A Continuous Block Integrator for the Solution of Stiff and Oscillatory Differential Equations

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: ,p-ISSN: X, Volume 8, Issue 3 (Sep. - Oct. 3), PP A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios J. Suday, Y. Skwame ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria. Departmet of Mathematical, Modibbo Adama Uiversity of echology, Yola, Nigeria. Abstract: I this paper, we preset a cotiuous block itegrator for direct itegratio of stiff ad oscillatory first-order ordiary differetial equatios usig iterpolatio ad collocatio techiques. he approximate solutio used i the derivatio is a combiatio of power series ad expoetial fuctio. he paper further ivestigates the properties of the block itegrator ad foud it to be zero-stable, cosistet ad coverget. he itegrator was also tested o some sampled stiff ad oscillatory problems ad foud to perform better tha some existig oes. Keywords: Approximate Solutio, Block Itegrator, Cotiuous, Oscillatory, Stiff AMS Subect Classificatio (): 65L5, 65L6, 65D3 I. Itroductio Nowadays, the itegratio of Ordiary Differetial Equatios (ODEs) could be carried out usig block itegrators. I this paper, we preset a cotiuous block itegrator for direct itegratio of stiff ad oscillatory problems of the form, y' f ( x, y), y( a) y, x [ a, b] () m m m f :, y, y, f satisfies Lipchitz coditio which guaratees the existece ad uiqueess of solutio of (). he developmet of umerical itegratio formulas for stiff as well as oscillatory differetial equatios has attracted cosiderable attetio i the past (Fatula, 98). A special problem arisig i the solutio of ODEs is stiffess. his problem occurs i sigle liear ad oliear ODEs, higher-order liear ad oliear ODEs ad systems of liear ad oliear ODEs (Hoffma, ). It is also importat to ote that mathematical models of physical situatios i kietic chemical reactios, process cotrol ad electrical circuit theory ofte results to stiff ODEs (Fatula, 98). Accordig to Saugi ad Evas (989), a iterestig ad importat class of IVPs which ca also arise i practice cosists of differetial equatios whose solutios are kow to be periodic or to oscillate with a kow frequecy. Examples of such problems ca be foud i the field of ecology, medical scieces ad oscillatory motio i a oliear force field. Almost ivariably, most covetioal umerical itegratio solvers caot efficietly cope with stiff ad oscillatory problems of the form () as they lack adequate stability characteristics (Fatula, 98). he degree of stiffess of a problem depeds o the defiitio of stiffess that is applied (Okuuga et al., 3). here are various defiitios of stiffess i the literature as regards to ODEs. Lambert (973) gave a simple defiitio of stiffess of a ODE i a such a maer that problem () possesses some stiffess if Re( i), i () m, is the eige value of the problem. A stiff equatio is a differetial equatio for which certai umerical methods for solvig the equatio are umerically ustable, uless the step size is take to be extremely small. he mai idea is that the equatio icludes some terms that ca lead to rapid variatio i the solutio. O the other had, a otrivial solutio (fuctio) of a ODE is called oscillatig if it does ot ted either to a fiite limit or to ifiity (i.e. if it has a ifiite umber of roots). he differetial equatio is called oscillatig, if it has at least oe oscillatig solutio (Borowski ad Borwei, 5). here are differet cocepts of the oscillatio of a solutio. he most widespread are oscillatio at a poit (usually take to ) ad oscillatio o a iterval. More recetly, authors like Butcher (3), Zaria, Mohammed, Kharil ad Zaariah (5), Awoyemi, Ademiluyi ad Amusegha (7), Okuuga, Akifewa ad Daramola (8), Abbas (9), Areo, Ademiluyi, ad Babatola, Ibiola, Skwame ad Kumleg, Akifewa, Yao ad Jator, Chollom, Olatubosu ad Omagu, Okuuga, Sofoluwe ad Ehigie (3), Yakubu, Madaki ad Kwami (3), Aie, Ikhile ad Oumayi (3), amog others, have all proposed block methods to geerate umerical solutio to (). hese authors proposed methods i which the approximate solutio rages from power series, Chebychev s, Lagrage s ad Laguerre s polyomials. I this paper, the derivatio of the cotiuous block itegrator is carried out usig a approximate solutio which is a combiatio of power series ad expoetial fuctio Page

2 A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios II. Methodology: Costructio of the Cotiuous Block Itegrator I derivig the itegrator, iterpolatio ad collocatio procedures are used by choosig iterpolatio poit s at a grid poit ad collocatio poits r at all poits givig rise to s r system of equatios whose coefficiets are determied by usig appropriate procedures. he approximate solutio to () is take to be a combiatio of power series ad expoetial fuctio give by, rs rs x y( x) a x ars ()! with the first derivative give by, rs rs x y '( x) a x ars (3) ( )! a, for ()4 ad yx ( ) is cotiuously differetiable. Let the solutio of () be sought o the partitio N : a x < x < x <... < x < x <...< x N = b, of the itegratio iterval ab, with a costat step-size h, give by, h x x,,,..., N. he, substitutig (3) i () gives, rs rs x f ( x, y) a x ars (4) ( )! Now, iterpolatig () at poit xs, s ad collocatig (4) at poits xr, r ()3, leads to the followig system of equatios, AX U (5) A [ a a a a a ] 3 4 U [ y f f f f ] 3 ad x x x x x x x! 3! 4! x x x 3x x! 3! x x X x 3x x! 3! x x x 3x x! 3! x3 x 3 x3 3x3 x3! 3! Solvig (5), for a ' s, ()4 ad substitutig back ito () gives a cotiuous liear multistep method of the form, 3 y( x) ( x) y h ( x) f (6) ad the coefficiets of f gives 76 Page

3 A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios ( t t t 4 t ) 4 ( t t t ) 4 ( t t t ) 4 ( ) 3 t t t 4 t ( x x ) h. Evaluatig (6) at t ()3 gives a cotiuous discrete block scheme of the form, A m hd hb m Y Ey f( y ) F( Y ) (8) y y y 3, y y y f f f f f f Y y m F( Y ), f( y ), A m 3, E, d (7) b III. Aalysis of Basic Properties of the New Block Itegrator 3.. Order of the New Block Itegrator L y( x); h associated with the block (8) be defied as, Let the liear operator L y( x); h A Y Ey hdf ( y ) hbf ( Y ) (9) m m expadig usig aylor series ad comparig the coefficiets of h gives, p p p p L y( x); h c y( x) c hy '( x) c h y ''( x)... c h y ( x) c h y ( x)... p p Defiitio 3. he liear operator L ad the associated cotiuous liear multistep method (6) are said to be of order p if c c c... cp ad cp. cp is called the error costat ad the local trucatio error is give by, ( p) ( p) p tk cp h y ( x ) O( h ) () For our method, f y y 4 f Ly( x); h y y h f y 3 y f () Expadig () i aylor series gives, 77 Page

4 A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios ( h) 9h ' h 9 5 y y y y () () (3)! 4! ( h) h ' h 4 y y y y () () (3)! 3! 3 3 (3 h) 3h ' h y y y y () () (3)! 8! c c c c3 c4, c5.64( ),.( ),3.75( ). herefore, the block Hece, itegrator is of order four. 3.. Zero Stability Defiitio 3. he block itegrator (8) is said to be zero-stable, if the roots z, s,,..., k of the first characteristic polyomial () z defied by s ( z) det( za E ) satisfies z ad every root satisfyig z have multiplicity ot exceedig the order of the differetial equatio. Moreover, as h, r ( z) z ( z ) is the order of the differetial equatio, r is the order of the matrices A ad E (see Awoyemi et al. (7) for details). For our itegrator, ( z) z. Hece, the block itegrator is zero-stable. ( z) z ( z ), z z, z Covergece he ew block itegrator is coverget by cosequece of Dahlquist theorem below. heorem 3. (Dahlquist, 956) he ecessary ad sufficiet coditios that a cotiuous LMM be coverget are that it be cosistet ad zerostable Regio of Absolute Stability Defiitio 3.3 (Ya, ) Regio of absolute stability is a regio i the complex z plae, z h z such that the umerical solutios of y' ysatisfy y as for ay iitial coditio. s (4). It is defied as those values of o determie the absolute stability regios of the block itegrators, we adopt the boudary locus method. his is achieved by substitutig the test equatio, y' y (5) ito the block formula (8). his gives, A Ym ( r) Ey ( r) hdy ( r) hby m( r) (6) hus, A Ym( r) Ey( r) hr () (7) Dy( r) BYm( r) Writig (7) i trigoometric ratios gives, A Ym( ) Ey( ) h( ) (8) Dy( ) BYm( ) r i e. Equatio (8) is our characteristic or stability polyomial. s 78 Page

5 Im(z) A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios which gives the stability regio show i fig. below. Fig.: Showig Stability Regio of the Cotiuous Block Itegrator Re(z) x Accordig to Fatula (988), stiff algorithms have ubouded RAS. Also, Lambert (973) showed that the stability regio for L-stable schemes must ecroach ito the positive half of the complex z plae. IV. Numerical Experimets We shall use the followig otatios i the tables below; ERR- Exact Solutio-Computed Result ESSI- Error i Skwame et al. Problem Cosider the highly stiff ODE y' f ( x, y) ( y F( x)) F '( x), y( x) y (9) which has the exact solutio x y( x) ( y F) e F( x) is a positive costat ad Fx ( ) is a smooth slowly varyig fuctio. Equatio exhibits two widely differet time scales: a rapidly chagig term associated with exp( x) ad a slowly varyig term associated with Fx ( ), (Gear, 97). Skwame et al. cosidered a special case of (9), F( x), x ad y. hey solved the problem (9) by adoptig a L-stable hybrid block Simpso s method of order six. Problem Cosider the oscillatory ODE y' si x ( y cos x), y () whose exact solutio is give by: y( x) cos x e x () hough Ya did ot solve this problem, he however observed that it has a solutio that oscillates ad grows expoetially i x. He further stated that most umerical methods do ot perform well o this problem. able : Showig the result for stiff problem x Exact solutio Computed solutio ERR ESSI e-7 6.8e e-7.88e e-7 3.6e e-7.6e e e e-7.45e e-7 5.e e-7.76e e-7.e e e Page

6 A Cotiuous Block Itegrator for the Solutio of Stiff ad Oscillatory Differetial Equatios able : Showig the result for oscillatory problem x Exact solutio Computed solutio ERR e e e e e e e e e e-6 V. Coclusio We have preseted a cotiuous block umerical itegrator for the solutio of stiff ad oscillatory firstorder ordiary differetial equatios. he approximate solutio (basis fuctio) adopted i this research produced a block itegrator with L-stable stability regio. his made it possible for the block itegrator to perform well o stiff ad oscillatory problems. he block itegrator proposed was also foud to be zero-stable, cosistet ad coverget. he itegrator was also foud to perform better tha some existig methods. Refereces [] I. J. Aie, M. N. O. Ikhile, ad P. Oumayi, A Family of ( ) A Stable Block Methods for Stiff ODEs, Joural of Mathematical Scieces, (), pp.73-98, 3. [] A. O. Akifewa, N. M. Yao, ad S. N. Jator, A Self-Startig Block Adams Methods for Solvig Stiff Ordiary Differetial Equatios, IEEE 4 th IteratioalCoferece i Computatioal Sciece ad Egieerig (CSE), pp.7-36,. [3] E. A. Areo, R. A. Ademiluyi, ad P. O. Babatola, hree Steps Hybrid Liear Multistep Method for the Solutio of First-Order Iitial Value Problems i Ordiary Differetial Equatios, Joural of Mathematical Physics, 9, pp.6-66,. [4] D. O. Awoyemi, R. A. Ademiluyi, ad W. Amusegha, Off-grids Exploitatio i the Developmet of More Accurate Method for the Solutio of ODEs, Joural of Mathematical Physics,, pp , 7. [5] E. J. Borowski, ad J. M. Borwei, Dictioary of Mathematics, Harper Collis Publishers, Glasgow, 5. [6] J. C. Butcher, Numerical Methods for Ordiary Differetial Equatios, West Sussex: Joh Wiley & Sos Ltd, 3. [7] J. P. Chollom, I. O. Olatubosu, ad S. Omagu, A Class of A-Stable Block Explicit Methods for the Solutio of Ordiary Differetial Equatios, Research Joural of Mathematics ad Statistics, 4(), pp.5-56,. [8] G. G. Dahlquist, Covergece ad Stability i the Numerical Itegratio of Ordiary Differetial Equatios, Math. Scad., 4, pp.33-5, 956. [9] S. O. Fatula, Numerical Itegrators for Stiff ad Highly Oscillatory Differetial Equatios, Mathematics of Computatio, 34(5), pp , 98. [] S. O. Fatula, Numerical Methods for Iitial Value Problems i OrdiaryDifferetial Equatios, Academic Press Ic, New York, 988. [] C. S. Gear, Numerical Iitial Value Problems i Ordiary Differetial Equatios,Pretice-Hall, Eglewood Cliffs, New Jersey, 97. [] J. D. Hoffma, Numerical Methods for Egieers ad Scietists, Marcel Dekker Ic. (secod editio),. [3] E. A, Ibiola, Y. Skwame, ad G. Kumleg, Formulatio of Hybrid Method of Higher Step-sizes hrough the Cotiuous Multistep Collocatio, America Joural of Scietific ad Idustrial Research, (), pp.6-73,. [4] J. D. Lambert, Computatioal Methods i Ordiary Differetial Equatios, Joh Willey ad Sos, New York, 973. [5] S. A. Okuuga, A. O. Akifewa, ad R. Daramola, Oe-Poit Variable Step Block Method for Solvig Stiff Iitial Value Problems, Proceedigs of Mathematical Associatio of Nigeria, pp.33-39, 8. [6] S. A. Okuuga, A. B. Sofoluwe, ad J. O. Ehigie, Some Block Numerical Schemes for Solvig Iitial Value Problems i ODEs Joural of Mathematical Scieces, (), pp.387-4, 3. [7] B. B. Saugi, ad J. D. Evas, he Numerical Solutio of Oscillatory Problems Iter. J. Computer Math., 3, pp.37-55, 989. [8] Y. Skwame, J. Suday, ad E. A. Ibiola, L-Stable Block Hybrid Simpso s Methods for Numerical Solutio of Iitial Value Problems i Stiff ODEs It. J. Pure Appl. Sci. echology, (), pp.45-54,. [9] D. G. Yakubu, A. G. Madaki, ad A. M. Kwami, Stable wo-step Ruge-Kutta Collocatio Methods for Oscillatory Systems of IVPs i ODEs, America Joural of Computatioal ad Applied Mathematics, 3(), pp.9-3, 3. [] Y. L. Ya, Numerical Methods for Differetial Equatios, City Uiversity of Hog Kog, Kowloo,. [] B. I. Zaria, S. Mohammed, I. Kharil, ad M. Zaariah, Block Method for Geeralized Multistep Adams Method ad Backward Differetiatio Formula i Solvig First-Order ODEs, Mathematika, pp.5-33, Page