Applied Mthemticl Sciences, Vol. 9, 25, no. 8, 353-368 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/.2988/ms.25.486 Adomin Decomposition Method with Green s Function for Solving Twelfth-Order Boundry Vlue Prolems Wleed Al-Hyni Deprtment of Mthemtics College of Computer Science nd Mthemtics University of Mosul, Irq Copyright 24 Wleed Al-Hyni. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. Astrct In this pper, the Adomin Decomposition Method with Green s function (Stndrd Adomin nd Modified Technique) is pplied to solve liner nd nonliner twelfth-order oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. The numericl results otined with smll mount of computtion re compred with the exct solutions to show the efficiency of the method. The results show tht the decomposition method is of high ccurcy, more convenient nd efficient for solving high-order oundry vlue prolems. Keywords: Adomin Decomposition Method; Adomin s polynomils; Twelfthorder oundry vlue prolems; Green s function. Introduction In the eginning of the 98 s, Adomin [-4] proposed new nd fruitful method (herefter clled the Adomin Decomposition Method or ADM) for solving liner nd nonliner (lgeric, differentil, prtil differentil, integrl, etc.) equtions. It hs een shown tht this method yields rpid convergence of the solutions series to liner nd nonliner deterministic nd stochstic equtions. A clss of chrcteristic-vlue prolems of higher order (s higher s 24) is known to rise in hydrodynmic nd hydromgnetic stility [5, 6]. Furthermore,
354 Wleed Al-Hyni it is widely well known tht when n infinite horizontl lyer of fluid is heted from elow nd is sujected to the ction of rottion, instility sets in. When this instility is s ordinry convection the ordinry differentil eqution is sixth order; when the instility sets in s overstility, it is modelled y n eighthorder ordinry differentil eqution. Suppose, now, tht uniform mgnetic field is lso pplied cross the fluid in the sme direction s grvity. When instility sets in now s ordinry convection, it is modelled y tenth-order oundry vlue prolem; when instility sets in s overstility, it is modelled y twelfth-order oundry vlue prolem (for detils, see [5]). Agrwl's ook [7] contins theorems which detil the conditions for existence nd uniqueness of solutions of the twelfth-order oundry vlue prolems. Different numericl nd semi nlyticl methods hve een proposed y vrious uthors to solve twelfth-order oundry vlue prolems. A few of them re; Twelfth-degree spline method [8], Modified Decomposition Method with the inverse opertor (MDM) [9], Thirteen-degree spline mehod [], Non-polynomil spline technique [], Vritionl Itertion Method (VIM) [2,5], Differentil Trnsform Method (DTM) [3] nd Homotopy Perturtion Method (HPM) [4]. The min ojective of this pper is to pply the Stndrd Adomin with Green's function (SAwGF) nd Modified Technique with Green's function (MTwGF) to liner nd nonliner twelfth-order oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. 2. Anlysis of the method Let us consider the generl BVP of twelfth-order y (2) (x) + g(x, y) = f(x), x () with oundry conditions y (i) () = α i, y (i) () = β i, i =,,2,3,4,5 (2) where y = y(x), g(x, y) is liner or nonliner function of y, nd f(x) re continuous functions defined in the intervl x [, ] nd α i, β i ; (i =,,2,3,4,5) re finite rel constnts. Applying the decomposition method s in [-4], Eq. () cn e written s Ly = f(x) Ny, where L = d2 is the liner opertor nd Ny = g(x, y) is the nonliner opertor. dx2 Consequently,
Adomin decomposition method with Green s function 355 y(x) = h(x) + G(x, ξ)f(ξ)dξ G(x, ξ)nydξ, (3) where h(x) is the solution of Ly = with the oundry conditions (2) nd G(x, ξ) is the Green s function [5] given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x The Adomin s technique consists of pproximting the solution of () s n infinite series y = y n, (4) n= nd decomposing the nonliner opertor N s Ny = A n, (5) n= where A n re polynomils (clled Adomin polynomils) of y, y,, y n [-4] given y d n A n = n! dλ n [N ( λi y i )] i= λ=, n =,,2,. The proofs of the convergence of the series n= y n nd n= A n re given in [3, 7-2]. Sustituting (4) nd (5) into (3) yields y n n= = h(x) + G(x, ξ)f(ξ)dξ G(x, ξ) A n dξ. (6) n= From (6), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy:
356 Wleed Al-Hyni y = h(x) + G(x, ξ)f(ξ)dξ y n+ = G(x, ξ)a n dξ, n =,,2,, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = G(x, ξ)f(ξ)dξ G(x, ξ)a dξ, y n+2 = G(x, ξ)a n+ dξ, n =,,2,. Thus ll components of y cn e clculted once the A n re given. We then define n the n-term pproximnt to the solution y y φ n [y] = i= y i with lim φ n [y] = n y. 3. Applictions nd numericl results In this section, the ADM with the Green s function (Stndrd Adomin nd Modified Technique) for solving liner nd nonliner twelfth-order oundry vlue prolems is illustrted in the following exmples. The computtions ssocited with the exmples were performed using Mple 3 pckge with precision of 4 dígits. Exmple Consider the following liner BVP of twelfth-order [,, 3, 5]: y (2) (x) + xy(x) = (2 + 23x + x 3 )e x, x (7) with oundry conditions y (i) () = i(2 i), y (i) () = i 2 e, i =,,2,3,4,5. (8) The exct solution of (7), (8) is y Exct (x) = x( x)e x. Applying the decomposition method, Eq. (7) cn e written s Ly = (2 + 23x + x 3 )e x xy(x), where L = d2 is the liner opertor. Consequently, dx2
Adomin decomposition method with Green s function 357 y = h(x) G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ G(x, ξ)ξy(ξ)dξ, (9) where h(x) is the solution of Ly = with the oundry conditions (8) given y h(x) = ( 97 8 24 e) x ( 753 2 847 8 e) x + ( 24 623 e) x9 8 2 ( 389 2 739 2 e) x8 + ( 2275 24 277 8 e) x7 ( 865 4 99 e) x6 24 8 x5 3 x4 2 x3 + x nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = ( x 584 + x 288 x9 296 + x8 52 x7 26 + x6 864! ) ξ + ( x 288 x 584 + x9 234 x8 26 + x7 3456 x6 44 + x! ) ξ + ( x 296 + x 234 x9 8 + x8 864 x7 44 + x6 576 x2 72576 ) ξ9 + ( x 52 x 26 + x9 864 x8 72 + x7 52 x6 432 + x3 2492 ) ξ8 + ( x 26 + x 3456 x9 44 + x8 52 x7 728 + x6 576 x4 296 ) ξ7 + ( x 864 x 44 + x9 576 x8 432 + x7 576 x6 44 + x5 864 ) ξ6 g 2 (x, ξ) = ( ξ 584 + ξ 288 ξ9 296 + ξ8 52 ξ7 26 + ξ6 864! ) x + ( ξ 288 ξ 584 + ξ9 234 ξ8 26 + ξ7 3456 ξ6 44 + ξ! ) x
358 Wleed Al-Hyni + ( ξ 296 + ξ 234 ξ9 8 + ξ8 864 ξ7 44 + ξ6 576 ξ2 72576 ) x9 + ( ξ 52 ξ 26 + ξ9 864 ξ8 72 + ξ7 52 ξ6 432 + ξ3 2492 ) x8 + ( ξ 26 + ξ 3456 ξ9 44 + ξ8 52 ξ7 728 + ξ6 576 ξ4 296 ) x7 + ( ξ 864 ξ 44 + ξ9 576 ξ8 432 + ξ7 576 ξ6 44 + ξ5 864 ) x6. Sustituting (4) in (9), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x) G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ, y n+ = G(x, ξ)ξy n (ξ)dξ, n =,,2, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = G(x, ξ)(2 + 23ξ + ξ 3 )e ξ dξ y n+2 = G(x, ξ)ξy n+ (ξ)dξ, n =,,2,. G(x, ξ)ξy (ξ)dξ, In Tle, we list the solute errors otined y SAwGF nd MTwGF. Compring them with the Thirteen-degree spline method [], Non-polynomil spline technique [], DTM [3] nd VIM [5] results. In [] the mximum solute error is 7.38E-9 with μ =, k = 22. In [] the mximum solute error is 4.72E-6. It cn e noticed tht the result otined y the present method (SAwGF) is very superior (lower error comined with less numer of itertions) to tht otined y the other mentioned methods. From Tle, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n.
Adomin decomposition method with Green s function 359 Tle : Comprison of solute errors for exmple SAwGF MTwGF DTM [3] VIM [5] x n = n = 2 n = n = 2 n = 2........ 2.3E-6 3.88E-29 2.35E-3 4.27E-26 7.5E-4 9.52E-3.2 6.56E-5.25E-27 7.49E-2.37E-24 2.77E-2.25E-3.3 3.43E-4 6.49E-27 3.86E- 7.5E-24.73E- 3.35E-3.4 7.79E-4.47E-26 8.69E-.62E-23 5.2E- 5.38E-3.5.2E-3.89E-26.2E- 2.9E-23 9.34E- 8.4E-3.6 8.2E-4.49E-26 8.85E-.65E-23.28E-.4E-2.7 3.7E-4 6.75E-27 4.E- 7.4E-24.39E- 3.93E-3.8 7.39E-5.32E-27 7.9E-2.45E-24.23E-.23E-3.9 2.38E-6 4.8E-29 2.53E-3 4.57E-26 7.49E- 8.25E-3......95E- 3.27E-3 Exmple 2 Consider the following liner BVP of twelfth-order [,, 3, 5]: y (2) (x) y(x) = 2(2x cos x + sin x), x () with oundry conditions y( ) = y() =, y () ( ) = y () () = 2 sin(), y (2) ( ) = y (2) () = 4 cos() 2 sin(), y (3) ( ) = y (3) () = 6 cos() 6 sin(), y (4) ( ) = y (4) () = 8 cos() + 2 sin(), y (5) ( ) = y (5) () = 2 cos() + sin(). } () The exct solution of (), () is y Exct (x) = (x 2 ) sin x. Applying the decomposition method, Eq. () cn e written s Ly = 2(2x cos x + sin x) + y(x), where L = d2 dx2 is the liner opertor. Consequently, y = h(x) 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ + G(x, ξ)y(ξ)dξ, (2) where h(x) is the solution of Ly = with the oundry conditions () given y
36 Wleed Al-Hyni h(x) = ( 6 95 sin 384 + ( 497 257 sin 92 + ( 649 739 sin 384 384 cos ) x ( 377 384 64 cos ) x7 ( 76 92 384 cos ) x3 ( 85 384 nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = ( 2 x 2 798336 ) ξ 587 sin cos ) x9 384 23 sin cos ) x5 92 8 sin cos ) x 28 324432 + x 9 53846 x 7 264384 + 2 x 5 47456 x 3 769472 + 2 x 29492 + ( 2 x 265428 + x 8 428768 2 x 6 47456 + x 4 884736 x 2 589824 + 2 x 72576 2 29492 ) ξ + ( x 53846 x9 8257536 + x 7 29492 x5 769472 + x3 79648 x 2 4552 + x 589824 ) ξ9 + ( x 428768 x 8 589824 + x6 769472 x4 589824 + x 3 48384 x2 79648 + 769472 ) ξ8 + ( x 264384 + x 9 29492 x7 884736 + x5 29492 x 4 2492 + x3 589824 x 884736 ) ξ7 + ( 2 x 47456 + x8 769472 x6 29492 + 2 x 5 728 x4 29492 + x2 769472 2 47456 ) ξ6 + ( 2 x 47456 x9 769472 + x7 29492 2 x 6 728 + x5 29492 x3 769472 + 2 x 47456 ) ξ5 + ( x 884736 x8 589824 + x 7 2492 x6 29492 + x4 884736 x 2 29492 + 264384 ) ξ4 + ( x 769472 + x9 79648 x 8 48384 + x7 589824 x5 769472 + x 3 589824 x 428768 ) ξ3 + ( x 589824 + x 9 4552 x8 79648 + x6 769472 x 4 29492 + x2 8257536 53846 ) ξ2 + ( 2 x 29492 2 x 72576 + x 9 589824 x 7 884736 + 2 x 5 47456 x 3 428768 + 2 x 265428 ) ξ + 2 x 2 x + x 8 2 x 6 + x 4 x 2 + 2 798336 29492 769472 47456 264384 53846 324432 g 2 (x, ξ) = ( 2 ξ 2 798336 ) x 324432 + ξ 9 53846 ξ 7 264384 + 2 ξ 5 47456 ξ 3 769472 + 2 ξ 29492 + ( 2 ξ 265428 + ξ 8 428768 2 ξ 6 47456 + ξ 4 884736 ξ 2 589824 + 2 ξ 72576 2 29492 ) x + ( ξ 53846 ξ9 8257536 + ξ 7 29492 ξ5 769472 + ξ3 79648 ξ 2 4552 + ξ 589824 ) x9 + ( ξ 428768 ξ 8 589824 + ξ6 769472 ξ4 589824 + ξ 3 48384 ξ2 79648 + 769472 ) x8 + ( ξ 264384 + ξ 9 29492 ξ7 884736 + ξ5 29492 ξ 4 2492 + ξ3 589824 ξ 884736 ) x7 + ( 2 ξ 47456 + ξ8 769472 ξ6 29492 + 2 ξ 5 728 ξ4 29492 + ξ2 769472 2 47456 ) x6
Adomin decomposition method with Green s function 36 + ( 2 ξ 47456 ξ9 769472 + ξ7 29492 2 ξ 6 728 + ξ5 29492 ξ3 769472 + 2 ξ 47456 ) x5 + ( ξ 884736 ξ8 589824 + ξ 7 2492 ξ6 29492 + ξ4 884736 ξ 2 29492 + 264384 ) x4 + ( ξ 769472 + ξ9 79648 ξ 8 48384 + ξ7 589824 ξ5 769472 + ξ 3 589824 ξ 428768 ) x3 + ( ξ 589824 + ξ 9 4552 ξ8 79648 + ξ6 769472 ξ 4 29492 + ξ2 8257536 53846 ) x2 + ( 2 ξ 2 ξ + ξ 9 ξ 7 + 2 ξ 5 ξ 3 + 2 ξ ) x 29492 72576 589824 884736 47456 428768 265428 + 2 ξ 2 ξ + ξ 8 2 ξ 6 + ξ 4 ξ 2 + 2. 798336 29492 769472 47456 264384 53846 324432 Sustituting (4) in (2), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x) 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ y n+ = G(x, ξ)y n (ξ)dξ, n =,,2, nd the itertes defined using the Modified Technique [22] re determined in the following recursive wy: y = h(x), y = 2 G(x, ξ)(2ξ cos ξ + sin ξ)dξ y n+2 = G(x, ξ)y n+ (ξ)dξ, n =,,2,., + G(x, ξ)y (ξ)dξ, In Tle 2, we present the solute errors otined y SAwGF nd MTwGF. Compring them with the Thirteen-degree spline method [], Non-polynomil spline technique [], DTM [3] nd VIM [5] results. In [] the mximum solute error is 4.69E-5 with μ =, k = 22. In [] the mximum solute error is 4.67E-7. It cn e noticed tht the result otined y the present method (SAwGF) is very superior to tht otined y the four previous mentioned methods. From Tle 2, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n. Tle 2: Comprison of solute errors for exmple 2 SAwGF MTwGF DTM [3] VIM [5] x n = n = 2 n = n = 2 n = 2...... 2.78E-7..22E- 8.8E-22 2.28E-9.52E-9.64E-5.39E-5.2 2.3E-.34E-2 3.79E-9 2.5E-9 2.8E-3 8.33E-6
362 Wleed Al-Hyni.3 2.2E-.45E-2 4.3E-9 2.7E-9 3.44E-2.65E-4.4.8E-.8E-2 3.4E-9 2.22E-9 2.46E- 2.6E-5.5.4E- 7.39E-22 2.6E-9.39E-9.E-.38E-4.6 5.27E-2 3.37E-22 9.98E- 6.33E-2 3.67E- 2.22E-5.7.57E-2 9.89E-23 2.98E-.86E-2 9.89E-.6E-4.8 2.2E-3.37E-23 4.2E- 2.58E-2 2.28E-9.26E-4.9 5.32E-5 3.27E-25.2E-2 6.4E-23 4.68E-9.99E-5..... 8.72E-9 9.33E-5 In results not presented here, we hve seen tht the solute errors otined y SAwGF nd MTwGF in the intervl [,] re the sme s for the intervl [,]. Exmple 3 Finlly, we consider the following nonliner BVP of twelfth-order [9, 2-4]: y (2) (x) = 2e x y 2 (x) + y (x), < x < (3) with two sets of oundry conditions y (i) () = ( ) i, y (i) () = ( ) i e, i =,,2,3,4,5 (4) y (2i) () =, y (2i) () = e, i =,,2,3,4,5. (5) The exct solution of (3) with ((4) or (5)) is y Exct (x) = e x. Applying the decomposition method, Eq. (3) cn e written s Ly = 2e x Ny + y (x), where L = d2 dx 2 is the liner opertor nd Ny = y2 is the nonliner opertor. Consequently, y = h(x) + G(x, ξ){2e ξ Ny(ξ) + y (ξ)}dξ, (6) where h(x) is the solution of Ly = with the oundry conditions (4) given y
Adomin decomposition method with Green s function 363 889 497e h(x) = x 98989 2698e x 2 2 4372 8846e + x 9 24 48797 32644e x 8 27595 75e + x 7 3739 86276e x 6 24 24 2 5! x5 + 4! x4 3! x3 + 2! x2 x +, nd G(x, ξ) is the Green s function given previously in exmple. For the oundry conditions (5), h(x) is given y h(x) = e x +!! x + ( e 435456 2726 ) x9 + 8! x8 + ( 37e 844 59 2268 ) x7 + 6! x6 + ( 2863e 844 24 34 ) x5 + 4! x4 + ( 2723e 92 34 4725 ) x3 + 2! x2 + ( 4556793e 4556793 6426 ) x +, 467775 nd G(x, ξ) is the Green s function given y G(x, ξ) = { g 2(x, ξ) if x ξ g (x, ξ) if ξ x where g (x, ξ) = (! x! ) ξ + ( 27728 x3 72576 x2 + x) ξ9 8864 + ( 648 x5 296 x4 + 972 x3 x) ξ7 2268 + ( 648 x7 864 x6 + 432 x5 324 x3 + x) ξ5 567 + ( 27728 x9 2492 x8 + 972 x7 324 x5 + 7 x3 x) ξ3 2835
364 Wleed Al-Hyni + (! x! x + + 2 x) ξ 93555! ξ 8864 x9 2268 x7 + 567 x5 2835 x3 g 2 (x, ξ) = (! ) x + ( 27728 ξ3 72576 ξ2 + + ( 648 ξ5 296 ξ4 + 972 ξ3 ξ) x7 2268 + ( 648 ξ7 864 ξ6 + 432 ξ5 324 ξ3 + ξ) x5 567 + ( 27728 ξ9 2492 ξ8 + 972 ξ7 324 ξ5 + 7 ξ3 ξ) x3 2835 + (! ξ! ξ + + 2 ξ) x 93555 8864 ξ9 ξ) x9 8864 2268 ξ7 + 567 ξ5 2835 ξ3 Sustituting (4) nd (5) in (6), the itertes defined using the Stndrd Adomin Method re determined in the following recursive wy: y = h(x), y n+ = G(x, ξ){2e ξ A n + y (ξ)}dξ, n =,,2,. For the nonliner term Ny = y 2 = n= A n the corresponding Adomin polynomils re: A = y 2, A = 2y y, A 2 = 2y y 2 + y 2, n A n = y i y n i, n i, n =,, 2,. i= In Tles 3A nd 3B we give the solute errors for the prolem (3) with oundry conditions (4) nd (5) respectively otined y SAwGF. Compring it with the MDM [9], DTM [3] nd HPM [4] results, it cn e seen esily tht the result otined y the present method (SAwGF) is very superior to tht otined
Adomin decomposition method with Green s function 365 y the three previous mentioned methods. From Tles 3A nd 3B, it cn e deduced tht, the error decresed monotoniclly with the increment of the integer n. Tle 3A: Comprison of solute errors for exmple 3 x SAwGF, n = DTM [3].... 6.99E-6 4.E-5.2 2.9E-4.3E-3.3.E-3 6.75E-3.4 2.46E-3.53E-2.5 3.2E-3.98E-2.6 2.42E-3.57E-2.7.8E-3 7.7E-3.8 2.9E-4.42E-3.9 6.58E-6 4.6E-5...22E-5 Tle 3B: Comprison of solute errors for exmple 3 SAwGF MDM [9] VIM [2] DTM [3] HPM [4] x n = n = 2 n = 3 n = 3...... --. 2.64E-7.4E-2.6E-7.6E-7.6E-7 7.47E-7.2 5.3E-7 2.7E-2 3.7E-7 3.7E-7 3.7E-7.42E-6.3 6.92E-7 2.99E-2 4.22E-7 4.22E-7 4.22E-7.96E-6.4 8.4E-7 3.52E-2 4.97E-7 4.97E-7 4.97E-7 2.3E-6.5 8.55E-7 3.7E-2 5.22E-7 5.22E-7 5.22E-7 2.42E-6.6 8.4E-7 3.53E-2 4.97E-7 4.97E-7 4.96E-7 2.3E-6.7 6.92E-7 3.E-2 4.22E-7 4.22E-7 4.22E-7.96E-6.8 5.3E-7 2.9E-2 3.7E-7 3.7E-7 3.7E-7.42E-6.9 2.64E-7.5E-2.6E-7.6E-7.6E-7 7.5E-7... 2.E- 2.E-.E-6 --- It is cler from the Tles 3A nd 3B tht the numericl results corresponding to prolem (3) with oundry conditions (4) re superior to those with oundry conditions (5). 4. Conclusions The ADM with Green s function (Stndrd Adomin nd Modified Technique) hs een pplied for solving liner nd nonliner twelfth-order
366 Wleed Al-Hyni oundry vlue prolems with oundry conditions defined t even-order nd odd-order derivtives s well. Comprison of the results otined y the present method with those otined y the Twelfth-degree spline method, Modified decomposition method with the inverse opertor, Thirteen-degree spline method, Non-polynomil spline technique, Vritionl itertion method, Differentil trnsform method nd Homotopy perturtion method hs reveled tht the present method is superior ecuse of the lower error nd fewer required itertions. It hs een shown tht error is monotoniclly reduced with the increment of the integer n. References []. G. Adomin, Stochstic Systems. Acdemic Press, New York, 983. [2]. G. Adomin, Nonliner Stochstic Opertor Equtions. Acdemic Press, New York, 986. http://dx.doi.org/.6/978--2-44375-8.524- [3]. G. Adomin, Nonliner Stochstic Systems Theory nd Applictions to Physics. Kluwer Acdemic Pulishers, Dordrecht, 989. http://dx.doi.org/.2/zmm.99788 [4]. G. Adomin, Solving Frontier Prolems of Physics: The Decomposition Method. Kluwer Acdemic Pulishers, Dordrecht, 994. [5]. S. Chndrsekhr, Hydrodynmic Hydromgnetic Stility. Clrendon Press, Oxford, 96 (Reprinted: Dover Books, New York, 98). http://dx.doi.org/.7/s222622592 [6]. K. Djidjeli, E. H. Twizell nd A. Boutye, Numericl methods for specil nonliner oundry vlue prolems of order 2m. J. Comput. Appl. Mth. 47 () (993), 35-45. http://dx.doi.org/.6/377-427(93)988-s [7]. R. P. Agrwl, Boundry vlue prolems for higher-order differentil equtions. World Scientific, Singpore, 986. http://dx.doi.org/.42/266 [8]. S. S. Siddiqi nd E. H. Twizell, Spline solutions of liner twelfth-order oundry-vlue prolems. J. Comput. Appl. Mth. 78 (997), 37-39. http://dx.doi.org/.6/s377-427(96)64- [9]. A. M. Wzwz, Approximte solutions to oundry vlue prolems of higher order y the modified decomposition method. Comput. Mth. Appl. 4 (2), 679-69. http://dx.doi.org/.6/s898-22()87-5
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368 Wleed Al-Hyni [2]. S. Guelll nd Y. Cherruult, Prcticl formul for clcultion of Adomin's polynomils nd ppliction to the convergence of the decomposition method. Int. J. Biomed. Comput. 36 (3) (994), 223-228. http://dx.doi.org/.6/2-7(94)957-4 [22]. A. M. Wzwz, A relile modifiction of Adomin decomposition method. Appl. Mth. Comput. 2 () (999), 77-86. http://dx.doi.org/.6/s96-33(98)24-3 Received: Octoer 7, 24; Pulished: Jnury 5, 25