M. Mechee, 1,2 N. Senu, 3 F. Ismail, 3 B. Nikouravan, 4 and Z. Siri Introduction

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1 Hndaw Publhng Corporaton Mathematcal Problem n Engneerng Volume 23, Artcle ID , 7 page Reearch Artcle A Three-Stage Ffth-Order Runge-Kutta Method for Drectly Solvng Specal Thrd-Order Dfferental Equaton wth Applcaton to Thn Flm Flow Problem M. Mechee,,2 N. Senu, 3 F. Imal, 3 B. Nkouravan, 4 and Z. Sr Inttute of Mathematcal Scence, Unverty of Malaya, 563 Kuala Lumpur, Malaya 2 Department of Mathematc, Faculty of Mathematc and Computer Scence, Unverty of Kufa, Najaf, Iraq 3 Department of Mathematc and Inttute for Mathematcal Reearch, Unvert Putra Malaya, 434Serdang,Selangor,Malaya 4 Department of Phyc, Faculty of Scence, Unverty of Malaya, 563 Kuala Lumpur, Malaya Correpondence hould be addreed to N. Senu; norazak@cence.upm.edu.my Receved 23 March 23; Accepted 23 May 23 Academc Edtor: Ebrahm Momonat Copyrght 23 M. Mechee et al. Th an open acce artcle dtrbuted under the Creatve Common Attrbuton Lcene, whch permt unretrcted ue, dtrbuton, and reproducton n any medum, provded the orgnal work properly cted. In th paper, a three-tage ffth-order Runge-Kutta method for the ntegraton of a pecal thrd-order ordnary dfferental equaton (ODE) contructed. The zero tablty of the method proven. The numercal tudy of a thrd-order ODE arng n thn flm flow of vcou flud n phyc dcued. The mathematcal model of thn flm flow ha been olved ung a new method and numercal comparon are made when the ame problem reduced to a frt-order ytem of equaton whch are olved ung the extng Runge-Kutta method. Numercal reult have clearly hown the advantage and the effcency of the new method.. Introducton A pecal thrd-order dfferental equaton (ODE) of the form y (x) =f(x,y(x)), y(x )=α, y (x )=β, y (x )=γ, x x () whch not explctly dependent on the frt dervatve y (x) and the econd dervatve y (x) of the oluton frequently found n many phycal problem uch a electromagnetc wave, thn flm flow, and gravty drven flow. The oluton to () can be obtaned by reducng t to an equvalent frt-order ytem whch three tme the dmenon and can be olved ung a tandard Runge-Kutta method or a multtep method. Mot reearcher, centt, and engneer olve problem () by convertng the problem to a ytem of frt-order equaton. However, there are alo tude on numercal method whch olve () drectly.suchworkcanbeeennawoyem [], Waeleh et al. [2], Zanuddn [3], and Jator [4]. Awoyem and Idowu [5] and Jator[6] propoed a cla of hybrd collocaton method for the drect oluton of hgher-order ordnary dfferental equaton (ODE). Samat and Imal [7] developed an embedded hybrd method for olvng pecal econd-order ODE. Waeleh et al. [2] developedablock multtep method whch can drectly olve general thrdorder equaton; on the other hand, Ibrahm et al. [8] developed a multtep method that can drectly olve tff thrd-order dfferental equaton. All of the method dcued above are multtep method; hence, they need tartngvaluewhenuedtoolveodeucha(). Senu et al. [9] derved the Runge-Kutta-Nytröm method for olvng econd-order ODE drectly. Mechee et al. [] contructednew Runge-Kutta method for olvng (). In th paper, we are concerned wth a one-tep method, partcularly the three-tage ffth-order Runge-Kutta method, for drectly olvng pecal thrd-order ODE. Accordngly, we have developed a drect Runge-Kutta method (RKD) whch can be drectly ued to olve (). The advantage of the new method over multtep method that t ntale telf. The method produce y n+, y n+,andy n+ to approxmate y(x n+ ), y (x n+ ),andy (x n+ ),wherey n+ the computed oluton and y(x n+ ) the exact oluton.

2 2 Mathematcal Problem n Engneerng 2. Dervaton of RKD Method The general form of RKD method wth tage for olvng ntal value problem ()canbewrttena where y n+ =y n +hy n + h2 2 y n +h3 b k, = y n+ =y n +hy n +h2 b k, = y n+ =y n +h b = k, (2) k =f(x n,y n ), (3) k =f(x n +c h, y n +hc y n + h2 2 c2 y n +h3 a j k j ) (4) for =2,3,...,. The parameter of the RKD method are c,a j,b,b,b for =,2,...,and j =,2,...,areaumedtobereal. If a j =for j, t an explct method, and otherwe t an mplct method. The RKD method can be expreed n the Butcher notaton ung the table of coeffcent a follow: c j= A b T b T b T (5) To determne the coeffcent of the method gven by (2) (4), the RKD method expreon expanded ung Taylor ere expanon. After ome algebrac manpulaton, th expanon equated to the true oluton that gven by Taylor ere expanon. General order condton for the RKD method can be found from the drect expanon of the local truncaton error. Th dea baed on the dervaton of order condton for the Runge-Kutta method ntroduced n Dormand []. The RKD formulae n (2) may be expreed a y n+ =y n +hφ(x n,y n ), y n+ =y n +hφ (x n,y n ), y n+ =y n +hφ (x n,y n ), where the ncrement functon are Φ(x n,y n )=y n + h 2 y n +h2 b k, = = Φ (x n,y n )=y n +h b k, Φ (x n,y n )= b = k, (6) (7) and k defnednformula(4). If Δ the Taylor ere ncrement functon, then the local truncaton error of the oluton, the frt dervatve, and the econd dervatve may be obtaned by ubttutng the true oluton y(x) of ()nto the RKD ncrement functon. Th gve t n+ =h[φ Δ], t n+ =h[φ Δ ], t n+ =h[φ Δ ]. Thee expreon are bet gven n term of elementary dfferental and the Taylor ere ncrement may be wrtten a Δ=y + 2 hy + 6 h2 F (3) + 24 h3 F (4) +O(h 4 ), Δ =y + 2 hf(3) + 6 h2 F (4) + 24 h3 F (5) +O(h 4 ), Δ =F (3) + 2 hf(4) + 6 h2 F (5) +O(h 3 ), where for the calar cae the frt few elementary dfferental are F (3) =f, F (4) =f x +f y y, F (5) =f xx +2f xy y +f y y +f yy (y ) 2. (8) (9) () Ung the above term, the ncrement functon Φ, Φ, and Φ for the RKD formula become = = b = b k = b k = k = = + 2 b f+ = = +O(h 3 ), = + 2 b c (f x +f y y )h b c 2 (f xx +2f xy y +f y y +f yy (y ) 2 )h 2 b f+ b c (f x +f y y )h = = +O(h 3 ), b = + 2 b c2 (f xx +2f xy y +f y y +f yy (y ) 2 )h 2 b = f+ c (f x +f y y )h b = +O(h 3 ). c 2 (f xx +2f xy y +f y y +f yy (y ) 2 )h 2 ()

3 Mathematcal Problem n Engneerng 3 The expreon for the local truncaton error n the y oluton, the frt dervatve, and the econd dervatve are t n+ =h 3 [ t n+ =h2 [ t n+ =h[ b = = = b k ( 6 F(3) + 24 F(4) + )], b k ( 2 F(3) + 6 F(4) + )], k (F (3) + 2 F(4) + 6 h2 F (5) + )]. (2) Subttutng () nto(2) and expandng a a Taylor expanon ung MAPLE 4 oftware a ntroduced by Gander and Gruntz [2], the error equaton or the order condton for -tage x-order RKD method can be wrtten a follow. Order condton for y: Order 3 Order 4 Order 5 Order 6 b = 6. (3) b c = 24. (4) b c 2 = 6. (5) b c 3 = 2, b a j = 72. (6) Order condton for y : Order 2 Order 3 Order 4 Order 5 b = 2. (7) b c = 6. (8) b c2 = 2. (9) b c3 = 2, b a j = 2. (2) Order 6 b c4 = 3, b a jc j = 72, b c a j = (2) 8. Order condton for y : Order Order 2 Order 3 Order 4 Order 5 b =. (22) b c = 2. (23) b c 2 = 3. (24) b c 3 = 4, b a j = 24. (25) b c 4 = 5, b a j c j = 2, b c a j = 3. (26) Order 6 b c 2 a j = 36, b a j c 2 j + b c a j c j = 7 72, b c 5 = 6, b a j c 2 j = 36, b c a j c j = 44, 2 b a j c 2 j + b c a j c j = 2. (27) All ndce are from to. To obtan the ffth-order RKD method, the followng mplfyng aumpton ued n order to reduce the number of equaton to be olved: b =b ( c ), =,...,. (28) To derve the three-tage ffth-order RKD method, we ue the algebrac condton up to order fve (3) (5), (7) (2), and (22) (26). The reultng ytem of equaton cont of 6 nonlnear equaton wth 3 unknown varable for whch we need to olve. Conequently, there no oluton nce the number of equaton exceed the number of unknown to be olved for. To overcome th, the mplfyng aumpton (28) mpoed. Th wll reduce the number of equaton to wth unknown makng the ytem olvable. Th ytem ha no free parameter but content and yeld a unque oluton. The coeffcent of the method are gven below (ee The RKD5 method n (29)). The error norm for y n, y n,and y n are gven by τ(6) 2 = , τ (6) 2 = , τ (6) 2 = ,repectvely, where τ (6), τ (6),andτ (6) are error term of the xthorder condton for y n, y n,andy n,repectvely.

4 4 Mathematcal Problem n Engneerng The RKD5 Method. Conder the followng: (29) Next, we wll dcu the zero tablty of the method whch one of the crtera for the method to be convergent. Zero tablty an mportant tool for provng the tablty and convergence of lnear multtep method. The ntereted reader referred to the textbook by Lambert [3] and Butcher [4] n whch zero tablty dcued. Zero tablty ha been dcued n Harer et al. [5] to determne an upper bound on the order of convergence of lnear multtep method. In tudyng the zero tablty of the RKD method, we can wrte method (2) a follow: ( )( y n+ hy n h 2 y n+ 2 )=( )( y n+ hy n ), h 2 y n+ ξ 2 p (ξ) = det [Iξ A] = det ( ξ ). ξ Thu, the charactertc polynomal (3) p (ξ) = (ξ ) 3. (3) Hence, the method zero table nce the root, ξ=,are le than or equal to one. 3. Problem Teted and Numercal Reult In th ecton, we wll apply the new method to ome thrdorder dfferental equaton problem. The followng explct RK method are elected for the numercal comparon. () RKD5: the three-tage ffth-order RKD method dervednthpaper. () RK4: the four-tage fourth-order RK method a n Butcher [4]. () DOPRI: the even-tage ffth-order Runge-Kutta derved by Dormand []. Problem. Conder d 3 y (x) dx 3 = 3 8 (+x) 5, y() =, y () = 2, Exact oluton: y(x) = +x. Problem 2. Conder y () = 4. (32) d 3 y (x) dx 3 =y 2 (x) + co 2 (x) co (x), y() =, Exact oluton: y(x) = n(x). y () =, y () =. 4. An Applcaton to a Problem n Thn Flm Flow (33) In th ecton, we wll apply the propoed method to a wellknown problem n phyc regardng the thn flm flow of a lqud. Many problem have dcued th problem; ee, for example, [6 23]. In a urvey paper, Tuck and Schwartz [6] dcuedtheflowofathnflmofvcoufludoveraold urface. Tenon and gravty a well a vcoty are taken nto account. The problem wa formulated ung thrd-order ordnary dfferental equaton (ODE) a follow: d 3 y dx 3 =f(y) (34) for the flm profle y(x) n a coordnate frame movng wth the flud. The form of f(y) vare accordng to the phycal context. Dfferent form of the functon f are tuded n Tuck and Schwartz [6]. For dranage down a dry urface, the form of f(y) wa gven a d 3 y dx 3 = + y 2. (35) When the urface prewetted by a thn flm wth thckne δ > (where δ > very mall), the functon f gven by f (y) = + +δ+δ2 y 2 δ+δ2 y 3. (36) Problem concernng the flow of thn flm of vcou flud wth a free urface n whch urface tenon effect play a role typcally lead to thrd-order ordnary dfferental equaton governng the hape of the free urface of the flud, y= y(x). Accordng to Tuck and Schwartz [6], one uch equaton y =y k, x x (37)

5 Mathematcal Problem n Engneerng 5 Table : Table comparng value of the numercal oluton, a ffthorder Runge-Kutta method (DOPRI), fourth-order Runge-Kutta method (RK4), and our new RKD5 method at x [,.2,.4,.6,.8,.] takng h =. and k = 2 wth the ntal condton y() = y () = y () =. x Exact oluton DOPRI RK4 RKD Table 2: Table comparng value of the numercal oluton, a ffthorder Runge-Kutta method (DOPRI), fourth-order Runge-Kutta method (RK4), and our new RKD5 method at x [,.2,.4,.6,.8,.] takng h =. and k = 2 wth the ntal condton y() = y () = y () =. x Exact oluton DOPRI RK4 RKD Table 3: Table comparng value of the numercal oluton ung a ffth-order Runge-Kutta method (DOPRI), fourth-order Runge- Kutta method (RK4), and our new RKD5 method at x [,.2,.4,.6,.8,.] takng h =. and k=3wth the ntal condton y() = y () = y () =. x RK4 DOPRI RKD Table 4: Table comparng value of the numercal oluton ung a ffth-order Runge-Kutta method (DOPRI), fourth-order Runge- Kutta method (RK4), and our new RKD5 method at x [,.2,.4,.6,.8,.] takng h =. and k = 3 wth the ntal condton y() = y () = y () =. x RK4 DOPRI RKD wth ntal condton y(x )=α, y (x )=β, y (x )=γ, (38) where α, β, and γ are contant, of partcular mportance becaue t decrbe the dynamc balance between urface and vcouforcenathnfludlayerntheabence(orneglect) of gravty. For comparon purpoe, we wll ue Runge-Kutta method whch are fourth-order (RK4) and ffth-order (DOPRI) method, repectvely. To ue Runge-Kutta method we wrte () a a ytem of three frt-order equaton. Followng Bazar et al. [7], we can wrte (37)athefollowngytem: where dy dx =y 2 (x), dy 2 dx =y 3 (x), dy 3 dx =y k (x), (39) y () =, y 2 () =, y 3 () =, (4) we have taken x =and α=β=γ=. Unfortunately, for general k,(37) cannot be olved analytcally. We can, however, ue thee reducton to determne an effcent way to olve () numercally.wefocuonthecae k=2and k=3. The reult are dplayed n Table and 2 for the cae k=2and Table 3 and 4 for the cae k=3. For comparon purpoe, we preent effcency curve where the common logarthm of the maxmum global error along the ntegraton veru the functon evaluaton a hownnfgure and 2.FromFgure and 2,weoberved that the method RKD5 perform better when ntegratng thrd-order ODE compared to RK4 and DOPRI method. From Table and 2 we oberve that the numercal reult ung RKD5 are correct to fve decmal place. Applyng the fourth- and ffth-order RK method (RK4 and DOPRI) to (39) for k = 2 alo yeld fve-decmal-place accuracy. Whle from Table 3 and 4 we oberved the numercal reult for the new method, RKD5 agree to eght decmal place when compared wth the fourth- and ffth-order RK method (RK4 and DOPRI). Th content wth the reult dplayed n Table and 2.InFgure3 and 4,weplot the numercal oluton, y for k=2and k=3,repectvely, wth h =.. Fgure 5 how that the new RKD5 method requre le functon evaluaton than the RK4 and DOPRI method. Th becaue when problem(37) olved ung RK4 and DOPRI method, t need to be reduced to a ytem of frt-order equaton whch three tme the dmenon. 5. Dcuon and Concluon In th paper, we have derved the order condton for a Runge-Kutta method whch can be ued to drectly olve pecal thrd-order ordnary dfferental equaton. A threetage ffth-order RKD method ha been derved and appled to the thn flm flow problem. Numercal reult how that

6 6 Mathematcal Problem n Engneerng log (MAXERR) RKD5 RK4 DOPRI 5 Functon evaluaton 2 Fgure : The effcency curve of the RKD5 method and t comparon for Problem wth x end =and h=/2, =,...,5. log (MAXERR) RKD5 RK4 DOPRI Functon evaluaton Fgure 2: The effcency curve of the RKD5 method and t comparon for Problem 2 wth x end =πand h=/2, =,..., y y.2.4 Fgure 4: Plot of the oluton y for problem (37)fork=3, h =.. log (functon evaluaton) RKD5 DOPRI RK4 3 x 2.5 log (h) Fgure 5: Plot of graph for functon evaluaton agant tep ze, h for k=3, h = /, =,...,4. the new method agree very well wth well-known extng method n the lterature and requred le functon evaluaton. A uch, th method more cot effectve n term of computaton tme than other extng method. y x y Fgure 3: Plot of the oluton y for problem (37)fork=2, h =.. Reference [] D. O. Awoyem, A P-table lnear multtep method for olvng general thrd order ordnary dfferental equaton, Internatonal Journal of Computer Mathematc,vol.8,no.8,pp , 23. [2] N. Waeleh, Z. A. Majd, and F. Imal, A new algorthm for olvng hgher order IVP of ODE, Appled Mathematcal Scence,vol.5,no.53 56,pp ,2. [3] N. Zanuddn, 2-pont block backward dfferentaton formula for olvng hgher order ODE [Ph.D. the], UnvertPutra Malaya. [4] S. N. Jator, Solvng econd order ntal value problem by a hybrd multtep method wthout predctor, Appled Mathematc and Computaton,vol.27,no.8,pp ,2.

7 Mathematcal Problem n Engneerng 7 [5] D. O. Awoyem and O. M. Idowu, A cla of hybrd collocaton method for thrd-order ordnary dfferental equaton, Internatonal Journal of Computer Mathematc, vol.82,no.,pp , 25. [6] S. N. Jator, On a cla of hybrd method for y = f(x,y,y ), Internatonal Journal of Pure and Appled Mathematc, vol.59, no. 4, pp , 2. [7] F. Samat and F. Imal, An embedded explct hybrd for ordnary dfferental Equaton, Journal of Mathematc and Stattc,vol.8,pp.32 36,22. [8] Z. B. Ibrahm, K. I. Othman, and M. Suleman, Implct r-pont block backward dfferentaton formula for olvng frt-order tff ODE, Appled Mathematc and Computaton, vol. 86, no., pp , 27. [9] N. Senu, M. Suleman, and F. Imal, An embedded explct rungekutta-nytröm method for olvng ocllatory problem, Phyca Scrpta,vol.8,no.,ArtcleID55,7page,29. [] M. S. Mechee, F. Imal, N. Senu, and Z. Sr, Runge-kutta method for olvng pecal thrd order dfferental equaton y =f(x,y)drectly, n Proceedng of the 8th Mathematc andphycalscencegraduatecongre,chulalongkornunverty,bangkok,thaland,22. [] J. R. Dormand, Numercal Method for Dfferental Equaton, A Computatonal Approach, CRC Pre, Boca Raton, Fla, USA, 996. [2] W. Gander and D. Gruntz, Dervaton of numercal method ung computer algebra, SIAM Revew, vol. 4, no. 3, pp , 999. [3] J. D. Lambert, Numercal Method for Ordnary Dfferental Sytem, The Intal Value Problem, John Wley & Son, Chcheter, UK, 99. [4] J. C. Butcher, Numercal Method for Ordnary Dfferental Equaton, John Wley & Son, Chcheter, UK, 2nd edton, 28. [5] E. Harer, S. P. Nørett, and G. Wanner, Solvng Ordnary Dfferental Equaton I: Nontff Problem,vol.8,Sprnger,Berln, Germany, 2nd edton, 993. [6]E.O.TuckandL.W.Schwartz, Anumercalandaymptotc tudy of ome thrd-order ordnary dfferental equaton relevant to dranng and coatng flow, SIAM Revew,vol.32,no.3, pp ,99. [7] J. Bazar, E. Babolan, and R. Ilam, Soluton of the ytem of ordnary dfferental equaton by Adoman decompoton method, Appled Mathematc and Computaton, vol. 47, no. 3, pp.73 79,24. [8] E.MomonatandF.M.Mahomed, Symmetryreductonand numercal oluton of a thrd-order ODE from thn flm flow, Mathematcal & Computatonal Applcaton, vol.5,no.4,pp , 2. [9] F. Bern and L. A. Peleter, Two problem from dranng flow nvolvng thrd-order ordnary dfferental equaton, SIAM JournalonMathematcalAnaly,vol.27,no.2,pp , 996. [2] B. R. Duffy and S. K. Wlon, A thrd-order dfferental equaton arng n thn-flm flow and relevant to Tanner law, Appled Mathematc Letter, vol., no. 3, pp , 997. [2] T. G. Myer, Thn flm wth hgh urface tenon, SIAM Revew,vol.4,no.3,pp ,998. [22] E. Momonat, Symmetre, frt ntegral and phae plane of a thrd-order ordnary dfferental equaton from thn flm flow, Mathematcal and Computer Modellng,vol.49,no.-2,pp , 29. [23] E. Momonat, Numercal nvetgaton of a thrd-order ODE from thn flm flow, Meccanca,vol.46,no.2,pp ,2.

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