Appled Mathematcs, 6, 7, 5-4 Publshed Onlne Jul 6 n ScRes. http://www.scrp.org/journal/am http://.do.org/.436/am.6.77 umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton Method Ignatus. joseh, Ebmene J. Mamadu Department of Mathematcs and Computer Scence, Delta State Unverst, Abraka, gera Department of Mathematcs, Unverst of Ilorn, Ilorn, gera Receved 6 Ma 6; accepted 8 Jul 6; publshed Jul 6 Coprght 6 b authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). http://creatvecommons.org/lcenses/b/4./ Abstract In ths paper, a new approach called Power Seres Approxmaton Method (PSAM) s developed for the numercal soluton of a generalzed lnear and non-lnear hgher order Boundar Value Problems (BVPs). The proposed method s effcent and effectve on the expermentaton on some selected thrteen-order, twelve-order and ten-order boundar value problems as compared wth the analtc solutons and other exstng methods such as the Homotop Perturbaton Method (HPM) and Varatonal Iteraton Method (VIM) avalable n the lterature. A convergence analss of PSAM s also provded. Kewords Power Seres, Lnear and onlnear Problems, Boundar Value Problem (BVP), umercal Smulaton. Introducton Hgher order boundar value problems n lnear and non-lnear form have been a major concern n recent ears. Ths s due to ts applcablt n man areas of Mathematcal Phscs and other scences n ts precse analss of nonlnear phenomena such as computaton of radowave attenuaton n the atmosphere, nterface condtons determnaton n electromagnetc feld, potental theor and determnaton of wave nodes n wave propagaton. Most conventonal analtc methods for hgher order boundar value problems are prone to roundng-off and computaton errors. As a result, the analtcs methods are less dependent n seekng the soluton of hgher order boundar values problems n most cases, especall the non-lnear tpe. Thus, numercal methods have ganed How to cte ths paper: joseh, I.. and Mamadu, E.J. (6) umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton Method. Appled Mathematcs, 7, 5-4. http://.do.org/.436/am.6.77
I.. joseh, E. J. Mamadu momentum n seekng the soluton of hgher order boundar value problems. Over the ears, several numercal technques have been developed, such as the Varatonal Iteraton Method (VIM) [], Homotop Perturbaton Method (HPM) [], Splne-Collocaton Approxmatons Method (SCAM) [3], Splne Method [4], etc. that possess an elaborate procedure and structurall complex, whch nevertheless elds effcent results. Sddq and Iftkhar [5] worked on a numercal soluton of hgher order boundar value problems. Also, Sddq and Iftkhar [6] adopted the technque of varaton of parameter methods for the soluton of seventh order boundar value problems. Iftkhar et al. [7] solved the thrteenth order value problems b Dfferental transform method. Akram and Rehman [8] presented a numercal soluton of eghth order boundar value problems n reproducng kernel space. Wu et al. [9] presented a precse and rgorous work on nonlnear functonal analss of boundar value problems: novel theor, methods and applcatons. Mamadu and joseh [] have proposed a method whch effcentl fnds exact solutons and s used to solve lnear Volterra ntegral equatons. In ths present work, the Power Seres Approxmaton Method (PSAM) s a new approach developed for the numercal soluton of a generalzed th order boundar value problems. The proposed method s structurall smple wth well posed Mathematcal formulae. It nvolves transformng the gven boundar value problems nto sstem of ODEs together wth the boundar condtons prescrbed. Thereafter, the coeffcents of the power seres soluton are unquel obtaned wth a well posed recurrence relaton along the boundar ξ, whch leads to the soluton. The unknown parameters n the soluton are determned at the other boundar ξ. Ths fnall leads to a sstem of algebrac equatons, whch on solvng elds the requred approxmate seres soluton. The method s accurate and effcent n obtanng the approxmate solutons of lnear and non-lnear boundar value problems. The method requres no dscretzaton and lnearzaton or perturbaton. Also, computatonal and roundng-off errors are avoded. The method has an excellent rate of convergence as compared wth exstng methods n [] [] and the exact solutons avalable n the lterature. The rest of ths paper wll be organzed as follows: Secton of ths work gve detaled Mathematcal formulaton of th order BVPs usng PSAM. Secton 3 presents the error analss and convergence theorem of the method. Secton 4 offers numercal stmulaton of the method on some selected thrteen-order, twelve-order and ten-order boundar value problems. Fnall, the concluson s presented n Secton 5.. Power Seres Approxmaton Method (PSAM) We consder the th order BVP of the form wth the boundar condtons where f ( x), g( x ) and m=,,,3,,( n ) ( ), ξ ξ x + f x x = g x < x< () ( m ) ( ξ ) = λ = ( ), m,,,3,, n, () m ( m ) ( ξ ) = β, =,,,3,,( ) m n (3) m x are assumed real and contnuous on ξ x ξ, λ m and β m, are fnte real constants. The gven nth order BVP (), () and (3) are transformed to sstems of ODEs such that we have wth the boundar condtons =, =, = 3, = g( x) f ( x) ( x), (4) 6
I.. joseh, E. J. Mamadu and ξ = λ, ξ = λ, 3 ξ = λ,, n ξ = λn (5) ξ = β,, ξ = β, 3 ξ = β,, n ξ = βn. (6) Let the seres approxmaton of (), () and (3) be gven as x = ax, <, (7) where a, = are unknown constants to be determned and x [ ξ, ξ] ow, we estmate the unknown constants a,, x ξ whch s as follows: We consder the frst dervatve of At = wrt to x as,.e.,. = at = b substtutng (7) n (4) successvel, = = (8) = ax = a + ax =. ξ = λ, we have, a + aξ = λ a = λ aξ. Thus (8) becomes ext: d. (9) = = = λ ξ + = = () a a x x = = ( ) ax = a + ( ) ax = () = = 3 = we obtan, a + ( ) aξ = λ a = λ ( ) aξ () ξ λ, Thus () becomes Carrng on the above sequental approach to the = 3 = 3 (3) = λ aξ + a x = 3 = 3 th n order we obtan the followng recursve formulae at x ξ a n a n =, n n = n!, n! λ ξ (4) = n+ n n n = n+ = n+ (5) n = λ n! aξ + n! a x, n, Here, the choce of s equvalent to the order of the BVP consdered. 3. Error Analss and Convergence Theorem An error estmate for the approxmate soluton (7) of (), () and (3) s obtaned here. Let n e = x x as the error functon of ( x ) to ( x ) ; where Hence, ( x ) satsfes the followng problems: x s the exact soluton of (), () and (3). 7
I.. joseh, E. J. Mamadu ( ) [ ] x = g x f x x + H x, x ξ, ξ, (6) ( m ) ( ξ ) = λ, =,,,,( ) m n (7) m ( m ) ( ξ ) = β, =,,,,( ) m n (8) m The perturbaton term H ( x ) can be obtaned b substtutng the computed soluton ( ) ( H x ( x) g( x) f ( x) ) ( x) x to obtan = + (9) We then transform (6), (7) and (8) nto sstems of ordnar dfferental equatons and proceed to fnd an approxmate en, ( x ) to the error functon en ( x ) n the same wa as we dd before for the soluton of the problem (), () and (3). Thus, the error functon satsfes the problem wth the homogeneous condtons 3.. Convergence Theorem ( ) [ ] e x g x + f x e x = H x, x ξ, ξ, () n n ( m ) ξ =, m=,,,, () ( m ) ξ =, m=,,,, () We now prove that f the soluton seres b PSAM s convergent, t must be an exact soluton b ncreasng the order of approxmaton. Theorem : If the soluton seres = converges t must be an exact soluton b ncreasng the order of ap- = proxmaton. Proof: Let the seres We have Usng Equaton (3), Usng Equaton (4), x ax ax be convergent. Then = = Snce a n Equaton (7), we have x = ax (3) = ( x) lm = (4) ax xa x (5) = ax xa x = lm x = (6) ax xa x = ax = = (7) If the value of s so large or approaches nfnt as n (4) and (5), ax = (8) = 8
I.. joseh, E. J. Mamadu and ths completes the proof. ax a x = = = + + = 4. umercal Examples To mplement the method developed, three examples are consdered. Example Consder the followng thrteenth-order problem [] The exact soluton s ( 3 ) x = cos x sn x, (9) ( ) =, ( ) =, ( ) =, ( 3 ) =, ( 4 ) =, ( 5 ) =, ( 6 ) =, ( ) =, ( ) =, ( ) =, ( 3 ) =, ( 4 ) =, ( 5 ) =, ( x) = sn x+ cos x. The gven 3th order BVP (9) are transformed to sstems of ODEs such that we have wth the boundar condtons at x = ξ = =, =, = 3, = cos x sn x, 9
I.. joseh, E. J. Mamadu =, =, =, =, =, =, 3 4 5 6 =, = a, = b, =, = e, = f. 7 8 9 3 The seres approxmaton of (9) s gven as Equaton (7) a, = are unquel determned b Equaton (4). Snce, ξ =, we have Equaton (4) as where the unknown constants Usng Equaton (3) for n = an, we have the followng: λn =, n (3) n! a =, a =, a =, a3 =, a4 =, a5 =, a6 =, 6 4 7 a b c d e a7 =, a8 =, a9 =, a =, a =. 54 43 3688 3688 39968 Substtutng (3) nto Equaton (7) for = () we obtan x x d xc xb xa x 3688 3688 43 54 7 9 8 7 6 = + + + + + + + + 4 6 39968 5 4 3 x x x x x x e Usng boundar condton at x = ξ = n Equaton (3) we obtan the values of a, b, c, d and e, as a =, b =, c =, d =.999997 and e =. The above values of abcd,,, and e concde wth the results n [], where Varatonal Iteraton Method s used for the same problem consdered. Thus, the fnal approxmaton soluton of BVP (9) can be wrtten as x E x x x x 3688 43 54 7 9 8 7 =.75573655 + + 7 4 6 6 5 4 3 x + x + x x x + x+ The comparson of the approxmate soluton of example obtaned wth the help of PSAM and the approxmate soluton usng VIM obtaned n [] s gven n Table. From the numercal results, t s clear that the PSAM s more effcent and accurate. B ncreasng the order of approxmaton more accurac can be obtaned. Example Consder the followng lnear tenth-order problem [] ( ) x (3) (3) x = e x, a x b. (33) wth the followng boundar condtons ( k ) =, k =,,,3, 4. (34) ( k ) =, k =,,,3, 4. (35) The exact soluton s ( x ) = e x. The gven th order BVP (33) s transformed to sstems of ODEs such that we have
I.. joseh, E. J. Mamadu wth the boundar condtons at x = ξ = Snce, ξ =, we have Equaton (4) as Hence for n = 9 we have =, =, = 3, x = e ( x) = = = = =,,,,, 3 4 5 = a, = b, = c, =, = e. 6 7 8 9 an λn =, n. n! a a =, a =, a =, a3 =, a4 =, a5 =, 6 4 b c d e a6 =, a7 =, a8 =, a9 =. 7 54 43 3688 Hence, substtutng the above values of a, n n = 9 n (7), we obtan 3 4 5 6 7 8 9 ( x) = + x + x + x + x + ax + bx + cx + + ex (36) 6 4 7 54 43 3688 Usng boundar condton at x = ξ = on equaton (36) we obtan the values of a, b, c, d and e, as a =.933, b =.999799, c =.8535, d =.973568663, and e =.888 Thus, the fnal approxmaton soluton of the BVP (33) can be wrtten as 3 4 5 6 ( x) = + x+ x + x + x +.8333577767x +.38848789x 6 4 +.989774x +.4463587x + 3.3575847E x 7 8 7 9 The comparson of the approxmate soluton of Example obtaned wth the help of PSAM and the approxmate soluton usng HPM [] s gven n Table. From the numercal results, t s clear that the PSAM s more effcent and accurate. B ncreasng the order of approxmaton more accurac can be obtaned. Example 3 Consder the followng twelve-order problem wth the followng boundar condtons 3 x = e x x + x, a x b. (37) ( k ) =, k =,,,3, 4,5 (38) The exact soluton s ( k ) =, k =,,,3, 4,5 e e x x =. The gven th order BVP (37) s transformed to sstems of ODEs such that we have (39)
I.. joseh, E. J. Mamadu =, =, = 3, = ( x) + x ( 3 ) x e, wth the boundar condtons (at x = ξ = ) =, ( ) = a, 3 =, 4( ) = b, 5 =, 6( ) = c, 7 =, =, =, = e, () = and = f. 8 9 Snce, ξ =, we have Equaton (4) as an Hence for n = we obtan the followng λn =, n. n! b c d a =, a = aa, =, a3 =, a4 =, a5 =, a6 =, a7 =, 6 4 7 54 e f a8 =, a9 =, a =, a =. 43 3688 3688 39968 Hence, substtutng the above values of a, n n = ( ) n (7), we obtan x ax x bx x cx x 6 4 7 54 8 9 + x + ex + x + fx 43 3688 3688 39968 3 4 5 6 7 = + + + + + + + Usng boundar condton at x = ξ = n Equaton (4) we obtan the values of a, b, c, d, e and f, as a =.99999493, b =.58885, c =.9994994, d =.5758, e =.9434337955 and f =.63555. Thus, substtutng the values a, b, c, d, e and f n (4), the fnal approxmaton soluton of BVP (37) can be wrtten as 3 4 ( x) =.99999493x+ x.66676488x + x 4.8384945x + x.99548667x + x 7 43.5998567x + x 4.88864E x 3688 5 6 7 8 9 8 The comparson of the approxmate soluton of Example 3 obtaned wth the help of PSAM and the approxmate soluton usng HPM [] s gven n Table 3. From the numercal results, t s clear that the PSAM s more effcent and accurate. B ncreasng the order of approxmaton more accurac can be obtaned. 5. Concluson In ths paper, the Power Seres Approxmaton Method has been appled to obtan the numercal soluton of lnear and nonlnear generalzed th order boundar value problems. The PSAM requres no dscretzaton, lnea-rzaton or perturbaton. B ncreasng the order of approxmaton more accurac can be obtaned. Comparson of the results obtaned wth exstng technques [] [] shows that the PSAM s more effcent and accurate. Hence, t s easer and more economcal to appl PSAM n solvng BVPs. (4)
I.. joseh, E. J. Mamadu Table. Comparson of results of PSAM wth Varatonal Iteraton Method (VIM). X Exact Soluton PSAM VIM......948376.948376.99454..787359.787359.93864.3.58567.58568.769356.4.34793.348.78469.5.3578.357.65987.6.389978.389989.5375.7.49599.4994.387.8.4468.44457.474.9.449369.4557.96..387733.38676. Table. Comparson of results of PSAM wth HPM. X Exact Soluton PSAM HPM......476.476.4846.4.49847.49847.4983358.6.888.8878.87686.8.55493.55455.554643..78883.787885.788799 Table 3. Comparson of results of PSAM wth HPM. X Exact Soluton PSAM HPM..... 8.873753 8.873873 8.873873.4 6.7346 6.73854 6.73854.6 5.488636 5.48834449 5.488445.8 4.4938964 4.4933834 4.49389646. 3.6787944 3.67879448 3.678794453 References [] Adeosun, T.A., Fenuga, O.J., Adelana, S.O., John, A.M., Olalekan, O. and Alao, K.B. (3) Varatonal Iteraton Methods Solutons for Certan Thrteenth Order Ordnar Dfferental Equatons. Journal of Appled Mathematcs, 4, 45-4. http://.do.org/.436/am.3.49 [] Othman, M.I.A., Mah, A.M.S. and Farouk, R.M. () umercal Soluton of th Order Boundar Value Problems b Usng Homotop Perturbaton Method. Journal of Mathematcs and Computer Scence,, 4-7. [3] Watson, L.M. and Scott, M.R. (987) Solvng Splne-Collocaton Approxmatons to onlnear Two-Pont Boundar Value Problems b a Homotop Method. Journal of Mathematcs and Computaton, 4, 333-357. http://.do.org/.6/96-33(87)95-4 [4] Sddq, S.S. and Twzell, E.H. (998) Splne Soluton of Lnear Tenth-Order Boundar Value Problems. Internatonal Journal of Computer Mathematcs, 68, 345-36. http://.do.org/.8/76988847 3
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