Five Steps Block Predictor-Block Corrector Method for the Solution of ( )

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1 Applied Mathematics, 4,, -66 Published Olie May 4 i SciRes. Five Steps Block Predictor-Block Corrector y = f x, y, y Method for the Solutio of ( ) Mathew Remileku Odekule, Michael Otokpa Egwurube, Adetola Olaide Adesaya, Mfo Oko Udo Departmet of Mathematics, Modibbo Adama Uiversity of Techology, Yola, Nigeria Departmet of Mathematics ad Statistics, Cross River Uiversity of Techology, Calabar, Nigeria mfudo4sure@yahoo.com, remi_odekule@yahoo.com Received February 4; revised March 4; accepted March 4 Copyright 4 by authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract Theory has it that icreasig the step legth improves the accuracy of a method. I order to affirm this we icreased the step legth of the cocept i [] by oe to get k =. The techique of collocatio ad iterpolatio of the power series approximate solutio at some selected grid poits is cosidered so as to geerate cotiuous liear multistep methods with costat step sizes. Two, three ad four iterpolatio poits are cosidered to geerate the cotiuous predictor-corrector methods which are implemeted i block method respectively. The proposed methods whe tested o some umerical examples performed more efficietly tha those of []. Iterestigly the cocept of self startig [] ad that of costat order are reaffirmed i our ew methods. Keywords Step Legth, Power Series, Block Predictor, Block Corrector, Costat Order, Step Size, Grid Poits, Self Startig, Efficiecy. Itroductio I this paper we examie the solutio to geeral secod order iitial value problem of the form y = f x, y, y, y x = y, y x = y () ( ) ( ) ( ) I literature, it has bee stated clearly the ourey of the developmet of direct methods to offset the burde of reductio []-[6]. Various methods have bee proposed by scholars for solvig higher order ordiary differetial equatio (ODE). Notable authors like [] [7]-[] have developed direct methods of solvig geeral secod order ODE s to cater for the burde iheret i the method of reductio. Now writig computer code is less bur- How to cite this paper: Odekule, M.R., et al. (4) Five Steps Block Predictor-Block Corrector Method for the Solutio of y = f x, y, y. Applied Mathematics,, ( )

2 M. R. Odekule et al. desome sice it o loger requires special ways to icorporate the subroutie to supply the startig values. As a result, this leads to computer time ad huma effort coservatio. The ew methods are cotiuous i ature with the advatage of possible evaluatio at all poits withi the itegratio iterval. We have take advatage of the works of [7] []-[] who proposed direct block methods as predictor i the form where ( ) ( ) ( i) ( i) m = + i + i m i= ( ) ( ) A Y ey h d f y b f y () Y = y y y y m r ( ) = [ ] T f y f f f f m r ( ) = [ ] T f y f f f f ei = r r matrix, A = r r idetity martix. Ad also the discrete block formula as corrector i the form where ( ) A = r r idetity matrix ( ) ( i) ( k) ( ) ( i) m = m + m + m + m A Y A Y A Y h B f B f f = f f f f m Y = y y y y m [ ] T Y = y y y y y m + k f = f f f m s with the aim to cater for some of the setbacks of predictor-corrector method [6] [7]. The fact that iterpolatio poit caot exceed the order of the differetial equatio for block methods is worrisome [9]. Also vital to this paper is the cocept of block predictor-corrector method (Mile approach). This method formed a bridge betwee the predictor-corrector method ad block method [4] [] []. I [] we stated that results geerated at a overlappig iterval affect the accuracy of the method ad the ature of the model caot be determied at the selected grid poits. I this paper as i [], we developed a method usig the Mile approach but the corrector was implemeted at a o overlappig iterval. The umerical experimet compared the results geerated at differet step legths, whe k = 4 ad whe k = respectively.. Methodology.. Developmet of the Cotiuous Liear Multistep Methods We cosider a power series approximate solutio i the form where r ad s are the umber of iterpolatio ad collocatio poits respectively. The secod derivative of (4) gives Substitutig () ito () gives ( ) y x r+ s = ax (4) = r+ s () = ( ) ( ) y x = ax r+ s (6) = (,, ) = ( ) f xyy ax ()

3 M. R. Odekule et al. Iterpolatig (4) ad collocatig (6) at some selected grid poits gives a system of o liear equatios i the form where AX = U (7) A= a a a a a + r s U = y y y f f f + + r + + s r+ s x x x x r+ s x+ x+ x+ x+ r+ s x+ r x+ r x+ r x+ r X = r+ s 6x ( s+ r )( s+ r ) x r+ s 6x+ ( s+ r )( s+ r ) x+ r+ s 6x+ s ( s+ r )( s+ r ) x + s Solvig (7) for the ukow costats as usig Guassia elimiatio method ad substitutig back ito (4) gives a cotiuous liear multistep method i the form where α ( t) ad ( t) are polyomials,... Developmet of the Block Predictor Iterpolatig (4) at x, r, r Solvig for the ukow costats Equatio (8) reduced to r ( ) = α ( ) + + ( ) + (8) y t t y h t f = = x x f+ = ( fx + h, y ( x + h), y ( x + h) ), t = h + = ad collocatig (6) at x s ( ) s +, =. the parameters i (7) becomes s A= [ a ] T a a a a4 a a6 a7 U = y y f f f f f f x x x x x x x x x x x x x x X = x x x x x 4 x+ x+ x+ x+ x+ 4 x+ x+ x+ x+ x+ 4 x+ x+ x+ x+ x+ 4 x+ 4 x+ 4 x+ 4 x+ 4 4x+ 4 4 x+ x+ x+ x+ x+ as usig Guassia elimiatio method ad substitutig ito (4), makes ( ) = α ( ) + + ( ) + (9) y t t y h t f = = 4

4 M. R. Odekule et al. where α = t α = t = ( t 7 4t 6 7t 7t 4 86t 4t 46t) = ( t 7 96t 6 49t 9t 4 84t 4t) = ( t 7 8t 6 9t 74t 4 4t t) = ( t 7 68t 6 9t 7t 4 8t 94t) 4 = + 8 ( t 7 4t 6 86t t 4 t 68t) = + 8 ( t 7 8t 6 47t t 4 6t 7t) Solvig for the idepedet solutio i (9) ad simplifyig gives where i ( h ) ( i) σ ( ) () y = y + h t f + + i= i! = σ = ( t 7 4t 6 7t 7t 4 86t 4t ) σ = ( t 7 96t 6 49t 9t 4 84t ) σ = ( t 7 8t 6 9t 74t 4 4t ) σ = ( t 7 68t 6 9t 7t 4 8t ) σ 4 = ( t 7 4t 6 86t t 4 t ) σ = + 8 ( t 7 8t 6 47t t 4 6t ) Evaluatig () at the selected grid poits, the parameters i () gives the followig I) Whe i = ( ) A = idetity matrix ( ) m = Y y y y y y

5 M. R. Odekule et al. e =, e = 4 II) Whe i = d = ( i), b = [ ] T m e =, d = b = Developmet of the Block Corrector Here there are three cases (I, II ad III) to be cosidered. Developmet of the Block Corrector for Case I x, r Iterpolatig (4) at + r = ( ) ad collocatig (6) at x ss ( ) A= [ a ] T a a a a4 a a6 a7 a8 +, =, makes Equatio (7) reduced to 6

6 M. R. Odekule et al. Solvig for the ukow costats Equatio (8) reduced to where U = y y y y f f f f f x x x x x x x x x x x x x x x x x x x x x x x x X = 6x x x x 4x 6x x x x x x x x+ x+ x+ x+ 4x+ 6x+ 4 6 x+ x+ x+ x+ x+ x+ 4 6 x+ 4 x+ 4 x+ 4 x+ 4 x+ 4 x x+ x+ x+ x+ x+ x+ as usig Guassia elimiatio method ad substitutig ito (4), makes ( ) = α ( ) + + ( ) + () y t t y h t f = = α = ( t t t t t t t ) α = t t + t t + t t + t ( ) α = ( t 8 6t 7 476t 6 89t 86t 4 6t 99t) = ( 4t 8 t 7 89t 6 797t 7988t 4 786t 84t 679t) = ( 67t 8 6t t t 644t t 444t) = ( 6t 8 47t 7 64t 6 99t 98877t 4 69t 78t) = ( 44t 8 66t 7 844t 6 4t 948t 4 488t 6696t) 4 = ( 78t 8 t 7 94t t t 4 474t t) = Evaluatig () at t = ( ) gives the followig ( 6t 8 88t 7 88t 6 7t 6t 4 696t 7t) 9 4 y+ = y y+ + y+ h y+ 4 = y y+ + y+ ( 7 f 78 f 4998 f 78 f 47 f f ) h ( 77 f 864 f 74 f 44 f 69 f 8 f ) () () 7

7 M. R. Odekule et al y+ = y y+ + y+ 44 h 4 ( 6 f f 46 f 6 f 6 f 47 f ) Evaluatig the first derivatives of () at t =, gives the followig hy = y + y+ y h ( 999 f 4668 f 948 f 44 f 69 f 44 f ) hy + = y y+ + y+ 6 6 h ( f 496 f 668 f 674 f 49 f 6 f ) Writig Equatios () to (6) i block form, the parameters i () gives the followig (4) () (6) A ( i) ( ) A = idetity matrix m m 4 [ ] T m + ( ) = F Y f f f f f m ( ) = F y f f f f f 4 =, B ( ) A ( k ) = =,

8 M. R. Odekule et al. B ( i) = I a similar way the results for cases II ad III are summarized as: Developmet of the Block Corrector for Case II ( ) A = idetity matrix m m 4 [ ] T m + ( ) = F Y f f f f f m ( ) = F y f f f f f 4 A ( i) =, A ( k ) = B ( ) =

9 M. R. Odekule et al. B ( i) = Developmet of Block Corrector case III ( ) A = idetity matrix m m 4 [ ] T m + ( ) = F Y f f f f f m ( ) = F y f f f f f 4 A ( i) =, ( k ) A = B ( ) =

10 M. R. Odekule et al. B ( i) = Aalysis of the Properties of the Methods.. Order of the Methods.. Order of the Block Predictor Whe i =, if we take a Taylor series expasio, we get = = ( h) ( ) y y hy! 4 + h ( + ) y =! ( h) ( ) 7 y y hy! 6 ( ) ( ) ( ) ( 4) ( ) + h ( + ) y =! ( h) ( ) ( h) ( ) ( ) ( ) ( ) ( 4) ( ) y y hy =! 4 = = = + h ( + ) y =! y y 4hy! ( ) ( ) ( ) ( 4) ( ) + h ( + ) y =! ( h) ( ) y y hy! 8 ( ) ( ) ( ) ( 4) ( ) + h ( + ) y =! ( ) ( ) ( ) ( 4) ( ) Collectig coefficiets i powers of h, we see that the order of the method is six ad the error costat is Also whe i = The order of the method is six ad the error costat is T 6

11 M. R. Odekule et al T... Order of the Block Corrector for Case I Takig a Taylor series expasio gives ( h) ( ) ( ) ( ) y y hy y = =! h ( + ) y ( ) ( ) ( ) ( 4) ( ) + + =! ( h) ( ) 6 ( y y hy = ) 7 y ( ) hy =! h ( + ) y ( ) ( ) ( ) ( 4) ( ) =! ( h ) ( ) 7 87 ( ) 999 y ( ) y hy y = =! = h ( + ) y ( ) + ( ) + ( ) + ( 4) ( ) =! ( 4h) ( ) 89 6 ( ) 9496 y ( ) y hy y = =! h ( + ) y ( ) ( ) = + + ( ) ( 4) ( )! ( h) ( ) ( ) 694 y ( ) y hy y = =! h ( + ) y ( ) ( ) ( ) ( 4) = ( )! ad the order of our method is seve with error costat as I a similar way, we compute ad summarize the order for cases II ad III as follows.... Order of the Block Corrector for Case II I this case the order of our method is eight with error costat as Order of the Block Corrector for Case III Also usig the same approach, the order of our method is ie with error costat as Cosistecy of the Method A block method is said to be cosistet if it has order p [9]. T T T 6

12 M. R. Odekule et al. From the above, it clearly shows that our methods are cosistet... Zero Stability A block method is said to be zero stable if, r; = k of the first characteristics polyomials ( ) k ρ ( R) =, that is ρ ( R) = det A R = satisfyig R must have multiplicity equal to uity [9]. Applyig this rule, we have that ρ ( r) h the root ( ) = = where R =,,,, for each method. Hece the methods are zero stable 4. Numerical Experimet 4.. Implemetatio We implemet the proposed methods to verify their efficacies over existig methods. To be cosidered are, two cases for k = 4 [] ad three cases for k =. Four examples were cosidered at h =. ad h =.. All computatios were made with the usage of MATLAB (Ra). A error (Err) is defied i this paper as the absolute value of the differece betwee the computed ad expected values. The followig keys are used i displayig our results o the tables for clearity. CASE : Two iterpolatio poits. CASE II: Three iterpolatio poits. CASE III: Four iterpolatio poits Test Problem I Cosider the o-liear ODE Exact Solutio: y( x) 4... Test Problem Cosider the o-liear iitial value problem ( ) ( ) ( ) y x y =, y =, y =, h=. + x = + l x ( y ) y = y; y π π y =, y =, h= Exact solutio: y( x) ( si ( x) ) = Test Problem Cosider the iitial value ODE 4x =. e x Exact solutio: y( x) = + ; ( ) ( ) y y xe x y =, y =, h=.. 6

13 M. R. Odekule et al Test Problem 4 Cosider the iitial value problem Exact solutio: y( x) = cos x+ si x.. Discussio y = 4y; y( ) =, y ( ) =, h=.. We have cosidered two o-liear ad two liear secod order iitial value problems i this paper as show i Table to Table 4. I [] we compared our method with the existig methods like the block ad block predictor-corrector ad the results re-affirms the claim of [] that though block predictor-corrector method takes loger time to implemet, it gives better approximatio tha the block method. I this paper we exteded the step legth cosidered i [] ad cosidered varyig the umber of iterpolatio poits to observe the effect o the performace of the method. Table. Comparig results for differet iterpolatio poits. Err for k = 4 Err for k = Case I Case II Case I Case II Case III e.797e e 6.977e e..448e.669e.4768e.4687e.47468e. 4.78e e.9699e.969e e e.9946e 6.94e 7.66e 7.67e. 9.96e e.864e.779e.7484e e 9.78e 9.6e.766e.4767e e 9.949e 9.949e.986e.9947e.8 4.6e 9.777e 9.e.9796e.e e e e 9.e 9.869e e e e e e 9 Table. Comparig results for differet iterpolatio poits. Err for k = 4 Err for k = Case I Case II Case I Case II Case III e 8.e 9.4e 8.498e e e 8.97e 9.46e 8.497e 8.477e e e e e 8.644e e e 9.e 8.8e 8.6e e e 8.4e 8.44e 8.49e e e e e e e 7.876e e e e e 7.44e 8 4.e 8 4.4e e e e 8 4.7e e 8 4.7e e 7 4.9e e e e 8 64

14 M. R. Odekule et al. Table. Comparig results for differet iterpolatio poits. Err for k = 4 Err for k = Case I Case II Case I Case II Case III e 6.487e 4.7e e e e.47e.849e.87e.87e..7888e.778e 4.664e 4.667e e e 4.96e 9.787e 9.787e e. 7.4e e e e.744e e.44e.9444e.9444e.94487e e.646e 4.77e 4.77e 4.774e.8.869e.9e 7.69e 7.69e 7.676e.9.798e.7e.79e.79e.778e e 4.679e.6679e.6679e.66769e Table 4. Comparig results for differet iterpolatio poits. Err for k = 4 Err for k = Case I Case II Case I Case II Case III..4469e.476e e 6.668e e..6899e.894e.777e 4.e 4.e e.7e.97e 4.979e 4.97e e.4869e.747e.7e.747e. 4.6e.778e.67996e.66977e.66977e.6.49e.46e.6e.64e.84e.7.74e e.484e.484e.484e e 6.76e e.8676e.8676e.9.6e.69e e 4.774e 4.774e..978e 4.4e.96e.9474e.9474e 6. Coclusio/Recommedatio I this paper we have proposed the varyig of the step legth from k = 4 [] to k =. Block methods which have the properties of evaluatio at all poits withi the iterval of itegratio are adopted to give idepedet solutios at o overlappig itervals as predictors to the correctors. The ew method k = performed better tha that of k = 4. Thus it has bee cofirmed that varyig the step legth improves the accuracy of the method. However, icreasig the umber of iterpolatio poits does ot sigificatly improve the result. We therefore, recommed the block predictor-block corrector method for use i the quest for solutios to secod order iitial value problems of ordiary differetial equatios. Refereces [] Odekule, M.R., Egwurube, M.O., Adesaya, A.O. ad Udo, M.O. (4) Body Math Four Steps Block Predictory = f xyy,,. Joural of Advaces i Mathematics,, Block Corrector. Method for the Solutio of ( ) [] Jator, S.N. ad Li, J. (9) A Self Startig Liear Multistep Method for the Direct Solutio of Geeral Secod Order 6

15 M. R. Odekule et al. Iitial Value Problems. Iteratioal Joural of Computer Mathematics, 86, [] Adesaya, A.O., Odekule, M.R. ad Adeyeye, A.O. () Cotiuous Block Hybrid-Predictor-Corrector Method for y = f xyy,,. Iteratioal Joural of Mathematics ad Soft computig,, -4. the Solutio of ( ) [4] Adesaya, A.O., Odekule, M.R. ad Udo, M.O. () Four Steps Cotiuous Method for the Solutio of y = f xyy,,. America Joural of Computatioal Mathematics,, ( ) [] Awoyemi, D.O. ad Kayode, S.J. () A Maximal Order Collocatio Method for Direct Solutio of Iitial Value Problems of Geeral Secod Order Ordiary Differetial Equatio. Proceedigs of the Coferece Orgaised by the Natioal Mathematical Cetre, Abua. [6] Jator, S.N. (7) A Sixth Order Liear Multistep Method for Direct Solutio of y f( xyy) al of Pure ad Applied Mathematics, 4, =,,. Iteratioal Jour- [7] Awoyemi, D.O. () A New Sixth Order Algorithm for Geeral Secod Order Ordiary Differetial Equatio. Iteratioal Joural of Computer Mathematics, 77, 7-4. [8] Awoyemi, D.O., Adebile, E.A., Adesaya, A.O. ad Aake, T.A. () Modifid Block Method for the Direct Solutio of Secod Order Ordiary Differetial Equatio. Iteratioal Joural of Applied Mathematics ad Computatio,, [9] Lambert, J.D. (97) Computatioal Methods i ODES. Joh Wiley ad Sos, New York. [] Adesaya, A.O., Aake, T.A. ad Udo, M.O. (8) Improved Cotiuous Method for Direct Solutio of Geeral Secod Order Ordiary Differetial Equatio. Joural of the Nigeria Associatio of Mathematical Physics,, 9-6. [] Udo, M.O., Olayi, G.A. ad Ademiluyi, R.A. (7) Liear Multistep Method for Solutio of Secod Order Iitial Value Problems of Ordiary Differetial Equatios: A Trucatio Error Approach. Global Joural of Mathematical Scieces, 6, 9-6. [] Zaria, B.I., Mohamed, S. ad Iskala, I.O. (9) Direct Block Backward Differetiatio Formulas for Solvig Secod Order Ordiary Differetial Equatio. Joural of Mathematics ad Computatio Scieces,, -. [] James, A.A., Adesaya, A.O. ad Suday, J. () Cotiuous Block Method for the Solutio of Secod Order Iitial Value Problems of Ordiary Differetial Equatios. Joural of Mathematics ad Computatio Scieces, 8, [4] Awoyemi, D.O. ad Idowu, M.O. () A Class of Hybrid Collocatio Method for Third Order Ordiary Differetial Equatio. Iteratioal Joural of Computer Mathematics, 8, [] Awoyemi, D.O., Udo, M.O. ad Adesaya, A.O. (6) No-Symmetric Collocatio Method for Direct Solutio of Geeral Secod Order Iitial Value Problems of Ordiary Differetial Equatios. Joural of Natural ad Applied Scieces, 7, -7. [6] Awoyemi, D.O. () A p-stable Liear Multistep Method for Solvig Third Order Ordiary Differetial Equatio. Iteratioal Joural of Computer Mathematics, 8, [7] Yahaya, Y.A. ad Badmus, A.M. (9) A Class of Collocatio Methods for Geeral Secod Order Differetial Equatio. Africa Joural of Mathematics ad Computer Research,,