Some experimente with interval shooting methods for two point boundary value problems in ordinary differential equations

Pub . Ma . UAB N° 22 No . 1980 Ac es VII JMH1 SOME EXPERIMENTE WITH INTERVALSHOOTINGMETHODS FORTWO POINT BOUNDARY VALÚE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS . Ma ía Raquel G .P . Valenga A ea de Ma emá ica Uni e sidade do Minho Abs ac The pu pose o his communica ion is o gi e some discussion, suppo ed by nume ical esul s o in e alshoo ing echniques o 2 p B .V .P . wi h  a commen a y on he ela i e ad an ages and disa an ages o hese me hods . 1 . The Aim o In e alAnalysis The aim o in e al analysis ís he simul aneous machine compu a ion o gua an eed bounds on he exac solu ion o a p oblem . Quan i ies a e ep esen ed by in e als con aining hei exac alue and echniques a e de eloped such ha h oughou compu a ion ound-o , un- ca ionand p opaga ion o ini ial e o a e all accoun ed o . 2 . In e al A i hme ic In he sequel we use he no a ion x I o an in e al wi h lowe and uppe bounds x_ , x such ha xI=  V x  , x i  ,  x 6 x  x_ .x  E  . The wid h o an in e al is deno ed by W(X I ) = x - x " A i hme icope a ions wi h in e als a e de ined in such a way ha he esul ing in e al will always con ain he exac esul o he ope a ion, Moo e (3) In a ious con ex s we shall ha e o compu e in e als which a e unc ions o in e als,Moo e (3) . 3 . Shoo ing Me hods o Linea P oblems We conside i s he Linea p oblem o second o de gi en by Y''=p(x)y'+q(x)Y+ (x),x E [a,b]  (3 .1) oge he wi hbounda y condi ions a ly(a) - a2y'  (a) =d- , bl Y(b)+b 2 y'  (b) =/ 1 3 (3 .2) whe e ai, bi, i=1,2,y'(a), y'(a),y(b),y'(b),o¿ .q3  may deno e in e als . We assume ha p (x) , q(x) and (x) a e con inuous on (a,b) and ha he homogeneous p oblemde i ed om (3 .1), (3 .2) has only he i ial solu ion y(x) =0 in which case (3 .1), (3 .2) has a unique solu ion . In heo y we can ind his by combining he solu ions o wo ini ial alue p oblems o sa is y he pai o bounda y condi ions . A s aigh , o wa d in e al ex ension o he classical shoo ing echnique was de eloped,Valenga (5) . The associa ed ini ial alue p oblems a e sol ed by in e al ini ial alue me hods,Moo e (3), Valenga (5), Hansen (2) . An es ima e o he o de o accu acy ob ained is ound Valenga (5) in e ms o he o de o he in e al ini ial alue me hods used . h=0 .1 max W'(Y I (x)) = Nhk Whe e N is a cons an , h he in eg a ion s ep and K he o de o he ini iál alue me hóds . Example 3 .1 . Y(2) =0 .Y(3)=0 . A ou h o de in e al ini ial alue me hod p oducedwi h w(y I )< 2 .0x10 -5 max y (x ) :z 0 . 55 The co esponding esul s wi h h=0,2 had a wid hwhich we e a mul iple e y close o 14 .6 o hose o h=0 .1 . This gi es easonable p oo o 0(h 4 ) accu acy .  ' In ce ain cí cuns ances, howe e he compu edin e al solu ionsmay la gely o e es ima e he e o in he solu ion . The me hod is uns able i he ini ial alue p oblems ha e solu ions ha inc ease apidly in absolu e alue while he solu ion o he bounda y alue p oblem emains almos s a ione y . Example in Valenga (5) . I he ini ial alue p oblems ha e oscilla o y solu ions a echnique o educe heso called "w apping e ec ",Moo e (3), mus be used . We no e howe e ha his is somewha unusual o well condi ioned bounda y alue p oblems . 4 . Shoo ing Me hods o NonLinea P oblems . We conside he nonlinea di e en ial equa ion o second o de y,'= (X,y,y,  (4 .1) oge he wi h linea bounda y condi ions (3 .2) We assume   con inuous and Lipschi zian onAaja,<x ~ b,y?+y'<~  Y  )0, I  Ó  C  M on R ; a ¡ - >, O,b i > 0,i=1,2, a l +a2 7 0, b 1 +b 2 7 0, a l +b l 70 . a y~ An i e a i e scheme which is essen ially an in e al ex ension o he 1 classical shoo ing echnique was de elopd Valenga (5) and condi ions o con e gence s a ed . Example 4 .1 max y(x )y0 .13 ,x=0 .5 . 5 . Conclusions y''=-1-0 .49y'2 , y(0)=0,y(1)=0 . Wi h s a ing in e als o wid h o o de 10-1 and h= 0 .1 using a ou h o de in e al ini ial alue me hod he algo i hm con e ges apidly p oducing a e h ee "s eps" an in e al solu ion o max w(y1)<3.5x10-5 In e alshoo ing me hods a e na u ally dependen on he pa icula i ies o in e al ini ial alue me hods . We we e able o compu e accu a eboundinj solu ions o some p oblems whe eas in o he cases e ypessímis ic bounds can be ob ained . One o he causes o he bad esul s p oduced in hese cases occu s in he classicalme hod as well . P ecau ions mus also be aken when he associa ed ini ial alue p oblems a e such ha he in luence o he "w apping e ec " is conside able . We no e ha mo e es ic ionsa e imposed wi h in e alshoo ing me hods han wi h he co esponding classical me hods bu his a e all may be he p ice o he knowledge o gua an eed e o bounding solu ions . BIBLIOGRAPHY : Fox, L, and Valenga M .R "Some expe imen s wi h In e al Me hods o 2 p B .V .P . in O .D.E ." o be published (2) Hansen, E . (ed) "Topics in In e al Analysis"Cla endon P ess . Moo e, R . E . "In e al Analysis" P en iceHall Nickel, K . "The applica ion o In e al Analysis o he Nume ical Solu ion o Di e en ial Equa ions" - In e ne Be ich des Ins . F . In o ma ik 69/9 . Uni e si ae Ka ls uhe Valenga, M . R . "In e al Me hods o O dina yDi e en ial Equa ions" D .Phil Thesis, Ox o d (1978) . A pa icipagáo da au o a nas VII Jo nadas Ma emá icas oi subsidiada pelo Ins i u o Nacional de In es igagáoCien i ica,Po ugal .