Direct Solution of Initial Value Problems of Fourth Order Ordinary Differential Equations Using Modified Implicit Hybrid Block Method

Transcription

1 Joural of Scietific Research & Reports (): 79-8, ; Article o. JSRR...7 ISSN: 7 SCIENCEDOMAIN iteratioal Direct Solutio of Iitial Value Probles of Fourth Order Ordiary Differetial Equatios Usig Modified Iplicit Hybrid Block Method S. J. Kayode, M. K. Duroola ad Bolariwa Bolai * Departet of Matheatical Scieces, Federal Uiversity of Techology, Akure, Nigeria. Departet of Coputer Scieces, Sale Uiversity, Lokoa, Nigeria. Authors cotributios This work was carried out i cooperatio betwee all authors. Author SJK desiged the study, wrote the cocept of the article, take care of forattig ad provided fiishig touch to this auscript. Author MKD aaged the experietal process ad wrote the first draft of the auscript. Author BB too aaged the experietal process, alog with the literature searches ad aalyses of the study. All authors read ad approved the fial auscript. Article Iforatio DOI:.97/JSRR//9 Editor(s): () Narcisa C. Apreutesei, Techical Uiversity of Iasi, Roaia. Reviewers: () Hesa-Eldie Derili Gheralar, Departet of Matheatics, Kara brach, Islaic Azad Uiversity, Kara, Ira. () Aoyous, Maipal Uiversity, Idia. () Aoyous, Baha-Ud-Di Uiversity, Multa, Pakista. Peer review History: Origial Research Article Received th Jue Accepted th July Published st Septeber ABSTRACT Our focus i this article is the derivatio; aalysis ad ipleetatio of a ew odified iplicit hybrid block ethod for the direct solutio of iitial value probles of fourth order ordiary differetial equatios. I the derivatio of the ethod, we adopted the approach of collocatio approxiatio to obtai the ai schee with cotiuous coefficiets. Fro the ai schee, additioal schees were developed. The ipleetatio strategy of the ew ethod is by cobiig the ai schee ad the additioal schees as *Correspodig author: E-ail: bolariwa.bolai@yahoo.co;

2 Kayode et al.; JSRR, Article o. JSRR...7 siultaeous itegrator to iitial value proble of fourth order ordiary differetial equatios. As required of ay uerical ethod, the properties aalysis of the block was doe ad the result showed that it is cosistet, coverget, zero stable ad absolutely stable. We the test our ethod with uerical exaples solved usig existig ethod ad were foud to give better results. Keywords: Iterpolatio; cotiuous coefficiets; block ethod; uerical itegratio; fourth order ordiary differetial equatios.. INTRODUCTION Soe epirical probles ad physical. Pheoea i sciece ad egieerig, such as echaical systes without dissipatio, celestial echaics, cotrol theory, coputer aided desigs whe odeled result to higher order ordiary differetial equatios of the for: y ( ) = f ( x, y, y, y, y ), y( x ) = η, ( x ) = η, y ( x) = η, y ( x) = η y iv Covetioally, to solve () uerically, we first reduce it to syste of first order ordiary differetial equatios ad the apply ay other existig first order ethod to solve it. May literature abouds o this [,]. The drawback of this ethod is that it is tie cosuig, cubersoe to solve, ad take uch coputer space. To circuvet these draw backs, ay researchers have solved () directly, they iclude: [,,,6] who developed block ethods for uerical solutio of fourth order ordiary differetial equatios. The works of [,6] serve as iproveet o the work of [] who developed Liear ultistep ethod for the solutio of fourth order ordiary differetial equatios whose ipleetatio is Predictor Corrector ode. We are otivated to advace the course of research work by cotiuig with the propositio of block ethod which have bee show to eliiate the drawbacks of Predictor corrector ethod as discussed i [,7,8,9,,] i their works have proposed sigle Step hybrid ethods for the direct uerical solutio of iitial value probles of secod order ad third order Ordiary differetial equatios respectively. I all Cases, their ethods of ipleetatio are block ode with the proposed ethods beig efficiet, adequate ad suitable towards caterig for the class of probles for which they were desiged. Cosequetly, our otivatio i this work is the success story of the adoptio of sigle step ethod to solvig higher order ordiary differetial equatios. Thus, i this work, we are proposig a sigle step ethod for the direct uerical solutio of fourth order ordiary differetial equatios, which eliiates the use of predictors by providig sufficietly accurate siultaeous differece equatios fro a sigle cotiuous forula ad its derivatives. Accordig to [], the geeral block forula is give by: Y ( y ) + h bf( y ) = ey + h df () () 79

3 Kayode et al.; JSRR, Article o. JSRR...7 Where e is s s vector, d is r - vector ad b is r r vector, s is the iterpolatio poits th ad r is the collectio poits. F is a k vector whose J etry is f + = f ( t+, y+ ), is the order of the differetial equatio. Give a predictor equatio i the for: Y () = ey + h df ( y ). () By Puttig () i () we have: Y ( y ) + h bf ( ey + h dfy ). = ey + h df () Equatio () is called a self startig block-predictor-corrector ethod because the predictio equatio is gotte directly fro the block forula [,]. Cosequetly, our focus i this paper is the propositio of a iproved iplicit cotiuous hybrid algorith for the solutio of iitial value probles of fourth order ordiary differetial equatios.. DERIVATION OF THE METHODS We take our basis fuctio to be a power series of the for: y r ( ) + x = = s a x () The third derivative of () gives: y r + s ( a x (6) = ( iv) x) = ( )( )( ) By puttig (6) ito () we have the differetial syste: r s + = ( )( )( ) ' = f ( x, y( x), y ( x), y ( x), y ( x)) a x (7) Where a the paraeters to be deteried are, while r+s deotes the uber of collocatio ad iterpolatio poits. By collocatig (7) at the esh poits x = x, ( + = ), ad iterpolatig () at x = x +, =,,, yields a syste of equatios: 79

4 Kayode et al.; JSRR, Article o. JSRR...7 r+ s a x = y+ s = r+ s + = ( )( )( ) a x = f r (9) By puttig these syste of equatios i atrix for ad the solved to obtai the values of Paraeters a s, =,,. Which whe substituted i (), yields, after soe aipulatio, a hybrid liear ethod with cotiuous coefficiets of the for: y( x) = y+ ( x) + h = = The co efficiet of α(x) ad are: f + ( x) α () α ( t ) = ( t + t + t) ( t) α = ( t + t t ) + α ( t ) = ( t + t + t) ( ) α t = ( t + t + t ) α 6 ( ) t = ( t + t + t ) ( t) = ( t + t t + t t + t t ), + t ( t ) = (t t t + 9t t 9t + ), (8) ( t) = ( t 6t t t 7t + t t 6), ( t) = (t t + t t t t + t ), ( t) = ( t + 9t t + 7t 6t t + t 9), () Where x x t = h 79

5 Kayode et al.; JSRR, Article o. JSRR...7. Derivatio of the Block The geeral block forula proposed by Awoyei et al. [], i the Noralized for is give by: A () Y = ey + h df λ ( y ) + h bf( y ) λ () By evaluatig () at t = ; the first, secod ad the third derivative at x = x +, i = ( ) ad substitutig ito () gives its coefficiets as: 96 d = T e = T A = idetity atrix B = T 796

6 Kayode et al.; JSRR, Article o. JSRR...7. ANALYSIS OF THE PROPERTIES OF THE BLOCK I this sectio we carry out the aalysis of the Basic properties of the ew ethod.. Order of the Method The liear operator of the block () is defied as: L λ λ { y( x) h} = Y ey + h df ( y ) + h bf( y ) : () By expadig y( x + ih) ad ( x h) f + i Taylor series, () becoes: L { y( x) : h} = C y( x) + Chy ( x) + C h y ( x) +... p ( p) + C h y ( x) + () The block () ad associated liear operator are said to have order p if C C =... = C p =, C. = + p+ p The ter C p+ is called the error costat ad iplies that the local trucatio error is give by: ( p+ ) ( p+ ) ( p+ ) ( ) t h () + k = C p+ h y x + Hece the block () has order 8 with error costat: C p +. Zero Stability of the Block 9 9 7,,, ,,,, = 8 9,,,, ,,, The block () is said to be Zero stable if the roots ρ satisfies z ad the root z = polyoial ( z) = det( za E), T, z s =,,..., N of the characteristic has ultiplicity ot r exceedig the order of the differetial equatio. Moreover as h, ρ( z) = z ( λ ), 797

7 Kayode et al.; JSRR, Article o. JSRR...7 Where is the order of the differetial equatio, for the block (), r = 6, = ( ) = λ ( λ ) ρ z Hece our ethod is Zero stable.. Covergece = λ =,,,,,,,,,,,,,,, The ecessary ad sufficiet coditio for a uerical ethod to be coverget is for it to be Zero stable ad has order p, Sice our ethod has bee show to be zero stable ad has order 8, it satisfied the above coditio, thus our ethod is coverget.. NUMERICAL EXPERIMENTS To test the accuracy, workability ad suitability of the ethod, we adopted our ethod to solvig soe iitial value probles of fourth order ordiary differetial equatios. Test proble. We cosider a o liear fourth order proble: ( ) = ( y ) y( y ) x + e( x + x ) x, y( ) =, y ( ) =, y ( ) =, y ( ) y iv h = x Whose exact solutio is give by: y ( x) = x + e. The result is as show i Table. Table. Showig results for proble XVAL ERC NRC ERR E E E E E E E E E E - Test proble. We cosider special fourth order proble: y iv ( ) =, y ( ) =, y ( ) =, y ( ) =,. = x; y h = =, 798

8 Kayode et al.; JSRR, Article o. JSRR...7 = x Whose exact solutio is: y ( x) + x Our ethod was used to solve the proble ad result copared with []. The result is as show i Table. Table. Showig results for proble XVAL ERC NRC ERR ERR i [] E- 7.E E E E -.999E E -.E E E E -.8E E -.E E -.E E -.88E E -.6E -8. Nuerical Results We ake use of the followig Notatios i the table of results: XVAL: Value of the idepedet variable where uerical value is take. ERC: Exact result at XVAL NRC: Our Nuerical result at XVAL ERR: Error of our result at XVAL.. CONCLUSION I this paper, we have proposed a odified Iplicit Hybrid Block algorith for the uerical solutio of iitial value probles of fourth order ordiary differetial equatios. For better perforace of the ethod, step size is chose withi the stability iterval. The results of our ew ethod whe copared with the block ethod proposed by [] showed that our ethod is ore accurate. COMPETING INTERESTS Authors have declared that o copetig iterests exist. REFERENCES. Labert JD. Coputatioal ethods i ordiary differetial equatios. Joh Wiley ad Sos Ic, New York; 97.. Fatula SO. Nuerical ethods for iitial value probles i ordiary differetial equatio. Acadeic Press Ic, Harcourt Brace Jovaovich publisher;

9 Kayode et al.; JSRR, Article o. JSRR...7. Oar Z. Developig parallel block ethod for solvig higher orders ODES directly. PhD Thesis, Uiversity putra, Malaysia; Kayode SJ. A zero stable ethod for direct solutio of fourth order ordiary differetial equatio. Aerica Joural of Applied Scieces. 8;(): Olabode BT. A six-step schee for the solutio of fourth order ordiary differetial equatios. Pacific Joural of Sciece ad Techology. 9;(); Adesaya AO. Block ethods for the solutios of geeral higher order iitial value probles of ordiary differetial equatios. A Ph.D thesis i Matheatical Scieces departet of Federal Uiversity of Techology, Akure, Nigeria;. 7. Yusuph Y. Soe theories ad applicatio of liear ultistep ethods for Ordiary differetial equatios. Ph.D Thesis Uiversity of Jos, Nigeria;. 8. Aake. Cotiuous Iplicit hybrid oe-step ethods for solutios of iitial value probles of geeral secod order ordiary differetial equatios. Ph.D thesis, Coveat Uiversity;. 9. Bolariwa Bolai. Iplicit hybrid block ethods for the uerical solutios of iitial value proles of third order ordiary differetial equatios. A Ph.D Thesis i Matheatical Scieces Departet of Federal Uiversity of Techology, Akure; Nigeria;.. Bolariwa Bolai, Adeiluyi RA, Awoyei DO, Ogudele JO. A sigle step iplicit hybrid uerical ethod for the uerical itegratio of iitial value probles of third order ordiary differetial equatios. Caadia Joural o Sciece ad Egieerig Matheatics. ;-.. Bolariwa Bolai, Adeiluyi RA, Olasei Tude, Duroola MK. A ew iplicit hybrid block ethod for the direct solutio iitial value probles of geeral third order ordiary differetial equatios. Caadia Joural o Sciece ad Egieerig Matheatics. ;:.. Awoyei DO, Adebile EA, Adesaya AO, Aake TA. Modified block ethod for direct solutio of secod order ordiary differetial equatio. Iteratioal Joural of Applied Matheatics ad Coputatio. ;(): Shapie LF, Watts HA. Block iplicit oe step ethods. Joural of Math of Coputatio. 969;(8):7-7.. Mohaed U. A six step block ethod for solutio of fourth order ordiary differetial equatios. The Pacific Joural of Sciece ad Techology. ;():8-6. Kayode et al.; This is a Ope Access article distributed uder the ters of the Creative Coos Attributio Licese ( which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Peer-review history: The peer review history for this paper ca be accessed here: 8