Abstract
Authors
Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
C. Mustăţa, Approximation by spline functions of the solutions of a linear bilocal problem, Rev. Anal. Numér. Théor. Approx. 26 (1997) 1-2. 137-148.
About this paper
Journal
Revue d’Analyse Numer.Theor.Approx.
Publisher Name
Publishing Romanian Academy
Print ISSN
2457-6794
Online ISSN
2501-059X
[1] P. Blaga and G. Micula” Polynoníal natural spline of even degree, Stuclia Univ. “Babeç- Bolyai”, Mathematica 38, 2 (1993),3140.
[2] P. Blaga, R. Gorenflo and G. Micul4 Evar degree spline teclmiquefor numerical solution oJ delay differential equations, Froie Univorsität Borlin, Preprint No. A-15 (1996), Sorie A-Mathernatik.
[3] R. L. Btuden and T. Douglas Fafuæ, Nunøìcal Analysis, Third Blition, PWS-KENT Publiúing Company, Boston, 1985,
[4] G. Micuta, P. Blaga and M. Micula, On even degree polyomíal splinefunctions with applicatíons to numerical solution of diffirentiøl equations with retarded argument, Technischo Hochschule Darmstadt, Preprint No. 1771, Fachbereich Matheinatik (1995).
Paper (preprint) in HTML form
APPROXIMATION BY SPLINE FUNCTIONS OF THE SOLUTION OF A BILOCAL LINEAR PROBLEM
COSTICĂ MUSTĂȚA
In the last years the theory of spline functions has become an important tool in the numerical solving of some problems for differential equations (see, for instance, [1], [2], [4]).
In this paper we shall define a space of spline functions of degree 5 which can be used to approximate the solution of a bilocal linear problem.
Let n >= 3n \geq 3 be a natural number and let
Delta_(n):-oo=t_(-1) < a=t_(0) < t_(1) < dots < t_(n)=b < t_(n+1)=+oo\Delta_{n}:-\infty=t_{-1}<a=t_{0}<t_{1}<\ldots<t_{n}=b<t_{n+1}=+\infty
be a division of the real axis.
Denote by S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) the set of all functions s: RrarrR\mathbf{R} \rightarrow \mathbf{R} having the following properties:
1^(0)s inC^(4)(R)1^{0} s \in C^{4}(\mathrm{R}).
2^(0)s|_(I_(k))inP_(5),I_(k)=[t_(k-1),t_(k)),k=1,2,dots,n\left.2^{0} s\right|_{I_{k}} \in \mathscr{P}_{5}, I_{k}=\left[t_{k-1}, t_{k}\right), k=1,2, \ldots, n.
3^(0)s|_(I_(0))inP_(3),s|_(I_(n+1))inP_(3),I_(0)=(t_(-1),t_(0)),I_(n+1)=[t_(n),t_(n+1))\left.3^{0} s\right|_{I_{0}} \in \mathscr{P}_{3},\left.s\right|_{I_{n+1}} \in \mathscr{P}_{3}, I_{0}=\left(t_{-1}, t_{0}\right), I_{n+1}=\left[t_{n}, t_{n+1}\right).
THEOREM 1. If s inS_(5)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right), then
Denote by S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) the set of all functions s: RrarrR\mathbf{R} \rightarrow \mathbf{R} having the following properties:
1^(0)s inC^(4)(R)1^{0} s \in C^{4}(\mathrm{R}).
2^(0)s|_(I_(k))inP_(5),I_(k)=[t_(k-1),t_(k)),k=1,2,dots,n\left.2^{0} s\right|_{I_{k}} \in \mathscr{P}_{5}, I_{k}=\left[t_{k-1}, t_{k}\right), k=1,2, \ldots, n.
3^(0)s|_(I_(0))inP_(3),s|_(I_(n+1))inP_(3),I_(0)=(t_(-1),t_(0)),I_(n+1)=[t_(n),t_(n+1))\left.3^{0} s\right|_{I_{0}} \in \mathscr{P}_{3},\left.s\right|_{I_{n+1}} \in \mathscr{P}_{3}, I_{0}=\left(t_{-1}, t_{0}\right), I_{n+1}=\left[t_{n}, t_{n+1}\right).
THEOREM 1. If s inS_(5)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right), then
{:(1)s(t)=sum_(i=0)^(3)A_(i)t^(i)+sum_(k=0)^(n)a_(k)(t-t_(k))_(+)^(5)","quad t inR",":}\begin{equation*}
s(t)=\sum_{i=0}^{3} A_{i} t^{i}+\sum_{k=0}^{n} a_{k}\left(t-t_{k}\right)_{+}^{5}, \quad t \in \mathrm{R}, \tag{1}
\end{equation*}
where
{:(2)sum_(k=0)^(n)a_(k)=0quad" and "quadsum_(k=0)^(n)a_(k)t_(k)=0:}\begin{equation*}
\sum_{k=0}^{n} a_{k}=0 \quad \text { and } \quad \sum_{k=0}^{n} a_{k} t_{k}=0 \tag{2}
\end{equation*}
Proof. Let s inS_(5)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right). If t >= bt \geq b, then s^((4))(t)=0s^{(4)}(t)=0 so that
0=5!sum_(k=0)^(n)a_(k)(t-t_(k))_(+)=5!sum_(k=0)^(n)a_(k)(t-t_(k))_(+)"because "0=5!\sum_{k=0}^{n} a_{k}\left(t-t_{k}\right)_{+}=5!\sum_{k=0}^{n} a_{k}\left(t-t_{k}\right)_{+} \text {because }
(t-t_(k))_(+)={[0," for "t < t_(k)],[t-t_(k)," for "t >= t_(k)].:}\left(t-t_{k}\right)_{+}=\left\{\begin{array}{ll}
0 & \text { for } t<t_{k} \\
t-t_{k} & \text { for } t \geq t_{k}
\end{array} .\right.
Consequently sum_(k=0)^(n)a_(k)=0\sum_{k=0}^{n} a_{k}=0 and sum_(k=0)^(n)a_(k)t_(k)=0\sum_{k=0}^{n} a_{k} t_{k}=0.
THEOREM 2. a) If f:RrarrRf: \mathbf{R} \rightarrow \mathbf{R} verifies the conditions
(3)
THEOREM 2. a) If f:RrarrRf: \mathbf{R} \rightarrow \mathbf{R} verifies the conditions
(3)
f(a)=alpha_(1),f(b)=beta_(1),f^('')(t_(k))=lambda_(k),k=0,1,dots,n,f(a)=\alpha_{1}, f(b)=\beta_{1}, f^{\prime \prime}\left(t_{k}\right)=\lambda_{k}, k=0,1, \ldots, n,
then there exists a unique spline function s_(f)inS_(5)(Delta_(n))s_{f} \in S_{5}\left(\Delta_{n}\right) such that
{:(4)s_(f)(a)=alpha_(1)","s_(f)(b)=beta_(1)","s_(f)^('')(t_(k))=lambda_(k)","k=0","1","dots","n.:}\begin{equation*}
s_{f}(a)=\alpha_{1}, s_{f}(b)=\beta_{1}, s_{f}^{\prime \prime}\left(t_{k}\right)=\lambda_{k}, k=0,1, \ldots, n . \tag{4}
\end{equation*}
b) If h:RrarrRh: \mathbf{R} \rightarrow \mathbf{R} verifies the conditions
{:(5)h(a)=alpha_(2)","h^(')(a)=beta_(2)","h^('')(t_(k))=mu_(k)","k=0","1","dots","n",":}\begin{equation*}
h(a)=\alpha_{2}, h^{\prime}(a)=\beta_{2}, h^{\prime \prime}\left(t_{k}\right)=\mu_{k}, k=0,1, \ldots, n, \tag{5}
\end{equation*}
then there exists a unique spline function s_(h)inS_(5)(Delta_(n))s_{h} \in S_{5}\left(\Delta_{n}\right) such that
{:(6)s_(h)(a)=alpha_(2)","s_(h)^(')(a)=beta_(2)","s_(h)^('')(t_(k))=mu_(k)","k=0","1","dots","n.:}\begin{equation*}
s_{h}(a)=\alpha_{2}, s_{h}^{\prime}(a)=\beta_{2}, s_{h}^{\prime \prime}\left(t_{k}\right)=\mu_{k}, k=0,1, \ldots, n . \tag{6}
\end{equation*}
Proof. a) Using the representation (1) and taking into account conditions (4), we obtain the system
{:[A_(0)+A_(1)a+A_(2)a^(2)+A_(3)a^(3)=alpha_(1)],[A_(0)+A_(1)b+A_(2)b^(2)+A_(3)b^(3)+sum_(k=0)^(n-1)a_(k)(b-t_(k))^(5)=beta_(1)],[(7)2A_(2)+6A_(3)t_(j)+20sum_(k=0)^(n)a_(k)(t_(j)-t_(k))_(+)^(3)=lambda_(j);j= bar(0,n)],[sum_(k=0)^(n)a_(k)=0;quadsum_(k=0)^(n)a_(k)t_(k)=0]:}\begin{gather*}
A_{0}+A_{1} a+A_{2} a^{2}+A_{3} a^{3}=\alpha_{1} \\
A_{0}+A_{1} b+A_{2} b^{2}+A_{3} b^{3}+\sum_{k=0}^{n-1} a_{k}\left(b-t_{k}\right)^{5}=\beta_{1} \\
2 A_{2}+6 A_{3} t_{j}+20 \sum_{k=0}^{n} a_{k}\left(t_{j}-t_{k}\right)_{+}^{3}=\lambda_{j} ; j=\overline{0, n} \tag{7}\\
\sum_{k=0}^{n} a_{k}=0 ; \quad \sum_{k=0}^{n} a_{k} t_{k}=0
\end{gather*}
of n+5n+5 equations with n+5n+5 unknowns: A_(0),A_(1),A_(2),A_(3),a_(0),a_(1),dots,a_(n)A_{0}, A_{1}, A_{2}, A_{3}, a_{0}, a_{1}, \ldots, a_{n}.
The system (7) has a unique solution if and only if the associated homogeneous system (obtained for alpha_(1)=beta_(1)=0,lambda_(k)=0,k=0,1,dots,n\alpha_{1}=\beta_{1}=0, \lambda_{k}=0, k=0,1, \ldots, n ) has only the trivial solution. Suppose that s inS_(5)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right) verifies the homogeneous conditions (4) (i.e., {:alpha_(1)=beta_(1)=0,lambda_(k)^('')=0,k=0,1,dots,n)\left.\alpha_{1}=\beta_{1}=0, \lambda_{k}^{\prime \prime}=0, k=0,1, \ldots, n\right). Then we have
The system (7) has a unique solution if and only if the associated homogeneous system (obtained for alpha_(1)=beta_(1)=0,lambda_(k)=0,k=0,1,dots,n\alpha_{1}=\beta_{1}=0, \lambda_{k}=0, k=0,1, \ldots, n ) has only the trivial solution. Suppose that s inS_(5)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right) verifies the homogeneous conditions (4) (i.e., {:alpha_(1)=beta_(1)=0,lambda_(k)^('')=0,k=0,1,dots,n)\left.\alpha_{1}=\beta_{1}=0, \lambda_{k}^{\prime \prime}=0, k=0,1, \ldots, n\right). Then we have
{:[int_(a)^(b)[s^((4))(t)]^(2)dt=int_(a)^(b)s^((4))(t)*(s^(''')(t))^(')dt=-int_(a)^(b)s^((5))(t)*s^(''')(t)=],[=-sum_(k=1)^(n)int_(t_(k-1))^(t_(k))s^((5))(t)*s^(''')(t)dt=-sum_(k=1)^(n)c_(k)int_(t_(k-1))^(t_(k))s^(''')(t)dt=],[=-sum_(k=1)^(n)c_(k)[s^('')(t_(k))-s^('')(t_(k-1))]=0]:}\begin{gathered}
\int_{a}^{b}\left[s^{(4)}(t)\right]^{2} \mathrm{~d} t=\int_{a}^{b} s^{(4)}(t) \cdot\left(s^{\prime \prime \prime}(t)\right)^{\prime} \mathrm{d} t=-\int_{a}^{b} s^{(5)}(t) \cdot s^{\prime \prime \prime}(t)= \\
=-\sum_{k=1}^{n} \int_{t_{k-1}}^{t_{k}} s^{(5)}(t) \cdot s^{\prime \prime \prime}(t) \mathrm{d} t=-\sum_{k=1}^{n} c_{k} \int_{t_{k-1}}^{t_{k}} s^{\prime \prime \prime}(t) \mathrm{d} t= \\
=-\sum_{k=1}^{n} c_{k}\left[s^{\prime \prime}\left(t_{k}\right)-s^{\prime \prime}\left(t_{k-1}\right)\right]=0
\end{gathered}
where c_(k)=s^((5))(t)|_(I_(k)^('))k= bar(1,n)c_{k}=\left.s^{(5)}(t)\right|_{I_{k}^{\prime}} k=\overline{1, n}.
It follows that s^((4))=0s^{(4)}=0, for all t in[a,b]t \in[a, b]. Since s inP_(3)s \in \mathscr{P}_{3} on I_(0)I_{0} and on I_(n+1)I_{n+1} and s inC^(4)(R)s \in \mathbf{C}^{4}(\mathbf{R}), it follows that s^((4))(t)=0s^{(4)}(t)=0 for all t inRt \in \mathbf{R}, implying s^('')inP_(1)s^{\prime \prime} \in \mathscr{P}_{1}. As s^('')(t_(k))=0s^{\prime \prime}\left(t_{k}\right)=0, k=0,1,dots,n,(n >= 3)k=0,1, \ldots, n,(n \geq 3) we conclude that s^('')(t)=0s^{\prime \prime}(t)=0 for all t inRt \in \mathbf{R}.
Finally, taking into account the equalities s(a)=s(b)=0s(a)=s(b)=0, one obtains s(t)=0s(t)=0 for all t inRt \in \mathbf{R}, implying that all the coefficients in representation (1) are null. This shows that the homogeneous system associated to (7) has only the trivial solution.
Assertion b) can be proved similarly, supposing that the function ss given by (1) verifies conditions (6).
COROLLARY 3. There exist the systems of functions
{:[S={s_(0),s_(1),S_(0),S_(1),dots,S_(n)}subS_(5)(Delta_(n))],[U=(u_(0),u_(1),U_(0),U_(1),dots,U_(n))subS_(5)(Lambda_(n))]:}\begin{aligned}
& \mathscr{S}=\left\{s_{0}, s_{1}, S_{0}, S_{1}, \ldots, S_{n}\right\} \subset S_{5}\left(\Delta_{n}\right) \\
& \mathscr{U}=\left(u_{0}, u_{1}, U_{0}, U_{1}, \ldots, U_{n}\right) \subset S_{5}\left(\Lambda_{n}\right)
\end{aligned}
verifying the conditions
{:[s_(0)(a)=1","s_(0)(b)=0","s_(0)^('')(t_(k))=0","k= bar(0,n)","],[s_(1)(a)=0","s_(1)(b)=1","s_(1)^('')(t_(k))=0","k= bar(0,n)","],[S_(k)(a)=0","S_(k)(b)=0","k= bar(0,n);S_(k)^('')(t_(j))=delta_(kj)","k","j= bar(0,n)","],[u_(0)(a)=1","u_(0)^(')(a)=0","u_(0)^('')(t_(k))=0","k= bar(0,n)","],[u_(1)(a)=0","u_(1)^(')(a)=1","u_(1)^('')(t_(k))=0","k= bar(0,n)","],[U_(kappa)(a)=0","U_(k)^(')(a)=0","k= bar(0,n);U_(k)^('')(t_(j))=delta_(kj)","k","j= bar(0,n).]:}\begin{gathered}
s_{0}(a)=1, s_{0}(b)=0, s_{0}^{\prime \prime}\left(t_{k}\right)=0, k=\overline{0, n}, \\
s_{1}(a)=0, s_{1}(b)=1, s_{1}^{\prime \prime}\left(t_{k}\right)=0, k=\overline{0, n}, \\
S_{k}(a)=0, S_{k}(b)=0, k=\overline{0, n} ; S_{k}^{\prime \prime}\left(t_{j}\right)=\delta_{k j}, k, j=\overline{0, n}, \\
u_{0}(a)=1, u_{0}^{\prime}(a)=0, u_{0}^{\prime \prime}\left(t_{k}\right)=0, k=\overline{0, n}, \\
u_{1}(a)=0, u_{1}^{\prime}(a)=1, u_{1}^{\prime \prime}\left(t_{k}\right)=0, k=\overline{0, n}, \\
U_{\kappa}(a)=0, U_{k}^{\prime}(a)=0, k=\overline{0, n} ; U_{k}^{\prime \prime}\left(t_{j}\right)=\delta_{k j}, k, j=\overline{0, n} .
\end{gathered}
If f,h:RrarrRf, h: \mathbf{R} \rightarrow \mathbf{R} verify the conditions of Theorem 2, then the functions s_(f)s_{f} and s_(h)s_{h} admit the representations
{:[(8)s_(f)(t)=s_(0)(t)*f(a)+s_(1)(t)*f(b)+sum_(k=0)^(n)S_(k)(t)*f^('')(t_(k))","quad t inR","],[(9)s_(h)(l)=u_(0)(t)*h(a)+u_(1)(t)*h^(')(a)+sum_(k=0)^(n)U_(k)(t)*h^('')(t_(k))","quad t inR.]:}\begin{align*}
& s_{f}(t)=s_{0}(t) \cdot f(a)+s_{1}(t) \cdot f(b)+\sum_{k=0}^{n} S_{k}(t) \cdot f^{\prime \prime}\left(t_{k}\right), \quad t \in \mathbf{R}, \tag{8}\\
& s_{h}(l)=u_{0}(t) \cdot h(a)+u_{1}(t) \cdot h^{\prime}(a)+\sum_{k=0}^{n} U_{k}(t) \cdot h^{\prime \prime}\left(t_{k}\right), \quad t \in \mathbf{R} . \tag{9}
\end{align*}
Remark 1. By Corollary 3 it follows that the set S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) is a (real) linear space of dimension n+3n+3 and S\mathscr{S} and U\mathscr{U} are two bases in S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right).
Some properties of the space S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) will be presented in what follows.
Let
Let
{:(10)W_(2)^(4)(Delta_(n)):={[g:[a","b]rarrR","g^(''')" abs.cont.on "I_(k)","k= bar(1,n)],[" and "g^((4))inL_(2)[a","b]]}",":}W_{2}^{4}\left(\Delta_{n}\right):=\left\{\begin{array}{c}
g:[a, b] \rightarrow \mathbf{R}, g^{\prime \prime \prime} \text { abs.cont.on } I_{k}, k=\overline{1, n} \tag{10}\\
\text { and } g^{(4)} \in L_{2}[a, b]
\end{array}\right\},
(11) quadW_(2,f)^(4)(Delta_(n)):={g inW_(2)^(4)(Delta_(n)):g^('')(t_(k))=f^('')(t_(k)),quad k= bar(0,n)}\quad W_{2, f}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2}^{4}\left(\Delta_{n}\right): g^{\prime \prime}\left(t_{k}\right)=f^{\prime \prime}\left(t_{k}\right), \quad k=\overline{0, n}\right\},
(12) quadW_(2,f,D)^(4)(Delta_(n)):={g inW_(2,f)^(4)(Delta_(n)):g(t_(0))=f(t_(0)),g(t_(n))=f(t_(n))}\quad W_{2, f, D}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2, f}^{4}\left(\Delta_{n}\right): g\left(t_{0}\right)=f\left(t_{0}\right), g\left(t_{n}\right)=f\left(t_{n}\right)\right\},
(13) W_(2,h,C)^(4)(Delta_(n)):={g inW_(2,h)^(4)(Delta_(n)):g(t_(0))=h(t_(0)),g^(')(t_(0))=h^(')(t_(0))}W_{2, h, C}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2, h}^{4}\left(\Delta_{n}\right): g\left(t_{0}\right)=h\left(t_{0}\right), g^{\prime}\left(t_{0}\right)=h^{\prime}\left(t_{0}\right)\right\}.
(12) quadW_(2,f,D)^(4)(Delta_(n)):={g inW_(2,f)^(4)(Delta_(n)):g(t_(0))=f(t_(0)),g(t_(n))=f(t_(n))}\quad W_{2, f, D}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2, f}^{4}\left(\Delta_{n}\right): g\left(t_{0}\right)=f\left(t_{0}\right), g\left(t_{n}\right)=f\left(t_{n}\right)\right\},
(13) W_(2,h,C)^(4)(Delta_(n)):={g inW_(2,h)^(4)(Delta_(n)):g(t_(0))=h(t_(0)),g^(')(t_(0))=h^(')(t_(0))}W_{2, h, C}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2, h}^{4}\left(\Delta_{n}\right): g\left(t_{0}\right)=h\left(t_{0}\right), g^{\prime}\left(t_{0}\right)=h^{\prime}\left(t_{0}\right)\right\}.
Then we have
THEOREM 4. a) If s inS_(5)(Delta_(n))nnW_(2,f,D)^(4)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right) \cap W_{2, f, D}^{4}\left(\Delta_{n}\right), then
(14)
(14)
||s^((4))||_(2) <= ||g^((4))||_(2)," for all "g inW_(2,f,D)^(4)(Delta_(n))". "\left\|s^{(4)}\right\|_{2} \leq\left\|g^{(4)}\right\|_{2}, \text { for all } g \in W_{2, f, D}^{4}\left(\Delta_{n}\right) \text {. }
b) If s inS_(5)(Delta_(n))nnW_(2,h,c)^(4)(Delta_(n))s \in S_{5}\left(\Delta_{n}\right) \cap W_{2, h, c}^{4}\left(\Delta_{n}\right), then
(15)
(15)
||s^((4))||_(2) <= ||g^((4))||_(2)," for all "g inW_(2,h,C)^(4)(Delta_(n))\left\|s^{(4)}\right\|_{2} \leq\left\|g^{(4)}\right\|_{2}, \text { for all } g \in W_{2, h, C}^{4}\left(\Delta_{n}\right)
Proof. We have
But
{:[0 <= ||g^((4))-s^((4))||_(2)^(2)=int_(a)^(b)[g^((4))(t)-s^((4))(t)]dt=],[=int_(a)^(b)[g^((4))(t)]^(2)dt-int_(a)^(b)[s^((4))(t)]^(2)dt-2int_(a)^(b)s^((4))(t)[g^((4))(t)-s^((4))(t)]dt]:}\begin{gathered}
0 \leq\left\|g^{(4)}-s^{(4)}\right\|_{2}^{2}=\int_{a}^{b}\left[g^{(4)}(t)-s^{(4)}(t)\right] \mathrm{d} t= \\
=\int_{a}^{b}\left[g^{(4)}(t)\right]^{2} \mathrm{~d} t-\int_{a}^{b}\left[s^{(4)}(t)\right]^{2} \mathrm{~d} t-2 \int_{a}^{b} s^{(4)}(t)\left[g^{(4)}(t)-s^{(4)}(t)\right] \mathrm{d} t
\end{gathered}
{:[int_(a)^(b)s^((4))(t)[g^((4))(t)-s^((4))(t)]dt=s^((4))(t)[g^(''')(t)-s^(''')(t)]|_(a)^(b)-],[-int_(a)^(b)s^((5))(t)[g^(''')(t)-s^(''')(t)]dt=-int_(a)^(b)s^((5))(t)[g^(''')(t)-s^(''')(t)]dt=],[=-sum_(k=1)^(n)int_(t_(k-1))^(t_(k))s^((5))(t)[g^(''')(t)-s^(''')(t)]dt=],[=-sum_(k=1)^(n)C_(k)[g^('')(t_(k))-s^('')(t_(k))-(g^('')(t_(k-1))-s^('')(t_(k-1)))]=0]:}\begin{gathered}
\int_{a}^{b} s^{(4)}(t)\left[g^{(4)}(t)-s^{(4)}(t)\right] \mathrm{d} t=\left.s^{(4)}(t)\left[g^{\prime \prime \prime}(t)-s^{\prime \prime \prime}(t)\right]\right|_{a} ^{b}- \\
-\int_{a}^{b} s^{(5)}(t)\left[g^{\prime \prime \prime}(t)-s^{\prime \prime \prime}(t)\right] \mathrm{d} t=-\int_{a}^{b} s^{(5)}(t)\left[g^{\prime \prime \prime}(t)-s^{\prime \prime \prime}(t)\right] \mathrm{d} t= \\
=-\sum_{k=1}^{n} \int_{t_{k-1}}^{t_{k}} s^{(5)}(t)\left[g^{\prime \prime \prime}(t)-s^{\prime \prime \prime}(t)\right] \mathrm{d} t= \\
=-\sum_{k=1}^{n} C_{k}\left[g^{\prime \prime}\left(t_{k}\right)-s^{\prime \prime}\left(t_{k}\right)-\left(g^{\prime \prime}\left(t_{k-1}\right)-s^{\prime \prime}\left(t_{k-1}\right)\right)\right]=0
\end{gathered}
where C_(k)=s^((5))(t)|_(I_(k)),k=1,2,dots,nC_{k}=\left.s^{(5)}(t)\right|_{I_{k}}, k=1,2, \ldots, n.
It follows ||g^((4))||_(2)^(2)-||s^((4))||_(2)^(2) >= 0\left\|g^{(4)}\right\|_{2}^{2}-\left\|s^{(4)}\right\|_{2}^{2} \geq 0, which is equivalent to (14).
Inequalities (15) can be proved by a similar argument.
It follows ||g^((4))||_(2)^(2)-||s^((4))||_(2)^(2) >= 0\left\|g^{(4)}\right\|_{2}^{2}-\left\|s^{(4)}\right\|_{2}^{2} \geq 0, which is equivalent to (14).
Inequalities (15) can be proved by a similar argument.
THEOREM 5. a) If f inW_(2)^(4)(Delta_(n))f \in W_{2}^{4}\left(\Delta_{n}\right) and s_(f)inS_(5)(Delta_(n))s_{f} \in S_{5}\left(\Delta_{n}\right) verify conditions (4) from Theorem 2, then
{:(16)||s_(f)^((4))-f^((4))||_(2) <= ||s^((4))-f^((4))||_(2)","" for all "s inS_(5)(Delta_(n)):}\begin{equation*}
\left\|s_{f}^{(4)}-f^{(4)}\right\|_{2} \leq\left\|s^{(4)}-f^{(4)}\right\|_{2}, \text { for all } s \in S_{5}\left(\Delta_{n}\right) \tag{16}
\end{equation*}
b) If h inW_(2)^(4)(Delta_(n))h \in W_{2}^{4}\left(\Delta_{n}\right) and s_(h)inS_(5)(Delta_(n))s_{h} \in S_{5}\left(\Delta_{n}\right) verify conditions (6) from Theorem 2, then
{:(17)||s_(h)^((4))-h^((4))||_(2) <= ||s^((4))-h^((4))||_(2)","" for all "s inS_(5)(Delta_(n)):}\begin{equation*}
\left\|s_{h}^{(4)}-h^{(4)}\right\|_{2} \leq\left\|s^{(4)}-h^{(4)}\right\|_{2}, \text { for all } s \in S_{5}\left(\Delta_{n}\right) \tag{17}
\end{equation*}
Proof. In order to prove (16), we use the identity
{:[||s^((4))-f^((4))||_(2)^(2)=int_(a)^(b)[s^((4))(t)-s_(f)^((4))(t)]^(2)dt+int_(a)^(b)[s_(f)^((4))(t)-f^((4))(t)]^(2)dt+],[+2int_(a)^(b)[s^((4))(t)-s_(f)^((4))(t)]*[s_(f)^((4))(t)-f^((4))(t)]dt]:}\begin{gathered}
\left\|s^{(4)}-f^{(4)}\right\|_{2}^{2}=\int_{a}^{b}\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right]^{2} \mathrm{~d} t+\int_{a}^{b}\left[s_{f}^{(4)}(t)-f^{(4)}(t)\right]^{2} \mathrm{~d} t+ \\
+2 \int_{a}^{b}\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right] \cdot\left[s_{f}^{(4)}(t)-f^{(4)}(t)\right] \mathrm{d} t
\end{gathered}
and prove that
T=int_(a)^(b)[s^((4))(t)-s_(f)^((4))(t)]*[s_(f)^((4))(t)-f^((4))(t)]dt=0T=\int_{a}^{b}\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right] \cdot\left[s_{f}^{(4)}(t)-f^{(4)}(t)\right] \mathrm{d} t=0
Indeed, integrating by parts, we find
{:[T=[s^((4))(t)-s_(f)^((4))(t)]*[s_(f)^(''')(t)-f^(''')(t)]]_(a)^(b)-],[-int_(a)^(b)[s^((5))(t)-s_(f)^((5))(t)]*[s_(f)^(''')(t)-f^(''')(t)]dt=],[=-sum_(k=1)^(n)c_(k)(s)[s_(k)^('')(t_(k))-f^('')(t_(k))]-[s_(f)^('')(t_(k-1))-f^('')(t_(k-1))]=0","]:}\begin{gathered}
\left.T=\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right] \cdot\left[s_{f}^{\prime \prime \prime}(t)-f^{\prime \prime \prime}(t)\right]\right]_{a}^{b}- \\
-\int_{a}^{b}\left[s^{(5)}(t)-s_{f}^{(5)}(t)\right] \cdot\left[s_{f}^{\prime \prime \prime}(t)-f^{\prime \prime \prime}(t)\right] \mathrm{d} t= \\
=-\sum_{k=1}^{n} c_{k}(s)\left[s_{k}^{\prime \prime}\left(t_{k}\right)-f^{\prime \prime}\left(t_{k}\right)\right]-\left[s_{f}^{\prime \prime}\left(t_{k-1}\right)-f^{\prime \prime}\left(t_{k-1}\right)\right]=0,
\end{gathered}
where c_(k)(s)=s^((5))(t)-s_(f)^((5))(t),t inI_(k)k= bar(1,n)c_{k}(s)=s^{(5)}(t)-s_{f}^{(5)}(t), t \in I_{k} k=\overline{1, n}. (We have used the fact that (s^((4))-s_(f)^((4)))(a)==(s^((4))-s_(f)^((4)))(b)=0\left(s^{(4)}-s_{f}^{(4)}\right)(a)= =\left(s^{(4)}-s_{f}^{(4)}\right)(b)=0.)
Therefore,
{:(18)||s^((4))-f^((4))||_(2)^(2)=||s^((4))-s_(f)^((4))||_(2)^(2)+||s_(f)^((4))-f^((4))||_(2)^(2)",":}\begin{equation*}
\left\|s^{(4)}-f^{(4)}\right\|_{2}^{2}=\left\|s^{(4)}-s_{f}^{(4)}\right\|_{2}^{2}+\left\|s_{f}^{(4)}-f^{(4)}\right\|_{2}^{2}, \tag{18}
\end{equation*}
implying that inequality (16) holds.
Similarly, in the identity
Similarly, in the identity
{:[||s^((4))-h^((4))||_(2)^(2)=int_(a)^(b)[s^((4))(t)-s_(h)^((4))(t)]^(2)dt+int_(a)^(b)[s_(h)^((4))(t)-h^((4))(t)]^(2)dt+],[+2int_(a)^(b)[s^((4))(t)-s_(h)^((4))(t)]*[s_(h)^((4))(t)-h^((4))(t)]dt]:}\begin{aligned}
\left\|s^{(4)}-h^{(4)}\right\|_{2}^{2} & =\int_{a}^{b}\left[s^{(4)}(t)-s_{h}^{(4)}(t)\right]^{2} \mathrm{~d} t+\int_{a}^{b}\left[s_{h}^{(4)}(t)-h^{(4)}(t)\right]^{2} \mathrm{~d} t+ \\
& +2 \int_{a}^{b}\left[s^{(4)}(t)-s_{h}^{(4)}(t)\right] \cdot\left[s_{h}^{(4)}(t)-h^{(4)}(t)\right] \mathrm{d} t
\end{aligned}
we have (integrating by parts)
Q=int_(a)^(b)[s^((4))(t)-s_(h)^((4))(t)]*[s_(h)^((4))(t)-h^((4))(t)]dt=0Q=\int_{a}^{b}\left[s^{(4)}(t)-s_{h}^{(4)}(t)\right] \cdot\left[s_{h}^{(4)}(t)-h^{(4)}(t)\right] \mathrm{d} t=0
implying that
(^(**))quad||s^((4))-h^((4))||_(2)^(2)=||s^((4))-s_(h)^((4))||_(2)^(2)+||s_(h)^((4))-h^((4))||_(2)^(2)\left(^{*}\right) \quad\left\|s^{(4)}-h^{(4)}\right\|_{2}^{2}=\left\|s^{(4)}-s_{h}^{(4)}\right\|_{2}^{2}+\left\|s_{h}^{(4)}-h^{(4)}\right\|_{2}^{2}.
From this equality it follows (17).
(^(**))quad||s^((4))-h^((4))||_(2)^(2)=||s^((4))-s_(h)^((4))||_(2)^(2)+||s_(h)^((4))-h^((4))||_(2)^(2)\left(^{*}\right) \quad\left\|s^{(4)}-h^{(4)}\right\|_{2}^{2}=\left\|s^{(4)}-s_{h}^{(4)}\right\|_{2}^{2}+\left\|s_{h}^{(4)}-h^{(4)}\right\|_{2}^{2}.
From this equality it follows (17).
COROLLARY 6. Iff, h inW_(2)^(4)(Delta_(n))h \in W_{2}^{4}\left(\Delta_{n}\right) and s_(p),s_(h)inS_(5)(Delta_(n))s_{p}, s_{h} \in S_{5}\left(\Delta_{n}\right) verify conditions (4) and (6) from Theorem 2, then
{:[(19)||f^((4))||_(2)^(2)=||s_(f)^((4))||_(2)^(2)+||f^((4))-s_(f)^((4))||_(2)^(2)","],[(20)||h^((4))||_(2)^(2)=||s_(h)^((4))||_(2)^(2)+||h^((4))-s_(h)^((4))||_(2)^(2)","]:}\begin{align*}
& \left\|f^{(4)}\right\|_{2}^{2}=\left\|s_{f}^{(4)}\right\|_{2}^{2}+\left\|f^{(4)}-s_{f}^{(4)}\right\|_{2}^{2}, \tag{19}\\
& \left\|h^{(4)}\right\|_{2}^{2}=\left\|s_{h}^{(4)}\right\|_{2}^{2}+\left\|h^{(4)}-s_{h}^{(4)}\right\|_{2}^{2}, \tag{20}
\end{align*}
{:(29:मे ,760ig brom)||s_(f)^((4))||_(2) <= ||f^((4))||_(2)",":}\begin{equation*}
\left\|s_{f}^{(4)}\right\|_{2} \leq\left\|f^{(4)}\right\|_{2}, \tag{29:मे ,760ig brom}
\end{equation*}मे
{:(21)||s_(h)^((4))||_(2) <= ||h^((4))||_(2)",":}\begin{equation*}
\left\|s_{h}^{(4)}\right\|_{2} \leq\left\|h^{(4)}\right\|_{2}, \tag{21}
\end{equation*}
{:(22)||f^((4))-s_(f)^((4))|| <= ||f^((4))||_(2)",":}\begin{equation*}
\left\|f^{(4)}-s_{f}^{(4)}\right\| \leq\left\|f^{(4)}\right\|_{2}, \tag{22}
\end{equation*}
(24)
{:(23)||h^((4))-s_(h)^((4))||_(2) <= ||h^((4))||_(2).:}\begin{equation*}
\left\|h^{(4)}-s_{h}^{(4)}\right\|_{2} \leq\left\|h^{(4)}\right\|_{2} . \tag{23}
\end{equation*}
Proof. Equalities (19) and (20) follow from (18) and (*) for s-=0s \equiv 0. The remaining inequalities follow from (19) and (20).
Application. Consider the bilocal linear problem
(D) y^('')=p(t)*y+q(t),t in[a,b]y^{\prime \prime}=p(t) \cdot y+q(t), t \in[a, b],
y(a)=alpha,y(b)=betay(a)=\alpha, y(b)=\beta.
If p,qp, q are continuous functions on [a,b][a, b] and p(t) > 0,t in[a,b]p(t)>0, t \in[a, b], then the problem (D)(D) has a unique solution yy (see [3], Theorem 10.1, p. 519).
(D) y^('')=p(t)*y+q(t),t in[a,b]y^{\prime \prime}=p(t) \cdot y+q(t), t \in[a, b],
y(a)=alpha,y(b)=betay(a)=\alpha, y(b)=\beta.
If p,qp, q are continuous functions on [a,b][a, b] and p(t) > 0,t in[a,b]p(t)>0, t \in[a, b], then the problem (D)(D) has a unique solution yy (see [3], Theorem 10.1, p. 519).
Consider the Cauchy problems
(C_(1))y^('')=p(t)y+q(t),t in[a,b]\left(C_{1}\right) y^{\prime \prime}=p(t) y+q(t), t \in[a, b],
y(a)=alpha,y^(')(a)=0y(a)=\alpha, y^{\prime}(a)=0,
(C_(2))y^('')=p(t)y,t in[a,b]\left(C_{2}\right) y^{\prime \prime}=p(t) y, t \in[a, b],
y(a)=0,y^(')(a)=1y(a)=0, y^{\prime}(a)=1.
(C_(1))y^('')=p(t)y+q(t),t in[a,b]\left(C_{1}\right) y^{\prime \prime}=p(t) y+q(t), t \in[a, b],
y(a)=alpha,y^(')(a)=0y(a)=\alpha, y^{\prime}(a)=0,
(C_(2))y^('')=p(t)y,t in[a,b]\left(C_{2}\right) y^{\prime \prime}=p(t) y, t \in[a, b],
y(a)=0,y^(')(a)=1y(a)=0, y^{\prime}(a)=1.
The Cauchy problems have unique solutions y_(1),y_(2)y_{1}, y_{2}, respectively (see [3], Theorem 5.15, p. 263), and the function
{:(25)y(t)=y_(1)(t)+(beta-y_(1)(b))/(y_(2)(b))y_(2)(t)" with "y_(2)(b)!=0","quad t in[a","b]",":}\begin{equation*}
y(t)=y_{1}(t)+\frac{\beta-y_{1}(b)}{y_{2}(b)} y_{2}(t) \text { with } y_{2}(b) \neq 0, \quad t \in[a, b], \tag{25}
\end{equation*}
is the solution of the problem ( DD ) (see [3]).
Applying Theorem 2b) to the solutions y_(1),y_(2)y_{1}, y_{2} of the problems ( C_(1)C_{1} ), ( C_(2)C_{2} ), it follows that there exist the functions s_(y_(1)),s_(y_(2))inS_(5)(Delta_(n))s_{y_{1}}, s_{y_{2}} \in S_{5}\left(\Delta_{n}\right) such that
Applying Theorem 2b) to the solutions y_(1),y_(2)y_{1}, y_{2} of the problems ( C_(1)C_{1} ), ( C_(2)C_{2} ), it follows that there exist the functions s_(y_(1)),s_(y_(2))inS_(5)(Delta_(n))s_{y_{1}}, s_{y_{2}} \in S_{5}\left(\Delta_{n}\right) such that
{:[s_(y_(1))(a)=alpha","quads_(y_(1))^(')(a)=0","quads_(y_(1))^('')(t_(k))=y_(1)^('')(t_(k))","quad k= bar(0,n)","],[(**)s_(y_(2))(a)=0","quads_(y_(2))^(')(a)=1","quads_(y_(2))^('')(t_(k))=y_(2)^('')(t_(k))","quad k= bar(0,n).]:}\begin{align*}
& s_{y_{1}}(a)=\alpha, \quad s_{y_{1}}^{\prime}(a)=0, \quad s_{y_{1}}^{\prime \prime}\left(t_{k}\right)=y_{1}^{\prime \prime}\left(t_{k}\right), \quad k=\overline{0, n}, \\
& s_{y_{2}}(a)=0, \quad s_{y_{2}}^{\prime}(a)=1, \quad s_{y_{2}}^{\prime \prime}\left(t_{k}\right)=y_{2}^{\prime \prime}\left(t_{k}\right), \quad k=\overline{0, n} . \tag{**}
\end{align*}
We call the functions s_(y_(1)),s_(y_(2))s_{y_{1}}, s_{y_{2}} spline solutions in S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) of the problems (C_(1))\left(C_{1}\right), (C_(2))\left(C_{2}\right), and the function
{:(26)s_(y)(t)=s_(y_(1))(t)+(beta-s_(y_(1))(b))/(s_(y_(2))(b))s_(y_(2))(t)","s_(y_(2))(b)!=0","t in[a","b]",":}\begin{equation*}
s_{y}(t)=s_{y_{1}}(t)+\frac{\beta-s_{y_{1}}(b)}{s_{y_{2}}(b)} s_{y_{2}}(t), s_{y_{2}}(b) \neq 0, t \in[a, b], \tag{26}
\end{equation*}
is called a spline solution in S_(5)(Delta_(n))S_{5}\left(\Delta_{n}\right) of the problem (D)(D).
THEOREM 7. Consider the problem
THEOREM 7. Consider the problem
{:[(C)y^('')=p(t)y+q(t)","quad t in[a","b]],[y(a)=alpha","y^(')(a)=gamma","]:}\begin{gather*}
y^{\prime \prime}=p(t) y+q(t), \quad t \in[a, b] \tag{C}\\
y(a)=\alpha, y^{\prime}(a)=\gamma,
\end{gather*}
where p(t) > 0,t in[a,b]p(t)>0, t \in[a, b] and p,qp, q are continuous on [a,b][a, b].
If y inW_(2)^(4)(/_\_(n))y \in W_{2}^{4}\left(\triangle_{n}\right) is the exact solution of (C)(C) and s_(y)inS_(5)(Delta_(n))s_{y} \in S_{5}\left(\Delta_{n}\right) is its spline solution (cf. Theorem 2b)), then we have
If y inW_(2)^(4)(/_\_(n))y \in W_{2}^{4}\left(\triangle_{n}\right) is the exact solution of (C)(C) and s_(y)inS_(5)(Delta_(n))s_{y} \in S_{5}\left(\Delta_{n}\right) is its spline solution (cf. Theorem 2b)), then we have
{:(27)||y^('')-s^('')||_(oo) <= sqrt2||Delta_(n)||^(3//2)*||y^((4))||_(2)",":}\begin{equation*}
\left\|y^{\prime \prime}-s^{\prime \prime}\right\|_{\infty} \leq \sqrt{2}\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2}, \tag{27}
\end{equation*}
where ||Delta_(n)||=max{t_(k)-t_(k-1),k= bar(1,n)}\left\|\Delta_{n}\right\|=\max \left\{t_{k}-t_{k-1}, k=\overline{1, n}\right\}
Proof. We have
Proof. We have
y^('')(t_(i))-s^('')y(t_(i))=0,quad i= bar(0,n)y^{\prime \prime}\left(t_{i}\right)-s^{\prime \prime} y\left(t_{i}\right)=0, \quad i=\overline{0, n}
so that, by Rolle's theorem, there exist t_(i)^((1))in(t_(i),t_(i+1)),i= bar(0,n-1)t_{i}^{(1)} \in\left(t_{i}, t_{i+1}\right), i=\overline{0, n-1} such that
y^(''')(t_(i)^((1)))-s_(y)^(''')(t_(i)^((1)))=0,quad i=0,n-1y^{\prime \prime \prime}\left(t_{i}^{(1)}\right)-s_{y}^{\prime \prime \prime}\left(t_{i}^{(1)}\right)=0, \quad i=0, n-1
Applying again Rolle's theorem, it follows the existence of t_(i)^((2))in(t_(i)^((1)),t_(i+1)^((1)))t_{i}^{(2)} \in\left(t_{i}^{(1)}, t_{i+1}^{(1)}\right), i= bar(0,n-2)i=\overline{0, n-2} such that
y^((4))(t_(i)^((2)))-s_(y)^((4))(t_(i)^((2)))=0,i= bar(0,n-2).y^{(4)}\left(t_{i}^{(2)}\right)-s_{y}^{(4)}\left(t_{i}^{(2)}\right)=0, i=\overline{0, n-2} .
The inequalities
|t_(i+1)^((1))-t_(i)^((1))| <= 2||Delta_(n)||" and "|t_(i+1)^((2))-t_(i)^((2))| <= 3||Delta_(n)||\left|t_{i+1}^{(1)}-t_{i}^{(1)}\right| \leq 2\left\|\Delta_{n}\right\| \text { and }\left|t_{i+1}^{(2)}-t_{i}^{(2)}\right| \leq 3\left\|\Delta_{n}\right\|
hold for i=0,n-2i=0, n-2 and i= bar(0,n-3)i=\overline{0, n-3}, respectively.
For every t in[a,b]t \in[a, b] there is an index i_(0)in{0,1,dots,n-1}i_{0} \in\{0,1, \ldots, n-1\} such that |t-t_(i_(0))^((1))| <= 2||Delta_(n)||\left|t-t_{i_{0}}^{(1)}\right| \leq 2\left\|\Delta_{n}\right\| so that, taking into account (24), we have
For every t in[a,b]t \in[a, b] there is an index i_(0)in{0,1,dots,n-1}i_{0} \in\{0,1, \ldots, n-1\} such that |t-t_(i_(0))^((1))| <= 2||Delta_(n)||\left|t-t_{i_{0}}^{(1)}\right| \leq 2\left\|\Delta_{n}\right\| so that, taking into account (24), we have
{:[|y^(''')(t)-s^(''')y(t)|=|int_(t_(i_(0))^((1)))^(t)(y^((4))(u)-s_(y)^((4))(u))du| <= ],[ <= |int_(t_(i_(0))^((1)))^(t)(d)u|^(1//2)*|int_(t_(i_(0)^((1)))^((1)))^(t)[y^((4))(u)-s_(y)^((4))]^(2)(d)u|^(1//2) <= ],[ <= sqrt(2||Delta_(n)||)*|int_(a)^(b)[y^((4))(u)-s_(y)^((4))(u)]^(2)(d)u|^(1//2)","],[ <= sqrt2*||Delta_(n)||^(1//2)*||y^((4))||_(2).]:}\begin{gathered}
\left|y^{\prime \prime \prime}(t)-s^{\prime \prime \prime} y(t)\right|=\left|\int_{t_{i_{0}}^{(1)}}^{t}\left(y^{(4)}(u)-s_{y}^{(4)}(u)\right) \mathrm{d} u\right| \leq \\
\leq\left|\int_{t_{i_{0}}^{(1)}}^{t} \mathrm{~d} u\right|^{1 / 2} \cdot\left|\int_{t_{i_{0}^{(1)}}^{(1)}}^{t}\left[y^{(4)}(u)-s_{y}^{(4)}\right]^{2} \mathrm{~d} u\right|^{1 / 2} \leq \\
\leq \sqrt{2\left\|\Delta_{n}\right\|} \cdot\left|\int_{a}^{b}\left[y^{(4)}(u)-s_{y}^{(4)}(u)\right]^{2} \mathrm{~d} u\right|^{1 / 2}, \\
\leq \sqrt{2} \cdot\left\|\Delta_{n}\right\|^{1 / 2} \cdot\left\|y^{(4)}\right\|_{2} .
\end{gathered}
Similarly, for every t in[a,b]t \in[a, b] there exist j_(0)in{0,1,dots,n-1}j_{0} \in\{0,1, \ldots, n-1\} such that |t-t_(j_(0))^((1))| <= ||Delta_(n)||\left|t-t_{j_{0}}^{(1)}\right| \leq\left\|\Delta_{n}\right\|, implying
{:[|y^('')(t)-s_(y)^('')(t)|=|int_(t_(s_(0))^((1)))^(t)[y^(''')(u)-s_(y)^(''')(u)du]| <= ],[ <= ||y^(''')-s_(y)^(''')||_(oo)*||Delta_(n)||]:}\begin{gathered}
\left|y^{\prime \prime}(t)-s_{y}^{\prime \prime}(t)\right|=\left|\int_{t_{s_{0}}^{(1)}}^{t}\left[y^{\prime \prime \prime}(u)-s_{y}^{\prime \prime \prime}(u) \mathrm{d} u\right]\right| \leq \\
\leq\left\|y^{\prime \prime \prime}-s_{y}^{\prime \prime \prime}\right\|_{\infty} \cdot\left\|\Delta_{n}\right\|
\end{gathered}
It follows that inequality (27) holds.
COROLLARY 8. If y inW_(2)^(4)(Delta_(n))y \in W_{2}^{4}\left(\Delta_{n}\right) is the exact solution of the problem ( CC ), then
COROLLARY 8. If y inW_(2)^(4)(Delta_(n))y \in W_{2}^{4}\left(\Delta_{n}\right) is the exact solution of the problem ( CC ), then
{:(28)||y-s_(y)||_(oo) <= sqrt2(b-a)^(2)||Delta_(n)||^(3//2)*||y^((4))||_(2).:}\begin{equation*}
\left\|y-s_{y}\right\|_{\infty} \leq \sqrt{2}(b-a)^{2}\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2} . \tag{28}
\end{equation*}
Proof. For every t in[a,b]t \in[a, b] we have
|y(t)-s_(y)(t)|=|int_(a)^(t)(y^(')(u)-s_(y)^(')(u))du| <= (b-a)*||y^(')-s_(y)^(')||_(oo)\left|y(t)-s_{y}(t)\right|=\left|\int_{a}^{t}\left(y^{\prime}(u)-s_{y}^{\prime}(u)\right) \mathrm{d} u\right| \leq(b-a) \cdot\left\|y^{\prime}-s_{y}^{\prime}\right\|_{\infty}
and
|y^(')(t)-s_(y)^(')(t)|=|int_(a)^(t)(y^('')(u)-s_(y)^('')(u))du| <= (b-a)*||y^('')-s_(y)^('')||_(oo).\left|y^{\prime}(t)-s_{y}^{\prime}(t)\right|=\left|\int_{a}^{t}\left(y^{\prime \prime}(u)-s_{y}^{\prime \prime}(u)\right) \mathrm{d} u\right| \leq(b-a) \cdot\left\|y^{\prime \prime}-s_{y}^{\prime \prime}\right\|_{\infty} .
From these inequalities and from (20) we obtain (28).
Remark 2. From the proof of Corollary 8 it follows that the inequality
{:(29)∣y^(')-s_(y)^(')||_(oo) <= sqrt2(b-a)||Delta_(n)||^(3//2)*||y^((4))||_(2):}\begin{equation*} \mid y^{\prime}-s_{y}^{\prime}\left\|_{\infty} \leq \sqrt{2}(b-a)\right\| \Delta_{n}\left\|^{3 / 2} \cdot\right\| y^{(4)} \|_{2} \tag{29} \end{equation*}
holds, too.
The approximative determination of the values of the spline solution s_(y)s_{y} of the problem ( DD ) on the nodes of the division Delta_(n)\Delta_{n}
The approximative determination of the values of the spline solution s_(y)s_{y} of the problem ( DD ) on the nodes of the division Delta_(n)\Delta_{n}
First observe that the exact solution y inW_(2)^(4)(Delta_(n))y \in W_{2}^{4}\left(\Delta_{n}\right) of the problem ( DD ) and its spline solution s_(y)inS_(5)(Delta_(n))s_{y} \in S_{5}\left(\Delta_{n}\right) given by (26) verify
{:[|y(t)-s_(y)(t)|=|y_(1)(t)+(beta-y_(1)(b))/(y_(2)(b))y_(2)(t)-s_(y_(1))(t)-(beta-s_(y_(1))(b))/(s_(y_(2))(b))*s_(y_(2))(t)| <= ],[ <= |y_(1)(t)-s_(y_(1))(t)|+|(beta-y_(1)(b))/(y_(2)(b))y_(2)(t)-(beta-s_(y_(1))(b))/(s_(y_(2))(b))*s_(y_(2))(t)|]:}\begin{aligned}
\left|y(t)-s_{y}(t)\right|= & \left|y_{1}(t)+\frac{\beta-y_{1}(b)}{y_{2}(b)} y_{2}(t)-s_{y_{1}}(t)-\frac{\beta-s_{y_{1}}(b)}{s_{y_{2}}(b)} \cdot s_{y_{2}}(t)\right| \leq \\
& \leq\left|y_{1}(t)-s_{y_{1}}(t)\right|+\left|\frac{\beta-y_{1}(b)}{y_{2}(b)} y_{2}(t)-\frac{\beta-s_{y_{1}}(b)}{s_{y_{2}}(b)} \cdot s_{y_{2}}(t)\right|
\end{aligned}
for every t in[a,b]t \in[a, b], where s_(y_(1))s_{y_{1}} and s_(y_(2))s_{y_{2}} are determined by the conditions (**).
Using (28), we obtain
Using (28), we obtain
(beta-y_(1)(b))/(y_(2)(b))=(beta-s_(y_(1))(b))/(s_(y_(2))(b))+O(||Delta_(n)||^(3//2)),\frac{\beta-y_{1}(b)}{y_{2}(b)}=\frac{\beta-s_{y_{1}}(b)}{s_{y_{2}}(b)}+O\left(\left\|\Delta_{n}\right\|^{3 / 2}\right),
showing that
||y(t)-s_(y)(t)||=O(||Delta_(n)||^(3//2)).\left\|y(t)-s_{y}(t)\right\|=O\left(\left\|\Delta_{n}\right\|^{3 / 2}\right) .
a) The approximative determination of the solution s_(y_(1))s_{y_{1}} on the nodes of the division Delta_(n)\Delta_{n}
Representation (9) yields
{:[s_(y_(1))(t)=u_(0)(t)*alpha+sum_(k=0)^(n)U_(k)(t)*y_(1)^('')(t_(k))=],[=u_(0)(t)*alpha+sum_(k=0)^(n)U_(k)(t)[p(t_(k))*y_(1)(t_(k))+q(t_(k))].]:}\begin{aligned}
& s_{y_{1}}(t)=u_{0}(t) \cdot \alpha+\sum_{k=0}^{n} U_{k}(t) \cdot y_{1}^{\prime \prime}\left(t_{k}\right)= \\
= & u_{0}(t) \cdot \alpha+\sum_{k=0}^{n} U_{k}(t)\left[p\left(t_{k}\right) \cdot y_{1}\left(t_{k}\right)+q\left(t_{k}\right)\right] .
\end{aligned}
Letting
{:[v_(i):=s_(y_(i))(t_(i))","quad i= bar(0,n)],[e_(i):=y_(1)(t_(i))-s_(y_(1))(t_(i))","quad i= bar(0,n)]:}\begin{gathered}
v_{i}:=s_{y_{i}}\left(t_{i}\right), \quad i=\overline{0, n} \\
e_{i}:=y_{1}\left(t_{i}\right)-s_{y_{1}}\left(t_{i}\right), \quad i=\overline{0, n}
\end{gathered}
one obtains the system
{:[s_(y_(1))(t_(i))=u_(0)(t_(i))alpha+sum_(k=0)^(n)U_(k)(t_(i))[p(t_(k))(e_(k)+v_(k))+q(t_(k))]=],[=u_(0)(t_(i))alpha+sum_(k=0)^(n)U_(k)(t_(i))[p(t_(k))v_(k)+q(t_(k))]+o(||Delta_(n)||^(3//2))","],[i= bar(0,n)]:}\begin{aligned}
s_{y_{1}}\left(t_{i}\right)= & u_{0}\left(t_{i}\right) \alpha+\sum_{k=0}^{n} U_{k}\left(t_{i}\right)\left[p\left(t_{k}\right)\left(e_{k}+v_{k}\right)+q\left(t_{k}\right)\right]= \\
& =u_{0}\left(t_{i}\right) \alpha+\sum_{k=0}^{n} U_{k}\left(t_{i}\right)\left[p\left(t_{k}\right) v_{k}+q\left(t_{k}\right)\right]+o\left(\left\|\Delta_{n}\right\|^{3 / 2}\right), \\
i & =\overline{0, n}
\end{aligned}
The approximative values of the spline solution s_(y_(1))s_{y_{1}} on the nodes of Delta_(n)\Delta_{n} are the solutions nu_(kappa)\nu_{\kappa} of the linear system
v_(i)=u_(0)(t_(i))alpha+sum_(k=0)^(n)U_(k)(t_(i))[p(t_(k))v_(k)+q(t_(k))],quad i= bar(0,n)v_{i}=u_{0}\left(t_{i}\right) \alpha+\sum_{k=0}^{n} U_{k}\left(t_{i}\right)\left[p\left(t_{k}\right) v_{k}+q\left(t_{k}\right)\right], \quad i=\overline{0, n}
b) The approximative determination of the solution s_(y_(2))s_{y_{2}} on the nodes of Delta_(n)\Delta_{n} Using again representation (9), one obtains
{:[s_(y_(2))(t)=u_(1)(t)+sum_(k=0)^(n)U_(k)(t)*y_(2)^('')(t_(k))=],[=u_(1)(t)+sum_(k=0)^(n)U_(k)(t)*p(t_(k))*y_(2)(t_(k)).]:}\begin{aligned}
& s_{y_{2}}(t)=u_{1}(t)+\sum_{k=0}^{n} U_{k}(t) \cdot y_{2}^{\prime \prime}\left(t_{k}\right)= \\
& =u_{1}(t)+\sum_{k=0}^{n} U_{k}(t) \cdot p\left(t_{k}\right) \cdot y_{2}\left(t_{k}\right) .
\end{aligned}
Letting
{:[w_(i):=s_(y_(2))(t_(i))","quad i= bar(0,n)],[ bar(e)_(i):=y_(2)(t_(i))-s_(y_(2))(t_(i))","quad i= bar(0,n)]:}\begin{gathered}
w_{i}:=s_{y_{2}}\left(t_{i}\right), \quad i=\overline{0, n} \\
\bar{e}_{i}:=y_{2}\left(t_{i}\right)-s_{y_{2}}\left(t_{i}\right), \quad i=\overline{0, n}
\end{gathered}
it follows that w_(i)w_{i} are the solutions of the system
w_(i)=u_(1)(t_(i))+sum_(k=0)^(n)U_(k)(t_(i))p(t_(k))w_(k)+O(||Delta_(n)||^(3//2))w_{i}=u_{1}\left(t_{i}\right)+\sum_{k=0}^{n} U_{k}\left(t_{i}\right) p\left(t_{k}\right) w_{k}+O\left(\left\|\Delta_{n}\right\|^{3 / 2}\right)
Therefore, the approximative values of s_(y_(2))(t_(i))s_{y_{2}}\left(t_{i}\right) can be obtained from the linear system
{:(30)w_(i)=u_(1)(t_(i))+sum_(k=0)^(n)U_(k)(t_(i))p(t_(k))*w_(k)","i= bar(0,n):}\begin{equation*}
w_{i}=u_{1}\left(t_{i}\right)+\sum_{k=0}^{n} U_{k}\left(t_{i}\right) p\left(t_{k}\right) \cdot w_{k}, i=\overline{0, n} \tag{30}
\end{equation*}
The approximative values of the spline solution s_(y)inS_(5)(Lambda_(n))s_{y} \in S_{5}\left(\Lambda_{n}\right) on the nodes of the division Delta_(n)\Delta_{n} are given by
(31)
(31)
s_(y)(t_(i))=v_(i)+(beta-v_(n))/(w_(n))w_(i),quad i= bar(0,n)s_{y}\left(t_{i}\right)=v_{i}+\frac{\beta-v_{n}}{w_{n}} w_{i}, \quad i=\overline{0, n}
A numerical example. The problem
(D) y^('')=4y,t in[0,1]y^{\prime \prime}=4 y, t \in[0,1]
y(0)=1,y(1)=e^(-2)y(0)=1, y(1)=e^{-2}
has the exact solution y=e^(-2t)y=e^{-2 t}.
The associated Cauchy problems are
(C_(1))y^('')=4y,t in[0,1]\left(C_{1}\right) y^{\prime \prime}=4 y, t \in[0,1]
y(0)=1,y^(')(0)=0y(0)=1, y^{\prime}(0)=0
( C_(2)C_{2} ) y^('')=4y,t in[0,1]y^{\prime \prime}=4 y, t \in[0,1]
y(0)=0,y^(')(0)=1y(0)=0, y^{\prime}(0)=1
and have the exact solutions
(D) y^('')=4y,t in[0,1]y^{\prime \prime}=4 y, t \in[0,1]
y(0)=1,y(1)=e^(-2)y(0)=1, y(1)=e^{-2}
has the exact solution y=e^(-2t)y=e^{-2 t}.
The associated Cauchy problems are
(C_(1))y^('')=4y,t in[0,1]\left(C_{1}\right) y^{\prime \prime}=4 y, t \in[0,1]
y(0)=1,y^(')(0)=0y(0)=1, y^{\prime}(0)=0
( C_(2)C_{2} ) y^('')=4y,t in[0,1]y^{\prime \prime}=4 y, t \in[0,1]
y(0)=0,y^(')(0)=1y(0)=0, y^{\prime}(0)=1
and have the exact solutions
{:[y_(1)(t)=(1)/(2)[e^(2t)+e^(-2t)]],[y_(2)(t)=(1)/(4)[e^(2t)-e^(-2t)].]:}\begin{aligned}
& y_{1}(t)=\frac{1}{2}\left[e^{2 t}+e^{-2 t}\right] \\
& y_{2}(t)=\frac{1}{4}\left[e^{2 t}-e^{-2 t}\right] .
\end{aligned}
For n=5n=5, let
Delta_(5):={t_(0)=0,t_(1)=0.2,t_(2)=0.4,t_(3)=0.6,t_(4)=0.8,t_(5)=1}\Delta_{5}:=\left\{t_{0}=0, t_{1}=0.2, t_{2}=0.4, t_{3}=0.6, t_{4}=0.8, t_{5}=1\right\}.
Using representation (1), one obtains Table 1 for the coefficients of s_(y_(1))s_{y_{1}} and s_(y_(2))s_{y_{2}}.
Delta_(5):={t_(0)=0,t_(1)=0.2,t_(2)=0.4,t_(3)=0.6,t_(4)=0.8,t_(5)=1}\Delta_{5}:=\left\{t_{0}=0, t_{1}=0.2, t_{2}=0.4, t_{3}=0.6, t_{4}=0.8, t_{5}=1\right\}.
Using representation (1), one obtains Table 1 for the coefficients of s_(y_(1))s_{y_{1}} and s_(y_(2))s_{y_{2}}.
Table 1
| n=5n=5 | s_(y_(1))s_{y_{1}} | s_(y_(2))s_{y_{2}} |
| A_(0)A_{0} | 1 | 0 |
| A_(1)A_{1} | 0 | 1 |
| A_(2)A_{2} | 2 | 0 |
| A_(3)A_{3} | 0.1576268148 | 0.6678202118 |
| a_(0)a_{0} | 0.8446081893 | 0.1257524863 |
| a_(1)a_{1} | -0.6853940844 | 0.07800170745 |
| a_(2)a_{2} | -0.6014825811 | -0.1770186129 |
| a_(3)a_{3} | 3.020199906 | 0.7804651905 |
| a_(4)a_{4} | -5.717416684 | -1.970643804 |
| a_(5)a_{5} | 3.139485253 | 1.163443033 |
For the values of s_(y)s_{y} on the nodes of Delta_(5)\Delta_{5}, one uses
s_(y)(t_(i))=s_(y_(1))(t_(i))+(e^(-2)-s_(y_(1))(1))/(s_(y_(2))(1))s_(y_(2))(t_(i)),i= bar(0,5)s_{y}\left(t_{i}\right)=s_{y_{1}}\left(t_{i}\right)+\frac{e^{-2}-s_{y_{1}}(1)}{s_{y_{2}}(1)} s_{y_{2}}\left(t_{i}\right), i=\overline{0,5}
Table 2 contains the values s_(y)(t_(i)),i= bar(0,5)s_{y}\left(t_{i}\right), i=\overline{0,5}, and the errors
E_(i)=|y(t_(i))-s_(y)(t_(i))|,quad i= bar(0,5)E_{i}=\left|y\left(t_{i}\right)-s_{y}\left(t_{i}\right)\right|, \quad i=\overline{0,5}
Table 2
| t_(i)\boldsymbol{t}_{i} | s_(y)(t_(i))\boldsymbol{s}_{y}\left(\boldsymbol{t}_{i}\right) | E_(i)\boldsymbol{E}_{i} |
| 0 | 1 | 0 |
| 0.2 | 0.6708587727 | 0.5387267*10^(-3)0.5387267 \cdot 10^{-3} |
| 0.4 | 0.4506125215 | 0.12835574*10^(-2)0.12835574 \cdot 10^{-2} |
| 0.6 | 0.303315766 | 0.21215541*10^(-2)0.21215541 \cdot 10^{-2} |
| 0.8 | 0.204249434 | 0.2352916*10^(-2)0.2352916 \cdot 10^{-2} |
| 1 | 0.135335284 | 0.8*10^(-9)0.8 \cdot 10^{-9} |
REFERENCES
- P. Blaga and G. Micula, Polynomial natural spline of even degree, Studia Univ. "BabesBolyai", Mathematica 38, 2 (1993), 31-40.
- P. Blaga, R. Gorenflo and G. Micula, Even degree spline technique for numerical solution of delay differential equations, Freic Universität Berlin, Preprint No. A-15 (1996), Serie A-Mathematik.
- R. L. Burden and T. Douglas Faires, Numerical Analysis, Third Edition, PWS-KENT Publishing Company, Boston, 1985.
- G. Micula, P. Blaga and M. Micula, On even degree polynomial spline functions with applications to numerical solution of differential equations with retarded argument, Technische Hochschule Darmstadt, Preprint No. 1771, Fachbereich Mathematik (1995).
Received May 15, 1996
"Tiberiu Popoviciu" Institute of Numerical Analysis
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"Tiberiu Popoviciu" Institute of Numerical Analysis
P.O. Box 68
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