Symmetric implicit multiderivative numerical integrators for direct solution of fifth-order differential equations

Transcription

1 Tammasat Iteratioal Joural of Sciece ad Tecology Vol.9, No., April-Jue 04 Symmetric implicit multiderivative umerical itegrators for direct solutio of fift-order differetial equatios S. J. Kayode* Departmet of Matematical Scieces, Scool of Scieces, Federal Uiversity of Tecology, P. M. B 70 4, Aure, Nigeria Abstract A direct metod of solutio of fift order ordiary differetial equatios (odes) is proposed i tis paper. Collocatio of te differetial system is tae at selected grid poits to reduce te umber of fuctios to be evaluated per iteratio. A umber of predictors of te same order of accuracy wit te mai metod for te estimatio of y -fuctios ad teir derivatives i te mai metod are geerated. Te symmetric implicit multiderivative algoritm [SIMA is suitable for umerical itegratio of o-stiff ad mildly-stiff fift order equatios. Test eamples are solved wit te metod to cofirm its efficiecy. Keywords: Symmetric; multiderivative; implicit; iterval of absolute stability; error costat.. Itroductio I tis article, a direct solutio of fift order ordiary differetial equatios (odes) of te form () () 4 y f (, y, y,..., y ), [ a, b () togeter wit iitial coditios () i y ( a) yi, i 0()4, is eplored. Tis class of problems as a lot of applicatios i applied scieces, mecaical, civil ad aerospace egieerig especially were vibratio of structures due to passage of movig loads uder dampig forces are of paramout importace. Abar ad Siddique [ ivestigated te aalytical solutio of fift order wealy oliear oscillatory systems of type (). I practice, te umerical itegratio process of problem () ivolves a reductio to systems of first order equatios wic may te be solved wit ay ow metods for first order equatios. Tis approac as its ieret setbacs, Awoyemi [. Differet autors ave proposed metods for direct solutio of iger order problems. Awoyemi [, Parad ad Hoati[, Saravai ad Mirraei [4, Golbabai ad Arabsai [, Kayode[6, [7, all cosidered secod order equatios usig differet step-legts ad step-umbers to obtai metods of various order of accuracy. Kayode ad Adeyeye [8 ad Kayode ad Obarua [9 also developed umerical ybrid metods for direct solutio of geeral secod order differetial equatios by usig differet basis fuctios to geerate collocatio ad iterpolatio equatios,. I teir wor, Awoyemi ad Idowu [0 cosidered direct umerical solutio of tird order equatios. Awoyemi [ ad Kayode [ ivestigated te direct solutio of fourt order differetial equatios. However, all tese metods are ot suitable to solve fift order problems of type () witout reductio to lower order problems. Kayode ad Awoyemi [ proposed a -step *Correspodece: suyay06@gmail.com; sayode@futa.edu.g

2 Tammasat Iteratioal Joural of Sciece ad Tecology Vol.9, No., April-Jue 04 multiderivative collocatio metod i a predictor-corrector mode for problem (). Te metod is o-symmetric ad may ot be suitable for oscillatory problems. Kayode [4 proposed a eplicit metod for solvig () directly. Te aim of tis article is to propose a symmetric implicit multiderivative algoritm [SIMA wit icreased stepumber as well as icreased order of accuracy (p) for te approimate solutio of () directly.. Metodology Te basis fuctio for te approimate solutio of problem () is tae to be te partial sum of power series of a sigle variable i te form were y( ) () 0 m R, yc ( a, b) P( ). Te fift derivative of () substituted i () is give as () () () (4) ( )( )( )( 4) f (, y, y, y, y, y ) 0 () Collocatig () at, 0() ad iterpolatig () at, 0() yield te followig collocatio ad iterpolatio matri A T 0, X T 6 4, B f f f y y y y y y T is te matri traspose, iv f f (, y, y, y, y, y ), r 0,,,..., r r r r r r r y i y( ). i Solvig te matri (4) for ' s ad usig t simplificatio, te cotiuous metod i () yielded, after 4 () y ( t) ( t) y ( t) f. 0 0 Te cotiuous coefficiets () t ad () t i (), we is tae to be 6, are foud to be: 00 0( t) [600t 908t 846t t 60t 8t 6 t t, were AX B (4) 040 ( t) [40t 6798t 796t t 60t 8t 6 t t, 0 ( t) [40t 8t 6t 0t 60t 8t 6 t t, 0 ( t) [960t 86t 46t 89t 60t 8t 6 t t, 040 4( t) [660 t 7t 896t 68t 60t 8t 6 t t, 00 ( t) [ t 764t 044t 47t 60t 8t 6 t 8 t, 4 6 7

3 Vol.9, No., April-Jue 04 Tammasat Iteratioal Joural of Sciece ad Tecology 7600 ( t) [90t 7044t 66668t 79t 4760t 4t 48t t, ( t) [490t 98964t 7876t 76t 40t 4t 86t 8 t, ( t) [040t 068t t 6769t 960t t 8 t t. A discrete implicit multiderivative form ad its derivatives arisig from (), we t, are obtaied to be y6 4y y4 y 4 y y [ f6 f f 7 p = 7 ad cp is ; (6) y 6 [6 y 7 y4 440 y 660 y 4440 y 06 y 40 4 [76 f6 670 f 707 f, 780 p = 7, cp ; (7) y 6 [ y y4 y y y y [9 f 776 f f, p = 7, cp ; 6 0 (8) y 6 [ 46 y 9 y4 0 y 660 y 480 y y 0 87 [89 f 4 f f, p = 7, c p ; (9) iv y6 4 [ 0 y 0 y4 0 y 0 y 0 y y 0 06 [069 f 780 f f, p = 7, c 0. p 97. Defiitios: [Lambert [ (0) (a) Order ad error costat Let L be te liear differece operator associated wit () give by L 0 v [ y( ); [ y( ) y ( ) () were y ( ) is a arbitrary fuctio, cotiuously differetiable o a iterval [a, b. Assumig tat y ( ) as as may iger derivatives as may be required, te epadig () by Taylor series gives () () ( ) L p 0 p [ y( ); C y( ) C y ( ) C y ( ) C y ( ) () were C, 0 0 C, p C ( ) ( 0 ),! p p p p Cp ( ) ( ), p,4,. p! ( p )! Te order of accuracy of () is p if C C C C C 0, C 0 0 p p p C ad its error costat is p. Applyig te defiitios of order ad error costat above to (6), (7), (8), (9) ad (0) produces order of accuracy p = 7 for eac of tem ad error costats Cp to be , , , 0.098, 0.97, respectively. (b) Symmetry, Cosistecy zero stability ad covergece Cosider te caracteristic equatio associated wit () give by ( r, ) ( r) ( r),, () f were is te eigevalue(s) of te y Jacobia of (). ad are te first ad secod caracteristic polyomials of () respectively. Tey are give by

4 Tammasat Iteratioal Joural of Sciece ad Tecology Vol.9, No., April-Jue ( r) r r 4r r r 4r 0 6 ( r) r ( r r ) 0 7 (4) () y 6 [704 y y4 70 y 640 y 99 y 4080 y 40 4 [9 f 64 f 9 f, 40 p = 7, cp , () Te liear multistep metod () is cosistet if te order of accuracy p, r 0, 0 ( r) ( r) ( r) ( r) 0 ad ( r )! ( r ) for te pricipal root r =. Zero stability property of a liear multistep metod () requires tat te roots of (4) must satisfy r, ad every root wit r must ave multiplicity 6. Te ecessary ad sufficiet coditios for a liear multistep metod to be coverget are cosistecy ad zero stability, [see Herici [6 for te proof. Applyig tese defiitios to (6) sows tat te metod is symmetric, cosistet, zerostable ad coverget.. Geeratig te Startig Values Implemetatio of te implicit discrete metod (6) requires some startig values for te evaluatio of ( r y ), ()6; r 0()4 i te f fuctio. I tis paper, te eplicit metod of order seve of Kayode [4 is adopted as te mai predictor to implemet y 6 ad its required derivatives. Tese are reproduced as follows: y6 4y y4 y 4 y y [ f 0 f f 6 p = 7 ad cp 0., () y [078 y 600 9y 8760 y 870 y 497 y 060 y 6 4 [749 f f 4760 f, p = 7, c 0. p 80064, () y 6 [ y 6 y4 70 y 080 y 6 y 690 y 0 [0f 0 f 7 f, 600 p = 7, c 0. p , (4) iv y 6 4 [49 y 0 y4 40 y 400 y 4 y 8 y [89f 6 f 40 f, 600 p = 7, c p = () ( r) Te startig values for y, 0,,...,4 r i Kayode ad Awoyemi [9 are adopted for f i () (). 4. Numerical Eamples Two o-liear umerical eamples are solved to demostrate te accuracy ad usability of te ew metod. Problem () () () (4) () () y y y yy y y 8 ( ) e, 0, y y y y y () () () (4) (0), (0), (0), (0), (0), 0.0. Teoretical solutio is y() = e. 4

5 Vol.9, No., April-Jue 04 Tammasat Iteratioal Joural of Sciece ad Tecology Table. Results of problem. -value y-eact y-computed Absolute Errors Errors i [ e-0.06e e e e e e e e e e e e e e e e e e e e e e e e e e e e-007 Problem () () () () () y 6{( y ) 6 yy y y y },, y y y y y Teoretical solutio is y( ). () () () (4) (), (), (), () 6, () 4, 0..

6 Tammasat Iteratioal Joural of Sciece ad Tecology Vol.9, No., April-Jue 04 Table. Results of problem. -value y-eact y-computed Absolute Errors Errors i [ e e e e e e e e e e e e e e e e e e e e e e e e e-00.96e e e-00.67e-004 6

7 Vol.9, No., April-Jue 04. Coclusio A si-step collocatio metod as bee proposed to directly solve fift order odes of te form () witout te burde of reductio to system of lower order problems. Te metod ad its associated derivatives are of order seve. Te startig values ad teir associated derivatives for te implemetatio of te metod are obtaied to be same order seve. Two oliear fift order problems were solved wit te ew metod as test problems. Te accuracy of te results is compared wit Kayode [4 as sow i Tables ad above. Te results obtaied sowed te usability of te metod for solvig fift order odes especially oliear problems. However, stiff problems ave ot bee addressed i tis wor. I future researc wor, efforts will be directed at proposig umerical metods capable of solvig geeral stiff fift order odes directly. 6. Refereces [ Abar, M.A. ad Siddique, T.A., Perturbatio solutios of fift order oscillatory oliear system. Noliear Aal. Mod. Cotrol, Vol. 6, No., pp. 4, 0. [ Awoyemi, D.O., A P-stable liear multistep metod for solvig geeral tird order ordiary differetial equatios. It. J. Comp. Mat. Vol. 80, No. 8, pp , 00. [ Parad, K. ad Hoati, G., Solvig volterra s populatio model usig ew secod derivative multistep metods. Am. J. Appl. Sci. Vol., pp. 09 0, 008. [4 Saravai, M.F. ad Mirraei, S.R., Numerical solutio of liear ordiary differetial equatios i quatum cemistry by Clesaw Tammasat Iteratioal Joural of Sciece ad Tecology metod. World Acad. Sci. Eg. Tecol., Vol. 49, pp. 0 04, 009. [ Golbabai ad Arabsai, M. M., Gettig started wit Numerov s metod. Comp. Pys., Vol. 0, No., pp. 4, 00. [6 Kayode, S.J., A zero-stable optimal order metod for direct solutio of secod-order differetial equatios. J. Mat. Stat., Vol. 6, No., pp. 67 7, 00. [7 Kayode, S.J., A class of o-poit zero-stable cotiuous ybrid metods for direct solutio of secod-order differetial equatios. Africa J. Mat. Comp. Sci. Res. Vol. 4, No., pp. 9 99, 0. [8 Kayode, S.J. ad Adeyeye, O., A - step ybrid metod for direct solutio of secod order iitial value problems. Australia J. Basic Appl. Sci., Vol., No., pp. 6, 0. [9 Kayode S.J. ad Obarua, F.O., Cotiuous y fuctio ybrid metods for direct solutio of differetial equatios. It. J. Diff. Eqs. Appl., Vol., No., pp. 7-48, 0. [0 Awoyemi, D. O. ad Idowu, O. M., A class of ybrid collocatio metods for tird order ordiary differetial equatios. It. J. Comp. Mat., Vol. 8, No. 0, pp. 87-9, 00. [ Awoyemi, D.O., Algoritmic collocatio approac for direct solutio of fourt order iitial value problems of ordiary differetial 7

8 Tammasat Iteratioal Joural of Sciece ad Tecology Vol.9, No., April-Jue 04 equatios. It. J. Comp. Mat. Vol. 8, No., pp. 9, 00. [ Kayode S.J., A order si zerostable metod for direct solutio of fourt order ordiary differetial equatios. America J. Appl. Sci., Vol. No., pp , 008. [ Kayode, S.J. ad Awoyemi, D.O., A multiderivative collocatio metod for t order ordiary differetial equatios. J. Mat. Stat., Vol. 6, No., pp. 60 6, 00. [4 Kayode S.J., A eplicit cotiuous algoritm for direct solutio of geeral fift order ordiary differetial equatios. It. J. Diff. Eqs. Appl., ISSN -87 (I Prit) [ Lambert, J. D., Computatioal metods i ordiary differetial equatio. Jo Willey & Sos Ic., New Yor, 97. [6 Herici, P., Discrete variable metods i Odes. Jo Wiley ad Sos, N. Yor, 96. 8