Research Article Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

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1 Hndaw Publhng Corporaton Mathematcal Problem n Engneerng Volume 205, Artcle ID , page Reearch Artcle Runge-Kutta Type Method for Drectly Solvng Specal Fourth-Order Ordnary Dfferental Equaton Kam Huan,,2 Fudzah Imal,,3 and Norazak Senu,3 Department of Mathematc, Faculty of Scence, Unvert Putra Malaya (UPM), Serdang, Selangor, Malaya 2 Department of Mathematc, College of Scence, Al-Mutanrya Unverty, Baghdad, Iraq 3 Inttute for Mathematcal Reearch, Unvert Putra Malaya (UPM), Serdang, Selangor, Malaya Correpondence hould be addreed to Fudzah Imal; Receved 25 May 205; Reved 4 July 205; Accepted 2 July 205 Academc Edtor: Yan-Jun Lu Copyrght 205 Kam Huan 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. A Runge-Kutta type method for drectly olvng pecal fourth-order ordnary dfferental equaton (ODE) whch denoted by RKFD method contructed. The order condton of RKFD method up to order fve are derved; baed on the order condton, three-tage fourth- and ffth-order Runge-Kutta type method are contructed. Zero-tablty of the RKFD method proven. Numercal reult obtaned are compared wth the extng Runge-Kutta method n the centfc lterature after reducng the problem nto a ytem of frt-order ODE and olvng them. Numercal reult are preented to llutrate the robutne and competency of the new method n term of accuracy and number of functon evaluaton.. Introducton Th paper deal wth the numercal ntegraton of the pecal fourth-order ordnary dfferental equaton (ODE) of the form wth ntal condton y (V) (x) f(x, y), () y(x 0 )y 0, y (x 0 )y 0, y (x 0 )y 0, y (x 0 )y 0, n whch the frt, econd, and thrd dervatve do not appear explctly. Th type of problem often are n many feld of appled cence uch a mechanc, atrophyc, quantum chemtry, and electronc and control engneerng. The general approach for olvng the hgher-order ordnary dfferental equaton (ODE) by tranformng t nto an equvalent frt-order ytem of dfferental equaton and (2) then applyng the approprate numercal method to olve the reultng ytem (ee [ 5]).However,the applcaton of uch technque take a lot of computatonal tme (ee [6, 7]). Drect ntegraton method propoed to avod uch computatonal encumbrance and ncreae the effcency of the method. Many author have propoed everal numercal method for drectly approxmatng the oluton for the hgher-order ODE; for example, Kayode [8] propoed zero-table predctor-corrector method for olvng fourthorder ordnary dfferental equaton. Majd and Suleman [9] derved one pont method to olve ytem of hgherorder ODE. Jan et al. [0] contructed fnte dfference method for olvng fourth-order ODE. Waeleh et al. [] contructed a new block method for olvng drectly hgherorder ODE. Awoyem and Idowu [2] propoedahybrd collocaton method for olvng thrd-order ODE. Hybrd lnearmulttepmethodwththreeteptoolveecondorder ODE wa ntroduced by Jator [3],andallthemethod dcued above are multtep n nature. Th paper prmarly am to contruct a one-tep method of order four and fve to olve pecal fourth-order ODE drectly; thee new method are elf-tartng n nature. The paper organzed a follow. In Secton 2, the dervaton

2 2 Mathematcal Problem n Engneerng of the order condton of RKFD method preented. In Secton 3, the zero-tablty of RKFD method gven. In Secton 4, three-tage RKFD method of order four and order fve are contructed. In Secton 5, numercal example are gven to how the effectvene and competency of the new RKFDmethodacomparedwththewellknownRunge- Kutta method from the centfc lterature. Concluon are gven n Secton Dervaton of the RKFD Method The general form of RKFD method wth -tage for drectly olvng pecal fourth-order ODE () can be wrtten a follow: where y n+ y n +hy n + h2 2 y n + h3 6 y n +h4 b k, (3) y n+ y n +h2 y n + h2 2 y n +h3 b k, (4) y n+ y n +hy y n+ y n +h n +h2 k, (5) k, (6) k f(x n,y n ), (7) k f(x n +c h, y n +hc y n + h2 2 c2 y n + h3 6 c3 y n +h 4 j a j k j ). 2, 3,...,. All parameter b, b, b,, a j,andc of the RKFD method areuedfor, j, 2,...,and are uppoed to be real. The RKFD method an explct method f a j 0for jand an mplct method f a j 0for j. The RKFD method canberepreentedbybutchertableauafollow: c A b T b T T T (8) (9) approach depend on the dervaton of order condton for the Runge-Kutta method propoed n Dormand [4]. The RKFD method n (3) (6) can be wrtten a follow: y n+ y n +hψ(x n,y n ), y n+ y n +hψ (x n,y n ), y n+ y n +hψ (x n,y n ), y n+ y n +hψ (x n,y n ), where the ncrement functon are ψ(x n,y n )y n + h 2 y n + h2 6 y n +h3 b k, ψ (x n,y n )y n + h 2 y n +h2 b k, ψ (x n,y n )y n ψ (x n,y n ) +h k, k, (0) () where k gven n (8). The frt few elementary dfferental for the calar equaton are F (4) y (V) f, F (5) f x +f y y, F (6) f xx + 2f xy y +f y y +f yy (y F (7) f xxx + 3f xxy y + 3f xyy (y ) 2 + 3f xy y + 3f yy y y +f yyy (y ) 3 +f y y. (2) We aume that the Taylor ere ncrement functon Δ. The local truncaton error of y(x), y (x), y (x), and y (x) can be obtaned after ubttutng the exact oluton of () nto () a follow: To determne the parameter of the RKFD method gven n (3) (8), the RKFD method expreon expanded ung the Taylor ere expanon. After performng ome algebrac manpulaton, th expanon equated to the true oluton that gven by the Taylor ere expanon. The drect expanon of the local truncaton error ued to derve the general order condton for the RKFD method. Th τ n+ h[ψ Δ], τ n+ h[ψ Δ ], τ n+ h[ψ Δ ], τ n+ h[ψ Δ ]. (3)

3 Mathematcal Problem n Engneerng 3 The Taylor ere ncrement functon of y(x), y (x), y (x), and y (x) canbeexpreedafollow: Δy + 2 hy + 6 h2 y + 24 h3 F (4) + 20 h4 F (5) h5 F (6) +O(h 6 ), Δ y + 2 hy + 6 h2 F (4) + 24 h3 F (5) + 20 h4 F (6) h5 F (7) +O(h 6 ), Δ y + 2 hf(4) + 6 h2 F (5) + 24 h3 F (6) + 20 h4 F (7) h5 F (8) +O(h 6 ), Δ F (4) + 2 hf(5) + 6 h2 F (6) + 24 h3 F (7) h4 F (8) +O(h 5 ). (4) Subttutng (2) nto (), the ncrement functon ψ, ψ, ψ, and ψ for RKFD method become a follow: b k + 2 b f+ b k b k b F (4) 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 c hf (5) + b 2 c 2 h2 F (6) b f+ b c (f x +f y y )h + 2 b k b F(4) + b c2 (f xx + 2f xy y +f y y +f yy (y ) 2 )h 2 b c hf (5) + b 2 c2 h2 F (6) k + 2 k f+ c (f x +f y y )h c 2 (f xx + 2f xy y +f y y +f yy (y ) 2 )h 2 F (4) + k + 2 f+ k c hf (5) + c 2 2 h2 F (6) c (f x +f y y )h c 2 (f xx + 2f xy y +f y y +f yy (y ) 2 )h 2 F (4) + +O(h 3 ). c hf (5) + c 2 2 h2 F (6) (5) Ung () and (4), the local truncaton error (3) can be wrtten a follow: τ n+ h 4 [ τ n+ h3 [ τ b k ( 24 F(4) + b k ( 6 F(4) + 20 hf(5) 24 hf(5) + )], + )], n+ h2 [ k ( 2 F(4) + 6 hf(5) + )], τ n+ h[ k (F (4) + 2 hf(5) + 6 h2 F (6) + )]. (6)

4 4 Mathematcal Problem n Engneerng By offettng (5) nto (6) and expandng a a Taylor ere expanon ung computer algebra package MAPLE (ee [5]), the local truncaton error or the order condton for tage ffth-order RKFD method can be wrtten a follow. The order condton for y are fourth order: b 24, (7) thrd order: fourth order: c 2 3, (28) c 3 4, (29) ffth order: The order condton for y are thrd order: fourth order: ffth order: b c 20. (8) The order condton for y are econd order: thrd order: fourth order: ffth order: b 6, (9) b c 24, (20) b c2 60. (2) The order condton for y are frt order: econd order: 2, (22) c 6, (23) c 2 2, (24) c (25), (26) c 2, (27) ffth order:,j c 4 5, a j Zero-Stablty of the RKFD Method (30) In th ecton, we dcu the convergence of the RKFD method by ntroducng the concept of zero-tablty of the RKFD method. A good numercal method a method n whch the numercal approxmaton to the oluton converge, and zero-tablty a gnfcant crteron for convergence. The zero-tablty concept for thoe numercal method that are ued for olvng frt- and econd-order ODEcanbeeennLambert[6], Dormand [4], and Butcher [4]. The RKFD method (3) (8) can be expreed n the matrx form a follow: y n hy 2 6 y n n+ [ ] [ h 2 y 0 hy n 2 n+] [ ] [ h 3 y [ 0 0 [ h 2 y, (3) n ] ] n+] [ [ h 3 y n ] ] where I[ ] the dentty matrx coeffcent of y n+, hy n+, h2 y n+,andh3 y /2 /6 n+,repectvely,anda[ 0 /2 ] matrx coeffcent of y n, hy n, h2 y n,andh3 y n,repectvely. The charactertc polynomal of the RKFD method denoted by (ζ) whch can be wrtten a follow: ζ 2 6 (ζ) Iζ A 0 ζ 2. (32) 0 0 ζ ζ Hence, (ζ) (ζ ) 4. We fnd that all the root are ζ,,,. Generalzng the theorem propoed by Henrc [7] forolvngpecal econd-order ODE, therefore, the RKFD method zerotable nce all root are le than or equal to the value of.

5 Mathematcal Problem n Engneerng 5 4. Contructon of RKFD Method In th ecton, we proceed to contruct explct RKFD method baed on the order condton whch we have derved n Secton A Three-Stage RKFD Method of Order Four. Th ecton wll focu on the dervaton of a three-tage RKFD method of order four, where we ue the algebrac order condton (7), (9)-(20), (22) (24),and(26) (29),repectvely.Thereultng ytem of equaton cont of 0 nonlnear equaton wth 4 unknown varable to be olved; olvng the ytem multaneouly yeld a oluton wth four free parameter c 3, b, b,andb 3 a follow: c 2 3 4c 3 + 6c 3, 6c2 3 6c 3 + 6c 3 ( + 4c 3 ), b b + 5c 3 48b c 3 6c c , b 2 b b (33) Thu, thee free parameter can be choen by mnmzng the local truncaton error norm of the ffth-order condton accordng to Dormand et al. [8]. However, we have another three free parameter a 2, a 3,anda 32 that do not appear n fourth-ordercondtonbuttheyappearnthemnmzaton of error equaton for ffth-order condton of y. The error norm and global error of ffth-order condton are defned a follow: n p + τ(5) 2 (τ (5) n τ (5) p + (τ (5) c c c3 3 08c 3 50c , 72c c3 3 48c c, 3 6c2 3 6c 3 + 6c 3 ( + 4c 3 ), 2 20c 3 8c c c 3 50c , 72c3 3 n τ (5) p + (τ (5) 2 n τ (5) p + (τ (5) 2 τ(5) g 2 n p + (τ (5) ) 2 + np + (τ (5) ) 2 n + p + (τ (5) ) 2 + n p + (τ (5) (34) 3 +c 3 36c3 3 48c c, 3 b 2 b c b c c 3 2c (3 8c 3 + 6c 2 3 ), where τ (5), τ (5), τ (5),andτ (5) are the local truncaton error norm for y, y, y,andy,repectvely,andτ (5) g the global error. Conequently, we fnd the error norm of y, y,andy, repectvely, a follow: τ(5) (60b c3 b b c 3 360b + 960b 3 c 3 4c 3 ) 2 ( 2 + 3c 3 τ (5) (0c b c b c c 3 7) 2 ( 2 + 3c 3 (35) τ (5) 2 20 (3 2c c3 )2 ( 2 + 3c 3 ) 2.

6 6 Mathematcal Problem n Engneerng Ourgoaltochooethefreeparameterc 3, b, b,andb 3 uch that the error norm of ffth-order condton have mnmal value. By plottng the graph of τ (5) 2 veru c 3 and choong amallvalueofc 3 n the nterval [0.7, 3], wefndthatc 3 7/20 the optmal value whch yeld a mnmum value for τ (5) Subttutngthevalueofc 3 7/20 nto τ (5) 2 and τ (5) 2 we get τ(5) (3 60b + 24b 3 τ (5) ( + 544b )2. (36) Alo through plottng the graph of τ (5) 2 agant b and b 3 n the nterval [ 0., 0.5] and choong a mall value of b 3, we get that b 3 /20theoptmalvaluewhchgveb 7/200 and τ (5) Now, utlzng the ame technque where we draw the graph of τ (5) 2 veru b n the nterval [, ],wefndthat b /8thebetchoce,andwththvalueofb,weget τ (5) Therefore, the error equaton of the ffth-order condton of y a follow: τ (5) ( RKFD method are wrtten n Butcher tableau and denoted by method a follow: (39) 4.2. A Three-Stage RKFD Method of Order Fve. In th ecton, a three-tage RKFD method of order fve wll be derved. The algebrac order condton up to order fve ((7)- (8), (9) (2), (22) (25), and(26) (30)) need to be olved. The reultng ytem of equaton cont of ffteen nonlnear equaton, olvng the ytem multaneouly whch reult n a oluton wth three free parameter b, a 2,anda 3 a follow: a a 2a a 2 a a a a 3a a a a 32) /2. Conequently, the global error τ(5) g ( (37) c , c , 9, , , a a 2a a 2 a a a a 3a a a a 32 ) /2. (38) By mnmzng the error norm n (37) and global error n (38) wth repect to the free parameter a 2, a 3,anda 32,weget a 2 /5, a 3 9/25, and a 32 9/25, whch produce τ (5) and τ (5) g Fnally, all the coeffcent of three-tage fourth-order 9, , , b 8, b , b ,

7 Mathematcal Problem n Engneerng 7 b ( )b, τ (6) ( a a b ( )b, a 32 ( )a 2 a (40) Thu, thee free parameter can be choen by mnmzng the local truncaton error norm of the xth-order condton. The error norm and the global error of the xth-order condton are gven by n p + τ(6) 2 (τ (6) a 2 6a a a a a 2 a a a a2 2 )/2. Alo, the global error can be wrtten a τ(6) g (520a a a 2 6a a a a a 2 a a a a2 2 ). (43) (44) n τ (6) p + (τ (6) 2 n τ (6) p + (τ (6) 2 n τ (6) p + (τ (6) 2 τ(6) g 2 n p + (τ (6) ) 2 + np + (τ (6) ) 2 n + p + (τ (6) ) 2 + n p + (τ (6) (4) Now, mnmzng the error coeffcent n (43) and (44) wth repect to the free parameter a 2, a 3,weobtana /87793 and a 3 02/53225 whch gve a / Thu, the error equaton for y, y, y,andy arecomputedandgvenby τ(6) 2 0, and global error norm τ (6) , τ (6) , τ (6) , (45) where τ (6), τ (6), τ (6),andτ (6) are the local truncaton error norm for y, y, y,andy of the RKFD method, repectvely. τ (6) g the global error. The error equaton of xth-order condton for y wth repect to the free parameter b a follow: τ(6) ( + 080b ) 2. (42) The error equaton τ (6) 2 ha a mnmum value equal to zero at b 9/ whch lead to b 2 3/080 6/260 and b 3 3/ /260. The truncaton error norm of the xth-order condton of y, y, y,andy are calculated a follow: τ(6) 2 0, τ (6) 2 200, τ (6) ( a a a a ) /2, τ(6) g (46) Therefore, the parameter of the three-tage ffth-order RKFD method denoted by can be repreented n Butcher tableau a follow: Numercal Example (47) In th ecton, ome numercal example wll be olved to how the effcency of the new RKFD method of order four

8 8 Mathematcal Problem n Engneerng Log 0 (max error) Log 0 (max error) Log 0 (functon evaluaton) Log 0 (functon evaluaton) RK5B6 RK5N6 RK44 RK5B6 RK5N6 RK44 Fgure : The effcency curve for Example wth h0./2, 0, 2, 3, 4. Fgure 3: The effcency curve for Example 3 wth h0./2, 0, 2, 3, 4. Log 0 (max error) Log 0 (functon evaluaton) RK5B6 RK5N6 RK44 Fgure 2: The effcency curve for Example 2 wth h0./2, 0, 2, 3, 4. Log 0 (max error) Log 0 (functon evaluaton) RK5B6 RK5N6 RK44 Fgure 4: The effcency curve for Example 4 wth h0./2, 0, 2, 3, 4. and order fve, whch are denoted by and, repectvely. The comparon made wth the well known method n the centfc lterature. We ue n the numercal comparon the crtera baed on computng the maxmum error n the oluton (max error max( y(t n ) y n )) whch equal to the maxmum between abolute error of the trueolutonandthecomputedoluton. Fgure 7 how the effcency curve of Log 0 (max error) agant the computatonal effort meaured by Log 0 (functon evaluaton) requred by each method. The followng method are ued for comparon: () : the three-tage ffth-order RKFD method derved n th paper. () : the three-tage fourth-order RKFD method derved n th paper. () RK5B6: the x-tage ffth-order Runge-Kutta method gven n Butcher [4]. (v) RK5N6: the x-tage ffth-order Runge-Kutta method gven n Harer [5]. (v) RK44: the four-tage fourth-order Runge-Kutta method gven n Dormand [4].

9 Mathematcal Problem n Engneerng Log 0 (max error) Log 0 (max error) Log 0 (functon evaluaton) Log 0 (functon evaluaton) RK5B6 RK5N6 RK44 RK5B6 RK5N6 RK44 Fgure 5: The effcency curve for Example 5 wth h0./2, 0, 2, 3, 4. Fgure 7: The effcency curve for Example 7 wth h0.025/2, 0,, 2, 3. Example 2. The nonhomogeneou nonlnear problem a follow: Log 0 (max error) y (V) y 2 + co 2 (x) + n (x), y (0) 0, y (0), y (0) 0, y (0). (49) 0 The exact oluton gven by y(x) n(x). Theproblem ntegrated n the nterval [0, 0] Log 0 (functon evaluaton) Example 3. The homogeneou lnear problem wth noncontantcoeffcentafollow RK5B6 RK5N6 RK44 Fgure 6: The effcency curve for Example 6 wth h0./2, 0, 2, 3, 4. (v) : the four-tage fourth-order 3/8 rule Runge- Kutta method gven n Butcher [4]. y (V) (6x 4 48x 2 + 2)y, y (0), y (0) 0, y (0) 2, y (0) 0. (50) The exact oluton gven by y(x) e x2.theproblem ntegrated n the nterval [0, 3]. Example 4. The nonlnear problem a follow: Example. The homogeneou lnear problem a follow: y (V) y, (48) y (0) 0, y (0), y (0) 2, y (0) 2. y (V) 3n(y) (3 + 2n2 (y)), co 7 (y) y (0) 0, y (0), y (0) 0, y (0). (5) The exact oluton gven by y(x) e x n(x).theproblem ntegrated n the nterval [0, 0]. The exact oluton gven by y(x) arcn(x).theproblem ntegrated n the nterval [0,π/4].

10 0 Mathematcal Problem n Engneerng Example 5. The lnear ytem a follow: y (V) e 3x u, y (0), y (0), y (0), y (0). z (V) 6e x y, z (0), z (0) 2, z (0) 4, z (0). w (V) 8e x z, w (0), w (0), w (0) 9, w (0) 27. u (V) 256e x w. u (0), u (0), u (0) 6, u (0) 4. The exact oluton gven by ye x, ze 2x, we x, ue x. The problem ntegrated n the nterval [0, 2]. Example 6. The nonlnear ytem a follow: y (V) y+ y 2 +z 2 w 2 +u, 2 y (0), y (0) 0, y (0), y (0) 0. z (V) z + y 2 +z 2 w 2 +u, 2 z (0) 0, z (0), z (0) 0, z (0). w (V) 6w+ y 2 +z 2 w 2 +u, 2 w (0), w (0) 0, w (0), w (0) 0. u (V) 6u + y 2 +z 2 w 2 +u. 2 u (0) 0, u (0) 2, u (0) 0, u (0). The exact oluton gven by yco (x), zn (x), wco (2x), un (2x). The problem ntegrated n the nterval [0, 2]. (52) (53) (54) (55) Example 7. The nonlnear ytem a follow: y (V) z2 w, y (0), y (0), y (0), y (0). z (V) 6 w2 u, w (V) 8 u2 y 5, z (0), z (0) 2, z (0) 4, z (0) 8. w (0), w (0) 3, w (0) 9, w (0) 27. u (V) 256y 4. u (0), u (0) 4, u (0) 6, u (0) 64. The exact oluton gven by ye x, ze 2x, we 3x, ue 4x. Theproblemntegratednthenterval[0, 2]. 6. Concluon (56) (57) Th paper deal wth Runge-Kutta type method denoted by RKFD method for drectly olvng pecal fourth-order ODE of the form y (V) (x) f(x, y). Frt,wedervedthe order condton for RKFD method, whch were then ued to contruct three-tage fourth- and ffth-order RKFD method. The method are denoted by and, repectvely. We alo proved that the RKFD method zero-table. From the numercal reult, we oberved that the new RKFD method are more competent a compared wth the extng Runge-Kutta method n the centfc lterature. From the numercal reult, we conclude that the new RKFD method are computatonally more effcent n olvng pecal fourthorder ODE and outperformed the extng method n term of error precon and number of functon evaluaton. Conflct of Interet The author declare that there no conflct of nteret regardng the publcaton of th paper. Reference [] P. Onumany, U. W. Srena, and S. N. Jator, Contnuou fnte dfference approxmaton for olvng dfferental equaton, Internatonal Computer Mathematc, vol.72,no., pp.5 27,999.

11 Mathematcal Problem n Engneerng [2] D. Sarafyan, New algorthm for the contnuou approxmate oluton of ordnary dfferental equaton and the upgradng of the order of the procee, Computer & Mathematc wth Applcaton,vol.20,no.,pp.77 00,990. [3] G. Dahlqut, On accuracy and uncondtonal tablty of lnear multtep method for econd order dfferental equaton, BIT Numercal Mathematc,vol.8,no.2,pp.33 36,978. [4] J. C. Butcher, Numercal Method for Ordnary Dfferental Equaton, John Wley & Son, New York, NY, USA, 2nd edton, [5] E. Harer, S. P. Nørett, and G. Wanner, Solvng Ordnary Dfferental Equaton I: Nontff Problem,vol.8ofSprnger Sere n Computatonal Mathematc, Sprnger, Berln, Germany, 2nd edton, 993. [6]S.N.JatorandJ.L, Aelf-tartnglnearmulttepmethod for a drect oluton of the general econd-order ntal value problem, Internatonal Computer Mathematc, vol. 86, no. 5, pp , [7] D. O. Awoyem, A new xth-order algorthm for general econd order ordnary dfferental equaton, Internatonal Journal of Computer Mathematc,vol.77,no.,pp.7 24,200. [8] S. J. Kayode, An effcent zero-table numercal method for fourth-order dfferental equaton, Internatonal Mathematc and Mathematcal Scence, vol.2008,artcleid 36402,0page,2008. [9] Z. A. Majd and M. B. Suleman, Drect ntegraton mplct varable tep method for olvng hgher order ytem of ordnary dfferental equaton drectly, San Malayana, vol. 35,no.2,pp.63 68,2006. [0] M.-K.Jan,S.R.K.Iyengar,andJ.S.V.Saldanha, Numercal oluton of a fourth-order ordnary dfferental equaton, Journal of Engneerng Mathematc,vol.,no.4,pp ,977. [] 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 ,20. [2] D. O. Awoyem and O. M. Idowu, A cla of hybrd collocaton method for thrd-order ordnary dfferental equaton, Internatonal Computer Mathematc, vol.82,no.0,pp , [3] 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 ,200. [4] J. R. Dormand, Numercal Method for Dfferental Equaton. A Computatonal Approach, Lbrary of Engneerng Mathematc, CRCPre,BocaRaton,Fla,USA,996. [5] W. Gander and D. Gruntz, Dervaton of numercal method ung computer algebra, SIAM Revew, vol. 4, no. 3, pp , 999. [6] J. D. Lambert, Numercal Method for Ordnary Dfferental Sytem, The Intal Value Problem,JohnWley&Son,London, UK, 99. [7] P. Henrc, Element of Numercal Analy,JohnWley&Son, New York, NY, USA, 964. [8] J. R. Dormand, M. E. A. EL-Mkkawy, and P. J. Prnce, Famle of Runge-Kutta Nytrom formulae, IMA Numercal Analy,vol.7,pp ,987.

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