interpolatory set Uf is nonempty and the problem (3)
coresponding to Uf has unique solution for each f ∈ H m,2 [a, b].
2
Main Results
Let us consider the following initial value problem

 y (r) = f (x, y, y 0, . . . , y (r−1) )
(4)
 y (j) (x ) = y (j) , j = 0, r − 1
1
1
and suppose that f : D ⊂ Rr+1 → R satisfies all the conditions assuring the
existence and uniqueness of the solution y : [a, b] → R of the problem (4).
If y is the exact solution of the problem (4), denote by
Ũy = u ∈ H m,2 [a, b] | u(j) (x1 ) = y (j) (x1 ), j = 0, r−1, u(r) (xi ) = y (r) (xi ), i = 2, n .
8
A.M. Acu
From Lemma 2 follow that there exist, and it is unique Birkhoff-type
natural spline interpolation function, sy , coresponding to the interpolatory
set Ũy , namely
(j)
sy (x1 ) = y (j) (x1 ), j = 0, r − 1
(r)
sy (xi ) = y (r) (xi ), i = 2, n.
The spline function sy has the representation
(5)
sy (x) =
r−1
X
(j)
y1 s1,j (x)
+
j=0
n
X
(0)
(1)
(r−1)
f (xi , yi , yi , . . . , yi
)si,r (x)
i=2
(0)
(1)
(r−1)
where the unknown values yi , yi , . . . , yi
will be determined.
The functions sj,µj , j = 1, n, µj ∈ Ij , where I1 = {0, 1, . . . , r − 1},
Ii = {r} , i = 2, n are the fundamental Birkhoff-type natural spline interpolation functions and are given by
sj,µj (x) =
m−1
X
k=0
(j) (b
ak
r−1
n
X (j) (x − xi )2m−1−r
− x)k X (j) (x − x1 )2m−1−i
+
+
+
b1,i
+
bi,r
k!
(2m − 1 − i)!
(2m − 1 − r)!
i=0
i=2
(j)
(j)
(j)
where the coefficients ak , k = 0, m − 1, b1,i , i = 0, r − 1, bi,r , i = 2, n are
given by the following linear system
(6)
 m−1
k−i
X (j)

i (b − x1 )


a
(−1)
= δ1,j δi,µj , i = 0, r − 1

k

(k
−
i)!


k=i


r−1
m−1

k−r
2m−1−r−i
X
X

(j)
(j) (xν − x1 )

r (b − xν )

+
b
a
(−1)

1,i
k

(k − r)!
(2m − 1 − r − i)!
k=r
i=0
ν−1
2m−1−2r
X


(j) (xν − xi )+

+
bi,r
= δν,j δr,µj , ν = 2, n



(2m − 1 − 2r)!

i=2


r−1
n

l−i
l−r
X
X


(j) (b − x1 )
(j) (b − xi )

b
+
b
= 0, l = 0, m − 1,

1,i
i,r

(l − i)!
(l − r)!
i=0,i≤l
i=2,r≤l
Natural splines of Birkhoff type approximating...
9
Theorem 1 If y is the exact solution of the problem (4), y ∈ H m+1,2 [a, b]
and sy is the spline approximating solution (5), then the following estimations
y (k) −s(k)
y
∞
≤
√
m−r+1(m−r)(m−r−1) . . . (k+1−r) k∆n km−k+1/2 y (m+1)
2
hold, for k = r, m, and k∆n k := max {xi+1 − xi } .
i=0,n
Proof. Because y (r) (xi ) − s(r) (xi ) = 0, i = 2, n, there exist the points
(1)
(1)
(xi ), xi < xi < xi+1 , i = 2, n − 1, such that
(1)
(1)
y (r+1) (xi ) − s(r+1)
(xi ) = 0, i = 2, n − 1
y
hold.
(1)
(r+1)
Now, because y (r+1) (xi ) − sy
(2)
(1)
(2)
(1)
(1)
(xi ) = 0, i = 2, n − 1, there exist a
set of points (xi ), xi < xi < xi+1 , i = 2, n − 2 such that
(2)
(2)
y (r+2) (xi ) − s(r+2)
(xi ) = 0, i = 2, n − 2.
y
(k−r)
Continuing in the same manner, from the relations y (k) (xi
(k)
(k−r+1)
0, for i = 2, n − k + r, there exist a set of points (xi
(k−r+1)
xi
(k+1−r)
) − s(k+1)
(xi
y
) = 0, i = 2, n − k − 1 + r
(m−r)
hold. Finally, for k = m − 1, there exist a set of points (xi
(m−1−r)
< xi+1
(m−r)
(m−r)
) − s(m)
y (xi
(m−1−r)
), xi
, i = 2, n − m + r such that the relation
y (m) (xi
hold.
<
(k−r)
(k+1−r)
(m−r)
(k−r)
), xi
)=
< xi+1 , i = 2, n − k + r − 1, such that
y (k+1) (xi
xi
(k−r)
)−sy (xi
) = 0, i = 2, n − m + r
<
10
A.M. Acu
(k)
(k)
(0)
Because xi+1 − xi
≤ (k + 1) k∆n k for k = 0, m − r, xi
= xi it is
clear that for any x ∈ [a, b] there exist an index i0 such that
(m−r)
x − x i0
≤ (m − r + 1) k∆n k .
Thus
y
(m)
(x) −
s(m)
y (x)
=
≤
Z
Z
x
(m−r)
0
xi
(m+1)
y
(t) − s(m+1)
(t) dt
y
1/2
x
(m−r)
0
Z
dt
xi
x
(m−r)
0
xi
≤ {(m−r+1) k∆n k}
1/2
1/2
(m+1)
2
y
(t) − s(m+1)
(t)
dt
y
Z
b
a
y
(m+1)
= {(m − r + 1) k∆n k}1/2 y (m+1) − s(m+1)
y
≤ ((m − r + 1) k∆n k)1/2 y (m+1)
(m)
and y (m) − sy
∞
≤
√
p
m − r + 1 k∆n k y (m+1)
(t)−s(m+1)
(t) 2 dt 1/2
y
2
2
2
.
In the same manner, for any x ∈ [a, b] there exists an index i0 such that
(m−r−1)
x − x i0
≤ (m − r) k∆n k
holds, and
y
(m−1)
(x) −
sy(m−1) (x)
=
Z
x
(m−r−1)
0
xi
(m−r−1)
x − x i0
≤ y (m) − s(m)
y
∞
p
√
≤ m − r + 1 k∆n k · y (m+1)
(m−1)
That means y (m−1) − sy
∞
≤
√
2
y (m) (t) − s(m)
(t)
dt
y
(m − r) · k∆n k .
m − r + 1(m−r) k∆n k1+1/2 y (m+1)
For any x ∈ [a, b] there exist an index i0 such that
(m−r−2)
x − x i0
≤ (m − r − 1) k∆n k
2
.
Natural splines of Birkhoff type approximating...
11
holds, and
y
(m−2)
(x) −
(m−2)
sy
(x)
Z
=
x
(m−r−2)
0
xi
y (m−1) (t) − sy(m−1) (t) dt
√
(m−r−2)
x−xi0
≤ m−r+1(m−r) k∆n k1+1/2 y (m+1)
∞ √
·(m − r − 1) k∆n k = m − r + 1(m − r)(m − r − 1) k∆k2+1/2 · y (m+1)
(m−1)
≤ y (m−1) −sy
2
2
,
namely
y (m−2) − sy(m−2)
∞
≤
√
m − r + 1(m − r)(m − r − 1) k∆n k2+1/2 y (m+1)
2
.
In generally
y (k) − s(k)
y
∞
≤
·
√
m − r + 1(m − r)(m − r − 1) . . . (k + 1 − r)
k∆n km−k+1/2 y (m+1)
2
, k = r, m.
Corollary 1 If the exact solution of the problem (4) y ∈ H m+1,2 [a, b] and
sy is the spline approximating solution, then
√
(k)
y (k) − sy
≤ (b − a)r−k m − r + 1(m − r)! k∆n km−r+1/2 y (m+1)
∞
2
,
k = 0, r−1.
Proof. We have
x
(r)
(r)
(x) −
=
y (r) (t) − s(r)
− sy
y (t) dt ≤ |x − x1 | · y
x1
√
≤ (b − a) m − r + 1(m − r)! k∆n km−r+1/2 y (m+1) 2 .
y
(r−1)
(r−1)
sy (x)
Z
From the above relation we obtain
√
(r−1)
y (r−1) − sy
≤ (b − a) m − r + 1(m − r)! k∆n km−r+1/2 y (m+1)
∞
2
∞
.
In generally, for k = 0, r − 1
y (k) − s(k)
y
∞
√
≤ (b − a)r−k m − r + 1(m − r)! k∆n km−r+1/2 y (m+1)
2
.
12
A.M. Acu
Corollary 2 If the exact solution of the problem (4) y ∈ H m+1,2 [a, b] and
sy is the spline approximating solution then
lim
k∆n k→0
y (k) − s(k)
y
∞
= 0 for
k = 0, m
hold and the order of convergence is

 m − k + 1/2 for k = r, m,
 m − r + 1/2 for k = 0, r − 1.
Now the problem is to construct effectively the approximating spline
(j)
function (5), i.e. to find an algorithm for determination the value sy (xi ),
i = 2, n, j = 0, r − 1, considering that sy (x) ≈ y(x), for any x ∈ [a, b].
Let denote the error of the method, as usual by
e(x) = y(x) − sy (x), x ∈ [a, b].
Directly from the Corollary 1, if y ∈ H m+1,2 [a, b], we have
√
e(j) (x) ≤ (b − a)r−j m − r + 1(m − r)! k∆n km−r+1/2 · y (m+1)
(j)
namely e (x) = O k∆n k
Writing again
sy (x) =
r−1
X
(j)
m−r+1/2
y1 s1,j (x) +
j=0
2
,
(∀) x ∈ [a, b].
n
X
(0)
(1)
(r−1)
f xk , y k , y k , . . . , y k
sk,r (x)
k=2
(j)
and denoting sy (xi ) =: wij , e(j) (xi ) =: eji , i = 2, n, j = 0, r − 1, it follows
(7)
wij =
r−1
X
k=0
(k) (j)
y1 s1,k (xi ) +
n
X
k=2
(j)
sk,r (xi )f xk , e0k + wk0 , e1k + wk1 , . . . , ekr−1 + wkr−1 ,
Natural splines of Birkhoff type approximating...
13
i = 2, n, j = 0, r − 1.
We suppose that in (4) the function f : D ⊂ Rr+1 → R, D ⊂ [a, b] × Rr
∂f (x, u0 , u1 , . . . , ur−1 )
is a continuous function, and also that derivatives
,
∂ui
i = 0, r − 1 are continuous function on the domain D. Thus
(0)
(1)
(r−1)
f (xk , yk , yk , . . . , yk
) = f xk , e0k + wk0 .e1k + wk1 , . . . , ekr−1 + wkr−1
xk , wk0 , wk1 , . . . , wkr−1
=f
+
r−1
X
i=0
eik
∂f (xk , ξk0 , ξk1 , ξkr−1 )
,
∂ui
where min(wki , wki + eik ) < ξki < max(wki , wki + eik ).
We can write the system (7) in the form
wij =
r−1
X
(k) (j)
y1 s1,k (xi ) +
k=0
Eij
=
n
X
n
X
k=2
(j)
sk,r (xi )
k=2
(j)
sk,r (xi )f xk , wk0 , wk1 , . . . , wkr−1 + Eij , where
r−1
X
r−1
0 1
i ∂f (xk , ξk , ξk , . . . , ξk )
ek
,
∂ui
i=0
i = 2, n, j = 0, r − 1
∂f (t, u0 , u1 , . . . , ur−1 )
≤ M, i = 0, r − 1 on D. Obviously
∂ui
Eij → 0 for k∆n k → 0.
supposing that
We have o solve the following nonlinear system:
(8)
wij =
r−1
X
n
X
(k) (j)
y1 s1,k (xi )+
k=0
(j)
sk,r (xi )f (xk , wk0 , wk1 , . . . , wkr−1 ), i = 2, n, j = 0, r−1,
k=2
Denote by
w 0 = (w20 w30 · · · wn0 )
w 1 = (w21 w31 · · · wn1 )
..
.
w r−1 = (w2r−1 w3r−1 · · · wnr−1 )
14
A.M. Acu
W = (w 0 w 1 . . . w r−1 )
r−1
n
X
X
(k) (j)
(j)
j
Hi (W ) =
y1 s1,k (xi ) +
sk,r (xi )f (xk , wk0 , wk1 , . . . , wkr−1 ),
k=0
H(W ) =
= Hij (W )
k=2
0
0
H2 (W ) . . . Hn (W ), H21 (W ) . . . Hn1 (W ), H2r−1 (W ) . . . Hnr−1 (W )
j=0,r−1,i=2,n
and









A=








∂H20 (W )
∂w20
···
∂Hn0 (W )
∂w20
···
∂H2r−1 (W )
∂w20
···
∂Hnr−1 (W )
∂w20
···
..
.
..
.
..
.
∂H20 (W )
0
∂wn
∂Hn0 (W )
0
∂wn
∂H2r−1 (W )
0
∂wn
∂Hnr−1 (W )
0
∂wn
···
∂H20 (W )
∂w2r−1
···
∂H20 (W )
r−1
∂wn
···
∂Hn0 (W )
∂w2r−1
···
∂Hn0 (W )
r−1
∂wn
···
∂H2r−1 (W )
∂w2r−1
···
∂H2r−1 (W )
r−1
∂wn
···
∂Hnr−1 (W )
∂w2r−1
···
∂Hnr−1 (W )
r−1
∂wn
...
s2,r (x2 )


















We remark that A = SF , where

s (x )
 2,r 2
..


.


 s2,r (xn )

..

S=
.

 (r−1)
 s2,r (x2 )


..

.

(r−1)
s2,r (xn )
...
sn,r (x2 )
...
sn,r (x2 )






. . . sn,r (xn ) . . . s2,r (xn ) . . . sn,r (xn ) 





(r−1)
(r−1)
(r−1)
. . . sn,r (x2 ) . . . s2,r (x2 ) . . . sn,r (x2 ) 




(r−1)
(r−1)
(r−1)
. . . sn,r (xn ) . . . s2,r (xn ) . . . sn,r (xn )
Natural splines of Birkhoff type approximating...
15
and









F=







∂f (x2 ,w20 ,w21 ,...,w2r−1 )
∂w20
..
.
...
..
.
0
..
.
...
..
.
0
..
.
0
0
..
.
...
..
.
0
..
.
..
.
...
..
.
0
..
.
...
..
.
0
..
.
...
..
.
0
..
.
...
0
...
0 ,w 1 ,...,wr−1 )
∂f (xn ,wn
n
n
r−1
∂wn
0 ,w 1 ,...,wr−1 )
∂f (xn ,wn
n
n
0
∂wn
To obtain our results we will use the following theorem

















Theorem 2 Let Ω ⊂ Rn be a bounded domain and let H : Ω → Ω be a vector
function defined by (w1 , . . . , wn ) → (H1 (w1 , . . . , wn ), . . . , Hn (w1 , . . . , wn )).
Denote by A(w) the following matric
A(w) = A(w1 , w2 , . . . , wn ) :=
∂Hi (w)
∂wj
.
i,j=1,n
∂Hi
, i, j = 1, n are continuous in Ω, then there exist
∂wj
in Ω a fix-point w ∗ of the application H, i.e w ∗ = H(w ∗ ), which can be
If the functions H and
found by iteration w ∗ = lim w (n) , w (k) := H(w (k−1) ), k = 1, 2, . . . , w (0) ∈ Ω
n→∞
(arbitrary).
If in addition kAk ≤ q < 1, for any iteration w (k) , the following estimation holds
w (k) − w ∗ ≤
qk
w (1) − w (0) .
1−q
Theorem 3 Suppose that
∂f (x, u0 , u1 , . . . , ur−1 )
≤ M, i = 0, r − 1,
∂ui
|f (x, u0 , u1 , . . . , ur−1 )| ≤ N (∀) (x, u0 , u1 , . . . , ur−1 ) ∈ D.
16
A.M. Acu
If M < kSk−1 , then the system (8) has a solution which can be deter-
mined by iteration.
Proof. From (8) we can write
wij
≤
≤
r−1
X
k=0
r−1
X
k=0
(k)
y1
(k)
y1
·
(j)
s1,k (xi )
·
(j)
s1,k (xi )
+
n
X
k=2
+N
(j)
sk,r (xi ) · f (xk , wk0 , . . . , wkr−1 )
n
X
k=2
(j)
sk,r (xi ) , i = 2, n, j = 0, r − 1.
Now we can apply the Theorem 2, where the domain Ω is defined by
Ω = (w20 , . . . , wn0 , w21 , . . . , wn1 , . . . , w2r−1 , . . . , wnr−1 ) ⊂ Rr(n−1) |
)
r−1
n
X
X
(k)
(j)
(j)
wij ≤
y1 · s1,k (xi ) + N
sk,r (xi ) , j = 0, r − 1, i = 2, n .
k=0
k=2
It is clear that
kAk = kSF k ≤ kSk · kF k ≤ kSk · M ≤ 1,
and therefore all the conditions of Theorem 2 are satisfied. There exist a
solution w ∗ of (8), which can be found by iteration.
Remark 2 In the above considerations, for the matrix A = (ai,j )i,j=1,n we
can use one of the following norms
( n
)
( n
)
X
X
kAkr := max
|aij | , kAkc := max
|aij | .
i=1,n
j=1
j=1,n
i=1
The algorithm for obtaining of numerical solution of the differential
equation problem (4) has the following steps:
1. The determination of fundamental spline functions s1,j , si,r , j = 0, r − 1,
Natural splines of Birkhoff type approximating...
17
i = 2, n which are obtained from system (6).
2. The starting iteration point W (0) can be chosen using a numerical method
for differential equation.
3. From Theorem 2.3 follow that the solution of differential equation (4)
can be found by iteration W ∗ = lim W (n) , where W (k) = H(W (k−1) ).
n→∞
References
[1] P. Blaga, G. Micula , Natural spline functions of even degree, Studia
Univ. Babeş-Bolyai Cluj-Napoca, Series Math., 38, 1993, 31-40.
[2] P. Blaga, G. Micula, Polynomial spline functions of even degree approximating the solution of differential equations, (I), ZAMM, 76, 1996,
Suppl.1, 477-478.
[3] P. Blaga, G. Micula, Polynomial spline functions of even degree approximating the solution of (delay) diferential equations , PAMM, Vol. 1,
No.1, 2002, 516-517.
[4] B.D. Bojanov, H.A. Hakopian and A.A. Sahakian, Spline Functios and
Multivariate Interpolations, Kluwer Academic Publishers, 1993.
[5] A. Branga, Natural splines of Birkhoff type and optimal approximation,
Journal of Computational Analysis and Applications, Vol. 7, No.1, 2005,
81-88.
Ana Maria Acu,
”Lucian Blaga” University
Department of Mathematics
Str. Dr. I. Ratiu, No.5-7, 550012 - Sibiu, Romania
e-mail: acuana77@yahoo.com