The approximation by spline functions of the solutions of a singularly perturbed bilocal problem

Costica Mustata
(original), Approximation Theory, Numerical Analysis, ODEs (numerical), paper, splines

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Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Adrian Muresan
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Radu Mustata
Babes-Bolyai University, Romania

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C. Mustăţa, A. Mureşan and R. Mustăţa, The approximation by spline functions of the solutions of a singularly perturbed bilocal problem, Rev. Anal. Numér. Théor. Approx. 27 (1998).

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Revue d’Analyse Numer. Theor.Approx.

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https://ictp.acad.ro/jnaat/journal/article/view/1998-vol27-no2-art10

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2457-6794

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[2] P.Blaga and Gh.Micula, Polynomial natural spline of even degree, Studia Univ.”Babeş-Bolyai”, Mathematica 38, 2 (1993), pp. 31-40.
[3] W. Heinrichs, A stabilized Multidomain approach for singular perturbations Problems, Journal of Scientific Computing 7 (1992), https://doi.org/10.1007/bf01059944
[4] P. W. Hemker, A Numerical Study of Stiff Two-Point Boundary Problems, Mathematical Centre Tracts 80, Amsterdam, 1977.
[5] H. B. Keller, Numerical Methods for Two-point Boundary-value Problems, Blaisdell Publ. Comp., 1968.
[6] C. Mustăţa, Approximation by spline functions of the solution of a linear bilocal problem, Rev. Anal. Numér. Theor. approx. 26 (1997), 1-2, pp. 137-148.

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1998-Mustata-The approximation by spline functions of the solution-Jnaat

THE APPROXIMATION BY SPLINE FUNCTIONS OF THE SOLUTION OF A SINGULARLY PERTURBED BILOCAL PROBLEM

C. MUSTÃTA, A. C. MUREŞAN AND R. MUSTĂTA

The singularly perturbed bilocal problems admit exact solutions having both slowly and rapidly varying parts. There are thin transition layers where the solution can jump abruptly, having as effect strong oscillations of the approximate solutions obtained by the method of centered differences, spectral methods, etc.
We define a class of spline functions of degree 5 which are appropriate for these problems and obtain sufficiently good approximate solutions, with attenuated oscillations on the subintervals where the exact solution jumps rapidly.
Let n ≥ 3 , n ∈ N n ≥ 3 , n ∈ N n >= 3,n inNn \geq 3, n \in \mathbb{N}n≥3,n∈N, and let
(1) Δ n : − ∞ = t − 1 < a = t 0 < t 1 < … < t n = b < t n + 1 = + ∞ (1) Δ n : − ∞ = t − 1 < a = t 0 < t 1 < … < t n = b < t n + 1 = + ∞ {:(1)Delta_(n):-oo=t_(-1) < a=t_(0) < t_(1) < dots < t_(n)=b < t_(n+1)=+oo:}\begin{equation*} \Delta_{n}:-\infty=t_{-1}<a=t_{0}<t_{1}<\ldots<t_{n}=b<t_{n+1}=+\infty \tag{1} \end{equation*}(1)Δn:−∞=t−1<a=t0<t1<…<tn=b<tn+1=+∞
a division of the real axis.
Denote by O s ( Δ n ) O s Δ n O_(s)(Delta_(n))\mathscr{O}_{s}\left(\Delta_{n}\right)Os(Δn) the set of functions s : R → R s : R → R s:RrarrRs: \mathbb{R} \rightarrow \mathbb{R}s:R→R verifying the conditions:
1 ∘ s ∈ C 4 ( R ) ; 2 ∘ s | I k ∈ P 5 , I k = [ t k − 1 , t k ) , k = 1 , 2 , … , n ; 3 ∘ s | I 0 ∈ P 3 , s | I n + 1 ∈ P 3 , I 0 = [ t − 1 , t 0 ) , I n + 1 = [ t n , t n + 1 ) , 1 ∘ s ∈ C 4 ( R ) ; 2 ∘ s I k ∈ P 5 , I k = t k − 1 , t k , k = 1 , 2 , … , n ; 3 ∘ s I 0 ∈ P 3 , s I n + 1 ∈ P 3 , I 0 = t − 1 , t 0 , I n + 1 = t n , t n + 1 , {:[1^(@)s inC^(4)(R);],[2^(@)s|_(I_(k))inP_(5)","I_(k)=[t_(k-1),t_(k))","k=1","2","dots","n;],[3^(@)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))","]:}\begin{aligned} & 1^{\circ} s \in C^{4}(\mathbb{R}) ; \\ & \left.2^{\circ} s\right|_{I_{k}} \in \mathscr{P}_{5}, I_{k}=\left[t_{k-1}, t_{k}\right), k=1,2, \ldots, n ; \\ & \left.3^{\circ} 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), \end{aligned}1∘s∈C4(R);2∘s|Ik∈P5,Ik=[tk−1,tk),k=1,2,…,n;3∘s|I0∈P3,s|In+1∈P3,I0=[t−1,t0),In+1=[tn,tn+1),
where P m P m P_(m)\mathscr{P}_{m}Pm denotes the set of polynomials of degree m m mmm.
As concerns the behavior of functions in this class, one can prove
THEOREM 1. Every function s ∈ P 5 ( Δ n ) s ∈ P 5 Δ n s inP_(5)(Delta_(n))s \in \mathscr{P}_{5}\left(\Delta_{n}\right)s∈P5(Δn) can be written in the form
s ( t ) = ∑ i = 0 3 A i t i + ∑ k = 0 n a k ( t − t k ) + 5 , t ∈ R , s ( t ) = ∑ i = 0 3   A i t i + ∑ k = 0 n   a k t − t k + 5 , t ∈ R , s(t)=sum_(i=0)^(3)A_(i)t^(i)+sum_(k=0)^(n)a_(k)(t-t_(k))_(+)^(5),quad t inR,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 \mathbb{R},s(t)=∑i=03Aiti+∑k=0nak(t−tk)+5,t∈R,
where
(3) ∑ k = 0 n a k = 0 , ∑ k = 0 n a k t k = 0 (3) ∑ k = 0 n   a k = 0 , ∑ k = 0 n   a k t k = 0 {:(3)sum_(k=0)^(n)a_(k)=0","quadsum_(k=0)^(n)a_(k)t_(k)=0:}\begin{equation*} \sum_{k=0}^{n} a_{k}=0, \quad \sum_{k=0}^{n} a_{k} t_{k}=0 \tag{3} \end{equation*}(3)∑k=0nak=0,∑k=0naktk=0
and
( t − t k ) + = { 0 if t < t k , t − t k if t ≥ t k , k = 0 , 1 , … , n t − t k + = 0       if  t < t k , t − t k       if  t ≥ t k , k = 0 , 1 , … , n (t-t_(k))_(+)={[0," if "t < t_(k)","],[t-t_(k)," if "t >= t_(k)","k=0","1","dots","n]:}\left(t-t_{k}\right)_{+}= \begin{cases}0 & \text { if } t<t_{k}, \\ t-t_{k} & \text { if } t \geq t_{k}, k=0,1, \ldots, n\end{cases}(t−tk)+={0 if t<tk,t−tk if t≥tk,k=0,1,…,n
Proof. Let s ∈ S 5 ( Δ n ) s ∈ S 5 Δ n s inS_(5)(Delta_(n))s \in \mathscr{S}_{5}\left(\Delta_{n}\right)s∈S5(Δn). By definition s ( 4 ) ( t ) = 0 s ( 4 ) ( t ) = 0 s^((4))(t)=0s^{(4)}(t)=0s(4)(t)=0 for all t ≥ b t ≥ b t >= bt \geq bt≥b so that s ( 4 ) ( t ) = ∑ k = 0 n a k ( t − t k ) = 0 s ( 4 ) ( t ) = ∑ k = 0 n   a k t − t k = 0 s^((4))(t)=sum_(k=0)^(n)a_(k)(t-t_(k))=0s^{(4)}(t)=\sum_{k=0}^{n} a_{k}\left(t-t_{k}\right)=0s(4)(t)=∑k=0nak(t−tk)=0, for all t ≥ b t ≥ b t >= bt \geq bt≥b, showing that ∑ k = 0 n a k = 0 ∑ k = 0 n   a k = 0 sum_(k=0)^(n)a_(k)=0\sum_{k=0}^{n} a_{k}=0∑k=0nak=0 and ∑ k = 0 n a k t k = 0 ∑ k = 0 n   a k t k = 0 sum_(k=0)^(n)a_(k)t_(k)=0\sum_{k=0}^{n} a_{k} t_{k}=0∑k=0naktk=0.
THEOREM 2. Let f : R → R f : R → R f:RrarrRf: \mathbb{R} \rightarrow \mathbb{R}f:R→R be such that
(4) f ( a ) = α , f ( b ) = β , f ′ ′ ( t k ) = λ k , k = 0 , 1 , 2 , … , n , (4) f ( a ) = α , f ( b ) = β , f ′ ′ t k = λ k , k = 0 , 1 , 2 , … , n , {:(4)f(a)=alpha","quad f(b)=beta","quadf^('')(t_(k))=lambda_(k)","quad k=0","1","2","dots","n",":}\begin{equation*} f(a)=\alpha, \quad f(b)=\beta, \quad f^{\prime \prime}\left(t_{k}\right)=\lambda_{k}, \quad k=0,1,2, \ldots, n, \tag{4} \end{equation*}(4)f(a)=α,f(b)=β,f′′(tk)=λk,k=0,1,2,…,n,
where t k ( t 0 = 1 , t n = b ) , k = 0 , n ― t k t 0 = 1 , t n = b , k = 0 , n ¯ t_(k)(t_(0)=1,t_(n)=b),k= bar(0,n)t_{k}\left(t_{0}=1, t_{n}=b\right), k=\overline{0, n}tk(t0=1,tn=b),k=0,n―, are the knots of the division Δ n Δ n Delta_(n)\Delta_{n}Δn and α , β , λ k α , β , λ k alpha,beta,lambda_(k)\alpha, \beta, \lambda_{k}α,β,λk, k = 0 , n ― k = 0 , n ¯ k= bar(0,n)k=\overline{0, n}k=0,n― are given numbers.
Then there exists a unique function s f ∈ S 5 ( Δ n ) s f ∈ S 5 Δ n s_(f)inS_(5)(Delta_(n))s_{f} \in \mathscr{S}_{5}\left(\Delta_{n}\right)sf∈S5(Δn) such that
(5) s f ( a ) = α , s f ( b ) = β , s f ′ ′ ( t k ) = λ k , k = 0 , n ― (5) s f ( a ) = α , s f ( b ) = β , s f ′ ′ t k = λ k , k = 0 , n ¯ {:(5)s_(f)(a)=alpha","quads_(f)(b)=beta","quads_(f)^('')(t_(k))=lambda_(k)","quad k= bar(0,n):}\begin{equation*} s_{f}(a)=\alpha, \quad s_{f}(b)=\beta, \quad s_{f}^{\prime \prime}\left(t_{k}\right)=\lambda_{k}, \quad k=\overline{0, n} \tag{5} \end{equation*}(5)sf(a)=α,sf(b)=β,sf′′(tk)=λk,k=0,n―
Proof. Using the representation (2) and imposing the conditions (5), one obtains the system:
(6)
A 0 + A 1 a + A 2 a 2 + A 3 a 3 = α A 0 + A 1 b + A 2 b 2 + A 3 b 3 + ∑ k = 0 n − 1 a k ( b − t k ) 5 = β 2 A 2 + 6 A 3 t j + 20 ⋅ ∑ k = 0 n a k ( t j − t k ) t 3 = λ j , j = 0 , n ― ∑ k = 0 n a k = 0 ∑ k = 0 n a k t k = 0 A 0 + A 1 a + A 2 a 2 + A 3 a 3 = α A 0 + A 1 b + A 2 b 2 + A 3 b 3 + ∑ k = 0 n − 1   a k b − t k 5 = β 2 A 2 + 6 A 3 t j + 20 ⋅ ∑ k = 0 n   a k t j − t k t 3 = λ j , j = 0 , n ¯ ∑ k = 0 n   a k = 0 ∑ k = 0 n   a k t k = 0 {:[A_(0)+A_(1)a+A_(2)a^(2)+A_(3)a^(3)=alpha],[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],[2A_(2)+6A_(3)t_(j)+20*sum_(k=0)^(n)a_(k)(t_(j)-t_(k))_(t)^(3)=lambda_(j)","quad j= bar(0,n)],[sum_(k=0)^(n)a_(k)=0],[sum_(k=0)^(n)a_(k)t_(k)=0]:}\begin{gathered} A_{0}+A_{1} a+A_{2} a^{2}+A_{3} a^{3}=\alpha \\ 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 \\ 2 A_{2}+6 A_{3} t_{j}+20 \cdot \sum_{k=0}^{n} a_{k}\left(t_{j}-t_{k}\right)_{t}^{3}=\lambda_{j}, \quad j=\overline{0, n} \\ \sum_{k=0}^{n} a_{k}=0 \\ \sum_{k=0}^{n} a_{k} t_{k}=0 \end{gathered}A0+A1a+A2a2+A3a3=αA0+A1b+A2b2+A3b3+∑k=0n−1ak(b−tk)5=β2A2+6A3tj+20⋅∑k=0nak(tj−tk)t3=λj,j=0,n―∑k=0nak=0∑k=0naktk=0
of n + 5 n + 5 n+5n+5n+5 equations with n + 5 n + 5 n+5n+5n+5 unknowns A 0 , A 1 , A 2 , A 3 , a 0 , a 1 , … a n A 0 , A 1 , A 2 , A 3 , a 0 , a 1 , … a n A_(0),A_(1),A_(2),A_(3),a_(0),a_(1),dotsa_(n)A_{0}, A_{1}, A_{2}, A_{3}, a_{0}, a_{1}, \ldots a_{n}A0,A1,A2,A3,a0,a1,…an.
The system (6) has a unique solution if and only if the corresponding homogeneous system (obtained for α = β = 0 , λ j = 0 , j = 0 , n ― α = β = 0 , λ j = 0 , j = 0 , n ¯ alpha=beta=0,lambda_(j)=0,j= bar(0,n)\alpha=\beta=0, \lambda_{j}=0, j=\overline{0, n}α=β=0,λj=0,j=0,n― ) has only the null solution.
If s ∈ P 5 ( Δ n ) s ∈ P 5 Δ n s inP_(5)(Delta_(n))s \in \mathscr{P}_{5}\left(\Delta_{n}\right)s∈P5(Δn) verifies the homogeneous conditions (5) (i.e., with α = β = 0 α = β = 0 alpha=beta=0\alpha=\beta=0α=β=0, λ k = 0 , j = 0 , n ― λ k = 0 , j = 0 , n ¯ lambda_(k)=0,j= bar(0,n)\lambda_{k}=0, j=\overline{0, n}λk=0,j=0,n― ), then
∫ a b [ s ( 4 ) ( t ) ] 2 d t = ∫ a b s ( 4 ) ( t ) ( s ′ ′ ′ ( t ) ) ′ d t = s ( 4 ) ( t ) ⋅ s ′ ′ ′ ( t ) | a b − ∫ a b s ′ ′ ′ ( t ) ⋅ s ( 5 ) ( t ) ⋅ d t = = − ∑ k = 1 n ∫ t k − 1 t k s ( 5 ) ( t ) ⋅ s ′ ′ ′ ( t ) d t = − ∑ k = 1 n c k ∫ t k − 1 t k s ′ ′ ′ ( t ) d t = = − ∑ k = 1 n c k [ s ′ ′ ( t k ) − s ′ ′ ( t k − 1 ) ] = 0 ∫ a b   s ( 4 ) ( t ) 2 d t = ∫ a b   s ( 4 ) ( t ) s ′ ′ ′ ( t ) ′ d t = s ( 4 ) ( t ) ⋅ s ′ ′ ′ ( t ) a b − ∫ a b   s ′ ′ ′ ( t ) ⋅ s ( 5 ) ( t ) ⋅ d t = = − ∑ k = 1 n   ∫ t k − 1 t k   s ( 5 ) ( t ) ⋅ s ′ ′ ′ ( t ) d t = − ∑ k = 1 n   c k ∫ t k − 1 t k   s ′ ′ ′ ( t ) d t = = − ∑ k = 1 n   c k s ′ ′ t k − s ′ ′ t k − 1 = 0 {:[int_(a)^(b)[s^((4))(t)]^(2)dt=int_(a)^(b)s^((4))(t)(s^(''')(t))^(')dt=],[s^((4))(t)*s^(''')(t)|_(a)^(b)-int_(a)^(b)s^(''')(t)*s^((5))(t)*dt=],[=-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)\left(s^{\prime \prime \prime}(t)\right)^{\prime} \mathrm{d} t= \\ \left.s^{(4)}(t) \cdot s^{\prime \prime \prime}(t)\right|_{a} ^{b}-\int_{a}^{b} s^{\prime \prime \prime}(t) \cdot s^{(5)}(t) \cdot \mathrm{d} 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}∫ab[s(4)(t)]2 dt=∫abs(4)(t)(s′′′(t))′dt=s(4)(t)⋅s′′′(t)|ab−∫abs′′′(t)⋅s(5)(t)⋅dt==−∑k=1n∫tk−1tks(5)(t)⋅s′′′(t)dt=−∑k=1nck∫tk−1tks′′′(t)dt==−∑k=1nck[s′′(tk)−s′′(tk−1)]=0
where c k = s ( 5 ) ( t ) | I k , k = 1 , n ― c k = s ( 5 ) ( t ) I k , k = 1 , n ¯ c_(k)=s^((5))(t)|_(I_(k)),k= bar(1,n)c_{k}=\left.s^{(5)}(t)\right|_{I_{k}}, k=\overline{1, n}ck=s(5)(t)|Ik,k=1,n―.
It follows that s ( 4 ) ( t ) = 0 s ( 4 ) ( t ) = 0 s^((4))(t)=0s^{(4)}(t)=0s(4)(t)=0 for all t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b]. Since the restrictions of s s sss to the intervals I 0 , I n + 1 I 0 , I n + 1 I_(0),I_(n+1)I_{0}, I_{n+1}I0,In+1 are in P 3 P 3 P_(3)\mathscr{P}_{3}P3 and s ∈ C 4 ( R ) s ∈ C 4 ( R ) s inC^(4)(R)s \in C^{4}(\mathbb{R})s∈C4(R), it results that s ( 4 ) ( t ) = 0 s ( 4 ) ( t ) = 0 s^((4))(t)=0s^{(4)}(t)=0s(4)(t)=0 for all t ∈ R t ∈ R t inRt \in \mathbb{R}t∈R. Therefore s ′ ′ ∈ P s ′ ′ ∈ P s^('')inPs^{\prime \prime} \in \mathscr{P}s′′∈P and, taking into account the equalities s ′ ′ ( t k ) = 0 s ′ ′ t k = 0 s^('')(t_(k))=0s^{\prime \prime}\left(t_{k}\right)=0s′′(tk)=0, k = 0 , n , n ≥ 3 k = 0 , n , n ≥ 3 k=0,n,n >= 3k=0, n, n \geq 3k=0,n,n≥3 (verified by hypothesis), one obtains s ′ ′ ( t ) = 0 s ′ ′ ( t ) = 0 s^('')(t)=0s^{\prime \prime}(t)=0s′′(t)=0 for all t ∈ R t ∈ R t inRt \in \mathbb{R}t∈R. Since s ( a ) = s ( b ) = 0 s ( a ) = s ( b ) = 0 s(a)=s(b)=0s(a)=s(b)=0s(a)=s(b)=0, it follows s ( t ) = 0 s ( t ) = 0 s(t)=0s(t)=0s(t)=0 for all t ∈ R t ∈ R t inRt \in \mathbb{R}t∈R, which is equivalent to A 0 = A 1 = A 2 = A 3 = 0 A 0 = A 1 = A 2 = A 3 = 0 A_(0)=A_(1)=A_(2)=A_(3)=0A_{0}=A_{1}=A_{2}=A_{3}=0A0=A1=A2=A3=0 and a k = 0 , k = 0 , n ― a k = 0 , k = 0 , n ¯ a_(k)=0,k= bar(0,n)a_{k}=0, k=\overline{0, n}ak=0,k=0,n―.
COROLLARY 3. a) There exists a system B = { s 0 , s 1 , S 0 , S 1 , … , S n } B = s 0 , s 1 , S 0 , S 1 , … , S n B={s_(0),s_(1),S_(0),S_(1),dots,S_(n)}\mathscr{B}=\left\{s_{0}, s_{1}, S_{0}, S_{1}, \ldots, S_{n}\right\}B={s0,s1,S0,S1,…,Sn} of functions in S 5 ( Δ n ) S 5 Δ n S_(5)(Delta_(n))\mathscr{S}_{5}\left(\Delta_{n}\right)S5(Δn) verifying the conditions:
(7) s 0 ( a ) = 1 , s 0 ( b ) = 0 , s 0 ′ ′ ( t k ) = 0 , k = 0 , n ― s 1 ( a ) = 0 , s 1 ( b ) = 1 , s 1 ′ ′ ( t k ) = 0 , k = 0 , n ― S k ( a ) = 0 = S k ( b ) , k = 0 , n ― ; S k ′ ′ ( t j ) = δ k j , k , j = 0 , n ― (7) s 0 ( a ) = 1 , s 0 ( b ) = 0 , s 0 ′ ′ t k = 0 , k = 0 , n ¯ s 1 ( a ) = 0 , s 1 ( b ) = 1 , s 1 ′ ′ t k = 0 , k = 0 , n ¯ S k ( a ) = 0 = S k ( b ) , k = 0 , n ¯ ; S k ′ ′ t j = δ k j , k , j = 0 , n ¯ {:[(7)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)","k= bar(0,n);S_(k)^('')(t_(j))=delta_(kj)","k","j= bar(0,n)]:}\begin{gather*} s_{0}(a)=1, s_{0}(b)=0, s_{0}^{\prime \prime}\left(t_{k}\right)=0, k=\overline{0, n} \tag{7}\\ 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), k=\overline{0, n} ; S_{k}^{\prime \prime}\left(t_{j}\right)=\delta_{k j}, k, j=\overline{0, n} \end{gather*}(7)s0(a)=1,s0(b)=0,s0′′(tk)=0,k=0,n―s1(a)=0,s1(b)=1,s1′′(tk)=0,k=0,n―Sk(a)=0=Sk(b),k=0,n―;Sk′′(tj)=δkj,k,j=0,n―
b) If f : R → R f : R → R f:RrarrRf: \mathbb{R} \rightarrow \mathbb{R}f:R→R verifies the conditions (4) and s f ∈ O 5 ( Δ n ) s f ∈ O 5 Δ n s_(f)inO_(5)(Delta_(n))s_{f} \in \mathscr{O}_{5}\left(\Delta_{n}\right)sf∈O5(Δn) verifies the conditions (5) then
(8) s f ( t ) = s 0 ( t ) ⋅ f ( a ) + s 1 ( t ) ⋅ f ( b ) + ∑ k = 0 n s k ( t ) ⋅ f ′ ′ ( t k ) , t ∈ R (8) s f ( t ) = s 0 ( t ) ⋅ f ( a ) + s 1 ( t ) ⋅ f ( b ) + ∑ k = 0 n   s k ( t ) ⋅ f ′ ′ t k , t ∈ R {:(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:}\begin{equation*} 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 \mathbb{R} \tag{8} \end{equation*}(8)sf(t)=s0(t)⋅f(a)+s1(t)⋅f(b)+∑k=0nsk(t)⋅f′′(tk),t∈R
Remark 1. By Corollary 3, O 5 ( Δ n ) O 5 Δ n O_(5)(Delta_(n))\mathscr{O}_{5}\left(\Delta_{n}\right)O5(Δn) is a real linear space of dimension n + 3 n + 3 n+3n+3n+3, and the system B B B\mathscr{B}B is a basis in O 5 ( Δ n ) O 5 Δ n O_(5)(Delta_(n))\mathscr{O}_{5}\left(\Delta_{n}\right)O5(Δn).
Let us introduce now the notations:
(9) W 2 4 ( Δ n ) := { g : [ a , b ] → R , abs.cont.on I k , k = 1 , n ― and g ( 4 ) ∈ L 2 [ a , b ] } , (10) W 2 , f 4 ( Δ n ) := { g ∈ W 2 4 ( Δ n ) : g ′ ′ ( t k ) = f ′ ′ ( t k ) , k = 0 , n ― } , (11) W 2 , f , D 4 ( Δ n ) := { g ∈ W 2 , f 4 ( Δ n ) : g ( t a ) = f ( a ) , g ( b ) = f ( b ) } . (9) W 2 4 Δ n := g : [ a , b ] → R ,  abs.cont.on  I k , k = 1 , n ¯  and  g ( 4 ) ∈ L 2 [ a , b ] , (10) W 2 , f 4 Δ n := g ∈ W 2 4 Δ n : g ′ ′ t k = f ′ ′ t k , k = 0 , n ¯ , (11) W 2 , f , D 4 Δ n := g ∈ W 2 , f 4 Δ n : g t a = f ( a ) , g ( b ) = f ( b ) . {:[(9)W_(2)^(4)(Delta_(n)):={[g:[a","b]rarrR","" abs.cont.on "I_(k)","k= bar(1,n)],[" and "g^((4))inL_(2)[a","b]]}","],[(10)W_(2,f)^(4)(Delta_(n)):={g inW_(2)^(4)(Delta_(n)):g^('')(t_(k))=f^('')(t_(k)),k= bar(0,n)}","],[(11)W_(2,f,D)^(4)(Delta_(n)):={g inW_(2,f)^(4)(Delta_(n)):g(t_(a))=f(a),g(b)=f(b)}.]:}\begin{gather*} W_{2}^{4}\left(\Delta_{n}\right):=\left\{\begin{array}{c} g:[a, b] \rightarrow \mathbb{R}, \text { abs.cont.on } I_{k}, k=\overline{1, n} \\ \text { and } g^{(4)} \in L_{2}[a, b] \end{array}\right\}, \tag{9}\\ 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), k=\overline{0, n}\right\}, \tag{10}\\ W_{2, f, D}^{4}\left(\Delta_{n}\right):=\left\{g \in W_{2, f}^{4}\left(\Delta_{n}\right): g\left(t_{a}\right)=f(a), g(b)=f(b)\right\} . \tag{11} \end{gather*}(9)W24(Δn):={g:[a,b]→R, abs.cont.on Ik,k=1,n― and g(4)∈L2[a,b]},(10)W2,f4(Δn):={g∈W24(Δn):g′′(tk)=f′′(tk),k=0,n―},(11)W2,f,D4(Δn):={g∈W2,f4(Δn):g(ta)=f(a),g(b)=f(b)}.
THEOREM 4. If s ∈ S 5 ( Δ n ) ∩ W 2 , f , D 4 ( Δ n ) s ∈ S 5 Δ n ∩ W 2 , f , D 4 Δ n s inS_(5)(Delta_(n))nnW_(2,f,D)^(4)(Delta_(n))s \in \mathscr{S}_{5}\left(\Delta_{n}\right) \cap W_{2, f, D}^{4}\left(\Delta_{n}\right)s∈S5(Δn)∩W2,f,D4(Δn) and f ∈ W 2 4 ( Δ n ) f ∈ W 2 4 Δ n f inW_(2)^(4)(Delta_(n))f \in W_{2}^{4}\left(\Delta_{n}\right)f∈W24(Δn), then
a) ‖ s f ( 4 ) ‖ 2 ≤ ‖ g ( 4 ) ‖ 2 for all g ∈ W 2 , f , D 4 ( Δ n ) ;  a)  s f ( 4 ) 2 ≤ g ( 4 ) 2  for all  g ∈ W 2 , f , D 4 Δ n ;  " a) "||s_(f)^((4))||_(2) <= ||g^((4))||_(2)" for all "g inW_(2,f,D)^(4)(Delta_(n))"; "\text { a) }\left\|s_{f}^{(4)}\right\|_{2} \leq\left\|g^{(4)}\right\|_{2} \text { for all } g \in W_{2, f, D}^{4}\left(\Delta_{n}\right) \text {; } a) ‖sf(4)‖2≤‖g(4)‖2 for all g∈W2,f,D4(Δn); 
b) ‖ s f ( 4 ) − f ( 4 ) ‖ 2 ≤ ‖ s ( 4 ) − f ( 4 ) ‖ 2 s f ( 4 ) − f ( 4 ) 2 ≤ s ( 4 ) − f ( 4 ) 2 ||s_(f)^((4))-f^((4))||_(2) <= ||s^((4))-f^((4))||_(2)\left\|s_{f}^{(4)}-f^{(4)}\right\|_{2} \leq\left\|s^{(4)}-f^{(4)}\right\|_{2}‖sf(4)−f(4)‖2≤‖s(4)−f(4)‖2 for all s ∈ S S ( Δ n ) s ∈ S S Δ n s inS_(S)(Delta_(n))s \in \mathscr{S}_{S}\left(\Delta_{n}\right)s∈SS(Δn).
Proof. a) We have
0 ≤ ‖ g ( 4 ) − s ( 4 ) ‖ 2 2 = ∫ a b [ g ( 4 ) ( t ) − s ( 4 ) ( t ) ] 2 d t = = ∫ a b [ g ( 4 ) ( t ) ] 2 d t − ∫ a b [ s ( 4 ) ( t ) ] 2 d t − 2 ∫ a b s ( 4 ) ( t ) [ g ( 4 ) ( t ) − s ( 4 ) ( t ) ] d t 0 ≤ g ( 4 ) − s ( 4 ) 2 2 = ∫ a b   g ( 4 ) ( t ) − s ( 4 ) ( t ) 2 d t = = ∫ a b   g ( 4 ) ( t ) 2 d t − ∫ a b   s ( 4 ) ( t ) 2 d t − 2 ∫ a b   s ( 4 ) ( t ) g ( 4 ) ( t ) − s ( 4 ) ( t ) d t {:[0 <= ||g^((4))-s^((4))||_(2)^(2)=int_(a)^(b)[g^((4))(t)-s^((4))(t)]^(2)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]^{2} \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}0≤‖g(4)−s(4)‖22=∫ab[g(4)(t)−s(4)(t)]2 dt==∫ab[g(4)(t)]2 dt−∫ab[s(4)(t)]2 dt−2∫abs(4)(t)[g(4)(t)−s(4)(t)]dt
But
∫ a b s ( 4 ) ( t ) [ g ( 4 ) ( t ) − s ( 4 ) ( t ) ] d t = = s ( 4 ) ( t ) [ g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ] | a b − ∫ a b s ( 5 ) ( t ) [ g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ] d t = = − ∑ k = 1 n ∫ t k − 1 t k s ( 5 ) ( t ) [ g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ] ⋅ d t = = − ∑ k = 1 n c k ∫ t k − 1 t k [ g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ] ⋅ d t = ( − ∑ k = 1 n c k ) ⋅ 0 = 0 ∫ a b   s ( 4 ) ( t ) g ( 4 ) ( t ) − s ( 4 ) ( t ) d t = = s ( 4 ) ( t ) g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) a b − ∫ a b   s ( 5 ) ( t ) g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) d t = = − ∑ k = 1 n   ∫ t k − 1 t k   s ( 5 ) ( t ) g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ⋅ d t = = − ∑ k = 1 n   c k ∫ t k − 1 t k   g ′ ′ ′ ( t ) − s ′ ′ ′ ( t ) ⋅ d t = − ∑ k = 1 n   c k ⋅ 0 = 0 {:[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=],[=-sum_(k=1)^(n)int_(t_(k-1))^(t_(k))s^((5))(t)[g^(''')(t)-s^(''')(t)]*dt=],[=-sum_(k=1)^(n)c_(k)int_(t_(k-1))^(t_(k))[g^(''')(t)-s^(''')(t)]*dt=(-sum_(k=1)^(n)c_(k))*0=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= \\ =-\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] \cdot \mathrm{d} t= \\ =-\sum_{k=1}^{n} c_{k} \int_{t_{k-1}}^{t_{k}}\left[g^{\prime \prime \prime}(t)-s^{\prime \prime \prime}(t)\right] \cdot \mathrm{d} t=\left(-\sum_{k=1}^{n} c_{k}\right) \cdot 0=0 \end{gathered}∫abs(4)(t)[g(4)(t)−s(4)(t)]dt==s(4)(t)[g′′′(t)−s′′′(t)]|ab−∫abs(5)(t)[g′′′(t)−s′′′(t)]dt==−∑k=1n∫tk−1tks(5)(t)[g′′′(t)−s′′′(t)]⋅dt==−∑k=1nck∫tk−1tk[g′′′(t)−s′′′(t)]⋅dt=(−∑k=1nck)⋅0=0
where c k = s ( 5 ) | I k , k = 1 , 2 , … , n c k = s ( 5 ) I k , k = 1 , 2 , … , n c_(k)=s^((5))|_(I_(k)),k=1,2,dots,nc_{k}=\left.s^{(5)}\right|_{I_{k}}, k=1,2, \ldots, nck=s(5)|Ik,k=1,2,…,n.
Therefore
showing that (12) holds.
0 ≤ ‖ g ( 4 ) ‖ 2 2 − ‖ s ( 4 ) ‖ 2 2 0 ≤ g ( 4 ) 2 2 − s ( 4 ) 2 2 0 <= ||g^((4))||_(2)^(2)-||s^((4))||_(2)^(2)0 \leq\left\|g^{(4)}\right\|_{2}^{2}-\left\|s^{(4)}\right\|_{2}^{2}0≤‖g(4)‖22−‖s(4)‖22
b) Taking into account the identity
(14) ‖ s ( 4 ) − f ( 4 ) ‖ 2 2 = ∫ a b [ s ( 4 ) ( t ) − s f ( 4 ) ( t ) ] 2 d t + ∫ a b [ s f ( 4 ) ( t ) − f ( 4 ) ( t ) ] 2 d t + + 2 ⋅ ∫ a b [ s ( 4 ) ( t ) − s f ( 4 ) ( t ) ] 2 ⋅ [ s f ( 4 ) ( t ) − f ( 4 ) ( t ) ] 2 d t (14) s ( 4 ) − f ( 4 ) 2 2 = ∫ a b   s ( 4 ) ( t ) − s f ( 4 ) ( t ) 2 d t + ∫ a b   s f ( 4 ) ( t ) − f ( 4 ) ( t ) 2 d t + + 2 ⋅ ∫ a b   s ( 4 ) ( t ) − s f ( 4 ) ( t ) 2 ⋅ s f ( 4 ) ( t ) − f ( 4 ) ( t ) 2 d t {:[(14)||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+],[+2*int_(a)^(b)[s^((4))(t)-s_(f)^((4))(t)]^(2)*[s_(f)^((4))(t)-f^((4))(t)]^(2)dt]:}\begin{align*} \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+ \tag{14}\\ +2 & \cdot \int_{a}^{b}\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right]^{2} \cdot\left[s_{f}^{(4)}(t)-f^{(4)}(t)\right]^{2} \mathrm{~d} t \end{align*}(14)‖s(4)−f(4)‖22=∫ab[s(4)(t)−sf(4)(t)]2 dt+∫ab[sf(4)(t)−f(4)(t)]2 dt++2⋅∫ab[s(4)(t)−sf(4)(t)]2⋅[sf(4)(t)−f(4)(t)]2 dt
the inequality (13) will be a consequence of the equality
T = ∫ a b [ s ( 4 ) ( t ) − s f ( 4 ) ( t ) ] [ s f ( 4 ) ( t ) − f ( 4 ) ( t ) ] d t = 0 T = ∫ a b   s ( 4 ) ( t ) − s f ( 4 ) ( t ) s f ( 4 ) ( t ) − f ( 4 ) ( t ) d t = 0 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]\left[s_{f}^{(4)}(t)-f^{(4)}(t)\right] \mathrm{d} t=0T=∫ab[s(4)(t)−sf(4)(t)][sf(4)(t)−f(4)(t)]dt=0
Integrating by parts, we get
T = [ s ( 4 ) ( t ) − s f ( 4 ) ( t ) ] [ s f ′ ′ ′ ( t ) − f ′ ′ ′ ( t ) ] | a b − ∫ a b [ s ( 5 ) ( t ) − s f ( 5 ) ( t ) ] ⋅ [ s f ′ ′ ′ ( t ) − f ′ ′ ′ ( t ) ] d t = = − ∑ k = 1 n C k ( s ) ( [ s f ′ ′ ( t k ) − f ′ ′ ( t k ) ] − [ s f ′ ′ ( t k − 1 ) − f ′ ′ ( t k − 1 ) ] ) = 0 T = s ( 4 ) ( t ) − s f ( 4 ) ( t ) s f ′ ′ ′ ( t ) − f ′ ′ ′ ( t ) a b − ∫ a b   s ( 5 ) ( t ) − s f ( 5 ) ( t ) ⋅ s f ′ ′ ′ ( t ) − f ′ ′ ′ ( t ) d t = = − ∑ k = 1 n   C k ( s ) s f ′ ′ t k − f ′ ′ t k − s f ′ ′ t k − 1 − f ′ ′ t k − 1 = 0 {:[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_(f)^('')(t_(k))-f^('')(t_(k))]-[s_(f)^('')(t_(k-1))-f^('')(t_(k-1))])=0]:}\begin{gathered} T=\left.\left[s^{(4)}(t)-s_{f}^{(4)}(t)\right]\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(\left[s_{f}^{\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]\right)=0 \end{gathered}T=[s(4)(t)−sf(4)(t)][sf′′′(t)−f′′′(t)]|ab−∫ab[s(5)(t)−sf(5)(t)]⋅[sf′′′(t)−f′′′(t)]dt==−∑k=1nCk(s)([sf′′(tk)−f′′(tk)]−[sf′′(tk−1)−f′′(tk−1)])=0
Therefore
(15)
‖ s ( 4 ) − f ( 4 ) ‖ 2 2 = ‖ s ( 4 ) − s f ( 4 ) ‖ 2 2 + ‖ s f ( 4 ) − f ( 4 ) ‖ 2 2 s ( 4 ) − f ( 4 ) 2 2 = s ( 4 ) − s f ( 4 ) 2 2 + s f ( 4 ) − f ( 4 ) 2 2 ||s^((4))-f^((4))||_(2)^(2)=||s^((4))-s_(f)^((4))||_(2)^(2)+||s_(f)^((4))-f^((4))||_(2)^(2)\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}‖s(4)−f(4)‖22=‖s(4)−sf(4)‖22+‖sf(4)−f(4)‖22
implying
‖ s f ( 4 ) − f ( 4 ) ‖ 2 ≤ ‖ s ( 4 ) − f ( 4 ) ‖ 2 s f ( 4 ) − f ( 4 ) 2 ≤ s ( 4 ) − f ( 4 ) 2 ||s_(f)^((4))-f^((4))||_(2) <= ||s^((4))-f^((4))||_(2)\left\|s_{f}^{(4)}-f^{(4)}\right\|_{2} \leq\left\|s^{(4)}-f^{(4)}\right\|_{2}‖sf(4)−f(4)‖2≤‖s(4)−f(4)‖2
COROLLARY 5. If f ∈ W 2 4 ( Δ n ) f ∈ W 2 4 Δ n f inW_(2)^(4)(Delta_(n))f \in W_{2}^{4}\left(\Delta_{n}\right)f∈W24(Δn) and s f ∈ S 5 ( Δ n ) s f ∈ S 5 Δ n s_(f)inS_(5)(Delta_(n))s_{f} \in \mathscr{S}_{5}\left(\Delta_{n}\right)sf∈S5(Δn) is given by (8), then the following relations hold:
(16) ‖ f ( 4 ) ‖ 2 2 = ‖ s f ( 4 ) ‖ 2 2 + ‖ f ( 4 ) − s f ( 4 ) ‖ 2 2 (17) ‖ s f ( 4 ) ‖ 2 ≤ ‖ f ( 4 ) ‖ 2 (18) ‖ f ( 4 ) − s f ( 4 ) ‖ 2 ≤ ‖ f ( 4 ) ‖ 2 (16) f ( 4 ) 2 2 = s f ( 4 ) 2 2 + f ( 4 ) − s f ( 4 ) 2 2 (17) s f ( 4 ) 2 ≤ f ( 4 ) 2 (18) f ( 4 ) − s f ( 4 ) 2 ≤ f ( 4 ) 2 {:[(16)||f^((4))||_(2)^(2)=||s_(f)^((4))||_(2)^(2)+||f^((4))-s_(f)^((4))||_(2)^(2)],[(17)||s_(f)^((4))||_(2) <= ||f^((4))||_(2)],[(18)||f^((4))-s_(f)^((4))||_(2) <= ||f^((4))||_(2)]:}\begin{gather*} \left\|f^{(4)}\right\|_{2}^{2}=\left\|s_{f}^{(4)}\right\|_{2}^{2}+\left\|f^{(4)}-s_{f}^{(4)}\right\|_{2}^{2} \tag{16}\\ \left\|s_{f}^{(4)}\right\|_{2} \leq\left\|f^{(4)}\right\|_{2} \tag{17}\\ \left\|f^{(4)}-s_{f}^{(4)}\right\|_{2} \leq\left\|f^{(4)}\right\|_{2} \tag{18} \end{gather*}(16)‖f(4)‖22=‖sf(4)‖22+‖f(4)−sf(4)‖22(17)‖sf(4)‖2≤‖f(4)‖2(18)‖f(4)−sf(4)‖2≤‖f(4)‖2
1201 Proof. Since (15) holds for every s ∈ S 5 ( Δ n ) s ∈ S 5 Δ n s inS_(5)(Delta_(n))s \in \mathscr{S}_{5}\left(\Delta_{n}\right)s∈S5(Δn), one obtains (16) by taxing s ≡ 0 s ≡ 0 s-=0s \equiv 0s≡0 in (15).
The inequalities (17) and (18) are immediate consequences of the equality
Remark 2. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ The property expressed by the inequality (12) is called the minimum norm property.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The property exprimed by the inequality (13) is called the best approximation property.

APPLICATION

Consider the singularly perturbed bilocal problem
(D)
ε y ′ ′ = f ( t , y , y ′ ) , t ∈ [ a , b ] , ε > 0 y ( a ) = α , y ( b ) = β . ε y ′ ′ = f t , y , y ′ , t ∈ [ a , b ] , ε > 0 y ( a ) = α , y ( b ) = β . {:[epsiy^('')=f(t,y,y^('))","t in[a","b]","epsi > 0],[y(a)=alpha","y(b)=beta.]:}\begin{gathered} \varepsilon y^{\prime \prime}=f\left(t, y, y^{\prime}\right), t \in[a, b], \varepsilon>0 \\ y(a)=\alpha, y(b)=\beta . \end{gathered}εy′′=f(t,y,y′),t∈[a,b],ε>0y(a)=α,y(b)=β.
One supposes that the problem (D) has a unique solution.
THEOREM 6. If the exact solution y y yyy of the problem ( D ) ( D ) (D)(D)(D) belongs to W 2 4 ( Δ n ) W 2 4 Δ n W_(2)^(4)(Delta_(n))W_{2}^{4}\left(\Delta_{n}\right)W24(Δn) and s y ∈ S 5 ( Δ n ) s y ∈ S 5 Δ n s_(y)inS_(5)(Delta_(n))s_{y} \in \mathscr{S}_{5}\left(\Delta_{n}\right)sy∈S5(Δn) is the function given by ( 8 ), then
(19) ‖ y ( k ) − s y ( k ) ‖ ∞ ≤ 2 ⋅ ( b − a ) 2 − k ⋅ ‖ Δ n ‖ 3 / 2 ⋅ ‖ y ( 4 ) ‖ 2 , k = 0 , 1 , 2 , (19) y ( k ) − s y ( k ) ∞ ≤ 2 ⋅ ( b − a ) 2 − k ⋅ Δ n 3 / 2 ⋅ y ( 4 ) 2 , k = 0 , 1 , 2 , {:(19)||y^((k))-s_(y)^((k))||_(oo) <= sqrt2*(b-a)^(2-k)*||Delta_(n)||^(3//2)*||y^((4))||_(2)","quad k=0","1","2",":}\begin{equation*} \left\|y^{(k)}-s_{y}^{(k)}\right\|_{\infty} \leq \sqrt{2} \cdot(b-a)^{2-k} \cdot\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2}, \quad k=0,1,2, \tag{19} \end{equation*}(19)‖y(k)−sy(k)‖∞≤2⋅(b−a)2−k⋅‖Δn‖3/2⋅‖y(4)‖2,k=0,1,2,
where
‖ Δ n ‖ = max { t i + 1 − t i : i = 0 , n − 1 ― } . Δ n = max t i + 1 − t i : i = 0 , n − 1 ¯ . ||Delta_(n)||=max{t_(i+1)-t_(i):i= bar(0,n-1)}.\left\|\Delta_{n}\right\|=\max \left\{t_{i+1}-t_{i}: i=\overline{0, n-1}\right\} .‖Δn‖=max{ti+1−ti:i=0,n−1―}.
Proof. Since y ′ ′ ( t i ) − s y ′ ′ ( t i ) = 0 , i = 0 , n ― y ′ ′ t i − s y ′ ′ t i = 0 , i = 0 , n ¯ y^('')(t_(i))-s_(y)^('')(t_(i))=0,i= bar(0,n)y^{\prime \prime}\left(t_{i}\right)-s_{y}^{\prime \prime}\left(t_{i}\right)=0, i=\overline{0, n}y′′(ti)−sy′′(ti)=0,i=0,n―, by Rolle's theorem there exist t i ( 1 ) ∈ ( t i , t i + 1 ) , i = 0 , n − 1 ― t i ( 1 ) ∈ t i , t i + 1 , i = 0 , n − 1 ¯ 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}ti(1)∈(ti,ti+1),i=0,n−1― such that
y ′ ′ ′ ( t i ( 1 ) ) − s y ′ ′ ′ ( t i ( 1 ) ) = 0 , i = 0 , n − 1 ― . y ′ ′ ′ t i ( 1 ) − s y ′ ′ ′ t i ( 1 ) = 0 , i = 0 , n − 1 ¯ . y^(''')(t_(i)^((1)))-s_(y)^(''')(t_(i)^((1)))=0,quad i= bar(0,n-1).y^{\prime \prime \prime}\left(t_{i}^{(1)}\right)-s_{y}^{\prime \prime \prime}\left(t_{i}^{(1)}\right)=0, \quad i=\overline{0, n-1} .y′′′(ti(1))−sy′′′(ti(1))=0,i=0,n−1―.
Applying again Rolle's theorem, it follows the existence of the points t i ( 2 ) ∈ ( t i ( 1 ) , t i + 1 ( 1 ) ) , i = 0 , n − 2 ― t i ( 2 ) ∈ t i ( 1 ) , t i + 1 ( 1 ) , i = 0 , n − 2 ¯ t_(i)^((2))in(t_(i)^((1)),t_(i+1)^((1))),i= bar(0,n-2)t_{i}^{(2)} \in\left(t_{i}^{(1)}, t_{i+1}^{(1)}\right), i=\overline{0, n-2}ti(2)∈(ti(1),ti+1(1)),i=0,n−2― such that
y ( 4 ) ( t i ( 2 ) ) − s y ( 4 ) ( t i ( 2 ) ) = 0 , i = 0 , n − 2 ― y ( 4 ) t i ( 2 ) − s y ( 4 ) t i ( 2 ) = 0 , i = 0 , n − 2 ¯ y^((4))(t_(i)^((2)))-s_(y)^((4))(t_(i)^((2)))=0,quad i= bar(0,n-2)y^{(4)}\left(t_{i}^{(2)}\right)-s_{y}^{(4)}\left(t_{i}^{(2)}\right)=0, \quad i=\overline{0, n-2}y(4)(ti(2))−sy(4)(ti(2))=0,i=0,n−2―
obviously that
| t i + 1 ( 1 ) − t i ( 1 ) | ≤ 2 ‖ Δ n ‖ t i + 1 ( 1 ) − t i ( 1 ) ≤ 2 Δ n |t_(i+1)^((1))-t_(i)^((1))| <= 2||Delta_(n)||\left|t_{i+1}^{(1)}-t_{i}^{(1)}\right| \leq 2\left\|\Delta_{n}\right\||ti+1(1)−ti(1)|≤2‖Δn‖
and
| t i + 1 ( 2 ) − t i ( 2 ) | ≤ 3 ‖ Δ n ‖ . t i + 1 ( 2 ) − t i ( 2 ) ≤ 3 Δ n . |t_(i+1)^((2))-t_(i)^((2))| <= 3||Delta_(n)||.\left|t_{i+1}^{(2)}-t_{i}^{(2)}\right| \leq 3\left\|\Delta_{n}\right\| .|ti+1(2)−ti(2)|≤3‖Δn‖.
Since for every t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b] there exists i 0 ∈ { 0 , 1 , … , n − 1 } i 0 ∈ { 0 , 1 , … , n − 1 } i_(0)in{0,1,dots,n-1}i_{0} \in\{0,1, \ldots, n-1\}i0∈{0,1,…,n−1} such that | t − t i 0 ( 1 ) | ≤ 2 ‖ Δ n ‖ t − t i 0 ( 1 ) ≤ 2 Δ n |t-t_(i_(0))^((1))| <= 2||Delta_(n)||\left|t-t_{i_{0}}^{(1)}\right| \leq 2\left\|\Delta_{n}\right\||t−ti0(1)|≤2‖Δn‖, one obtains:
| y ′ ′ ′ ( t ) − s y ′ ′ ′ ( t ) | = | ∫ t i 0 ( 1 ) t ( y ( 4 ) ( u ) − s y ( 4 ) ( u ) ) d u | ≤ y ′ ′ ′ ( t ) − s y ′ ′ ′ ( t ) = ∫ t i 0 ( 1 ) t   y ( 4 ) ( u ) − s y ( 4 ) ( u ) d u ≤ |y^(''')(t)-s_(y)^(''')(t)|=|int_(t_(i0)^((1)))^(t)(y^((4))(u)-s_(y)^((4))(u))du| <=\left|y^{\prime \prime \prime}(t)-s_{y}^{\prime \prime \prime}(t)\right|=\left|\int_{t_{i 0}^{(1)}}^{t}\left(y^{(4)}(u)-s_{y}^{(4)}(u)\right) \mathrm{d} u\right| \leq|y′′′(t)−sy′′′(t)|=|∫ti0(1)t(y(4)(u)−sy(4)(u))du|≤
≤ | ∫ t i ( 1 ) t d u | 1 / 2 ⋅ | ⋅ ∫ t i ( 1 ) t [ y ( 4 ) ( u ) − s y ( 4 ) ( u ) ] 2 d u | 1 / 2 ≤ ≤ 2 ‖ Δ n ‖ ⋅ | ∫ t i ( 1 ) t [ y ( 4 ) ( u ) − s y ( 4 ) ( u ) ] d u | 1 / 2 = = 2 ‖ Δ n ‖ ⋅ ‖ y ( 4 ) − s y ( 4 ) ‖ 2 ≤ 2 ⋅ ‖ Δ n ‖ 1 / 2 ⋅ ‖ y ( 4 ) ‖ 2 ≤ ∫ t i ( 1 ) t   d u 1 / 2 ⋅ ⋅ ∫ t i ( 1 ) t   y ( 4 ) ( u ) − s y ( 4 ) ( u ) 2 d u 1 / 2 ≤ ≤ 2 Δ n ⋅ ∫ t i ( 1 ) t   y ( 4 ) ( u ) − s y ( 4 ) ( u ) d u 1 / 2 = = 2 Δ n ⋅ y ( 4 ) − s y ( 4 ) 2 ≤ 2 ⋅ Δ n 1 / 2 ⋅ y ( 4 ) 2 {:[ <= |int_(t_(i)^((1)))^(t)(d)u|^(1//2)*|*int_(t_(i)^((1)))^(t)[y^((4))(u)-s_(y)^((4))(u)]^(2)(d)u|^(1//2) <= ],[ <= sqrt(2||Delta_(n)||)*|int_(t_(i)^((1)))^(t)[y^((4))(u)-s_(y)^((4))(u)]du|^(1//2)=],[=sqrt(2||Delta_(n)||)*||y^((4))-s_(y)^((4))||_(2) <= sqrt2*||Delta_(n)||^(1//2)*||y^((4))||_(2)]:}\begin{aligned} & \leq\left|\int_{t_{i}^{(1)}}^{t} \mathrm{~d} u\right|^{1 / 2} \cdot\left|\cdot \int_{t_{i}^{(1)}}^{t}\left[y^{(4)}(u)-s_{y}^{(4)}(u)\right]^{2} \mathrm{~d} u\right|^{1 / 2} \leq \\ & \leq \sqrt{2\left\|\Delta_{n}\right\|} \cdot\left|\int_{t_{i}^{(1)}}^{t}\left[y^{(4)}(u)-s_{y}^{(4)}(u)\right] \mathrm{d} u\right|^{1 / 2}= \\ & =\sqrt{2\left\|\Delta_{n}\right\|} \cdot\left\|y^{(4)}-s_{y}^{(4)}\right\|_{2} \leq \sqrt{2} \cdot\left\|\Delta_{n}\right\|^{1 / 2} \cdot\left\|y^{(4)}\right\|_{2} \end{aligned}≤|∫ti(1)t du|1/2⋅|⋅∫ti(1)t[y(4)(u)−sy(4)(u)]2 du|1/2≤≤2‖Δn‖⋅|∫ti(1)t[y(4)(u)−sy(4)(u)]du|1/2==2‖Δn‖⋅‖y(4)−sy(4)‖2≤2⋅‖Δn‖1/2⋅‖y(4)‖2
(the last inequality follows from Corollary 5, (18)).
It follows that
‖ y ′ ′ ′ − s y ′ ′ ′ ‖ ∞ ≤ 2 ⋅ ‖ Δ n ‖ 1 / 2 ⋅ ‖ y ( 4 ) ‖ 2 . y ′ ′ ′ − s y ′ ′ ′ ∞ ≤ 2 ⋅ Δ n 1 / 2 ⋅ y ( 4 ) 2 . ||y^(''')-s_(y)^(''')||_(oo) <= sqrt2*||Delta_(n)||^(1//2)*||y^((4))||_(2).\left\|y^{\prime \prime \prime}-s_{y}^{\prime \prime \prime}\right\|_{\infty} \leq \sqrt{2} \cdot\left\|\Delta_{n}\right\|^{1 / 2} \cdot\left\|y^{(4)}\right\|_{2} .‖y′′′−sy′′′‖∞≤2⋅‖Δn‖1/2⋅‖y(4)‖2.
Similarly, for every t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b], there exists j 0 ∈ { 0 , 1 , … , n − 2 } j 0 ∈ { 0 , 1 , … , n − 2 } j_(0)in{0,1,dots,n-2}j_{0} \in\{0,1, \ldots, n-2\}j0∈{0,1,…,n−2} such that | t − t j 0 | ≤ ‖ Δ n ‖ t − t j 0 ≤ Δ n |t-t_(j_(0))| <= ||Delta_(n)||\left|t-t_{j_{0}}\right| \leq\left\|\Delta_{n}\right\||t−tj0|≤‖Δn‖, implying
| | y ′ ′ ( t ) − s y ′ ′ ( t ) | = | ∫ t j 0 t [ y ′ ′ ′ ( u ) − s y ′ ′ ′ ( u ) ] d u | ≤ y ′ ′ ( t ) − s y ′ ′ ( t ) = ∫ t j 0 t   y ′ ′ ′ ( u ) − s y ′ ′ ′ ( u ) d u ≤ ||y^('')(t)-s_(y)^('')(t)|=|int_(t_(j_(0)))^(t)[y^(''')(u)-s_(y)^(''')(u)]du| <= :}\left|\left|y^{\prime \prime}(t)-s_{y}^{\prime \prime}(t)\right|=\left|\int_{t_{j_{0}}}^{t}\left[y^{\prime \prime \prime}(u)-s_{y}^{\prime \prime \prime}(u)\right] \mathrm{d} u\right| \leq\right.||y′′(t)−sy′′(t)|=|∫tj0t[y′′′(u)−sy′′′(u)]du|≤
Therefore
‖ y ′ ′ − s y ′ ′ ‖ ∞ ≤ 2 ‖ Δ n ‖ 3 / 2 ⋅ ‖ y ( 4 ) ‖ 2 , y ′ ′ − s y ′ ′ ∞ ≤ 2 Δ n 3 / 2 ⋅ y ( 4 ) 2 , ||y^('')-s_(y)^('')||_(oo) <= sqrt2||Delta_(n)||^(3//2)*||y^((4))||_(2),\left\|y^{\prime \prime}-s_{y}^{\prime \prime}\right\|_{\infty} \leq \sqrt{2}\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2},‖y′′−sy′′‖∞≤2‖Δn‖3/2⋅‖y(4)‖2,
showing that (19) holds for k = 2 k = 2 k=2k=2k=2.
Taking into account the equalities y ( a ) − s y ( a ) = 0 y ( a ) − s y ( a ) = 0 y(a)-s_(y)(a)=0y(a)-s_{y}(a)=0y(a)−sy(a)=0 and y ( b ) − s y ( b ) = 0 y ( b ) − s y ( b ) = 0 y(b)-s_(y)(b)=0y(b)-s_{y}(b)=0y(b)−sy(b)=0, it follows (by Rolle's theorem) the existence of a point c ∈ ( a , b ) c ∈ ( a , b ) c in(a,b)c \in(a, b)c∈(a,b) such that y ′ ( c ) − s y ′ ( c ) = 0 y ′ ( c ) − s y ′ ( c ) = 0 y^(')(c)-s_(y)^(')(c)=0y^{\prime}(c)-s_{y}^{\prime}(c)=0y′(c)−sy′(c)=0. Then, for every t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b] one has
| y ′ ( t ) − s y ′ ( t ) | = | ∫ c 1 [ y ′ ′ ( u ) − s y ′ ′ ( u ) ] d u | ≤ ≤ ( b − a ) ‖ y ′ ′ − s y ′ ′ ‖ ∞ ≤ 2 ( b − a ) ‖ Δ n ‖ 3 / 2 ‖ y ( 4 ) ‖ 2 y ′ ( t ) − s y ′ ( t ) = ∫ c 1   y ′ ′ ( u ) − s y ′ ′ ( u ) d u ≤ ≤ ( b − a ) y ′ ′ − s y ′ ′ ∞ ≤ 2 ( b − a ) Δ n 3 / 2 y ( 4 ) 2 {:[|y^(')(t)-s_(y)^(')(t)|=|int_(c)^(1)[y^('')(u)-s_(y)^('')(u)]du| <= ],[ <= (b-a)||y^('')-s_(y)^('')||_(oo) <= sqrt2(b-a)||Delta_(n)||^(3//2)||y^((4))||_(2)]:}\begin{gathered} \left|y^{\prime}(t)-s_{y}^{\prime}(t)\right|=\left|\int_{c}^{1}\left[y^{\prime \prime}(u)-s_{y}^{\prime \prime}(u)\right] \mathrm{d} u\right| \leq \\ \leq(b-a)\left\|y^{\prime \prime}-s_{y}^{\prime \prime}\right\|_{\infty} \leq \sqrt{2}(b-a)\left\|\Delta_{n}\right\|^{3 / 2}\left\|y^{(4)}\right\|_{2} \end{gathered}|y′(t)−sy′(t)|=|∫c1[y′′(u)−sy′′(u)]du|≤≤(b−a)‖y′′−sy′′‖∞≤2(b−a)‖Δn‖3/2‖y(4)‖2
showing that
‖ y ′ − s y ′ ‖ ∞ ≤ 2 ⋅ ( b − a ) ⋅ ‖ Δ n ‖ 3 / 2 ⋅ ‖ y ( 4 ) ‖ 2 y ′ − s y ′ ∞ ≤ 2 ⋅ ( b − a ) ⋅ Δ n 3 / 2 ⋅ y ( 4 ) 2 ||y^(')-s_(y)^(')||_(oo) <= sqrt2*(b-a)*||Delta_(n)||^(3//2)*||y^((4))||_(2)\left\|y^{\prime}-s_{y}^{\prime}\right\|_{\infty} \leq \sqrt{2} \cdot(b-a) \cdot\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2}‖y′−sy′‖∞≤2⋅(b−a)⋅‖Δn‖3/2⋅‖y(4)‖2
i.e., (19) holds for k = 1 k = 1 k=1k=1k=1, too.
Finally, for every t ∈ [ a , b ] t ∈ [ a , b ] t in[a,b]t \in[a, b]t∈[a,b] one can write
| y ( t ) − s y ( t ) | = | ∫ a t [ y ′ ( u ) − s y ′ ( u ) ] d u + y ( a ) − s y ( a ) | = y ( t ) − s y ( t ) = ∫ a t   y ′ ( u ) − s y ′ ( u ) d u + y ( a ) − s y ( a ) = |y(t)-s_(y)(t)|=|int_(a)^(t)[y^(')(u)-s_(y)^(')(u)]du+y(a)-s_(y)(a)|=\left|y(t)-s_{y}(t)\right|=\left|\int_{a}^{t}\left[y^{\prime}(u)-s_{y}^{\prime}(u)\right] \mathrm{d} u+y(a)-s_{y}(a)\right|=|y(t)−sy(t)|=|∫at[y′(u)−sy′(u)]du+y(a)−sy(a)|=
= | ∫ a 1 [ y ′ ( u ) − s y ′ ( u ) ] d u | ≤ ( b − a ) ‖ y ′ − s y ′ ‖ ∞ ≤ = ∫ a 1   y ′ ( u ) − s y ′ ( u ) d u ≤ ( b − a ) y ′ − s y ′ ∞ ≤ =|int_(a)^(1)[y^(')(u)-s_(y)^(')(u)]du| <= (b-a)||y^(')-s_(y)^(')||_(oo) <==\left|\int_{a}^{1}\left[y^{\prime}(u)-s_{y}^{\prime}(u)\right] \mathrm{d} u\right| \leq(b-a)\left\|y^{\prime}-s_{y}^{\prime}\right\|_{\infty} \leq=|∫a1[y′(u)−sy′(u)]du|≤(b−a)‖y′−sy′‖∞≤
≤ 2 ⋅ ( b − a ) ⋅ ‖ Δ n ‖ 3 / 2 ⋅ ‖ y ( 4 ) ‖ 2 , ≤ 2 ⋅ ( b − a ) ⋅ Δ n 3 / 2 ⋅ y ( 4 ) 2 , <= sqrt2*(b-a)*||Delta_(n)||^(3//2)*||y^((4))||_(2),\leq \sqrt{2} \cdot(b-a) \cdot\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2},≤2⋅(b−a)⋅‖Δn‖3/2⋅‖y(4)‖2,
implying
‖ y − s y ‖ ∞ ≤ 2 ( b − a ) ⋅ ‖ Δ n ‖ 3 / 2 ⋅ ‖ y ( 4 ) ‖ 2 . y − s y ∞ ≤ 2 ( b − a ) ⋅ Δ n 3 / 2 ⋅ y ( 4 ) 2 . ||y-s_(y)||_(oo) <= sqrt2(b-a)*||Delta_(n)||^(3//2)*||y^((4))||_(2).\left\|y-s_{y}\right\|_{\infty} \leq \sqrt{2}(b-a) \cdot\left\|\Delta_{n}\right\|^{3 / 2} \cdot\left\|y^{(4)}\right\|_{2} .‖y−sy‖∞≤2(b−a)⋅‖Δn‖3/2⋅‖y(4)‖2.
COROLLARY 7. Under the hypotheses of Theorem 6 we have
lim ‖ Δ Δ ‖ → 0 ‖ y ( k ) − s y ( k ) ‖ ∞ = 0 , k = 0 , 1 , 2 . lim Δ Δ → 0   y ( k ) − s y ( k ) ∞ = 0 , k = 0 , 1 , 2 . lim_(||Delta_(Delta)||rarr0)||y^((k))-s_(y)^((k))||_(oo)=0,quad k=0,1,2.\lim _{\left\|\Delta_{\Delta}\right\| \rightarrow 0}\left\|y^{(k)}-s_{y}^{(k)}\right\|_{\infty}=0, \quad k=0,1,2 .lim‖ΔΔ‖→0‖y(k)−sy(k)‖∞=0,k=0,1,2.
In the following, we shall approximate the exact solution y y yyy of the problem ( D ) ( D ) (D)(D)(D) by the function s y s y s_(y)s_{y}sy given by
(20) s y ( t ) = s 0 ( t ) ⋅ y ( a ) + s 1 ( t ) ⋅ y ( b ) + ∑ k = 0 n S k ( t ) ⋅ y ′ ′ ( t k ) , t ∈ [ a , b ] . (20) s y ( t ) = s 0 ( t ) ⋅ y ( a ) + s 1 ( t ) ⋅ y ( b ) + ∑ k = 0 n   S k ( t ) ⋅ y ′ ′ t k , t ∈ [ a , b ] . {:(20)s_(y)(t)=s_(0)(t)*y(a)+s_(1)(t)*y(b)+sum_(k=0)^(n)S_(k)(t)*y^('')(t_(k))","t in[a","b].:}\begin{equation*} s_{y}(t)=s_{0}(t) \cdot y(a)+s_{1}(t) \cdot y(b)+\sum_{k=0}^{n} S_{k}(t) \cdot y^{\prime \prime}\left(t_{k}\right), t \in[a, b] . \tag{20} \end{equation*}(20)sy(t)=s0(t)⋅y(a)+s1(t)⋅y(b)+∑k=0nSk(t)⋅y′′(tk),t∈[a,b].
This choice is motivated by the fact that the parameter ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 is multiplied by y ′ ′ y ′ ′ y^('')y^{\prime \prime}y′′ and s y s y s_(y)s_{y}sy is determined by the interpolation conditions on the second derivative of y y yyy on the knots of Δ n Δ n Delta_(n)\Delta_{n}Δn, which give ε y ′ ′ ( t i ) = ε s y ′ ′ ( t i ) , i = 0 , n ― ε y ′ ′ t i = ε s y ′ ′ t i , i = 0 , n ¯ epsiy^('')(t_(i))=epsis_(y)^('')(t_(i)),i= bar(0,n)\varepsilon y^{\prime \prime}\left(t_{i}\right)=\varepsilon s_{y}^{\prime \prime}\left(t_{i}\right), i=\overline{0, n}εy′′(ti)=εsy′′(ti),i=0,n―.
We shall use the following notations:
y ( t i ) = y i , y ′ ( t i ) = y i ′ , i = 0 , n ― s y ( t i ) = u i , s y ′ ( t i ) = u i ′ , i = 0 , n ― e ( t ) = y ( t ) − s y ( t ) , e ′ ( t ) = y ′ ( t ) − s y ′ ( t ) , t ∈ [ a , b ] e i = y i − u i , e i ′ = y i ′ − u i ′ , i = 0 , n ― y t i = y i , y ′ t i = y i ′ , i = 0 , n ¯ s y t i = u i , s y ′ t i = u i ′ , i = 0 , n ¯ e ( t ) = y ( t ) − s y ( t ) , e ′ ( t ) = y ′ ( t ) − s y ′ ( t ) , t ∈ [ a , b ] e i = y i − u i , e i ′ = y i ′ − u i ′ , i = 0 , n ¯ {:[y(t_(i))=y_(i)","quady^(')(t_(i))=y_(i)^(')","quad i= bar(0,n)],[s_(y)(t_(i))=u_(i)","quads_(y)^(')(t_(i))=u_(i)^(')","quad i= bar(0,n)],[e(t)=y(t)-sy(t)","quade^(')(t)=y^(')(t)-s_(y)^(')(t)","quad t in[a","b]],[e_(i)=y_(i)-u_(i)","quade_(i)^(')=y_(i)^(')-u_(i)^(')","quad i= bar(0,n)]:}\begin{gathered} y\left(t_{i}\right)=y_{i}, \quad y^{\prime}\left(t_{i}\right)=y_{i}^{\prime}, \quad i=\overline{0, n} \\ s_{y}\left(t_{i}\right)=u_{i}, \quad s_{y}^{\prime}\left(t_{i}\right)=u_{i}^{\prime}, \quad i=\overline{0, n} \\ e(t)=y(t)-s y(t), \quad e^{\prime}(t)=y^{\prime}(t)-s_{y}^{\prime}(t), \quad t \in[a, b] \\ e_{i}=y_{i}-u_{i}, \quad e_{i}^{\prime}=y_{i}^{\prime}-u_{i}^{\prime}, \quad i=\overline{0, n} \end{gathered}y(ti)=yi,y′(ti)=yi′,i=0,n―sy(ti)=ui,sy′(ti)=ui′,i=0,n―e(t)=y(t)−sy(t),e′(t)=y′(t)−sy′(t),t∈[a,b]ei=yi−ui,ei′=yi′−ui′,i=0,n―
Using the representation (20), one obtains
(21) u i = s 0 ( t i ) α + s 1 ( t i ) β + ∑ k = 0 n S k ( t i ) ⋅ f ( t k , y k , y k ′ ) , i = 0 , n ― u i ′ = s 0 ′ ( t i ) α + s 1 ′ ( t i ) β + ∑ k = 0 n S k ′ ( t i ) ⋅ f ( t k , y k , y k ′ ) , i = 0 , n ― . (21) u i = s 0 t i α + s 1 t i β + ∑ k = 0 n   S k t i ⋅ f t k , y k , y k ′ , i = 0 , n ¯ u i ′ = s 0 ′ t i α + s 1 ′ t i β + ∑ k = 0 n   S k ′ t i ⋅ f t k , y k , y k ′ , i = 0 , n ¯ . {:[(21)u_(i)=s_(0)(t_(i))alpha+s_(1)(t_(i))beta+sum_(k=0)^(n)S_(k)(t_(i))*f(t_(k),y_(k),y_(k)^('))","quad i= bar(0,n)],[u_(i)^(')=s_(0)^(')(t_(i))alpha+s_(1)^(')(t_(i))beta+sum_(k=0)^(n)S_(k)^(')(t_(i))*f(t_(k),y_(k),y_(k)^('))","quad i= bar(0,n).]:}\begin{align*} & u_{i}=s_{0}\left(t_{i}\right) \alpha+s_{1}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}\left(t_{i}\right) \cdot f\left(t_{k}, y_{k}, y_{k}^{\prime}\right), \quad i=\overline{0, n} \tag{21}\\ & u_{i}^{\prime}=s_{0}^{\prime}\left(t_{i}\right) \alpha+s_{1}^{\prime}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}^{\prime}\left(t_{i}\right) \cdot f\left(t_{k}, y_{k}, y_{k}^{\prime}\right), \quad i=\overline{0, n} . \end{align*}(21)ui=s0(ti)α+s1(ti)β+∑k=0nSk(ti)⋅f(tk,yk,yk′),i=0,n―ui′=s0′(ti)α+s1′(ti)β+∑k=0nSk′(ti)⋅f(tk,yk,yk′),i=0,n―.

9

Proposition 8. If the real-valued function f ( t , u , v ) f ( t , u , v ) f(t,u,v)f(t, u, v)f(t,u,v) defined on D ⊂ [ a , b ] × R 2 D ⊂ [ a , b ] × R 2 D sub[a,b]xxR^(2)D \subset[a, b] \times \mathbb{R}^{2}D⊂[a,b]×R2 has continuous partial derivatives ∂ f ∂ u , ∂ f ∂ v ∂ f ∂ u , ∂ f ∂ v (del f)/(del u),(del f)/(del v)\frac{\partial f}{\partial u}, \frac{\partial f}{\partial v}∂f∂u,∂f∂v, then the unknowns u i , u i ′ , i = 0 , n ― u i , u i ′ , i = 0 , n ¯ u_(i),u_(i)^('),i= bar(0,n)u_{i}, u_{i}^{\prime}, i=\overline{0, n}ui,ui′,i=0,n―, can be obtained from the system
u i = s 0 ( t i ) α + s 1 ( t i ) β + ∑ k = 0 n S k ( t i ) ⋅ f ( t k , y k , y k ′ ) , i = 0 , n ― u i ′ = s 0 ′ ( t i ) α + s 1 ′ ( t i ) β + ∑ k = 0 n S k ′ ( t i ) ⋅ f ( t k , y k , y k ′ ) , i = 0 , n ― u i = s 0 t i α + s 1 t i β + ∑ k = 0 n   S k t i ⋅ f t k , y k , y k ′ , i = 0 , n ¯ u i ′ = s 0 ′ t i α + s 1 ′ t i β + ∑ k = 0 n   S k ′ t i ⋅ f t k , y k , y k ′ , i = 0 , n ¯ {:[u_(i)=s_(0)(t_(i))alpha+s_(1)(t_(i))beta+sum_(k=0)^(n)S_(k)(t_(i))*f(t_(k),y_(k),y_(k)^('))","quad i= bar(0,n)],[u_(i)^(')=s_(0)^(')(t_(i))alpha+s_(1)^(')(t_(i))beta+sum_(k=0)^(n)S_(k)^(')(t_(i))*f(t_(k),y_(k),y_(k)^('))","quad i= bar(0,n)]:}\begin{aligned} & u_{i}=s_{0}\left(t_{i}\right) \alpha+s_{1}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}\left(t_{i}\right) \cdot f\left(t_{k}, y_{k}, y_{k}^{\prime}\right), \quad i=\overline{0, n} \\ & u_{i}^{\prime}=s_{0}^{\prime}\left(t_{i}\right) \alpha+s_{1}^{\prime}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}^{\prime}\left(t_{i}\right) \cdot f\left(t_{k}, y_{k}, y_{k}^{\prime}\right), \quad i=\overline{0, n} \end{aligned}ui=s0(ti)α+s1(ti)β+∑k=0nSk(ti)⋅f(tk,yk,yk′),i=0,n―ui′=s0′(ti)α+s1′(ti)β+∑k=0nSk′(ti)⋅f(tk,yk,yk′),i=0,n―
Proof. By the hypotheses of the proposition we have
f ( t k , y k , y k ′ ) = f ( t k u k + e k , u k ′ + e k ′ ) = f ( t k , u k , u k ′ ) + + ∂ f ( t k , ξ k , ξ k ′ ) ∂ u e k + ∂ f ( t k , ξ k , ξ k ′ ) ∂ u e k ′ , k = 0 , n ― f t k , y k , y k ′ = f t k u k + e k , u k ′ + e k ′ = f t k , u k , u k ′ + + ∂ f t k , ξ k , ξ k ′ ∂ u e k + ∂ f t k , ξ k , ξ k ′ ∂ u e k ′ , k = 0 , n ¯ {:[f(t_(k),y_(k),y_(k)^('))=f(t_(k)u_(k)+e_(k),u_(k)^(')+e_(k)^('))=f(t_(k),u_(k),u_(k)^('))+],[quad+(del f(t_(k),xi_(k),xi_(k)^(')))/(del u)e_(k)+(del f(t_(k),xi_(k),xi_(k)^(')))/(del u)e_(k)^(')","quad k= bar(0,n)]:}\begin{aligned} & f\left(t_{k}, y_{k}, y_{k}^{\prime}\right)=f\left(t_{k} u_{k}+e_{k}, u_{k}^{\prime}+e_{k}^{\prime}\right)=f\left(t_{k}, u_{k}, u_{k}^{\prime}\right)+ \\ & \quad+\frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial u} e_{k}+\frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial u} e_{k}^{\prime}, \quad k=\overline{0, n} \end{aligned}f(tk,yk,yk′)=f(tkuk+ek,uk′+ek′)=f(tk,uk,uk′)++∂f(tk,ξk,ξk′)∂uek+∂f(tk,ξk,ξk′)∂uek′,k=0,n―
where
min ( u k , u k + e k ) < ξ k < max ( u k , u k + e k ) , k = 0 , n ― min ( u k ′ , u k ′ + e k ) < ξ k ′ < max ( u k ′ , u k ′ + e k ′ ) , k = 0 , n ― min u k , u k + e k < ξ k < max u k , u k + e k ,      k = 0 , n ¯ min u k ′ , u k ′ + e k < ξ k ′ < max u k ′ , u k ′ + e k ′ ,      k = 0 , n ¯ {:[min(u_(k),u_(k)+e_(k)) < xi_(k) < max(u_(k),u_(k)+e_(k))",",k= bar(0,n)],[min(u_(k)^('),u_(k)^(')+e_(k)) < xi_(k)^(') < max(u_(k)^('),u_(k)^(')+e_(k)^('))",",k= bar(0,n)]:}\begin{array}{ll} \min \left(u_{k}, u_{k}+e_{k}\right)<\xi_{k}<\max \left(u_{k}, u_{k}+e_{k}\right), & k=\overline{0, n} \\ \min \left(u_{k}^{\prime}, u_{k}^{\prime}+e_{k}\right)<\xi_{k}^{\prime}<\max \left(u_{k}^{\prime}, u_{k}^{\prime}+e_{k}^{\prime}\right), & k=\overline{0, n} \end{array}min(uk,uk+ek)<ξk<max(uk,uk+ek),k=0,n―min(uk′,uk′+ek)<ξk′<max(uk′,uk′+ek′),k=0,n―
Replacing these in (21), we obtain the system:
u i = s 0 ( t i ) α + s 1 ( t i ) β + ∑ k = 0 n S k ( t i ) ⋅ f ( t k , u k , u k ′ ) + E i , i = 0 , n ― u i ′ = s 0 ′ ( t i ) α + s 1 ′ ( t i ) β + ∑ k = 0 n S k ′ ( t i ) ⋅ f ( t k , u k , u k ′ ) + E i ′ , i = 0 , n ― u i = s 0 t i α + s 1 t i β + ∑ k = 0 n   S k t i ⋅ f t k , u k , u k ′ + E i , i = 0 , n ¯ u i ′ = s 0 ′ t i α + s 1 ′ t i β + ∑ k = 0 n   S k ′ t i ⋅ f t k , u k , u k ′ + E i ′ , i = 0 , n ¯ {:[u_(i)=s_(0)(t_(i))alpha+s_(1)(t_(i))beta+sum_(k=0)^(n)S_(k)(t_(i))*f(t_(k),u_(k),u_(k)^('))+E_(i)","quad i= bar(0,n)],[u_(i)^(')=s_(0)^(')(t_(i))alpha+s_(1)^(')(t_(i))beta+sum_(k=0)^(n)S_(k)^(')(t_(i))*f(t_(k),u_(k),u_(k)^('))+E_(i)^(')","quad i= bar(0,n)]:}\begin{aligned} & u_{i}=s_{0}\left(t_{i}\right) \alpha+s_{1}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}\left(t_{i}\right) \cdot f\left(t_{k}, u_{k}, u_{k}^{\prime}\right)+E_{i}, \quad i=\overline{0, n} \\ & u_{i}^{\prime}=s_{0}^{\prime}\left(t_{i}\right) \alpha+s_{1}^{\prime}\left(t_{i}\right) \beta+\sum_{k=0}^{n} S_{k}^{\prime}\left(t_{i}\right) \cdot f\left(t_{k}, u_{k}, u_{k}^{\prime}\right)+E_{i}^{\prime}, \quad i=\overline{0, n} \end{aligned}ui=s0(ti)α+s1(ti)β+∑k=0nSk(ti)⋅f(tk,uk,uk′)+Ei,i=0,n―ui′=s0′(ti)α+s1′(ti)β+∑k=0nSk′(ti)⋅f(tk,uk,uk′)+Ei′,i=0,n―
having 2 n + 2 2 n + 2 2n+22 n+22n+2 equations and 2 n + 2 2 n + 2 2n+22 n+22n+2 unknowns u i , u i ′ , i = 0 , n ― u i , u i ′ , i = 0 , n ¯ u_(i),u_(i)^('),i= bar(0,n)u_{i}, u_{i}^{\prime}, i=\overline{0, n}ui,ui′,i=0,n―.
By Theorem 6, the quantities
E i = ∑ k = 0 n S k ( t i ) ⋅ ∂ f ( t k , ξ k , ξ k ′ ) ∂ u e k + ∑ k = 0 n S k ( t i ) ⋅ ∂ f ( t k , ξ k , ξ k ′ ) ∂ v e k ′ E i ′ = ∑ k = 0 n S k ′ ( t i ) ⋅ ∂ f ( t k , ξ k , ξ k ′ ) ∂ u e k + ∑ k = 0 n S k ′ ( t i ) ⋅ ∂ f ( t k , ξ k , ξ k ′ ) ∂ v e k ′ E i = ∑ k = 0 n   S k t i ⋅ ∂ f t k , ξ k , ξ k ′ ∂ u e k + ∑ k = 0 n   S k t i ⋅ ∂ f t k , ξ k , ξ k ′ ∂ v e k ′ E i ′ = ∑ k = 0 n   S k ′ t i ⋅ ∂ f t k , ξ k , ξ k ′ ∂ u e k + ∑ k = 0 n   S k ′ t i ⋅ ∂ f t k , ξ k , ξ k ′ ∂ v e k ′ {:[E_(i)=sum_(k=0)^(n)S_(k)(t_(i))*(del f(t_(k),xi_(k),xi_(k)^(')))/(del u)e_(k)+sum_(k=0)^(n)S_(k)(t_(i))*(del f(t_(k),xi_(k),xi_(k)^(')))/(del v)e_(k)^(')],[E_(i)^(')=sum_(k=0)^(n)S_(k)^(')(t_(i))*(del f(t_(k),xi_(k),xi_(k)^(')))/(del u)e_(k)+sum_(k=0)^(n)S_(k)^(')(t_(i))*(del f(t_(k),xi_(k),xi_(k)^(')))/(del v)e_(k)^(')]:}\begin{aligned} & E_{i}=\sum_{k=0}^{n} S_{k}\left(t_{i}\right) \cdot \frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial u} e_{k}+\sum_{k=0}^{n} S_{k}\left(t_{i}\right) \cdot \frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial v} e_{k}^{\prime} \\ & E_{i}^{\prime}=\sum_{k=0}^{n} S_{k}^{\prime}\left(t_{i}\right) \cdot \frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial u} e_{k}+\sum_{k=0}^{n} S_{k}^{\prime}\left(t_{i}\right) \cdot \frac{\partial f\left(t_{k}, \xi_{k}, \xi_{k}^{\prime}\right)}{\partial v} e_{k}^{\prime} \end{aligned}Ei=∑k=0nSk(ti)⋅∂f(tk,ξk,ξk′)∂uek+∑k=0nSk(ti)⋅∂f(tk,ξk,ξk′)∂vek′Ei′=∑k=0nSk′(ti)⋅∂f(tk,ξk,ξk′)∂uek+∑k=0nSk′(ti)⋅∂f(tk,ξk,ξk′)∂vek′
have the order O ( ‖ Δ n ‖ 3 / 2 ) O Δ n 3 / 2 O(||Delta_(n)||^(3//2))O\left(\left\|\Delta_{n}\right\|^{3 / 2}\right)O(‖Δn‖3/2). qquad\qquad
Eliminating E i , E i ′ , i = 0 , n ― E i , E i ′ , i = 0 , n ¯ E_(i),E_(i)^('),i= bar(0,n)E_{i}, E_{i}^{\prime}, i=\overline{0, n}Ei,Ei′,i=0,n―, one obtains the system (22). ◻ ◻ ◻\square◻

A NUMERICAL EXAMPLE

We consider the following singularly perturbed problem:
{ − ε y ′ ′ ( x ) + y ′ ( x ) = 0 , x ∈ [ − 1 , 1 ] y ( − 1 ) = 1 , y ( 1 ) = 0 , − ε y ′ ′ ( x ) + y ′ ( x ) = 0 , x ∈ [ − 1 , 1 ] y ( − 1 ) = 1 , y ( 1 ) = 0 , {[-epsiy^('')(x)+y^(')(x)=0","x in[-1","1]],[y(-1)=1","y(1)=0","]:}\left\{\begin{array}{l} -\varepsilon y^{\prime \prime}(x)+y^{\prime}(x)=0, x \in[-1,1] \\ y(-1)=1, y(1)=0, \end{array}\right.{−εy′′(x)+y′(x)=0,x∈[−1,1]y(−1)=1,y(1)=0,
which may be regarded as a linearized one-dimensional version of a convectiondominated flow problem.
This problem has a unique solution y ( x ) = e x + 1 ε − e 2 ε 1 − e 2 ε y ( x ) = e x + 1 ε − e 2 ε 1 − e 2 ε y(x)=(e^((x+1)/(epsi))-e^((2)/(epsi)))/(1-e^((2)/(epsi)))y(x)=\frac{e^{\frac{x+1}{\varepsilon}}-e^{\frac{2}{\varepsilon}}}{1-e^{\frac{2}{\varepsilon}}}y(x)=ex+1ε−e2ε1−e2ε, which displays one boundary layer at the point x = 1 x = 1 x=1x=1x=1, of the length O ( ε ) O ( ε ) O(epsi)O(\varepsilon)O(ε).
Considering the solution y r ( x ) y r ( x ) y_(r)(x)y_{r}(x)yr(x) of the reduced problem
{ y r ′ ( x ) − 0 , x ∈ [ − 1 , 1 ] y r ( − 1 ) = 1 , y r ′ ( x ) − 0 , x ∈ [ − 1 , 1 ] y r ( − 1 ) = 1 , {[y_(r)^(')(x)-0","x in[-1","1]],[y_(r)(-1)=1","]:}\left\{\begin{array}{l} y_{r}^{\prime}(x)-0, x \in[-1,1] \\ y_{r}(-1)=1, \end{array}\right.{yr′(x)−0,x∈[−1,1]yr(−1)=1,
the following estimations holds (see [6]):
| y ( x ) − y r ( x ) | ≤ C ( ε + e x − 1 ε ) , x ∈ [ − 1 , 1 ] y ( x ) − y r ( x ) ≤ C ε + e x − 1 ε , x ∈ [ − 1 , 1 ] |y(x)-y_(r)(x)| <= C(epsi+e^((x-1)/(epsi))),x in[-1,1]\left|y(x)-y_{r}(x)\right| \leq C\left(\varepsilon+e^{\frac{x-1}{\varepsilon}}\right), x \in[-1,1]|y(x)−yr(x)|≤C(ε+ex−1ε),x∈[−1,1]
where C C CCC denotes an arbitrary constant independent of x x xxx and ε ε epsi\varepsilonε.
The exact solution is of the form
y ( x ) = y r ( x ) + u ( x ) , y ( x ) = y r ( x ) + u ( x ) , y(x)=y_(r)(x)+u(x),y(x)=y_{r}(x)+u(x),y(x)=yr(x)+u(x),
where u ( x ) u ( x ) u(x)u(x)u(x) will be approximated by
u ( x ) ≈ { 0 , x ∈ [ − 1 , 1 − p ε ] v ( x ) , x ∈ [ 1 − p ε , 1 ] , u ( x ) ≈ 0 ,      x ∈ [ − 1 , 1 − p ε ] v ( x ) ,      x ∈ [ 1 − p ε , 1 ] , u(x)~~{[0",",x in[-1","1-p epsi]],[v(x)",",x in[1-p epsi","1]","]:}u(x) \approx \begin{cases}0, & x \in[-1,1-p \varepsilon] \\ v(x), & x \in[1-p \varepsilon, 1],\end{cases}u(x)≈{0,x∈[−1,1−pε]v(x),x∈[1−pε,1],
and the function ν ( x ) ν ( x ) nu(x)\nu(x)ν(x) is the solution of the following problem:
{ − ε v ′ ′ ( x ) + v ′ ( x ) = 0 , x ∈ [ 1 − p ε , 1 ] v ( 1 − p ε ) = 0 , v ( 1 ) = − 1 − ε v ′ ′ ( x ) + v ′ ( x ) = 0 ,      x ∈ [ 1 − p ε , 1 ] v ( 1 − p ε ) = 0 ,      v ( 1 ) = − 1 {[-epsiv^('')(x)+v^(')(x)=0",",x in[1-p epsi","1]],[v(1-p epsi)=0",",v(1)=-1]:}\begin{cases}-\varepsilon v^{\prime \prime}(x)+v^{\prime}(x)=0, & x \in[1-p \varepsilon, 1] \\ v(1-p \varepsilon)=0, & v(1)=-1\end{cases}{−εv′′(x)+v′(x)=0,x∈[1−pε,1]v(1−pε)=0,v(1)=−1
Thus, we approximate the solution y ( x ) y ( x ) y(x)y(x)y(x) by the solution y r ( x ) y r ( x ) y_(r)(x)y_{r}(x)yr(x) on the domain [ − 1 , 1 − p ε ] [ − 1 , 1 − p ε ] [-1,1-p epsi][-1,1-p \varepsilon][−1,1−pε], so that the error would be O ( ε ) O ( ε ) O(epsi)O(\varepsilon)O(ε). In this way we obtain p = ln ⁡ 1 ε p = ln ⁡ 1 ε p=ln((1)/(epsi))p=\ln \frac{1}{\varepsilon}p=ln⁡1ε. We approximate the solution y ( x ) y ( x ) y(x)y(x)y(x) upon the domain [ 1 − p ε , 1 ] [ 1 − p ε , 1 ] [1-p epsi,1][1-p \varepsilon, 1][1−pε,1] by y r ( x ) + v ( x ) y r ( x ) + v ( x ) y_(r)(x)+v(x)y_{r}(x)+v(x)yr(x)+v(x).
We use elements from the space of spline functions O 5 ( Δ n ) O 5 Δ n O_(5)(Delta_(n))\mathscr{O}_{5}\left(\Delta_{n}\right)O5(Δn) in order to approximate v ( x ) v ( x ) v(x)v(x)v(x).
In Table 1 we present the error of approximation of solution y ( x ) y ( x ) y(x)y(x)y(x) by y r ( x ) + s v ( x ) y r ( x ) + s v ( x ) y_(r)(x)+s_(v)(x)y_{r}(x)+s_{v}(x)yr(x)+sv(x) upon the domain [1-p ε , 1 ε , 1 epsi,1\varepsilon, 1ε,1 ] for different values of ε ε epsi\varepsilonε and n n nnn. The linear system for determining the spline s v s v s_(v)s_{v}sv is solved by using a direct method.
Table 1
n / ε n / ε n//epsin / \varepsilonn/ε 10 − 1 10 − 1 10^(-1)10^{-1}10−1 10 − 2 10 − 2 10^(-2)10^{-2}10−2 10 − 3 10 − 3 10^(-3)10^{-3}10−3 10 − 4 10 − 4 10^(-4)10^{-4}10−4
(1) 3 13 ⋅ 10 − 2 13 ⋅ 10 − 2 13*10^(-2)13 \cdot 10^{-2}13⋅10−2 48 ⋅ 10 − 3 48 ⋅ 10 − 3 48*10^(-3)48 \cdot 10^{-3}48⋅10−3 14 ⋅ 10 − 2 14 ⋅ 10 − 2 14*10^(-2)14 \cdot 10^{-2}14⋅10−2 -
8 - 67 ⋅ 10 − 4 67 ⋅ 10 − 4 67*10^(-4)67 \cdot 10^{-4}67⋅10−4 46 ⋅ 10 − 4 46 ⋅ 10 − 4 46*10^(-4)46 \cdot 10^{-4}46⋅10−4 -
12 - - 12 ⋅ 10 − 4 12 ⋅ 10 − 4 12*10^(-4)12 \cdot 10^{-4}12⋅10−4 17 ⋅ 10 − 3 17 ⋅ 10 − 3 17*10^(-3)17 \cdot 10^{-3}17⋅10−3
20 - - - 14 ⋅ 10 − 3 14 ⋅ 10 − 3 14*10^(-3)14 \cdot 10^{-3}14⋅10−3
50 - - - 12 ⋅ 10 − 3 12 ⋅ 10 − 3 12*10^(-3)12 \cdot 10^{-3}12⋅10−3
n//epsi 10^(-1) 10^(-2) 10^(-3) 10^(-4) (1) 3 13*10^(-2) 48*10^(-3) 14*10^(-2) - 8 - 67*10^(-4) 46*10^(-4) - 12 - - 12*10^(-4) 17*10^(-3) 20 - - - 14*10^(-3) 50 - - - 12*10^(-3)| $n / \varepsilon$ | $10^{-1}$ | $10^{-2}$ | $10^{-3}$ | $10^{-4}$ | | :--- | :--- | :--- | :--- | :--- | | (1) 3 | $13 \cdot 10^{-2}$ | $48 \cdot 10^{-3}$ | $14 \cdot 10^{-2}$ | - | | 8 | - | $67 \cdot 10^{-4}$ | $46 \cdot 10^{-4}$ | - | | 12 | - | - | $12 \cdot 10^{-4}$ | $17 \cdot 10^{-3}$ | | 20 | - | - | - | $14 \cdot 10^{-3}$ | | 50 | - | - | - | $12 \cdot 10^{-3}$ |
In Fig. 1 there are displayed the exact solution y ( x ) y ( x ) y(x)y(x)y(x) for ε = 10 − 3 ε = 10 − 3 epsi=10^(-3)\varepsilon=10^{-3}ε=10−3 denoted by a continuous curve, and the approximated solution for n = 3 n = 3 n=3n=3n=3 denoted by a dotted curve.
ACKNOWLEDGEMENT. The authors would like to thank Dr. C. I. Gheorghiu for several interesting discussions.

REFERENCES

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Received July 15, 1996. "Popoviciu" Institute of Numerical Analysis
01-84 P.O. Box 68
3400 Cluj-Napoca
E-mail.tmustata@math.ubbcluj.ro
Received July 15, 1996. "Popoviciu" Institute of Numerical Analysis 01-84 P.O. Box 68 3400 Cluj-Napoca E-mail.tmustata@math.ubbcluj.ro | Received July 15, 1996. | "Popoviciu" Institute of Numerical Analysis | | :--- | :--- | | | 01-84 P.O. Box 68 | | | 3400 Cluj-Napoca | | | E-mail.tmustata@math.ubbcluj.ro | | | | | | |
  1. 1 1 ^(1){ }^{1}1 AMS Classification Code: 34B15, 34A50.
1998

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