A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS

Transcription

1 Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet of Mathematics Faculty of Sciece Uiversiti Tekologi Malaysia 8110 UTM Skudai Johor Darul Ta zim Ibu Sia Istitute for Fudametal Sciece Studies Faculty of Sciece Uiversiti Tekologi Malaysia 8110 UTM Skudai Johor Darul Ta zim 1 tehyuayig@yahoo.com.sg y@mel.fs.utm.my orma@ibusia.utm.my Abstract : I this report we have developed a ew class of -step ratioal multistep methods (RMMs) of a secod to a fifth order of accuracy. We have preseted the derivatios of these RMMs as well as the local trucatio error ad stability aalysis for each RMM that we have derived. Numerical experimets have show that all RMMs preseted i this report are suitable to solve iitial value problem of various dimesios ad also stiff problems. Keywords: ratioal multistep methods. 1 Itroductio Amog special umerical methods based o o-polyomial iterpolats we are particularly iterested to a special umerical method based o ratioal fuctios. We address this kid of method as ratioal method. Our ratioale is that ratioal methods ca solve a variety of problem icludig o-stiff problem stiff problem ad most importatly solvig problems whose solutios possess sigularities. Various formulatios of ratioal methods ca be foud i the followig articles or texts: [1 11]. However all of the ratioal methods metioed i these articles are all oe-step methods. Therefore we have oticed the lack of study o multistep method that based o ratioal iterpolat. I view of this we shall explore the possibility of derivig effective multistep method that based o ratioal iterpolat. We have addressed multistep methods that based o ratioal iterpolat as ratioal multistep methods or i brief as RMMs. I this report let us cosider the iitial value problem give by y f x y y a y f x y x a b (1) where f is assumed to satisfy all the coditios i order that (1) has a uique solutio. Suppose that we have solved (1) umerically up to a poit ad have obtaied a value as a approximatio of yx that o previous trucatio errors have bee made i.e. y yx as the approximatio of yx solutio yx of (1) give by where x which is the theoretical solutio of (1). From [1] ad [1] assumig the localizig assumptio j a j 01 k derivatives of yx. y we are iterested i obtaiig y. For that purpose we suggest a approximatio to the theoretical y k j0 ah j b h j bh 0 () are parameters that may cotai approximatios of y x ad higher RMM () is defied as -step p-th order or i brief as (p) with p. With the i () we associate the differece operator L defied by k j ; j () j0 L y x h y x h b h a h 010 Jural Karya Asli Loreka Ahli Matematik Published by Pustaka Ama Press Sd. Bhd.

2 where y x is a arbitrary fuctio cotiuously differetiable o x a b Yaacob et all. Expadig y x h as Taylor series ad collectig terms i () gives the followig geeral expressio: 0 1 k k 1 ; L y x h C h C h C h C h 0 1 k k1. () We ote that the C i () cotais correspodig parameters that eed to be determied i the derivatio processes. Therefore the order ad local trucatio error of based o () are defied as follows. Defiitio 1 The differece operator () ad the associated ratioal multistep method () are said to be of order pk 1 if i () C0 C1 Ck 1 0 Ck 0. Defiitio The local trucatio error at of () is defied to be the expressio L y x ; h x. The x give by () whe yx is the theoretical solutio of the iitial value problem (1) at a poit local trucatio error of () is the k k Ly x; h Ckh Oh. (5) -step Secod Order I order to derive a secod order we have to take k 1 i () expad y x h series ad the the followig expressio is obtaied: L y x; h a0 by x ha1 y x byx h yx by x (6) h y x by x O h. Followig from Defiitio 1 ad () it is readily to deduce that: C0 a0 by x C1 a1 y x byx C yx by x C y x by x. Imposig that C0 C1 C 0 we obtai a system of three simultaeous equatios which have the followig solutios: yx yx yx yx b a0 a1 y x. (7) y x y x y x If we substitute the parameters i (7) ito the we have Whe x i.e. C yx y x C y x. (8) y x y x is ow take as the theoretical solutio of the iitial value problem (1) at a poit y x y x where y yx ad the (7) may be writte i the form m m y y y y b a0 a1 y (9) y y y y y x m 1 by the localizig assumptio. O takig k 1 () becomes a0 a1h y bh 0. (10) b h We idicate (10) based o (9) as () give i the form of h y y y y hy provided y ad y do ot vaish simultaeously. From Defiitio ad (8) the local trucatio error (i brief as LTE) of () is give by ito 7

3 Jural KALAM Vol. No. Page 6-9 where m m y y x yy LTE() Ch Oh h y Oh y by the localizig assumptio. This LTE aalysis has cofirmed that m 1 () is a secod order method. If we apply () to the Dahlquist s test equatio yieldig the differece equatio y y Settig z Re 0 h y ad The roots of (1) are give by y h y y. (11) 1 h i (11) yield the characteristic equatio z 0. (1) 1 z 1 1 z 1 z 1() ad () 1 z 1 z. The regio of absolute stability of () is the regio of the complex plae for which 1() 1 ad () 1 hold. By takig i the roots of (1) we have plotted the regio of absolute stability of () i Figure 1. z x iy Figure 1 Stability regio of () The shaded regio i Figure 1 is the regio of absolute stability of () where the coditios: 1() 1 ad () 1 are satisfied. From Figure.1 we ca see that the regio of absolute stability of () cotais the whole left-had half plae which show that () is A- stable. -step Third Order I order to derive a third order we have to take Imposig that C0 C1 C C 0 we obtai a system of four simultaeous equatios which have the followig solutios: 8 k series ad the the followig expressio is obtaied: L y x; h a0 by x h a1 y x byx h a y x by x i () expad yx h 5 h y x by x h y x by x O h. Followig from Defiitio 1 ad () it is readily to deduce that: C0 a0 by x C1 a1 y x by x C a yx by x C y x by x C y x by x. ito (1)

4 Yaacob et all y x y x y x y x y x y x b a0 a1 y x a yx. y x y x y x y x If we substitute the parameters i (1) ito C the we have x y x y x C y x. (15) y x Whe y x is ow take as the theoretical solutio of the iitial value problem (1) at a poit i.e. yx yx the (1) may be writte i the form where y y y y y y y b a0 a1 y a y. (16) y y y y m m y x ad y y x m 1 by the localizig assumptio. O takig () becomes a0 a1h ah y b h. (17) We idicate (17) based o (16) as () give i the form of 6h y y y hy y hy provided ad do ot vaish simultaeously. From Defiitio ad (15) LTE of () is give by 5 yy 5 LTE() Ch Oh h y Oh y m where y m y x m 1 by the localizig assumptio. This LTE aalysis has cofirmed that y y bh0 () is a third order method. If we apply () to the Dahlquist s test equatio y y Re 0 yieldig the differece equatio Settig z h y ad The roots of (19) are give by y k (1) hh y y. (18) h 0 1 i (18) yield the characteristic equatio zz z 0. (19) zz zz 1() ad (). z z The regio of absolute stability of () is the regio of the complex plae for which 1() 1 ad () 1 hold. By takig z x iy i the roots of (19) we have plotted the regio of absolute stability of () i Figure. 9

5 Jural KALAM Vol. No. Page Figure Stability regio of () The shaded regio i Figure. is the regio of absolute stability of () where the coditios: 1() 1 ad () 1 are satisfied. From Figure we ca see that the regio of absolute stability of () is a bouded regio o the left-had half plae which show that () is ot A-stable. -step Fourth Order I order to derive a fourth order we have to take series ad the the followig expressio is obtaied: L y x; h a by x h a y x by x h a y x by x 0 1 k i () expad h a y x by x h y x by x h y x by x O h. 15 Followig from Defiitio 1 ad () it is readily to deduce that: C a by x C a y x by x C a y x by x y x h 5 C a y x by x C y x by x C5 y x by x. 15 Imposig that C0 C1 C C C 0 we obtai a system of five simultaeous equatios which have the followig solutios: y x y x y x y x y x y x y x b a0 a1 y x a yx y x y x y x y x (1) 8y x a y x. y x If we substitute the parameters i (1) ito C 5 the we have x 5 8y x y x C5 y x 15y. () x Whe y x is ow take as the theoretical solutio of the iitial value problem (1) at a poit i.e. yx yx the (1) may be writte i the form y 8 y y y y y y y b a 0 a 1 y a y a y () y y y y y m m y y x ad y y x m 1 by the localizig assumptio. O takig k () where becomes ito (0)

6 Yaacob et all a0 a1h ah ah y bh 0. () b h We idicate () based o () as () give i the form of 8h y y y hy h y y hy provided is give by where y ad y m m do ot vaish simultaeously. From Defiitio ad () LTE of () LTE yy 6 () C5h Oh h y O h 15y y y x m 15 by the localizig assumptio. This LTE aalysis has cofirmed that () is a fourth order method. If we apply () to the Dahlquist s test equatio yieldig the differece equatio y y Settig z Re 0 h y ad The roots of (6) are give by y y h 6h h y. (5) 6 h i (5) yield the characteristic equatio 6 9z 6z z 6 z 0. (6) 6 9z 6z z 6 9z 6z z 1() ad (). 6z 6 z The regio of absolute stability of () is the regio of the complex plae for which 1() 1 ad () 1 hold. By takig z x iy i the roots of (6) we have plotted the regio of absolute stability of () i Figure. Figure Stability regio of () The shaded regio i Figure is the regio of absolute stability of () where the coditios: 1() 1 ad () 1 are satisfied. From Figure we ca see that the regio of absolute stability of () is a bouded regio o the left-had half plae which show that () is ot A-stable. 5 -step Fifth Order I order to derive a fifth order we have to take series ad the the followig expressio is obtaied: k i () expad yx h ito 1

7 Jural KALAM Vol. No. Page 6-9 L y x ; h a0 by x ha1 y x by x h a yx by x h a y x by x h a y x by x h y x by x h y x by x O h Followig from Defiitio 1 ad () it is readily to deduce that: C a by x C a y x by x C a y x by x C a y x by x C a y x by x C5 y x by x C 6 y x by x Imposig that C0 C1 C C C C5 0 we obtai a system of six simultaeous equatios which have the followig solutios: 5y x 5y x y x 5y x y x 5y x y x b a0 a1 y x a y x y x y x y x y x (8) 10y x y x 5y x a y x a y x 5 5. y x y x If we substitute the parameters i (8) ito C 6 the we have x 6 y x Whe i.e. yx yx the (8) may be writte i the form y x (7) 5 y x y x C6 y x. (9) is ow take as the theoretical solutio of the iitial value problem (1) at a poit y 5y y 5y y 5y y 10y y b a a y a y a y y y y y y a where y yx 5 y y 5 y ad m m y y x m 15 by the localizig assumptio. O takig becomes a0 a1h ah ah ah y bh 0. (1) b h We idicate (1) based o (0) as (5) give i the form of 5 10h y 5 5y hy y y hy h y h y provided y ad y do ot vaish simultaeously. From Defiitio ad (9) LTE of (5) is give by y y 7 LTE(5) C6h Oh h y O 5 h 15 9y m where y m y x m 156 by the localizig assumptio. This LTE aalysis has cofirmed that (5) is a fifth order method. If we apply (5) to the Dahlquist s test equatio y y Re 0 yieldig the differece equatio y k (0) () 15 h 18h 8h h y. () 15 6h

8 Yaacob et all Settig z h y ad The roots of () are give by y 0 1 i () yield the characteristic equatio 15 z 18z 8z z 15 6z 0. () 15 z 18z 8z z 15 z 18z 8z z 1(5) ad (5). 15 6z 15 6z The regio of absolute stability of (5) is the regio of the complex plae form which 1(5) 1 ad (5) 1 hold. By takig i the roots of () we have plotted the regio of absolute stability of (5) i Figure. z x iy Figure Stability regio of (5) The shaded regio i Figure is the regio of absolute stability of (5) where the coditios: 1(5) 1 ad (5) 1 are satisfied. From Figure we ca see that the regio of absolute stability of (5) is a bouded regio o the left-had half plae which show that (5) is ot A-stable. 6 Numerical Experimets ad Comparisos I this sectio some test problems are used to check the performace of all ewly derived - step usig differet umber of itegratio steps. We choose the 6-stage fifth order Kutta- Nyström method show i page 1 of [1] as the startig method for -step of order util order 5. We preset the maximum absolute errors over the itegratio iterval give by y x y 0max where N is the umber of itegratio steps; ad absolute errors at the ed-poit N give by y x y y x ad represets the exact solutio ad umerical solutio N N. We ote that y of a test problem at poit. The umerical results obtaied from our ew proposed methods are compared with the umerical results obtaied from the RMMs of [1] ad [1]. These existig RMMs are: -step secod order method which is y y ; () y hy y h y y y -step third order method is y x y y hy y hy hy h yy y hy h y step fourth order method is ; (5)

9 Jural KALAM Vol. No. Page y y y h y y h y y y y h 19 y y y 19y y y y h y y y y y y y yy y y ; ad 5-step fifth order method is 6 5 y y y 10h y y 00h y y y y 1 500h y 6 y 6y y y y y 65h y 6y y y y 8 y y y y y y 6 y h 10 y 0y y y 60 y y y 5 y y y y y y y y y y The startig method used for () (7) is the same 6-stage fifth order Kutta-Nyström method metioed above. It is very clear that all methods i () (7) caot solve problem with iitial value equals to zero. Problem 1 [11] y x 100 y x 99e x y 0 0 x x 100x The exact solutio is give by y x e e Problem [15] y x 101y x 100 y x 0 y y0 01. x. 100x x The exact solutios is give by yx 0.01e e y 1x yx y x 01 ; y x 100 y1 x 101y x y 0 x The exact solutios of this system are give by x x y x yx e e. Problem [11] yx 1 yx y The exact solutio is yx ta x. Problem ca also be writte as a system i.e. x. x x y x y x e e (6) (7) ad. From the exact solutio we otice that the solutio becomes ubouded i the eighbourhood of the sigularity at x Table 1: Maximum absolute errors for various secod order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi () () (-0) (-0) (-0) (-0)

10 Yaacob et all Table : Absolute errors at the ed-poit for various secod order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi () () (-05) (-06) (-07) (-08) Table : Maximum absolute errors for various third order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (5) () (-0) (-0) (-0) (-05) Table : Absolute errors at the ed-poit for various third order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (5) () (-08) (-09) (-10) (-11) Table 5: Maximum absolute errors for various fourth order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (6) () (-0) (-0) (-05) (-06) Table 6: Absolute errors at the ed-poit for various fourth order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (6) () (-10) (-1) (-1) (-1) Table 7: Maximum absolute errors for various fifth order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (7) (5) (-0) (-0) 18.95(-05).7(-05) Table 8: Absolute errors at the ed-poit for various fifth order methods with respect to the umber of steps (Problem 1) N Okosu ad Ademiluyi (7) (5) (-1) (-1) (-1) (-1) Table 9: Maximum absolute errors for various secod order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi () () (-0) 8.570(-0) (-0).190(-0) (-05) 5.68(-05) 5

11 Jural KALAM Vol. No. Page 6-9 Table 10: Absolute errors at the ed-poit for various secod order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi () () (-0).961(-0) (-0) 8.178(-05) (-05).15000(-05) Table 11: Maximum absolute errors for various third order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (5) () (-0) (-0) (-0).8(-0) (-0) (-05) Table 1: Absolute errors at the ed-poit for various third order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (5) () (-0).18(-06) (-0).6079(-07) (-05) (-08) Table 1: Maximum absolute errors for various fourth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (6) () (-0) (-0) (-0).61(-0) 10.51(-0) 8.867(-05) Table 1: Absolute errors at the ed-poit for various fourth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (6) () (-0) (-09) (-06) (-10) (-07).67688(-11) Table 15: Maximum absolute errors for various fifth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (7) (5) 56.01(-0) (-0) (-0).57(-0) (-0) 8.861(-05) Table 16: Absolute errors at the ed-poit for various fifth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (7) (5) (-0) (-11) (-0) (-1) (-0).7588(-1) Table 17: Maximum absolute errors for various secod order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi () () (+01) (+01) (+00) 5.57(+00) (+01).9570(+01) (+01) 9.68(+00) (+01) 9.617(+00) (+01) 8.819(+00) 6

12 Yaacob et all Table 18: Absolute errors at the ed-poit for various secod order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi () () (+01) (+01) 6.610(+00).71(+00) (+00) (-01) (-01) (-01) (-0).8897(-0) (-0) 1.169(-0) Table 19: Maximum absolute errors for various third order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (5) () (+00).059(-01) 1.66(+00).911(-0) (+00) 1.585(-01) (-01) 1.780(-0) (-01) (-0) (-01).0688(-0) Table 0: Absolute errors at the ed-poit for various third order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (5) () (+00).059(-01) 7.168(-01).951(-0) (-0).9076(-0) (-0).58019(-0) (-0).17(-05) (-0) 5.517(-06) Table 1: Maximum absolute errors for various fourth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (6) () (+00) 1.67(-0).5758(-01) (-0) (-01).1198(-0) (-0) (-0) (-0).51557(-05) (-0) (-06) Table : Absolute errors at the ed-poit for various fourth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (6) () (+00) 1.67(-0).6867(-01) (-0) (-0).5610(-05) (-0).8060(-06) (-05) 1.767(-07) (-06) (-08) Table : Maximum absolute errors for various fifth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (7) (5) (+00).55(-0) (-01) (-05) (-0) 1.86(-05) (-0).01(-07) (-0) (-08) 51.19(-05) 7.716(-09) 7

13 Jural KALAM Vol. No. Page 6-9 Table : Absolute errors at the ed-poit for various fifth order methods with respect to the umber of steps (Problem ) N Okosu ad Ademiluyi (7) (5) (+00).55(-0) (-0) 9.665(-06) (-0).8518(-07) (-05) (-09) (-06).7516(-10) (-08) (-11) 7 Discussios ad Coclusios All -step of order util order 5 proposed above have o problem i solvig Problem 1 Problem ad Problem. As expected all RMMs proposed by [1] ad [1] caot solve Problem 1 with iitial value equals to zero while all do ot face such difficulty. Next i solvig Problem we observed that () gives smaller maximum absolute errors ad smaller absolute errors at the ed-poit compare to all existig RMMs of order ad order 5 usig ay umber of itegratio steps. Lastly we also foud out that all accurate tha existig RMMs of [1] ad [1] i solvig Problem as the superiority of ca be observed from Table 19 Table. I coclusios all of differet orders ca be used to solve iitial value problems of differet dimesio due to less computatioal effort sice they are all explicit methods. I this report we have showed the existece of -step variable order. If -step variable order do exist it is reasoable to deduce that r-step variable order are possible as well. From () we geeralize -step to r-step give by k j ah j j0 yr b h bh 0. (8) With the r-step i (8) we associate the differece operator L defied by where y x k j ; j (9) j0 L y x h y x rh b h a h is a arbitrary fuctio cotiuously differetiable o x a b y x rh as Taylor series ad collectig terms i (9) gives the followig expressios: 0 1 k k 1 ; L y x h C h C h C h C h. Expadig 0 1 k k1. (0) We ote that the C i (0) cotais correspodig parameters that eed to be determied i the derivatio processes. Therefore the order of r-step based o (8) is defied as follows. Defiitio The differece operator (9) ad the associated ratioal multistep method (8) are said to be of order pk 1 if i (0) C0 C1 Ck 1 0 C 0. I additio the local trucatio error of r-step p-th order based o (8) is defied as follows. Defiitio The local trucatio error at x rof (8) is defied to be the expressio L y x ; h x. give by (9) whe yx is the theoretical solutio of the iitial value problem (1) at a poit The local trucatio error of (8) is the k k Ly x; h C k h O h. (1) From Defiitio ad Defiitio we have oticed that the order of accuracy of a r-step is ot affected by the umber of step r. I other words there exist -step of order ad eve 5-step of order. However i the sese of cheaper computatioal cost but higher accuracy we foud that a with r greater tha the order possessed has less practical use. Below we show those r-step which have more value i computatioal practice i Table 5. k 8

14 Yaacob et all Table 5: Potetial r-step of order p r p Refereces [1] Lambert J. D. (197). Computatioal Methods i Ordiary Differetial Equatios. Lodo: Joh Wiley & Sos Ltd. [] Lambert J. D. (197). Two Ucovetioal Classes of Methods for Stiff Systems. I Willoughby R. A. (Ed.) Stiff Differetial Equatios. New York: Pleum Press. [] Wambecq A. (1976). Noliear Methods i Solvig Ordiary Differetial Equatios. Joural of Computatioal ad Applied Mathematics. (1) 7. [] Fatula S. O. (198). No Liear Multistep Methods for Iitial Value Problems. Computers ad Mathematics with Applicatios. 8() 1 9. [5] Fatula S. O. (1986). Numerical Treatmet of Sigular Iitial Value Problems. Computers ad Mathematics with Applicatios. 1B(5/6) [6] Va Niekerk F. D. (1987). No-liear Oe-step Methods for Iitial Value Problems. Computers ad Mathematics with Applicatios. 1() [7] Va Niekerk F. D. (1988). Ratioal Oe-step Methods for Iitial Value Problems. Computers ad Mathematics with Applicatios. 16(1) [8] Ikhile M. N. O. (001). Coefficiets for Studyig Oe-Step Ratioal Schemes for IVPs i ODEs: I. Computers ad Mathematics with Applicatios. 1(001) [9] Ikhile M. N. O. (00). Coefficiets for Studyig Oe-Step Ratioal Schemes for IVPs i ODEs: II. Computers ad Mathematics with Applicatios. (00) [10] Ikhile M. N. O. (00). Coefficiets for Studyig Oe-Step Ratioal Schemes for IVPs i ODEs: III. Computers ad Mathematics with Applicatios. 7(00) [11] Ramos H. (007). A No-stadard Explicit Itegratio Scheme for Iitial-value Problems. Applied Mathematics ad Computatio. 189(007) [1] Lambert J. D. (1991). Numerical Methods for Ordiary Differetial Systems. Chichester: Joh Wiley & Sos Ltd. [1] Okosu K. O. ad Ademiluyi R. A. (007a). A Two Step Secod Order Iverse Polyomial Methods for Itegratio of Differetial Equatios with Sigularities. Research Joural of Applied Scieces. (1) (007) [1] Okosu K. O. ad Ademiluyi R. A. (007b). A Three Step Ratioal Methods for Itegratio of Differetial Equatios with Sigularities. Research Joural of Applied Scieces. (1) (007) [15] Yaakub A. R. ad Evas D. J. (00). New L-stable Modified Trapezoidal Methods for the Iitial Value Problems. Iteratioal Joural of Computer Mathematics. 80(1)