ScanGate document
Trang 1VNU JOURNAL OF SCIENCE, Mathematics - Physics T.XIX, No3 - 2003
REMARKS ON THE SHOOTING METHOD FOR NONLINEAR
TWO-POINT BOUNDARY-VALUE PROBLEMS
Nguyen Trung Hieu
Department of Mathematics, College of Science, VNU
Abstract In this note, we prove a convergent theorem for the shooting method combin- ing the explicit Euler’s scheme with the Newton method for solving nonlinear two-point boundary problems (TPBVPs) Some illustrative numerical examples are also considered
A convergent result obtained before by T Jankowski is a particular case of our result,
when the boundary condition (BC) becomes linear 1 Introduction
The shooting method for TPBVPs has been studied throughly in many works (c.f [1-10]) However, the convergence of the method did not receive adequate attention of researchers In 1995, T Jankowski gave an adequate proof for a convergent theorem of a shooting method
In this paper, we will generalize this result for nonlinear ordinary differential equa-
tions (ODEs) with nonlinear boundary conditions
Consider the problem
y' = f(t.y), t€ J = [a,b] (la)
90),()) = 0, (1b)
where f : JxR’ — R” is contimuons in t and continously differentiable in y, @: R” xR” > R’ continuously differentiable in both variables
If we denote y = y(t; 3) a solution of (1a) subject to initial condition
ula) =, (ac)
then the problem is reduced to that of finding s = 5 which solves the equation (y(a; 5), U(b; s)) = Ó(s, y(b; s)) = 0 (2) This equation can be solved approximately by the Newton’s iteration
8341 = 8) — Ó(3;.9(b;s;)) '6(s;,w(b:3;)), 7 20, (3) where ¢'(s) = 61(s, y(b; s)) + $2(s, y(b; s)).y4(b; s) and ¢1,¢2 are partial derivatives of ó with respect to the first and the second variables, respectively In addition , Y(t; s) =
y,(t;s) can be found as a solution of the IVP
{ Y3) = f,(t-y(ts))-Y (G8)
Y(a;s) =I 6
Trang 2The remaining problem is to solve approximately the two IVPs (1 a,c) and (4) There are many methods for solving IVPs, but in this paper, we only use explicit Euler’s method to solve them ‘Thus we obtain a method using Euler’s scheme combined with Newton’s iteration In the following, this method will be discussed in detail Let the integration interval be subdivided by the points
(5)
First, applying the explicit Muler’s scheme mentioned above, we get
= Sh,j3 | Yn (bi+1s hj) = Yates Sng) +L (i Ya (tas 8n,9))5 (6a) 52 Ya(tiats Sng) = (E+ bSy (ti, yn (tis 8ng))]¥a (tis Saya)» (6b) i=0,1, ,N-1, p nts Sn0 = 80,80 € Ris an initial vector yn (tos $n, Yh(foiSn) ¢ Then we use the Newton method to improve the shooting vector Shj~L = Sỏu [61(Sn,9,Ơn(bs $n,3)) +Â02(8n,9+ Ya (0: 8n,3))¥n(b, 8h.9)] 7 'O($n9+ Ym (D5 Sn.) (6c) j=0,1, 2 Convergence
In this note, we use the terminologies and notations of [4] However, the definition
of isolated solution should be stated as follows
Definition 1 A solution y(t) of (1) is said to be isolated if the following linear TPBVP
{ 2 = fy(t.u(t))z
$x(u(a),y(t))2(a) + o2(y(a),4(6))2(6) = s
has only trivial solution z(t) = 0
Lemma 1 The isolated solution is locally unique
Proof Consider the space C! (J R”) equipped with the norm ||y|| = max{||yl]max; {ly'Ilmax}:
the space C(J,R”) xR” equipped with the norm {|(Z, v2)|| = |\Ellmax +|+a|, where |2||max =
Trang 3and if y* is an isolated solution of (1), then KerF’(y*) = {0} Moreover, the shooting matrix 1 (y*(a), y*(b))U (a) + d2(y* (a), y*(b))U(0) is nonsingular, where U/(t) is a funda-
mental solution
Ư() =5‡(L*)00)
U(a) =I
It implies that ImF '(y*) = CJ, R’) xR” According to the Banach inverse mapping
theorem, [F’(y*)]~! exists and is continuous The inverse function theorem ensures that
the equation F(y) = 0 has a locally unique solution n
Theorem 1 Let the BVP (1) have an isolated solution ọ Assưme thai
i) both f: Ix S-> R” and f; :J x SR” are continuous;
ii}the function f: Jx S74 R” has a bounded by a constant L > 0 derivative with respect to the second variable, and there exists a function 2: R, — Rx such that
IIZz(,#) — ;(t,®)|| < 9(z — 3|),
where the matria norm is consistent with the vector norm;
iti) Q is continuous, 2(0) = 0 and Q is non-decreasing;
iv) The Euler scheme is consistent with (1a), i.e there ewists a function «: H >
R*,H = |0,h*] for some h* > 0,c(h) — 0 as h 4 0 such that IF Ct, e() + #Œ) — v(t + h)|| < he(h), for t € [a,b — A];
v) There exists a function 6: H + Rt, 6(h) + 0 ash > 0, such that the following
condition holds
lữ +f¿(,£(9)|Y (,e(e)) — Y( + hịg(a))||< h: ố{h) for t € |a,b — hị;
vi) The function ¢ in the boundary condition has Lipschitz continuous partial derivatives
llới(, ì) — 1 (23, yo) ll S Ki (| — 51] + Ilya — voll); W4o(%, yx) — #i(#;, we)|| < K2(|# — a5l] + lyn — yoll)- Moreover |la(s, w)|| < K› Then for a sufficiently small h, the shooting method (6) is convergent Let Uh = yn(tni ng) — P(tn); T= Ya(tn, $n9 [80,5 — 9(@)] — wh; Ta := hf (tas @(ta)) + (ta) — (tai); AR = T+ hfi (tn, ualtns sny))s
đi = Ya(tns 81,3) — ¥ (tn, 9(a))3 Cr = MO“); Cp = (Cy — 1)/L- Introducing the norm ||¿|| = max: et) lu(t)||, : u € C(J,R”) and using the Gronwall-
Trang 4Lemma 2 Under the conditions in the theorem 1, we have a) ell < Cull + Gach)
ii) TRI S Ca: v(t, Heals
it) [lah] < C2: [Cys Q(Cy [fed] + Cse(h)) + 6(A)], where v(h, €) = A(C\E + Cze(h))(Ci€é + Cse(h)) + c(h)
Proof The proof is carried out analogously to the proof of the convergence theorem in
L1 ñ
Lermma 3 Lef
Quíu) = 6i (u, (bị )) + óa(u, (bị ))Yn (b, 4); Q(u) = br (u, y(b; u)) + óa(u, y(b:4)) Y (b, u)
Then
Qn(sna)es! = beTh, — l@(Œ) — O25) — ở (8) = xj)},
tphere #j = (sa,j Wh(b; sh,j));# = (@(4),€(b)) and te hauc an estimate l2) ~ ø(+,) = ở (,)(Œ ~ z;)|I < Called? + ah) Gl + a),
tphere cị: H — l.,: c¡(h) — 0 as h — 0,:i= 1,2 and C3 is a nonnegative constant Proof The above-mentioned relation is easily obtained Furthermore,
lø(Œ) - #(&;) ~ #'(/)Œ — #j)|l = II | [6 (xy + (@ — 25) — # (x5)|(@ — 25)
IA ew 4 ;ilg- ll?
A
S Fons ~ eCa)ll + ly(bs 90.3) = e(8)|)?
< Giljl? +e()ljl + ea(®) 0
Now, we go on proving the theorem 1 It is known that, assuming that ¢ is an isolated
sohition of (1) then the matrix Q(y(a)) is nonsingular(see, for example, [5]) Let ||Q7'(e(a))|| < 6) From Lemma 3 and condition (vi) in theorem 1, it follows
that
HQn(sn.3) — Q((2))|| = llới (sà,¿, (Bi sẽ,;)) — ớì ((8), #())+
Trang 5l-Therefore
Q7! (Pa) )Qn(sn.5) ~ Q(¢(a))]I S Bi{ Milled] + Mae(h) + Koll ll} < A{M, |ludl| + Moe(h) + KyC2[C,Q(Cy [val] + Coe(h)) + 6(A)]} =: P2) — (8) Since Q is continuous and 2(0) = 0, one can choose p = ||v$|| = ||so — #(4)|| << 1, such
that
Po(h) Sa < 1 and Me+7 "qa0@) +o9(Œ\ø) <Ø <1 for h << 1, (9)
where 9 = 3) KyC2C,/(1— a), M = ‘maxi M,, My = 8,C3/(1 — a)}
Using Lemma 4.4.14 from [1], one conclides that the matrix
14+ Q7!(¢(a))[Qu(sn,o) — Q(¢(a))]
is nonsingular Furthermore, Q),(so) is nonsingular and {Qj '(so)|| < G1/(1-a) Applying Lemma 3, one has
Hebl) < HQn(sn.0)~ "I {Ilbe(sn,0,4n( i sn o))I TRI + [o(z) — O25) — ở (z)Œ = z))l}
Pr ex(h) = uh, (B= A KeCa/( = a)), a ¢ 3 <8: u(h,p) + MpỀ + I “na + It can be proved that there exists a function €, : €(h) > 0 as h - 0 such that 8 + —a(0) -a set 3 8 Well < uh = Bu(h, p) + Mp? + ra palh) +7 < BopU(Cip) + Mp? # a Peat) + 6(h) <ap+c(h) < p:= ub le h “<1 By an induction and argue as above, one sees that Q7'(sp,j) is nonsingular Moreover a fog supe! <i, x *(sas)Il S mm
where uit! = By(h, uj) + Ms)? + Tượng tế, cị (h) + Py a(h)
The sequence {u}}, is nonincreasing and nonnegative, so up, = limjsoo uj, exits h 6 i h and up satisfies the equation
_ > 8
un = Bo(h, up) + Muy)? + = un ex(h) + A eo(h) 1
Moreover, if u, + u as h > 0, then u is a non-negative solution of
u = BQ(Cyu)(Cyu) + Mu? (10)
Since 0 < uy < ue = p, it implies u < p The estimate JoQ(C,u) + Mu < & < | ensures
that u = 0 is an unique solution of (10) This and (i) in Lemma 2 yield the
Trang 63 Numerical Experiments
In this section, we consider three examples In each example, step sizes and an initial vector are Lê ‘The convergence of the method in each example is expressed by
maxlyj() — y2~'(4,)|, where yp (t:) denotes yp (t) 8ny)-
Example 1 Consider the problem
y! = —4ty"/?, te (0,1),
‘ 1 1
sin(y(0)) + cos(y(1)) — sin avy — cos B5: =
This problem has been studied in [4] with the linear boundary condition (0) — 2(1) = In both cases, it has an exact solution y(t) = (?+1+V0))~ ? and y(0) = 0.1715 Tae
Furthermore, Silty = —6ty'/? is not bounded However, it is bounded in a bounded
neighbourhood of the isolated solution We can put Q(v) = 6v'/? Thus Q and the
function in the boundary condition satisfy the conditions in the theorem 1 N 8ÿ Time Accuracy 10 | 0.17173830227481 | O.11s 10-4 40 | 0.17161116830044 | 0.275 10-5 150| 0.17158289850619 | 0.885 TÚ”
Table 1: Numerical result of example 1 with initial vector so = 0.5 Starting from the initial vector so = 0.5 we get the improved shooting points given
in Table 1 The convergent speed of the method (N = 10) is given in Table 2 j max|yh (ts) — "(tI 1 5.00 x 107? 3 2.38 x 107? 4 6.07 x 1075 5 4.07 x 1071° 6 ` 111x10"!19
Table 2: The convergence of the method in example 1 with N = 10 Example 2 Consider the problem
Trang 7This problem has an exact solution y = e~', : z = —e~t and y(0) = 1,: z(0) = —1 It is
easy to show that the function in the boundary condition satisfies all the conditions in the theorem 1 N 3;(1) 3;(2) Time Accuracy 30 1 —0.99462269847685 1.81s 10-? 250 1 —0.99937163278951 2.588 10-4 2000 1 —0.99992170530796 10.93s 10-5 Table 3: Numerical result of example 2 with initial vector sọ = [1.5; —1.5] j max|y} (ts) — gh (ti) | max|zj (ti) — 2471 (ti)| 1 1.45 x 10° 1.45 x 10° 5 6.39 x 107? 9.82 x 1072 7 8.44 x 1078 1.26 x 1075 8 5.82 x 10714 8.62 x 10-11 9 7.22 x 10716 7.22 x 10716
Table 4: The convergence of the method in example 2 with N = 30 Choose the initial vector so = (1.5; —1.5] we have Table 3 The convergent speed of
the method (N = 30) is given in Table 4 Example 3 Consider the problem 3 y= 59", te (0,1), e70-00147(0) — 70.016 _ 0, e70-001y7(1) _ ¿7.0001 _ g Put z = y’, the equation becomes HỆ = [say te [0,1]
The partial derivative As is not bounded in the whole space but it is bounded in a neigh-
Trang 8This problem has an exact solution y = 4/(1 + t)? and y(0) = 4,: 2(0) = —8 Choose the initial vector sp = [4.5;—-8.5] we have Table 5 The convergent speed of the
method (N = 40) is shown in Table 6 j max|y, (ts) — ¿` (R)| max|z}(t,) — z¿—'(H)| 1 3.99 x 10° 7.54 x 10° 5 7.81 x 107? 1.60 x 107? 7 7.07 x 1078 1.44 x 1075 8 1.73 x 10719 3.50 x 10720 9 5.60 x 10-14 1.09 x 10718 Table 6: The convergence of the method in example 3 with N = 40 References uì | (3 ( a of = a (6) (7 {8 l9] I0
Ascher, U., Mattheu, R., Russel, R., Numerical Solution for Boundary Value Prob-
lems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, New
Jersey, 1998
Bernfeld, Stephen R., Lakshmikantham, V., An Introduction to Nonlinear Boundary Values Problems, Academic Press, Inc, 1974
Chow, K L., Enright, W H., Distributed Parallel Shooting for BVPODEs, Tech- nical Report of Department of Computer Science, University of Toronto, 1995 Jankowski, T., Approximate Solutions of Boundary-Value Problems for Systems of Ordinary Differential Equations, Comp Maths Math Phys., Vol 35, No 7(1995), pp 837-843
Keller, H B., Nummerical solution of two-point boundary value problems Soc Industr and Appl Math., 24(1976), 1-61