TWO-SIDED QUINTIC SPLINE APPROXIMATIONS FOR
TWO-POINT BOUNDARY VALUE PROBLEMS
著者
SAKAI Manabu
journal or
publication title
鹿児島大学理学部紀要. 数学・物理学・化学
volume
10
page range
1-17
別言語のタイトル
2点境界値問題の5次のスプラインによる両側近時に
ついて
URL
http://hdl.handle.net/10232/6358
TWO-POINT BOUNDARY VALUE PROBLEMS
著者
SAKAI Manabu
journal or
publication title
鹿児島大学理学部紀要. 数学・物理学・化学
volume
10
page range
1-17
別言語のタイトル
2点境界値問題の5次のスプラインによる両側近時に
ついて
URL
http://hdl.handle.net/10232/00010033
Rep. Fac. Sci Kagoshima Univ., 拡ath. Rhys･ Chem.) No. 10, p.ト17, 1977
TWO-SIDRD OUINTIG SPLINE APPROXIMATIONS
FOR TWO-POINT BOUNDARY VALUE PROBLEMS
By Manabu
(Received September 14, 1977)
Two-sided quintic spline approximations for two･point boundary value problems are considered. A selection of numerical results is illustrated in Tables 1-3.
I. Introduction
Splines are of much, use for approximating solutions of simple two-point boundary value problems for both, linear and nonlinear ordinary differential equations. Recently we have considered the two-sided cubic spline approximations of second order ([7]). This paper disOusses the two-sided quintic spline approximations of fourth order.
The two-point boundary value problems to be solved is a;〟(t) - f(t, x(t), ∬′W) (o≦t≦1)
A<x(OトB&′(0) -ォ,
Ajxll)+Bt∬′(1) - & , with boundary conditions
1.1
where f(t, x, y) is defined and four times continuously differentiable inやregion D of
(i,浴, 2/)-space intercepted by two planes ｣-Q and t-l.
Now making use of JS-spJine Q6(t)-(l/120)∑(｣y(i)(t-i)+, we consider spline
func-＼tion xk(t) (Jc-l, 2) of the form
*l{t)-∑αi^cQy - T Ink-1) sHi剰
x望w - ∑ βmm-i)
with undetermined coe鮎ients α and β (i--5, -4,..., w-1).
The above xk(t) (&-1,2) will be the approximate solution to the problem (1. 1)-(1. 3) if it satis丘es
弔(t) - Pkf(t, xk(t),鶴(0) (o ≦ i ≦ 1)
AMョトBo宛(0) - a ,
A^m+Bjx孟(1) - 6. with boundary conditions
1.4
Here the operator Pk(k-l, 2) is defined as follows:
(i) (Px g)(t) is the cubic spline function with the node t{(i-0, 1, -, n) such that
(Pi9M -g(ti) (ti-ih, i-0, 13-,n),
(Pi9)′(td - 9%) (サ- 0, n)
(u) (P2 g)(t) is the cubic spline function with the node ti(i-0, 1, ., n) such that
(P*9, Li)
-H9t-i+10gt+gi+1)/12 (* - 1, 2,- - -, n-1)
h(7go+3gi)l20+hHZgふ-2#)/60 ( '- 0) ､Wgサ+3gn-i)/20-h?(Sgムー29品-1)/60 (i - n).
and (P99)¥ti) - 9%) (i - 0, n) ,
where for any甲{t) and <y]p(t)牀L2[0, 1], let us denote1
J.甲{t)-m)dt by (甲,*).
For k-l, it follows that Eqs. (1. 4)-(l. 6) are equivalent to the following system of (n+5) equations :
F-2{a) - ^｡(a-5+260_4+ 66α- ,+26a一色+ α_1)/120
-.#｡(0-!+ loo-a-10a-4-α-5)/Uh-a - 0 ,
*-1(ォ)/* -トa-5+2a-4--2a-2+ a-i)/2h3
-/^O, (0-5+26a-4+66α-s+26α-2+ a-i)/120,トa-.,-10a-4+ 10a-2+ α-1)/2ih)
-/2(0,...)(a-i+10a一望-10a-4-a-5)/24/j
-/3(0, -)(a-,+2a-忠-6α-3+2(Z-4+a-5)/6｣a - 0 ,
JI,(a) - (a,-5+ 2a,-4-6a;-3+ 2a/-2+ a一l)/6h2
-m-, (a,-5+ 26α,--4+ 66a,-3+ 26a,--a+ a,--1)/120 ,
((*, -!+ 10a*-a-10αト4-a,-5)/2a) - 0 )
Fn+I(a)lh - (an-1-2aM望+ 2a舛-4-ォn-5)/2h3
-A(Mォ舛-i +26aォ会+66aサー3+26aォー4+ aォ-5)/120 ,
(an-x + 10aw-2- 10aw-.4- αの-5)l2ih)
-Mt仰, - )(αn-i+ lOaルー皇-10αtt-4- αn-5J/24h
-mny - ･)(αルーi+2aォ包-6a舛-3+2aォー4+a舛-6)/6A2 - 0 ,
Fn+Z(a) - Al(att-1+ 2Qan-i+ 66an-3+ 26an-i+ a舛-5)/120
+ Bllaの^X+100,,-2-lOoa-^ofl-s)/2ih-b - 0 , where fk(ョv x包, a>a) -1,サ2>ォ3) ∂a7k
(Je-1,2 and3).
､ノー-∫-∫--I---_ ノ■ ー 1 - 一 - 1 r l I-牀 T ､ ▲ す . I b TIもヤ･1-8
Two･Sided Quintic Spline Approximations for Two-Point Boundary Value Problems
Fork-2,Eq.(1.4)isequivalenttothefollowingsystemof(n+1)equations: ao(β)-fe(^トxJO))/h-x^OトWo+s/j/a0--2/i)/60, Gi(β)-(a!ォ(*f+iト2xSA+x2{ti-i))/*a-(/m+io/<+み1)/12 ォ-1,2,...,サー1), Gn(β)-(ォ,(!トxJl-h))Jh-xi(l)+h{7fn+Zfn-1)/20-h(3需-2f￡-i)/60, wherex2(t)-∑βmm- )サfi-ftU,Mfi)>Vziti))and Si-/i(ォ*ォ&i)>xk{U))+S&i,x&t),ォ;(*<))蝣x&ti)+sサ(ti,zSi),x' Si))弼ft) Inthepresentpaperweassumethattheproblem(1.1)-(1.3)hastheisolatedsolution &(t)satisfyingtheinternaltycondition U-{(t,x,y)恒-&(t)¥+¥y-#ョ¥≦8,t∈[0,11}⊂DforsomeS>0. Theobjectofthispaperistoshowthefollowingasymptoticexpansionofthe ● errorfunction: e*(*)(-*(ォ上房*(*))-dkh隼(t)+o(･)(h-0)(^-1,2), where^-1/720,d2-｣/240,%k(t)(k-l,2)isthesolutionoftheapproximateproblem (1.4ト(1.6)and¥jp(t)isthesolutionofthefollowingvariationequationof(1.1)-(1.3): 車〝{t)-mm盛′(t))w)+Mt,m,盛′(0)車′(ォ+*< >(*)(0≦t≦1), subjecttotheboundaryconditions AMOトB赫′(0)-0, Al車(l)+*i車′(l)-0. 2.SomePropertiesofSplineFunctions Inwhatfollows,foranycontinuousfunction<p(t),weshalldenoteitsmaximum normby¥¥<p¥¥∞andforany丘nitedimensionalvectorc,weshalldenoteitsmaximum normby附∞ForanysquarematrixA,weshalldenotethenorminducedbythe maximumvectornormby￨￨-4￨￨oo. Lemma1([1]).LetB-(bjj)bediagonallydominant,then HB 1Hco≦max[(￨&ォ￨-2￨&,y￨卜1]･ C｣2 AsanapplciationofthisLemma,weshallprovethefollowingLen皿as2-9. ● Lemma2.Letg(t)∈C4[0,ll,then H(I-Pk)9¥¥∞≦chfiW*)帖(*-1,2), whereIistheunitoperatorandcisaconstantindependentofh. Proof.Fork-l9see[11. For#-2,letusrewrite(/-P2kintheform: {I-P,)g - {g-Px9)+(Px-P望) 9 , 2.1
where I9-Pi9¥¥∞≦ l<7(4)ll∞. For the second part of (2. 1), we have
(Pl-PI9 - ∑ (Pi-γr) QSIh-i)
with Px9-∑βiQァ{tlh-i) and P2g-∑γiQAtlh-i). From the definition of the operator P2, it follows that
(∑ γ#*( /h-i), Lh) - h(gk+1+10gk+ flr*-1)/12 tk - 1, 2,-, n-1). Thus we have (γ :+26γか-i+66γa-2+26 γ*-サ+γ S-4)/120 2.2 (2.3) - (9k+i+^9k+gk-i)/12 (ft- 1,2,-,サーl). (2.4) Similarly we have (∑ γm-/h-%), Lo)/h - (4γ-3+32γ包+23γ-1+γ - (700+3&)/20+A 30ムー2<70/60 , (2.5 (∑ γ /h-サ ), ｣ )/* - (4γ.-1+32γ肘,+23 γfl-3+γ,-4)/60
- 7^+3^-i)/20+hト39品+29品-1)/60.
Since <p(t)-(Pl g)(t) is a polynomial of degree 3 on [ti9 ti+1],甲(t) is represented as follows :
甲(*) - PiLAt)+Pi+1 Li+1(t)+Pi TAt)+甲ォ+lTi+1(t), wher e TM) -{t-ttf/2-{t-tif[6h-h(t-*<)/3 (ti ≦ i ≦ ti+l) (t-tt-rf/6h-Mt-tト1)/ fe-1 ≦ i ≦ ti) (otherwise). Thuswehave (甲,Lk)lh-(<pk+1+i郎+pk-1)/6-hHI中農+1+16や芸+7甲芸-1)/360 (*-1,2,-,n-1). Similarlyweもave (甲,L｡)/h-(7甲.+3Pl)/20+M3甲ムー2甲0/60, (甲,Ln)lk-(7Pn+3甲n-1)/ト3甲￡+2甲左-1)/60･ Since(P2gY(h)-g孟エア孟(&-0,n),wehavetwoequationsfor ek{-γh-βk): 0_3-0-l-0andOサーi-0ォー9-0. 'サー3 2.7 2.10
From (2. 4)-(2. 10), we have the following system of equations for 9k(k--2, -1, -, n-2) : (0*+260*-1+ 660*_,+260*_.+ 0*-4)/120
- -(9k+i-tyk+ g*-1)/12+/Wや芸+1+ 16平屋+7甲芸-0/860
Two-Sided Quintic Spline Approximations for Two-Point Boundary Value Problems
(320-2+270-,+Oo)/60- - -甲0/30,
(320ォ一,+27^-3+ ^-4)/60 - %品-1-甲左-1)/30.
Since ¥g¥彬LPi･解'I≦cW一桝fIgW帖(*-0, 1, - n and m-l, 2), by Lemma 1 and (2. 10) we
lO*1 ≦cW lg(i)¥¥∞ h-O,l,---,n). Thus we
II(Pi-Pz)g¥¥∞ ≦ 11011∞ ≦ eft'llflfWlle｡.
Combining (2.2) and (2.12) yields也e desired result. Lemma 3. Let g(t)∈CTO, 1], then we have
H(I-Pk)g¥¥∞ ≦cJi ¥¥g'¥¥∞ (A;- 1,2) ｡ Pkoof. For k-l9 we have
(甲サ+l+4甲:･+甲;-i)/6 - (9i十-9i-i)/2h with Thus we 血礼ve 甲 -(Prf)W [(Pi+i-9l+i)+HPi-9i)+(Pi-i-9i-1)]/6 -(9i+i--9i-i)/^-(g' i+i+ig' i+g' i-i)/6. (2.12 2.13
From the definition of the operator Pl9 we have two additional equations :
Pa一gム- 0 and や￡一g￡ - 0. (2.14)
Applying Lemma 1 to (2. 13) and (2. 14), we have the desired result. For k-2, let us
rewrite (I-P包)g in the form:
(I-p望) g - (g-Pi9)+(Pi-P*) g ,
where
"(I-Pi)9¥¥∞ ≦ cWll∞. 2.15
In a similar manner analogous to Lemma 2, we have only to show:
((Pi-P%) 9, Li) - (Plg, Lhトh(gk+1+ lOgk+ gk-i)l^
- A(3Pk+l+ 4:?>k + 3<Pk-1)/20-h2(甲孟+1-甲長一1)/30-h(gh+1+ 10gh+ g^j)l12 -%*+i-2ff*+0*-i)/15-h2(甲孟+rP長一1)/30 (*- 1,2,-,n-¥). (2.16)
A simple calculation shows:
((*!-P2) g, L｡) - 0 and ((Px-PJ fr LJ - O. (2.17) Thus we have the desired result.
Lemma 4 ([4] ). Let g(t) ∈ C6 [0, 1], then there exists a quintic splinefunction of the
/<orm
so sin劉 (ii) 甲 -9(tt) -0,1,-,n)
9 -gW{U) (i-O,n and m-1,2)
"p(鮒)-g(解)￨￨c｡-0(Aォ- ) (m-0,l,2,3) (h-0)･Proof. By the use of consistency conditions, we shall prove this Lemma. In the case of the quntic spline, there are the following relationships between the first and
●
second derivatives of the spline:
(Pi･'包+26甲ォ+l+ 66甲 +26甲;--i+甲;-2)/120 - (Pi'包+loゃi+1-10甲i-rPi-2)/24A (t - 2, 3,-, w-2) (2.18) and (甲サ+2+26甲サ+l+66や1+26甲:-i+甲;-ォ)/120 -(甲,-+2+2甲詛+1-6Pi+2甲ト1+甲i-2)/6h* (i-2,3, ,n-2)･ 2.19) If ristricted on [0, t3],甲(t) depends upon 8 parameters. Therefore 9 quantitiesや;, Pi and PJ, (i-0, 1, 2, 3) are not mutually independent, there is a unique linear relation:
37g>0-54Pl+9甲2+8甲,+ 12h甲ム - h2ト23PS+354甲", +201甲芸+8甲 37や,-54甲か1+ 9Pn一望+ 8甲n-3-12hP左 - M-23平易+354甲芸-i+201や芸-2+8甲蒜-3)/20 ･ Similarly we 血礼ve Also we have and -235^｡+ 65甲1+ 155甲望+ 15^3 - 16ft2甲岩+h(lllPム+227甲;+79や; +3甲;) 235甲,-65Pn-1-155Pn-2- 15Pn-2 - - 9品+Mlll甲立+227巌-!+79甲;-2+3甲左-3 (2.20) 2.21 (2.22) (2.23) Since?r一g'i彬>-0(i-0,nandm-l,2),byTaylorseriesexpansionandLemma1we have tP:･一g' {￨-0(･)(ォ-0,1, ,n)from(2.18),(2.22),and(2.23)(2.24) and lP'i一g-¥-0(･)(サ-0,l,.-,n)丘0m(2.19),(2.20)and(2.21).(2.25) Thuswehavethedesiredresult. Lemma5.LetP(*)-∑aiQ$lh-i)>then a￨￨∞≦cIMI∞.
Two-Sided Quintic Spline Approximations for Two-Point Boundary Value Problems
Proof. As is readily seen, we have
3a-i+ ao -三691-8甲1/2/3+ (?>o+91)/3
(aトサ+4aト2+a,-i)/6 -甲 ( - 2, 3,-,サー2) 3a^3+ォォー4 - 6恥-8恥-1/2/3+(Pn+Pn-1)/3. From Lemma 1. it follows that
cLi ≦C"平日∞ ( --1,0,-,n-3).
Since ai-3 - 6甲.･-4a,--,-of-1 ( - 1, 1
aト1 - 6Pi-4α,--a-af--3 li - n-l, n) 9 we have
all∞ ≦cH甲tI00.
Lemma 6. Let P(t)-∑aiQMh-i) such that
甲w>-りi (サ-0,l,-,n),甲′(*o)-り-i/h and や′(U-‰+1/A. Then we have
IMI∞ ≦oJh=∞. Proof. By consistency relations, it follows that
(アサ+l+4甲;･ +甲;-i)/6 - ((pi+t-Pi-1)/2h
- (丸+1-りi-1)/2h
(*-1,2,-,n-1),甲ム-り-1/h and P￡ - v*+Jh
AOcording to I｣emma 1, we have
Iや;･I ≦cJ酬∞/ (詛-0,1,-,n).
Since P.･l≦J酬∞ by (2.26) we have
(2.26)
"pll∞≦可酬00.
The following Lemma 7 follows through the same arguments as in the proof of the
●
latter part of Lemma 2.
Lemma 7. Letや(t) be cubic spline function with the node ti {i-0, !, , w) such
that (甲,Li)-hr)i (i-09lr..9n), 甲′t0- で-1/h and P′ォ.)-‰Jh. Then we have 瞳Hoo ≦olIでII∞. Lemma 8 ([3]). Let g(t)∈G*[0, ll (Pi9)〟('ォ ) -/&トh*g?l¥2+o(h*) (h -0). Proof. Let us denote ^(t)-(^>i^)(')> *^en we ^ave
(甲サ+l+4甲;+95-0/6 - (w+1-2甲.･+甲ト1)/h2
Thus we have
[(Pi+i-9芸i)+4(甲'i一g芸)+(甲冨-1-g;-1)]/6
--h*g?l12+o(h*) (*-1,2,-,サーl). (2.28) 甲(t) is cubic on [t｡,｣j, we have
h(2Pld+甲'0/6 - {(px-^/h-甲ふ, from which follows
l2(甲3-93)+(甲1-91)1/3 - -h%サ(12+o(h*). Similarly we 血礼ve
[2(甲芸-9品)+ (甲芸-i-9品-1)]/3 --%<f712+o(/*2). From (2.28)-(2.30), we have the desired result.
Lemma 9. Let g(t) be continously differentiable on [0, 1], the誌 we ¥¥Pi9¥¥∞ ≦ c(¥¥9¥¥∞+Wll｡｡)
-Proof. Let (Pxg){t) - ∑ αmm-i), then ¥¥Pi9¥¥∞ ≦ ‖an∞. It follows from the definition of operator Px that
(ォーi-α-3)/2h - g占,
(<*, _,+4αi-2+αi-3)[6 -gi (i - O, l,'- -, n) , (αn-x-aォー3)/2A - g左.
Hence, from Lemma 1, we have
a ∞≦e(HgH∞+別g'帖).
Thus we have the desired result.
(2.29)
(2.30)
3. Existence and Conve鴫ence of Spline Approximations
In this section, using Kantorovich's theorem on Newton's method, we shall
●
prove the solvability of the determining equations F(α)-(*"-ォ(α),乱1(α), -, Fn+1(α), *Wα))-0 and G{β)-(*一蝣(β), F-i(β), GWβ), -, Gn{β), Fn+1(β), Fn暮2(β))-O.
Corresponding to庶(t), one can determine uniquely a quniti￠ spline funOtion &h(t), of the form
Ut) - ∑ bmtjh-i)
so that
*k{U) -Wi) (i-O, l,---,n)
虎㌘)(t.¥一盛(焔w (i-0,n and m-1,2). Since ｣(t) ∈ C6[0, 1] due to the assumption f{t, x, y) ∈ C4 (D), it is valid that
Two･Sided Quintic Spline Approximations for Two-Point Boundary Value Problems
For simplicity, we shall consider the special case whenf(t, x, y) is independent of y.
Let J,(α) be the Jacobian matrix with respect to αi (*--5, -4,-, n-1). In order to
investigate the properties of J^a), let us consider a linear system
JAM-り, 3.1
where ｣-(｣-,￨-4,-,｣_!) and で-(り_2,で-1,-,でn+2
Corresponding to ￨ andでwe consider quintic and cubic splines y^t) and y2(t) defined by
y&) - ∑ ZiQsw-i)
and y&d-り(*-0,l,-,n) y' So)-り-Ay' z{Q-りn+1/k. From(3.1),wehave y[3)(t｡)-[Uto,Uto))+Mto,Uh))釦(ォo)]yifoサ) +/2(ォO,**(ォo))yi(ォo)+yi(ォo), y'AU)-Mi,Wu))yM+y&i)(i-0,1,-,n), y¥z¥Q-[Utn,UQ+UKUtサ))鶴(tサ)]yi(tサ) +Mtサ,**(O)y' liQ+y' Sn). ^02/i(OトBoy' x(O)-ヤー2, Sinc e Alyl(l)+Bly' 1(l)-i]n+t. **(*, )-*(*<)(サ-0,!,...,サ),蝣&(ti)-虎′U.)(サ-0,w), thus we haveyl(t) - Pi[Mt, ｣(t)) yi(t)]+y*(t) (o ≦ i ≦ 1) ･ Equation (3.2) can be rewritten as follows:
y;-Mt,*)yi- yサ+R,
where i?--[/-Px](/2Vl).
SinOe #(｣) is isolated, there exists the Green function H(t, s) such that
1
yl- ∫ H{-,s)[y2(s)+R(s)]ds+甲,
0
whereや(t) is the solution of the following equation:
甲〝(i)-ut,m))甲 -0 (0≦t≦1) subject to the boundary conditions
Ao<p(OトBop′ り-2 , AM¥)+Bl<P′(1) -りn+2 '
A simple calculation shows也at
3.2)
fJp‖∞≦<hl(ヤー2,で伸+丑I-Throughoutthissection,c{(i-l,2, ,15)willdenotetheconstantindependentofh. From(3.3)wehavetheinequalityoftheform: IWI∞≦cMvzW∞+¥¥R帖+c3瞳=∞･ ApplicationofLemma3yields "R=∞≦oM¥¥y' i¥∞+IMI｡-)-Byviruteofthe′well-knowninequality: wy' i‖∞≦c,Hyi!l∞+ce¥¥y"i¥∞forsomec5andc6 wehave HRH∞≦cMlyiW∞+¥¥ylH∞) ≦cMyill∞+t困l∞+¥¥R‖∞)･ Hencewehavetheestimateofl122II∞oftheform HRH∞≦c9h(‖yュ"∞+112/2帖forsu鮎ientlysmallA. Thuswehave llyi帖≦clo(llォ/2瞳+り-2>77サ+2)￨￨) Sincefly^∞≧∞and￨￨tf￨￨￨∞≦Cl納Il∞wefinallyhavetheinequalityoftheform ∞≦<y酬∞foranyh≦h｡.(3.4) providedhoissu鮎ientlysmall. By(3.1),inequality(3.4)impliesthenon-singularityofJ^a)andinadditionthe inequality ITWI∞≦Gu(-cis)foranyh≦ho･ Letα-(α-sサa-i,'-,a*-i)andβ-(β-5,β-4?*' 3βサーi)be arbitraryvectorssuchthat､ l (aトi+26α,--4+66a4--3+26a,--負+ α,--1)/120-ォォ,) l ≦ 8 , ) (β, -5+ 26β -4+66β;-3+26βト2+ β,-1)/120-*fo)仁≦ 8 , (i-0,l,---,w). Hence by the means of the mean value theorem we have
¥¥Jl(αトJl(β)"∞ ≦ 15 α-P"∞ By Lemma 2, we have the equality of the form
¥¥m)¥¥∞ - 0(W).
3.5
3.6
The expressions (3.4)-(3.6) show that all the conditions of Kantorovich's theorem on Newton's method are fulfilled ([5]). Therefore the determining equation F(a)-0 has the solution a such that
Two-Sided Quintic Spline Approximations for Two-Point Boundar: Value Problems ll
na-&11∞ -00*). Thus we 血礼ve
"褒r利∞≦lI房¥-&h ∞+睡h一利∞-0m (k-0), where 房l(*) - ∑瓦mtik-i).
For the second derivative, we have
褒'i-盛〝 - Pi fit,房1トM *)
- Pl｢/Mi)-MW-ii-pjM虎). By the means of Lemma 9 we have
ll房;-虎〝ll∞ - 0(V). For the first derivative, it follows that･l
/
lI褒i-V ≦C51搾1-釧>+c61￨褒;一g'=∞-0{･) (*-0). Therefore we have
ll房<サ>-｣(*) ∞-0(･) (m-0,1,2) (h-0).
In the general case whenf(t, x, y) contains the component y, we can show the same result by reducing the equations (1.1)-(1.3) into the system of first order differential equations ([6,8]).
Thus we have the following
●
Theorem 1. In the sufficiently small neighbourhood of the isolated solution x(t)
of the problem (1. 1)-(1. 3), there exists a quintic spline function房^t) of the form: 房l(0 - ∑ォ# (ォ/*-サ)
such that
tJ房(m) #(*サ)瞳-0{･) (w-0,1,2) {h-0). In analogy with Theorem 1, we now have Theorem 2.
Theorem 2. Let the hypotheses of Theorem 1 hold. There exists a quinitic spline junction房5(｣) of the form
房M - ∑βiQSIh-i) so that
(i) the coefficient β-(β-5, β-4,-, β -!) satisfies the determining equation G(β)-O, (") II褒㌢'-｣<蝣 ∞-0(W) (m-0,1,2) (h-0).
Proof. For the operator P2, let G(β)-(F-Jβ), *-1(β), Go(β), , GJβ), Fn+1(β), Fn12(β) ) and J2(β) be the Jacobian matrix with respect to β. (t-=-5, -4,..., n-1). For simplicity, we shall consider the special Case when f(t,x,y)-f{t,x, 0). In a similar manner analogous to Theorem 1, let us consider a linear system:
Corresponding to J andり, We Oonsider quintic and cubic spline functions zx(t) and z2(t) denned by
ztf) - ∑ t<Q6W-i)
Effl割
(z2,Li) -hr)i (i-0, 1, ,w)
z(*o) - V-ilh , z(tサ) -りn'1/A.
From (3.7), we have (z'uLA-lP包SjJ,Li)+(z2,Lt)(サ蝣-0,1, ,サ), *nti)-iM′(ti)+z' Mi)(i-0,n). ApplyingLemma7totheequations(3.8)and(3.9)wehave zl-P^M+z望(0≦t≦1). ThusinasimilarwayasinTheorem1,wehavethedesiredresult.
4. Asymptotic Expansion of Error Function Ek(t) (k-l, 2)
By the means of the results of the previous sections, we shall prove in this section
the asymptotic expansion of the error function ek(t)-Jt(t)一房*(*) (*-1, 2).
Theorem 3. With the hypotheses of Theorem 1, then we have the asymptotic expansion:
ek{t)-dk･yUt)+o{･) (h-0) (&-1,2).
Proof. From Theorem 1, we have
e芸-f&虎,盛′)ek+Mt,｣,盆′) e孟+(i-Pk)M虎,虎′)+0m , (ft-1,2) (h-0),
ト5^(0-0,
Alehm+^i(l) - O.
Since A(t) is isolated, equations (4.1ト(4.3) can be rewritten in the form:
1
ek(t)-∫ Hit,s)(トPk)g(s)ds+O(h5) (k-1,2) (k-0)･
0 with where 9(t)-盛〝(t). For k-1, we have(I-PJgM -gti)-9z(t)+9z(t)-(Pi9)(t) for t∈ ¥pii ti+l] 3
gM) - gi LAt)+ gi+1 Li+1(t)+ g"iTi{t)+ gi+1 Ti+1(t).
n J u h リ 1 2 3 ･ ^ i ^ j H ^ w n u u J t ー n H J J t l
According to Lemma 8, we have
9z(t)-(Pi9)(t) - 19"i -(Prf)J] ZV(*)+ [tfi +1-(PriOi-j awt)
- (1/12) h¥g? Tffl+ gVUTt+M+oV*) >
from which follows, using the second mean value theorem on the definite integral,
Two-Sided Quintic Spline Approximations for Two-Point Boundary Value Problems
¥ H(t, S) [g3(s)-(Pig)(s)] ds
- W12) [57(4) ∫ H{t, s) Ti{s) ds+gfl+1 JH{t, s) Ti十,(s) ds] +o(h5) - -(1/288) ･[gfH{t, Si)+tflJI{t, %)]+o(fcサ)
for some & andでi∈[k,<<+J
Thus we have
1
J H(t, s) [gJs)-Pig<s)]ds - -n/144)･J H{t, s)gォHs) ds+o{W)
0 0
Futhermore, a simple calculation shows :
g(tト9z(t) - [{(<-tv-m-m/U- {Ut-tif-hHトti)}/12] gf+o(･)
for any t∈ [ti,ti+1¥ -Thus it follows that
1 1
J H(t, s) (g(s)一癖)) (fe - (fc*/120) ∫ H(t, s)g(iHs)ds+o(hi)
0 0
From (4.4) and (4.5), we have the following asymptotic expansion:
1 1
J H(t, s) (トPJg{8)d8 - (k*172Q)∫ H(t, s)g^(s) ds+o(hi).
0 0
On the other hand, we have
1
e&) - ∫ H(t,s)(I-P2)g(s)ds - EIi ,
0
where, for each i-09 1, ,n-l,
/, - ∫ H(t, s)(I-P2)g(s) ds
- f [H(t,ォ,.) L^)+H(t, ti+1) L,+1(s)] (トP2) g(s) ds+o(hォ)
- H(t, U) j L,is)(I-Pa)g(s) ds+H(t, ti+1) f Li+1(s)(I-P%)g(8) da+0m.
Thus we have
e&) - H{t, t｡) ∫ L｡(s)(I-P2) g(s) ds
･岩Hit, 0 ∫ L,(s)(/-P2)9(s) ds+H(t, tn) f Ln(s)(I-P2)g(s) ds+o{･)
13
4.4
(4.5)
- H(t, t｡) [J L｡(s) g(s) ds-*(7flr｡+3flrl)/20-Aォ(3flfi-2^;)/60)]
･ ∑ H(t, ti) [¥ Li(s) g(s) ds-Hg.i+i+^9i+ 9i-i)/12]
･ H(t, tn) ∫ LJs)gls) ds-hllgn+Zgn-j)/20+A2 (30品-29品-1)/60j.
By Taylor expansion, we have
1
e望(*) - - (･124:0) ∫ H(t) s)gii){s) ds+o(hi). 0
Thus we have the desired result.
As an immediate consequence of Theorem 3, we have the Corollary. With the hypotheses of Theorem 1, then we have
(4,7)
(3房i(ォ)+褒2(*))/4 - #(*)+o(A4) (h - 0).
Finally it should be remarked that for the type of the first derivatives absent the
approximate problem (k-2) (1.4ト(1.6) is identical with the welLknown differnce scheme as the Numerov formula. Thus the collocation method using quintic spline function gives the opposite approximation to the solution of the problem (1.1)-(1.3) as
com-pared with比e Numerov di鮎rence method.
5. Numerical Examples
In this section, we discuss numerical results obtained from some concrete examples. These numerical results conform the theoretical accuracies established in previous sections. In the case of examples 1 and 2, the approximate problems (k-2) (1.4)-(1.6)
are identical with the Numerov difference schemes. We now consider the numerical
solutions of particular examples (1.1)-(1.3).
Example 1 ([1]). As our first example, we consider the linear problem: a3〝-100a; 0≦t≦1)
∬0 -∬1 -1. The unique solution is x(t)-cosh (10ト5)/cosh 5.
/ Table 1.1 (el(t)) A=l/20 h=l/40 A=*l/80 1 q 一 3 4 5 ● ● ● ● ● o o o o o ) ) ) ) ) I Q I O O ォ D ォ D I I I I I ′ー Q O c O f c - Q O c O 0 0 c O 0 0 f c - 0 5
c
o
c
o
c
O
餌
7
● ● ● ● ● t H t H < O c O < M ｣ ご こ こ :-9. 8931ト7)
-7. 3228ト7)
-L 1708ト7)
-2. 3598ト7)
-1. 8040ト7)
-6. 2338 -4. 6184 -2. 6314 -1. 4891 -1. 1385 ) ) ヽ ノ ) ) O O 0 0 0 0 0 0 0 0 二 二 二 こ こ ′ t ー ′ t ( ′ t 1.5 (-5)-1.5×10-*.Two-Sided Quintic Spline Approximations for Two-Point Boundary Value Problems Table 1.2 (e2(t)) h-1/20 h-¥/40 I h-1/80 1 Q 一 C O t H サ O ● ● ● ● ● o o o o o
4. 7568ト5)
3. 5245ト5)
2. 0081ト5)
1. 1363ト5)
8. 6870ト6)
2. 9940ト6)
2. 2185ト6)
1.2641ト6)
7. 1537ト7)
5. 4691ト7)
1. 8747ト7)
1. 3891ト7)
7. 9151ト8)
4. 4793ト8)
3. 4245(-8)For this simple problem we have
(3房L(0.5)+房,(0-5))/4-虎(0.5ト2.25ト11) fai h.- l/80･
Example 2. Let us consider the nonlinear differential equation: a;〟-1.5が (0≦t≦1)
a<0)-4, x(l)-1. This problem has two isolated solutions such that
*(ォ)-4/(<+l)a and ｣(0.5)≒-10.53. Table 2.1 leAt) for虎(0.5)-16/9)
h-1/20 A-1/40 0 佃. 日日 ≡ iォ H ( N e O T H サ O O b -0 0 0 5 ● ● ● ● ● ● ● ● ● o o o o o o o o o -1. -1. -1. -1. -1. -8. -6. -4. -2. ＼ ノ ) ) O O O O O N t - t - N I I I I I I ! ー ( ′ t ′ l ′ t ( ー 9 1 7 d D ^ O ォ ゥ G < 一 7 8 功一O サCI CO H b> IO tH 3 q一句一 i O IO IO tH O c O C O r -I O O O O O -6. -8. -8. -7. -6. -5. -3. -2. -1. ) ) ) ) ) ) ) ) ) o o o o o o o o o o o o o o o o o o I f I i I I ー ー ′ l I S * ョ ( M ^ l > l > o J b -O J O C O f f 一 ォ O r H ^ f c -O * l a ^ c o * a c o r -i o o サ o < j q -4. -5. -5. -4. -3. -3. -9. -1. -8. ヽ t ノ ) ) ) O d O i C d C 5 C i O 5 O d O 5 ^ 1 I I I I I I I ( ( ( ( ( ( ( ( ( I O H O O O O O O M O O H 8 q 一 1 0 ^ t -O i r H C O < O C O H ォ O ゥ ! 一 O S O H H C O H 二 田 且 [ * V サ > J -r M < 軌 蝣 > 蝣 蝣 J -1 日 リ
Table 2.2 (e2(t) for #(0.5)-16/9)
A-l/20 /&-l/40 h-l/80 H N C O ' ^ l O O t - O O O S ● ● ● ● ● ● ● ● ● o o o o o o o o o 3. 4. 4. 3. 3. 2. 1. 1. 6. ) ) ) ) ) ) ) ) ) q ^ C O C O C O g O g O β n V 7 I I I I I ( ( ( ( ( ( ( ( ( ォ l - 0 > ^ サ 0 0 ォ O O c f l O I O O f c - O O O ^ O S O r H 6 β D c O H S O I O ^ c O C O r H O O S O O ^ O O C i 一 2 1. 9. 9. 9. 1. 1. 1. 7. 3. ヽノ)))777け蝣-fc-fc-fr.oo00 IIIIIIII ((′ー((′ー(ー′l ss3SS5icq｡ooiォo >Hfc-00ォ0 85Q一ゥH^lOrWO o*1010c<一O>IOrHb-Oi 1. 1. 1. 1. 1. 9. 7. 4. 2. ) ) ) ) ) ) ) ) ) o o o o o o o o o o o d O O d O d I I I I I I ( ( ( ( ( ( ( ( ( Q一7 7 功一io サo cq oi 10
3
S
S
3
S
g
S
$
3
3
< N I O I O ^ G S 一 ォ O ( M 0 0 t ォAs in the previous exmaple, we have
(3房 男,0.5 /4-<wO.5V- -3.5(-12) for A- 1/80.
Table 2.3 (元(t) for *(0.5)≒-10.53) h-1/20 ^-1/40 *-1/80 H M C O " * l O O t -0 0 0 5 ● ● ● ● ● ● ● ● ● o o o o o o o o o 0. -3. -6. -9. -10. -10. -8. -5. -2. 3 I O t - O J H 1 4 q 一 f c -t H 0 0 C O E U 7 Q 一 H O a i O O O O f f 一 4 0 0 I ' H t H O H t H t H 0 0 H O < 0 ( M C i 一 t o o a s o O O O O H O O ^ O O N O H N O C O W C O H O J ォ O O J ^ O e O O O T H O O O ^ 0. -3. -6. -9. 0 1 0 け - < M O J O O O 5 r H O サ O ゥ = 一 1 5 < ゥ o t -^ f O c O O I O ^ f f i 0 O H I O f f l サ c c < 一 4 O O O M > サ O O H C i 一 O t -0 0 < M ^ t -^ サ O O O ォ O 0. -3. -6. -9. -10. -10. -8. -5. -2. 6 5け^OiOr-i&一9 srォOoH餌ooio COco*Q戯 OHO5b-(NOJ00OSH O5HOlサOC`HWOOォ OOOOOOOOOfc-Oi-H b-OcOWCOHO15005 ^ocooo^ォo<サ^
T唱ble 2.4 (雇It) for鬼(0.5)≒-10.53)
A-l/20 h-:l/40 i-l/80 1 q 一 C O -* サ O ォ O N 一 8 9 ● ● ● ● ● ● ● ● ● o o o o o o o o o 0. 47898 -3. 00794 -6. 33080 -9. 03843 -10. 53616 -10. 41014 -8. 69747 -5. 86075 - 2049152 O I O S O H M O S ^ O C O O O O O H 0 月 S H C O N t H l L O 1 0 0. -3. -6. -9. -10. -10. -8. -5. -2. 1 08 9 1 9 8 8 0 7 0 3 4 0 3 ォ O ( M 0 0 R 乱 L c * M = リ 8 6 0 3 3 1 0 5 4 7 8 0 5 8 6 7 0 1 9 6 9 6 8 4 6 0 * ? O O a ^ H r ォ 0 0 H c O ( M o O O ' * t -O l > ォ 0 ^ " ? H O < M < j < i a a < ゥ ォ ゥ o o 0. -3. -6. -9. -10. -10. -8. -5. -2. 6 0 3 H 0 0 C 一 4 1 q 一 7 9 8 の召 O t-^H 00 00 OZ CO t-i-I SO &一句一4 3 O H O 5 N ( S 一 O i 0 0 O * r H O S H O i i O ( ? 一 1 5 8 A n V 0 0 0 0 O O O < ゥ O f c - O i -H b -O C O C O C O H O S C D O S ^ o o o i o ^ o o o ^
Example 3 ([2]). As our final exmaple, we consider the following nonlinear differential equation :
x〝 - xs-(1+cos ｣)3-cos t ,
x′(0)-0, ∬′ が(l)sinl/(l+cosl)3. The unique solution is #(｣)-!+ cos t.
Table 3.1 {el(t)) ft-l/16 h-1/32 冗 ｣ ｣ * 限 E ^ M * ^ K ￨ X * ^ ^ K H H C O H サ O c O t -9. 2. 2. 2. 2. 9. 2. 2. 1. ー))))))))000000000000000000 IIIIII ′ーー′ cO00OIO一句一OOiTHO ゥーーHH(f一Q一ooサo 稚tfJHOOO !>fc-OIO舶oilOCO <MOfc-1. 1. 1. 1. 1. 1. 1. 1. 1. I))))))))OdOSOdOJO5OiOiOdO5 IIIIIIIII ′t(′tt( ooォゥe<!fc-ca^H<サt-o q一t-HOHOO(M1005 r-1005< t-t-so蝣1T-HCOCO｡｡ 'oIOtH(N㈹ 1. 1. 1. 1. 1. 9. 8. 8. 6. ヽ ノ ) ) ) O O O O O r H r H r H r H i -I H H r l t -H t H r -H r H t H T L L ニ T T T ニ ニ ( ー ((( ー ((( O f c -サ 0 > O i -H 0 0 T -1 1 Q 0 0 N O サ O c O O H . 1 O T t -1 0 0 0 o o A O O i O 0 0
Two-Sided Quintic Spline Approximations for Two-Point Boundary Value Problems 17 Table 3.2 (e2(t)) h-1/ h-l/16 h-1/32 ^ ^ E ^ K s ^ E ^ E ^ 昭 S f c i 印 H H C O H I O C O b i -5. -6. -6. -7. -7. -6. -6. -5. -4. ) ) ) ) ) ) ) ) ) o o o o o o o o c o o o o o o o o o r 1 I I I ! I I ( ( ( ( ( ( ( ( ( I O C 5 5 b -^ O O O t -C ∂ 7 ^ H 0 0 O O O J H t H サ 0 0 5 0 5 0 1 O H ( M H C O O H O J O O J H O t -H G i 一 〇 -4. -4. -4. -4. -4. -4. -4. -3. -2. ) ) ) ) ) ) ) ) ) 0 5 O d O 5 0 5 0 5 0 2 O d O d O 5 I I I I I I I ( ( ( ( ( ( ( ( ( 00 rH CO G^一CO O lゥ 00 0 0 3 0 i ^ O H ォ D O 5 t - H C O O O O H H O ^ D t - O C O O t -t -O ^ O i O O l -2. -3. -3. -3. -2. -2. -2. -2. -1. ) ) ) ) ) ) ) ) ) o o o o o o o o o i -< r H t H r H t -I v -I t -H i -I t -I I I I I I I I I ′
^ O Tォ O CO ff!一ゥcO
b-O C O ^ o O G < 一 0 5 M 3 k O O O C Q I O M I O H H f f 一 3 0 > 0 0 0 0 5 < X ) ? O W O i サ
For this example, we have
(3房iO.5 +房2(0.5))/4-盛(0.5)- -1.7ト12) for h- 1/32.
References
[1] J. Ahlberg, E. Nilson and J. Walsh: The theory of splines and their applications. Academic Press, New York. 1967.
[21 P. CIARLET,班･ So王iultz and R. Varga: Numerical methods of high order accuracy for nonlinear boundary value problems. il. Nonlinear boundary conditions. Numer. Math. ll, 33ト345 (1968).
[3] J. Daniel and B. Swartz: Extrapolated collocation for two-point boundary value problems using cubic splines. J. Inst. Math Applies. 16, 16ト174 (1975).
[4] W. Hoskins and. 6. McMasteb,: Multipoint boundary expansions for spline interpolation. Proceeding of the second Manitoba conference on Numerical Mathematics. Utilitas Mathematica Publishing Incorporated. Winnipeg, 1972.
[5] L. Rall: Computational solution of nonlinear operator equation. Wiley, New, York,
1969.
[6] M. Sakai: Piecewise cubic interpolation and two-point boundary value problems. Publ. B>. I.M.S., Kyoto Univ. 26, 345-361 (1971).
Cubic spline interpolation and two-sided difference methods to two-point boundary
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Cubic spline function and difference method. Mem. Fac. Sci., Kyushu Univ.
