Applied Mathematics, 4,, -66 Published Olie May 4 i SciRes. http://www.scirp.org/oural/am http://dx.doi.org/.46/am.4.87 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 Email: 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). http://creativecommos.org/liceses/by/4./ 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,, -66. http://dx.doi.org/.46/am.4.87 ( )
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 ()
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 + + + + + 4 + x x x x x x x x x x x x x x 6 4 6 4 X = 6 4 6 4 6 6 4 4 6 7 4 6 7 + + + + + + + 4 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
M. R. Odekule et al. where α = t α = t = + + + 8 ( t 7 4t 6 7t 7t 4 86t 4t 46t) = + + 8 ( t 7 96t 6 49t 9t 4 84t 4t) = + + 4 ( t 7 8t 6 9t 74t 4 4t t) = + + 4 ( 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 = + + 8 ( t 7 4t 6 7t 7t 4 86t 4t ) σ = + + 8 ( t 7 96t 6 49t 9t 4 84t ) σ = + + 4 ( t 7 8t 6 9t 74t 4 4t ) σ = + + 4 ( t 7 68t 6 9t 7t 4 8t ) σ 4 = + + 8 ( 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 = + + + + 4 + Y y y y y y
M. R. Odekule et al. e =, e = 4 II) Whe i = d 4 7 6 = 4 76 8 ( i), b 86 76 94 4 7 6 4 4 8 44 7 6 8 6 6 9 87 9 9 = 4 4 44 76 68 6 6 6 87 6 6 7 6 4 8 8 6 [ ] T m + + + + 4 + e =, d 9 88 4 4 = 6 4 4 9 88 47 4 7 44 4 7 44 6 4 7 7 4 4 9 9 7 7 b = 6 8 8 6 6 64 8 64 4 4 4 4 9 96 44 44 96 88... 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
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 + + + + + + + 4 x x x x x x x x x x x x x x x x x x x x x x x x 6 4 6 X = 6x x x x 4x 6x 6 6 4 6 6 4 6 6 4 6 4 6 7 8 4 6 7 8 + + + + + + + + 4 6 7 8 + + + + + + + + 4 6 x x x x x x 4 6 + + + + + + 4 6 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+ 4 4 6 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 = 6 + 476 89 + 86 6 + 47 44 44 8 7 6 4 ( t t t t t t t ) α = t t + t t + t t + t ( 8 6 7 476 6 89 86 4 6 6 ) α = + + + 44 ( t 8 6t 7 476t 6 89t 86t 4 6t 99t) = + + + 768 ( 4t 8 t 7 89t 6 797t 7988t 4 786t 84t 679t) = + + + 768 ( 67t 8 6t 7 4848t 6 796699t 644t 4 8964t 444t) = + + + 84 ( 6t 8 47t 7 64t 6 99t 98877t 4 69t 78t) = + + 84 ( 44t 8 66t 7 844t 6 4t 948t 4 488t 6696t) 4 = + + + 768 ( 78t 8 t 7 94t 6 4789t t 4 474t t) = + + + 768 Evaluatig () at t = ( ) gives the followig ( 6t 8 88t 7 88t 6 7t 6t 4 696t 7t) 9 4 y+ = y y+ + y+ h + + + + + 4 9 6 7 y+ 4 = y y+ + y+ ( 7 f 78 f 4998 f 78 f 47 f f ) + + + + 4 + h + + + + + ( 77 f 864 f 74 f 44 f 69 f 8 f ) + + + + 4 + () () 7
M. R. Odekule et al. 66 79 9 y+ = y y+ + y+ 44 h 4 ( 6 f f 46 f 6 f 6 f 47 f ) + + + + 4 + Evaluatig the first derivatives of () at t =, gives the followig 47 6 99 hy = y + y+ y+ 44 44 h + + 7846 ( 999 f 4668 f 948 f 44 f 69 f 44 f ) + + + + 4 + 7 4 hy + = y y+ + y+ 6 6 h + + + 4 ( f 496 f 668 f 674 f 49 f 6 f ) + + + + 4 + Writig Equatios () to (6) i block form, the parameters i () gives the followig (4) () (6) A ( i) ( ) A = idetity matrix m + + + + 4 + m 4 [ ] T m + ( ) = F Y f f f f f m + + + + 4 + ( ) = F y f f f f f 4 =, B ( ) A ( k ) 7 99 76 76 6 86 86 7 87 = 76 76 89 6 86 86 7 687 76 76 679 7996 7 878 999 =, 9 9496 96 694 47966 8
M. R. Odekule et al. B ( i) 669 6 8 97 7 987 86994 89968 7996 7988 7978 6 844 487 78 469 784 784 469 784 8779 9 79 77 = 9 9666 9666 8664 9 464 98768 8498 86 96 784 96 8 784 6767 99787 74 86 477 47966 7988 7988 47966 47966 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 + + + + 4 + m 4 [ ] T m + ( ) = F Y f f f f f m + + + + 4 + ( ) = F y f f f f f 4 A ( i) =, A ( k ) = 794 6 977 96 6 96 76 687 4 6 6 6 446 696 66967 697 78 697 849 844 484 6 6 6 87 87 487 7 487 B ( ) 84667 989 867 476 8487 = 996 448 868 47 768 9
M. R. Odekule et al. B ( i) 979 867 89 4667 74 768 687784 687784 687784 768 = 96 96487 88 8999 64 868 868 868 476 868 9 69977 674947 776 497 487984 499 499 499 487984 46 476 9646 4976 7 8688 868 868 868 868 97 697 46 98 847 77 66 66 66 77 Developmet of Block Corrector case III ( ) A = idetity matrix m + + + + 4 + m 4 [ ] T m + ( ) = F Y f f f f f m + + + + 4 + ( ) = F y f f f f f 4 A ( i) =, 68487 8 848 84969 9 44896 448 9 9 46 6 44 9 669 9 6 ( k ) 69 4 84 9677 A = 786 46 786 786 479 7 9696 77776 987 669 9 987 4864 76 9 47 44896 44896 44896 46 B ( ) 4944 844 4784 464 9679 = 497 77488 464 697 6896 6
M. R. Odekule et al. B ( i) 78799 699 77 9 968 6896 844 669 6768448 844 7 698 8964 487 98 98 98 98 6 9499 4 67 699 = 9944 4786 4786 9944 497 787 494 868 88 98 98 487 487 464 68947 8898 447 66687 877 6896 844 844 6896 6896. 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 ( + ) 86 76 94 4 7 y =! + + 6 4 4 8 ( h) ( ) 7 y y hy! 6 ( ) ( ) ( ) ( 4) ( ) + h ( + ) 44 7 6 8 y =! + + 6 6 ( h) ( ) ( h) ( ) ( ) ( ) ( ) ( 4) ( ) y y hy =! 4 = = = + h ( + ) 9 87 9 9 y =! + + 4 4 4 76 y y 4hy! ( ) ( ) ( ) ( 4) ( ) + h ( + ) 44 76 68 6 6 y =! + + + 6 ( h) ( ) y y hy! 8 ( ) ( ) ( ) ( 4) ( ) + h ( + ) 87 6 6 7 y =! + + + + 6 4 8 8 6 ( ) ( ) ( ) ( 4) ( ) Collectig coefficiets i powers of h, we see that the order of the method is six ad the error costat is 99 9 4 8 7 49 94 448 89 49 Also whe i = The order of the method is six ad the error costat is T 6
M. R. Odekule et al. 86 7 9 8 7 648 78 4 94 96 T... Order of the Block Corrector for Case I Takig a Taylor series expasio gives ( h) ( ) ( ) ( ) 7 99 679 y y hy y = =! 76 76 7996 + h ( + ) 669 6 8 97 7 y ( ) ( ) ( ) ( 4) ( ) + + =! 987 86994 89968 7996 7988 ( h) ( ) 6 ( y y hy = ) 7 y ( ) hy =! 86 86 878 + h ( + ) 7978 6 844 487 78 y ( ) ( ) ( ) ( 4) ( ) =! + + 469 784 784 469 784 ( h ) ( ) 7 87 ( ) 999 y ( ) y hy y = =! 76 76 9 + = h ( + ) 8779 9 79 77 y ( ) + ( ) + ( ) + ( 4) ( ) =! 9 9666 9666 8664 9 ( 4h) ( ) 89 6 ( ) 9496 y ( ) y hy y = =! 86 86 96 + h ( + ) 464 98768 8498 86 y ( ) ( ) = + + ( ) ( 4) ( )! + 96 784 96 8 784 ( h) ( ) 7 687 ( ) 694 y ( ) y hy y = =! 76 76 47966 + h ( + ) 6767 99787 74 86 477 y ( ) ( ) ( ) ( 4) = + + + + ( )! 47966 7988 7988 47966 47966 ad the order of our method is seve with error costat as 977 469 4 48 67 644 49 6676 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 84969 44 7 97 47 8686 9497 996 887 46846464..4. Order of the Block Corrector for Case III Also usig the same approach, the order of our method is ie with error costat as 77 877 967 8474 74 884886 88 797 446.. Cosistecy of the Method A block method is said to be cosistet if it has order p [9]. T T T 6
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. 4... 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=.. 6 4 6 Exact solutio: y( x) ( si ( x) ) =. 4... Test Problem Cosider the iitial value ODE 4x =. e x Exact solutio: y( x) = + ; ( ) ( ) y y xe x y =, y =, h=.. 6
M. R. Odekule et al. 4..4. 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..48666e.797e 6.99988e 6.977e 6.9868e..448e.669e.4768e.4687e.47468e. 4.78e 4.7474e.9699e.969e.974887e.4.87796e.9946e 6.94e 7.66e 7.67e. 9.96e 9.76778e.864e.779e.7484e.6.4446e 9.78e 9.6e.766e.4767e.7.66484e 9.949e 9.949e.986e.9947e.8 4.6e 9.777e 9.e.9796e.e.9 8.8468e 9 6.846e 9.76466e 9.e 9.869e 9..4468e 8 9.7446e 9.7978e 9.8898e 9.864998e 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..48646e 8.e 9.4e 8.498e 8.4488e 8. 7.47e 8.97e 9.46e 8.497e 8.477e 8. 9.46e 8 4.978e 9.6446e 8.64489e 8.644e 8..484e 7 9.87e 9.e 8.8e 8.6e 8.4.678e 7.4497e 8.4e 8.44e 8.49e 8..4889e 7.9974e 8.88486e 8.88498e 8.8768e 8.6.764e 7.876e 8 4.844e 8 4.87e 8 4.8494e 8.7.677e 7.44e 8 4.e 8 4.4e 8 4.48e 8.8.6986e 7.94488e 8 4.7e 8 4.78e 8 4.7e 8..68684e 7 4.9e 8 4.88e 8 4.8987e 8 4.849e 8 64
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. 6.486478e 6.487e 4.7e 4 4.946e 4 4.48e 4..774e.47e.849e.87e.87e..7888e.778e 4.664e 4.667e 4.6677e.4 4.689e 4.96e 9.787e 9.787e 7.77894e. 7.4e 6.98969e 7.7449e 7.7449e.744e.6.48768e.44e.9444e.9444e.94487e.7.4647e.646e 4.77e 4.77e 4.774e.8.869e.9e 7.69e 7.69e 7.676e.9.798e.7e.79e.79e.778e. 4.668e 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 6.4994e 6.668e 6.4994e..6899e.894e.777e 4.e 4.e 4..7477e.7e.97e 4.979e 4.97e 4.4 4.89e.4869e.747e.7e.747e. 4.6e.778e.67996e.66977e.66977e.6.49e.46e.6e.64e.84e.7.74e 9.4798e.484e.484e.484e.8 7.4996e 6.76e.864686e.8676e.8676e.9.6e.69e 4.77486e 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,, 746-7. 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
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