Implicit One-Step Legendre Polynomial Hybrid Block Method for the Solution of First-Order Stiff Differential Equations

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1 British Joural of Mathematics & Computer Sciece 8(6): , 205, Article o.bjmcs ISSN: SCIENCEDOMAIN iteratioal Implicit Oe-Step Legedre Polyomial Hybrid Block Method for the Solutio of First-Order Stiff Differetial Equatios J. Suday *, Y. Skwame ad I. U. Huoma 2 Departmet of Mathematics, Adamawa State Uiversity, Mubi, Nigeria. 2 Departmet of Mathematics, Uiversity of Nigeria, Nsukka, Nigeria. Article Iformatio DOI: /BJMCS/205/6252 Editor(s): () Adre V. Plotikov, Departmet of Applied ad Calculus Mathematics ad CAD, Odessa State Academy of Civil Egieerig ad Architecture, Ukraie. (2) Zuomao Ya, Departmet of Mathematics, Hexi Uiversity, Chia. (3) Paul Bracke, Departmet of Mathematics, The Uiversity of Texas-Pa America Ediburg, TX 78539, USA. Reviewers: () Aoymous, Nigeria. (2) Aoymous, Turkey. (3) Mfo Oko Udo, Departmet of Mathematics/Statistics, CRUTECH, Calabar, Nigeria. Complete Peer review History: Origial Research Article Received: 9 Jauary 205 Accepted: 0 April 205 Published: 04 May 205 Abstract We formulate a implicit hybrid block method for the umerical solutio of stiff first-order Ordiary Differetial Equatios (ODEs) usig the Legedre polyomial as our basis fuctio via iterpolatio ad collocatio techiques. The paper further ivestigates the basic properties of the implicit hybrid block method ad foud it to be zero-stable, cosistet ad coverget. The method was also tested o some sampled stiff problems ad foud to perform better tha some existig oes with which we compared our results. Keywords: Hybrid; implicit; Legedre Polyomial; oe-step; Stiff. AMS Subect Classificatio (200): 65L05, 65L06, 65D30. Itroductio I this paper, we are cocered with the solutio of stiff first-order differetial equatios of the form, y ' f ( x, y), y( a), x a, b () *Correspodig author: oshuasuday2000@yahoo.com;

2 where f is assumed to be Lipchitz cotiuous i y ad is a give iitial value. Differet methods have bee proposed for the solutio of [] ragig from predictor-corrector methods to hybrid methods. Despite the success recorded by the predictor-corrector methods, its maor setback is that the predictors are i reducig order of accuracy especially whe the value of the step-legth is high ad moreover the results are at overlappig iterval [2]. Hybrid methods have advatage of icorporatig fuctio evaluatio at off-step poits which affords the opportuity of circumvetig the the Dahlquist zero stability barrier ad it is actually possible to obtai coverget k-step methods with order 2k up to k 7, [3]. The method is also useful i reducig the step umber of a method ad still remais zero-stable, []. Stiff differetial equatios were first ecoutered i the study of the motio of sprigs varyig stiffess, from which the problem derives its ame. Stiffess occurs whe some compoets of the solutio decay much more rapidly tha others. These problems are of frequet occurrece i the mathematical formulatio of physical situatios i cotrol theory ad mass actio kietics, where processes with widely varyig time costats are usually ecoutered. Historically, two chemical egieers, Curtis ad Hirschfelder i 952 proposed the first set of umerical itegratio formulas (both implicit backward differetiatio formulas) that are well suited for stiff differetial equatios. Stiff equatios pose stability problems for most umerical itegrators, [4]. Defiitio [5]: Legedre polyomials y( x) defied o [,] are give by, M 2m m (2 2 m)! x y( x) ( ) 2 m!( m)!( 2 m)! (2) m0 where M 2 or ( ) 2 whichever is a iteger. I particular, properties; y x y x x y x x The polyomials satisfy the followig 2 0( ), ( ), 2 ( ) (3 ) 2,... y ( x ) is a eve polyomial if is eve ad a odd polyomial if is odd, y ( ) x are orthogoal polyomials ad satisfy 0, m ym( x) y ( x) dx 2, m 2 y ( x) ( ) y ( x) Defiitio 2 [6]: A differetial equatio is said to be stiff if Re( i ) 0, i () m, where is the eige value of the differetial equatio. May scholars have proposed various forms of methods for the solutio of () by adoptig power series, Chebyshev ad Lagrage polyomials as basis fuctios. I this paper, we develop a implicit hybrid block method that gives better stability coditio by usig Legedre polyomial as our basis fuctio. 483

3 2 Formulatio of the Implicit Oe-Step Hybrid Block Method We cosider the first six terms of the Legedre polyomial as our basis fuctio. This is give by, ( ) y ( x) y ( x) y ( x) (2 ) xy ( x) y ( x) 4 (3) 0 Iterpolatig (3) at poit xs, s 0 ad collocatig its first derivative at poits xr, r 0 4, where s ad r are the umbers of iterpolatio ad collocatio poits respectively, leads to the followig system of equatios, where ad XA U (4) A [ a a a a a a ] T U [ y f f f f f ] T x 8x 50x 35x 63x x 50x 40x 35x 0 36x 50x 40x 35x X 0 36x 50x 40x 35x 0 36x 50x 40x 35x x 50x 40x x Solvig (4), for a ' s, 0()5 ad substitutig back ito (3) gives a cotiuous liear multistep method of the form, y( x) 0( x) y h ( x) f, (5) where the coefficiets of y ad f are give by, 484

4 0( t) ( t) (92t 600t 700t 375t 90 t) ( t) ( 384t 080t 040t 360 t ) ( t) (92t 480t 380t 90 t ) ( t) ( 384t 840t 560t 20 t ) ( t) (92t 360t 220t 45 t ) ad t ( x x ) h. Evaluatig (5) at t 4 4 form, Where (0) A m E hd hb m (6) gives a discrete hybrid block method of the Y y f ( y ) F( Y ) (7) Y T y y y y, y y y y y m F( Ym ) f f f 3 f, f ( y ) f 3 f f f (0) A, E 360, d b T T, T 485

5 3 Aalysis of Basic Properties of the Implicit Oe-Step Hybrid Block Method 3. Order of the Implicit Oe-step Hybrid Block Method Let the liear operator Ly( x); (0) ( ); m ( ) ( m) h associated with the block (7) be defied as, L y x h A Y Ey hdf y hbf Y (8) Expadig (8) usig Taylor series ad comparig the coefficiets of h gives, L y( x); h c y( x) c hy '( x) c h y ''( x)... c h y ( x) c h y ( x)... (9) 2 p p p p 0 2 p p Defiitio 3 [4]: The liear operator L ad the associated cotiuous liear multistep method (5) are said to be of order p if c c c... c p ad c p c is called the error costat ad the local trucatio error is give by, p ( p ) ( p ) p 2 t k c h y p ( x ) O( h ) (0) For implicit oe-step hybrid block method, L y( x); h y y y y y y y y f 29 3 f h f f f () 486

6 Expadig () i Taylor series gives, h 4 25h ' h y y y y () 0! ! h 2 29h ' h y y y y () 0 0! 360 0! h h ' h y y y y () 0! 80 0! ( h) 7h ' h y y y y () 0 0! 90 0! (2) Equatig the coefficiets of the Taylor series expasio to zero yields, c0 c c2 c3 c4 c5 0, c ( 06).248( ) ( 06) 5.670( 07) T. Therefore, the implicit oe-step hybrid block method is of uiform order five. 3.2 Zero Stability of the Implicit Oe-Step Hybrid Block Method Defiitio 4 [4]: The implicit oe-step hybrid block method (7) is said to be zero-stable, if the roots (0) z s k of the first characteristic polyomial ( z) defied by s,,2,..., ( z) det( za E ) satisfies zs ad every root satisfyig zs have multiplicity ot exceedig the order of the r differetial equatio. Moreover, as 0, differetial equatio, r is the order of the matrices For our method, h ( z) z ( z ) where is the order of the A (0) ad E, see [7] for details ( z) z (3). Hece, the implicit oe-step hybrid ( z) z z z ( z ) 0, z z2 z3 0, z4 block method is zero-stable. 3.3 Cosistecy ad Covergece of the Implicit Oe-Step Hybrid Block Method The implicit oe-step hybrid block method (7) is cosistet sice it has order p 5 ad it is also coverget by cosequece of Dahlquist theorem stated below. 487

7 Theorem 2 [8]: The ecessary ad sufficiet coditios that a cotiuous LMM be coverget are that it be cosistet ad zero-stable. 3.4 Covergece ad Regio of Absolute Stability of the Implicit Oe-Step Hybrid Block Method Defiitio 5 [9]: Regio of absolute stability is a regio i the complex z plae, where z h. It is defied as those values of z such that the umerical solutios of y ' y satisfy 0 as for ay iitial coditio. y We shall adopt the boudary locus method to determie the regio of absolute stability of the implicit oe-step hybrid block method. This gives the stability polyomial below, h( w) h w w h w w h w w h w w w w (4) The stability regio is show i the figure below Im(z) Re(z) Fig.. Stability Regio of the Implicit Oe-Step Hybrid Block Method The stability regio obtaied i Fig. above is A-Stable, sice it cotais the whole of the left-half complex plae, [6]. 488

8 4 Numerical Experimets The implicit oe-step hybrid block method formulated shall be tested o some set of stiff differetial equatios ad compare the results with solutios from some methods of similar derivatio. The umerical results were obtaied usig MATLAB. The followig otatios shall be used i the tables below; ERR- Exact Solutio-Computed Solutio EJA- Error i [0] EOK- Error i [] Problem A certai radioactive substace is kow to decay at the rate proportioal to the amout preset. A block of this substace havig a mass of 00g origially is observed. After 40hours, its mass reduced to 90 g. Test for the cosistecy of the implicit oe-step hybrid block method o this t 0,. problem for This stiff problem is modeled by the differetial equatio, dn dt N, , N(0) 00, t 0, (5) where N represets the mass of the substace at ay time t ad is a costat which specifies the rate at which this particular substace decays. The theoretical solutio to (5) is give by; N( t) 00 Source: [2] t e (6) This problem was solved usig Half-step hybrid method of order six. We preset the result for problem usig the ewly derived method i Table below. Problem 2 Cosider the highly stiff ODE 2 y ' 0( y ), y(0) 2 (7) with the exact solutio, y( t) 0 t (8) Source: []. This problem was earlier discussed by [9], he showed that may predictor-corrector ad block methods become ustable with this problem, icludig the hybrid methods. However, the ewly 489

9 derived block method is used for the itegratio of this problem withi the iterval 0 t 0.. The authors i [0] solved this stiff problem by adoptig a ew 3-poit block method with step size ratio at r ad of order five. We preset the result for problem 2 usig the ewly derived method i Table 2 below. Table. Showig the results for stiff problem t Exact solutio Computed solutio ERR EJA t / sec e e e e e e e e e e e e e e e e e e e e Table 2. Showig the result for stiff problem 2 t Exact solutio Computed solutio ERR EOK t / sec e e e e e e e e e e e e e e e e e e e-008.8e Coclusio We formulated a A-stable implicit oe-step hybrid block method for the solutio of stiff first-order ordiary differetial equatios where we adopted Legedre polyomial as our basis fuctio. The method developed was also foud to be zero-stable, cosistet ad coverget. The hybrid block method was also foud to perform better tha some existig methods i view of the umerical results obtaied. Competig Iterests Authors have declared that o competig iterests exist. Refereces [] Adesaya AO, Udoh MO, Aileye AM. A ew hybrid method for the solutio of geeral thirdorder iitial value problems of ordiary differetial equatios. Iteratioal J. of Pure ad Applied Mathematics. 203;86(2):

10 [2] Adesaya AO, Odekule MR, Udoh MO. Four-steps cotiuous method for the solutio of y ''' f ( x, y, y ', y ''). America J. of Computatioal Mathematics. 203;3: [3] Awoyemi DO, Ademiluyi RA, Amusegha W. Off-grids exploitatio i the developmet of more accurate method for the solutio of ODEs. Joural of Mathematical Physics. 2007;2: [4] Fatula SO. Numerical methods for iitial value problems i ordiary differetial equatios. Academic Press Ic, New York; 988. [5] Jai MK, Iyegar SRK, Jai RK. Numerical methods (for scietific ad egieerig computatio). New age iteratioal (p) limited publishers, New Delhi (5 th editio); [6] Lambert JD. Computatioal methods i ordiary differetial equatios. Joh Willey ad Sos, New York; 973. [7] Butcher JC. Numerical methods for ordiary differetial equatios i the 20 th cetury. Joural of Computatioal ad Applied Mathematics. 2000;25:-29. [8] Dahlquist GG. Covergece ad Stability i the Numerical Itegratio of Ordiary Differetial Equatios, Math. Scad. 956;4: [9] Ya YL. Numerical methods for differetial equatios. City Uiversity of Hog Kog, Kowloo; 20. [0] James AA, Adesaya AO, Suday J, Yakubu DG. Half-step cotiuous block method for the solutio of modeled problems of ordiary differetial equatios. America Joural of Computatioal Mathematics. 203;3: [] Okuuga SA, Sofoluwe AB, Ehigie JO. Some block umerical schemes for solvig iitial value problems i ODEs. Joural of Mathematical Scieces. 203;2(): [2] Suday J. O Adomia decompositio method for umerical solutio of ODEs arisig from the atural laws of growth ad decay. The Pacific Joural of Sciece ad Techology. 20;2(): Suday et al.; This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Peer-review history: The peer review history for this paper ca be accessed here (Please copy paste the total lik i your browser address bar) 49