Non polynomial spline solutions for special linear tenth-order boundary value problems

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1 ISSN England UK World Journal of Modellng and Smulaton Vol No. pp Non polynomal splne solutons for specal lnear tenth-order boundary value problems J. Rashdna R. Jallan 2 K. Farajeyan 3 School of Mathematcs Iran Unversty of Scence and Technology Narmak Tehran 6844 Iran 2 Department of Mathematcs Ilam Unversty P. O. Box Ilam Iran 3 Department of Mathematcs Islamc Azad Unversty of Bonab Iran Receved May 2009 Accepted January 200 Abstract. Non-polynomal splne s used for soluton of the tenth-order lnear boundary value problems. We obtaned the classes of numercal methods for a specfc choce of the parameters nvolved n non-polynomal splne. The end condtons consstent wth the boundary value problems are derved. Truncaton errors are gven. A new approach convergence analyss of the presented methods s dscussed. Two examples are consdered for the numercal llustraton. However t s observed that our approach produce better numercal solutons n the sense that max e s mnmum. Keywords: tenth-order boundary-value problem convergence analyss end condtons Introducton We consder n ths work the numercal approxmaton for the tenth-order boundary value problems of the form: y 0 x fxyx = gx x [a b] wth boundary condtons: { ya = A0 y 2 a = A y 4 a = A 2 y 6 a = A 3 y 8 a = A 4 yb = B 0 y 2 b = B y 4 b = B 2 y 6 b = B 3 y 8 b = B 4 2 Hgher order dfferental equatons arse n many felds. When nstablty sets n an ordnary convecton t s modeled by tenth-order boundary value problem [2]. Scott and Watts [9] descrbed several computer codes that were developed usng the superposton and orthonormalzaton technque and nvarant mbeddng. Twzell et al. [3] developed numercal methods for 8th-0th-and 2th-order egenvalue problems arsng n thermal nstablty. Sddq and Twzell [2] presented the soluton of 0th-order boundary value problem usng 0th degree splne. Sddq and Akram [0] developed the soluton of 0th-order boundary value problems usng non-polynomal splne. Sddq and Akram [] presented the soluton of 0th-order boundary value problem by usng eleventh degree splne. Rashdna et al. [8] developed numercal methods for 8th-order boundary value problem usng non-polynomal splne. Djdejel and Twzell [4] derved numercal method for specal nonlnear boundary-value problems of order 2m. Abdellah Lamn et al. [6] developed and analyzed numercal method for approxmatng solutons of some general lnear boundary value problems. Ramadan et al. [7] have been appled nonpolynomal splne functon for approxmatng solutons of 2µth order two pont BVPs. Followng [0] the splne functons proposed n ths paper have the form T = span{ x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 coskx snkx} where k s the frequency of the trgonometrc part of the splne functons Correspondng author. E-mal address: rezajallan@ust.ac.r rezajallan72@gmal.com. Publshed by World Academc Press World Academc Unon

2 World Journal of Modellng and Smulaton Vol No. pp whch can be real or pure magnary and whch wll be used to rase the accuracy of the method. Thus n each subnterval x x x we defne: Defnton. span{ x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 cos k x sn k x}. where k 0 we have the eleventh degree of splne []. [! kx3 lm snkx kx kx5 k o k 3! 5! [ 0! lm k o kx7 7! kx2 coskx kx4 kx6 kx8 k0 2! 4! 6! 8! ] kx9 = x 9! ] = x 0 span{ x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0 x } when k 0. In Secton 2 the new non-polynomal splne methods are developed for solvng Eq. along wth boundary condton Eq. 2. Development of the boundary formulas are consdered n Secton 3. Secton 4 s devoted to convergence analyss. In Secton 5 some numercal experments are consdered and numercal results are compared wth the method developed n [2]. 2 Development of the class of methods To develop the splne approxmaton for the tenth-order boundary-value problem Eqs. and 2 frst of all the nterval [a b] s dvded nto n equal subntervals usng the grd x = a h = 0 n where h = b a n. We consder the followng splne S on each subnterval [x x ] = 0 n S x = a cos kx x b sn kx x c x x 9 d x x 8 e x x 7 j x x 6 o x x 5 p x x 4 q x x 3 r x x 2 u x x w 3 where a b c d e j o p q r u and w are real coeffcents to be determned and k s free parameter. The splne s defned n terms of ts 2nd 4th 6th 8th and 0th dervatves and we denote these values at knots as = 0 n : S x = y S x = y S 2 x = m S 2 x = m S 4 x = M S 4 x = M S 6 x = z S 6 x = z S 8 x = v S 8 x = v S 0 x = l S 0 x = l. Algebrac manpulaton yelds the followng expressons whereby θ = kh and = 0 n : a = l k 0 b = cscθ cosθl l k 0 c = l l k 2 v k 2 v hk 2 d = l k 2 v 40320k 2 j = l k 4 z 720k 4 r = l k 8 m 2k 8 w = l k 0 y k 0 p = l k 6 M 24k 6 e = 6 2θ2 l 6 θ 2 l h 2 k 4 v v 6k 4 z z 30240hk 4 o = 43200hk 6 [360 20θ2 8θ 4 l θ 2 7θ 4 l 360k 6 M M h 4 k 6 v v 60h 2 k 6 2z z ] q = 90720hk 8 [ θ2 2θ 4 2θ 6 l θ 2 294θ 4 3θ 6 l WJMS emal for subscrpton: nfo@wjms.org.uk

3 42 J. Rashdna: Non polynomal splne solutons for specal lnear 520k 8 m m 2520h 2 k 8 2M M h 6 k 8 32v 3v h 4 k 8 336z 294z ] u = hk 0 [ θ2 05θ 4 0θ 6 θ 8 l θ 2 760θ 4 240θ 6 27θ 8 l 00800h 2 k 0 2m m h 4 k M 760M h 8 k 0 28v 27v k 0 y y h 6 k 0 280z 270z ]. Contnuty condton of the frst thrd ffth seventh and nnth dervatves that s S λ x = S λ x where λ = and 9 yelds the followng equatons respectvely: θ θ 6 7θ θ θ csc θ l l θ9 m 4m m 6h 8 2 θ θ [ 3 θ3 45 2θ5 945 θ cot θ l θ 9 7M 6M 7M 360h 6 27v 256v 27v h 2 y 2y y h 0 3z ] 64z 3z 520h 4 = 0 4 [ 520 θ θ 6 7θ θ csc θ l 2 θ θ 3 θ3 45 2θ cot θ l θ θ ] 6 7θ θ5 520 csc θ l θ [ m 2m m h 8 M 4M M h 6 55v 605v 284v 8h 2 427z ] 6z 7z 7h 4 = θ θ 6 7θ3 360 csc θ l l 720 θ θ 3 θ3 45 cot θ l θ 5 360M 2M M h 6 7v 6v 7v h 2 60z 4z z h 4 = θ θ 6 csc θ l l 2 θ θ 3 cot θ l θ 5 v 4v v h 2 6z 2z z h 4 =0 7 2 θ csc θ l l θ 2 cot θ l θv 4v v h 2 = 0. 8 In order to get useful consstency splne relaton n terms of 0th dervatve of splne we have to elmnate m M z v from Eqs. 4 8 therefor s replaced by ± 4 ± 3 ± 2 ± n each of the Eqs. 4 8 we have the followng useful consstency relaton n terms of tenth dervatve of splne l and y. h 0 αl 5 βl 4 γl 3 δl 2 ηl τl ηl δl 2 γl 3 βl 4 αl 5 = y 5 0y 4 45y 3 20y 2 20y 252y 20y 20y 2 45y 3 0y 4 y 5. 9 where = 5 6 n 5 and α = θ sn θ θ 2 20θ 4 6θ 6 θ sn θ θ 9 β = 25 cos θ 59 cos θ cos θ θ sn θ θ 2 60θ 4 γ = cos θ cos θ θ sn θ θ 2 δ = cos θ 7 cos θ θ sn θ θ cos θ 5θ 4 η = cos θ 0722 cos θ θ sn θ θ 2 cos θ 3θ cos θ 20θ cos θ 60θ cos θ θ cos θ 6θ 6 0 sn θ θ cos θ θ cos θ cos θ 3θ 6 θ cos θ 3θ cos θ θ 8 45 sn θ θ 9 20 sn θ θ 9 20 sn θ θ 9 WJMS emal for contrbuton: submt@wjms.org.uk

4 World Journal of Modellng and Smulaton Vol No. pp τ = cos θ 75 cos θ 4395 cos θ 3450 cos θ 240 cos θ 252 sn θ θ sn θ θ 2 30θ 4 3θ 6 θ 8 θ 9. Usng the splne Eq. 9 and also by usng the gven dfferental Eq. we have h 0 [αg 5 f 5 y 5 βg 4 f 4 y 4 γg 3 f 3 y 3 δg 2 f 2 y 2 ηg f y τg f y ηg f y δg 2 f 2 y 2 γg 3 f 3 y 3 βg 4 f 4 y 4 αg 5 f 5 y 5 ] = y 5 0y 4 45y 3 20y 2 20y 252y 20y 20y 2 45y 3 0y 4 y 5 = 5 6 n 5 In specal case f k 0 that s θ 0 then α β γ δ η τ And the consstency relaton of non-polynomal s reduced to consstency relaton of the eleventh polynomal splne functons derved n []. The local truncaton error correspondng to the method Eq. 0 can be obtaned as: t = 2α 2β 2γ 2δ 2η τh 0 y α 6β 9γ 4δ η α256β8γ6δη h 4 y α4096β729γ64δη α 65536β 656γ 256δ η h 8 y α β 59049γ 024δ η h 20 y α β 5344γ 4096δ η h 22 y α β γ 6384δ η h 24 y = 5 6 n 5.. h 2 y 2 0 h 6 y 6 By usng the above truncaton error to elmnate the coeffcents of varous powers of h we can obtan classes of the methods. For any choce of α β δ η and τ whose 2α 2β 2γ 2δ 2η τ = we can obtaned the followng class of methods. Remark. Second-order method for α = β = γ = δ = η = and τ = we obtan the second order method wth truncaton error t = 2 h2 y 2 x Oh 3. Remark 2. Fourth-order method for α = 0 β = 0 γ = 0 δ = 0 η = 5 2 and τ = 6 we obtan the fourth order method wth truncaton error t = 7 44 h4 y 4 x Oh 5. WJMS emal for subscrpton: nfo@wjms.org.uk

5 44 J. Rashdna: Non polynomal splne solutons for specal lnear Remark 3. Sxth-order method for α = 0 β = 0 γ = 0 δ = 7 44 η = 2 9 order method wth truncaton error t = h6 y 6 x Oh 7. and τ = 24 we obtan the sxth Remark 4. Eght-order method for α = 0 β = 0 γ = δ = η = eght order method wth truncaton error 30 and τ = 3024 we obtan the t = h8 y 8 x Oh 9. Remark 5. Tenth-order method for α = 0 β = γ = δ = η = 447 obtan the tenth order method wth truncaton error t = h20 y 20 x Oh and τ = we Remark 6. Twelfth-order method for α = β = γ = δ = η = τ = we obtan the twelfth order method wth truncaton error 69 t = h22 y 22 x Oh and 3 Development of the boundary formulas Lner system Eq. 0 consst of n unknown so that to obtan unque soluton we need eght more equatons to be assocate wth Eq. 0 so that we can develop the boundary formulas of dfferent orders but for sake of brefness here we develop the eght order boundary formulas so that we defne the followng dentty ā k y k bh 2 y 2 0 ch 4 y 4 0 dh 6 y 6 0 ēh 8 y 8 0 h 0 a k y k b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h 0 8 k= a k y k b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h 0 p k y 0 k t = k= 0 a k y k b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h 0 a k y n k b h 2 y 2 n c h 4 y 4 n d h 6 y 6 n e h 8 y 8 n h 0 p k y0 k t 2 = 0 3 k= p k y0 k t 3 = 0 4 k= k= a k y n k b h 2 y 2 n c h 4 y 4 n d h 6 y 6 n e h 8 y 8 n h 0 a k y n k b h 2 y 2 n c h 4 y 4 n d h 6 y 6 n e h 8 y 8 n h 0 a k y n k b h 2 y 2 n c h 4 y 4 n d h 6 y 6 n e h 8 y 8 n h 0 9 k= 8 k= p k y0 k t 4 = 0 5 p k y0 n k t n 4 = k= p k y0 n k t n 3 = 0 7 p k y0 n k t n 2 = 0 8 p k y0 n k t n = 0 9 WJMS emal for contrbuton: submt@wjms.org.uk

6 World Journal of Modellng and Smulaton Vol No. pp where all of the coeffcents are arbtrary parameters to be determned. In order to obtan the eght-order method we fnd that: ā 0 ā ā 2 ā 3 ā 4 ā 5 ā 6 = a 0 a a 2 a 3 a 4 a 5 a 6 = b c d ē = b c d e = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 = b c d e = b c d e = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 a 8 = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 a = b c d e = b c d e = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 0 = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 = a 0 a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 = b c d e = b c d e = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 0 p = p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 0 p = And also t = h8 y0 8 t 2 = h9 y0 9 t 3 = h20 y t 4 = h2 y t n 4 = h2 yn 2 t n 3 = h20 yn 20 WJMS emal for subscrpton: nfo@wjms.org.uk

7 46 J. Rashdna: Non polynomal splne solutons for specal lnear t n 2 = h9 y 9 n t n = h8 y 8 n. 4 Convergence analyss The approxmate soluton of boundary value problem Eqs. and 2 s determned by usng the system defned by Eq.0 and assocated wth the boundary Eqs The arsng lnear system n matrx form can be gven as: A Y = C T A Ỹ = C A E = T. 20 where vectors Y = y and Ỹ = ỹ are the exact and approxmate soluton of boundary value problem respectvely and also C = c T = t and E = ẽ = y ỹ are n -dmensonal column vectors. The matrx A s defned by where A B and F are n n matrces. A= B = ā ā 2 ā 3 ā 4 ā 5 ā 6 a a 2 a 3 a 4 a 5 a 6 a 7 A = A h 0 BF. a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a a 2 a 3 a 4 a 5 a 6 a 7 a 8 a a 9 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a a 8 a 7 a 6 a 5 a 4 a 3 a 2 a a 7 a 6 a 5 a 4 a 3 a 2 a a 6 a 5 a 4 a 3 a 2 a 2 p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 0 p p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 0 p β γ δ η τ η δ γ β α α β γ δ η τ η δ γ β α and F = dagf = 2 n. Moreover α β γ δ η τ η δ γ β α α β γ δ η τ η δ γ β p p 0 p 9 p 8 p 7 p 6 p 5 p 4 p 3 p 2 p p 0 p 9 p 8 p 7 p 6 p 5 p 4 p 3 p 2 p p 9 p 8 p 7 p 6 p 5 p 4 p 3 p 2 p p 8 p 7 p 6 p 5 p 4 p 3 p 2 p 22 WJMS emal for contrbuton: submt@wjms.org.uk

8 World Journal of Modellng and Smulaton Vol No. pp c = ā 0 y 0 bh 2 y 2 0 ch 4 y 4 0 dh 6 y 6 0 ēh 8 y 8 0 h0 p g p 2 g 2 p 3 g 3 p 4 g 4 p 5 g 5 p 6 g 6 p 7 g 7 p 8 g 8 c 2 = a 0y 0 b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h0 p g p 2g 2 p 3g 3 p 4g 4 p 5g 5 p 6g 6 p 7g 7 p 8g 8 p 9g 9 c 3 = a 0 y 0 b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h0 p g p 2g 2 p 3g 3 p 4g 4 p 5g 5 p 6g 6 p 7g 7 p 8g 8 p 9g 9 p 0g 0 c 4 = a y 0 b h 2 y 2 0 c h 4 y 4 0 d h 6 y 6 0 e h 8 y 8 0 h0 p g p 2 g 2 p 3g 3 p 4g 4 p 5 g 5 p 6 g 6 p 7 g 7 p 8 g 8 p 9 g 9 p 0g 0 p g c 5 = αh 0 f 0 y 0 h 0 αg 0 βg γg 2 δg 3 ηg 4 τg 5 ηg 6 δg 7 γg 8 βg 9 αg 0 c = h 0 αg 5 βg 4 γg 3 δg 2 ηg τg ηg δg 2 γg 3 βg 4 αg 5 =6 7 n 6 c n 5 = αh 0 f n y n h 0 αg n βg n γg n 2 δg n 3 ηg n 4 τg n 5 ηg n 6 δg n 7 γg n 8 βg n 9 αg n 0 c n 4 = a 0y n b h 2 y n 2 c h 4 y n 4 d h 6 y n 6 e h 8 y n 8 h 0 p g n p 2g n 2 p 3g n 3 p 4g n 4 p 5g n 5 p 6g n 6 p 7g n 7 p 8g n 8 p 9g n 9 p 0g n 0 p g n c n 3 = a 0 y n b h 2 y 2 n c h 4 y 4 n d h 6 y 6 h 0 p g n p 2 g n 2 p 3 g n 3 n e h 8 y n 8 p 4 g n 4 p 5 g n 5 p 6 g n 6 p 7 g n 7 p 8 g n 8 p 9 g n 9 p 0g n 0 c n 2 = a 0y n b h 2 y n 2 c h 4 y n 4 d h 6 y n 6 e h 8 y n 8 h 0 p g n p 2g n 2 p 3g n 3 p 4g n 4 p 5g n 5 p 6g n 6 p 7g n 7 p 8g n 8 p 9g n 9 c n = a 0y n b h 2 y n 2 c h 4 y n 4 d h 6 y n 6 e h 8 y n 8 h 0 p g n p 2g n 2 p 3g n 3 p 4g n 4 p 5g n 5 p 6g n 6 p 7g n 7 p 8g n 8. To explan the exstence of A h 0 BF we have to show A h 0 BF s nonsngular. We can easly show that A = p 5 s a monotone eleven band matrx of order n where P = p j s a trdagonal matrx defned by 2 = j = 2 n p j = j = 23 0 otherwse The matrx P s a monotone matrx so that the eleven band matrx A s also monotone and A s exst. Therefor A can be defned as: Defnton 2. A = A h 0 BF = AI h 0 A BF we need to show that the matrx I h 0 A BF s nonsngular. Frst of all we need to recall the followng lemma []. Lemma. If M s a square matrx of order n and M < then I M exsts and I M < where represents the norm n matrx vector. M Now to show that the matrx I h 0 A BF s nonsngular we wll prove the followng lemma. WJMS emal for subscrpton: nfo@wjms.org.uk

9 48 J. Rashdna: Non polynomal splne solutons for specal lnear Lemma 2. The matrx A h 0 BF s nonsngular f where F f = max a x b fx. f < b a 0 Proof. By usng Lemma f we provded a condton on fx so that h 0 A BF < then [I h 0 A BF ] exsts. Followng Henrc [5] we can obtan bounds for the elements of P. If P = p j the elements of p j s non negatve [5] and can be obtaned as p j = { jn n j n j n j 24 therefore the summaton of th row of P s n p j = j= j= jn n n j= n j n n Accordng [5] nequalty can be wrtten as P b a2 8h 2 26 then we get A b a h0 Now we can fnd a bound for B but n provrus secton the matrx B s varous for dfferent classes of the methods f we consder the eght-order method we can obtan Also by usng lemma we can get where B [I h 0 A BF ] < h 0 A BF h 0 A BF h 0 A B F b a f < then we have f < b a 0. Now we prove that A s exsts. By usng Eq. 20 we can get E = A T E A T. 29 To obtan a bound on error vector E we wll prove the followng theorem. WJMS emal for contrbuton: submt@wjms.org.uk

10 World Journal of Modellng and Smulaton Vol No. pp Theorem. Let yx be the exact soluton of the boundary value problem and we assume ỹ = 2 n s the numercal soluton obtaned by usng eght order method α = 0 β = 0 γ = δ = η = τ = assocated wth boundary Eqs. 2 9 and by solvng the system Eq. 20. Then we have provded E Oh 8 f < b a 0. Proof. Our man purpose s to drve a band on E usng Eq. 29 we have By usng Lemma we get E A T [I h 0 A BF ] A T. 30 E provded [h 0 A BF ] < also we can obtan A T [h 0 A BF ] 3 T h8 M where M 8 = max a ξ b y 8 ξ. Usng Eqs and 32 n Eq. 3 we obtan the followng bound as: where E φb a0 h 8 M 8 ψ ωb a 0 f Oh8 33 φ = ψ = ω = provded that f < b a 0 5 Numercal results Example. We consder the followng boundary-value problem [6 2] In [6 ] the boundary condtons are dfferent y 0 x xyx = 89 2x x 2 x 3 e x x y = y = 0 y 2 = y 2 = 2 e y4 = 4 e y4 = 20e y 6 = 8 e y6 = 42e y 8 = 40 e y8 = 72e The exact soluton for ths problem s yx = x 2 e x. Ths problem has been solved by [2]. We appled our methods descrbed n secton 2 3 to solve ths problem wth dfferent value of parameters α β γ δ η and τ and h = /9. The observed maxmum absolute errors are gven n Tab. and compared wth the method n [2]. It has been observed that our methods are more effcent. WJMS emal for subscrpton: nfo@wjms.org.uk

11 50 J. Rashdna: Non polynomal splne solutons for specal lnear Example 2. We Consder the followng boundary-value problem [ 2] In [] the boundary condtons are dfferent y 0 x yx = 02x snx 9 cosx x y = y = 0 y 2 = 4 sn 2 cos y 4 = 8 sn 2 cos y 6 = 2 sn 30 cos y 8 = 6 sn 56 cos y 2 = 4 sn 2 cos y 4 = 8 sn 2 cos y 6 = 2 sn 30 cos y 8 = 6 sn 56 cos. The exact soluton for ths problem s yx = x 2 cosx. Ths problem has been solved by [2]. We appled our presented methods descrbed n secton 2 3 to solve ths problem wth dfferent value of parameters α β γ δ η and τ and h = /9. The observed maxmum absolute errors are gven n Tab. 2. Table. Observed maxmum absolute errors n Example n = 8 Our second-order method Our fourth-order method Our sxth-order method Our eght-order method Our tenth-order method Our twelfth-order method Ref. [2] for x [x 5 x n 5 ] Table 2. Observed maxmum absolute errors n Example 2 n = 8 Our second-order method Our fourth-order method Our sxth-order method Our eght-order method Our tenth-order method Ref. [2] for x [x 5 x n 5 ] Example 3. We Consder the followng boundary-value problem y 0 x 5yx = 0 cosx 4 x snx 0 x y0 = y = 0 y 2 0 = 2 y 2 = 2 cos y 4 0 = 4 y 4 = 4 cos y 6 0 = 6 y 6 = 6 cos y 8 0 = 8 y 8 = 8 cos. The exact soluton for ths problem s yx = x snx. We appled our methods descrbed n secton 2 3 to solve ths problem wth dfferent value of parameters α β γ δ η and τ and h = 0. The observed maxmum absolute errors are gven n Tab. 3. λ Second-order method: α = β = γ = δ = η = τ = λ 2 Fourth-order method: α = 0 β = 0 γ = 0 δ = 0 η = 5/2 τ = /6. λ 3 Sxth-order method: α = 0 β = 0 γ = 0 δ = 7/44 η = 2/9 τ = /24. λ 4 Eght-order method: α = 0 β = 0 γ = 7/2096 δ = 9/224 η = 09/448 τ = 30/3024. λ 5 Tenth-order method: λ 6 Twelfth-order method: α = 0 β = / γ = 25/8440 δ = 93/22680 η = 447/8440 τ = 569/ α = / β = 6/ γ = 2203/ δ = 477/28520 η = 25687/ τ = 78069/ WJMS emal for contrbuton: submt@wjms.org.uk

12 World Journal of Modellng and Smulaton Vol No. pp Table 3. Observed maxmum absolute errors n Example 3 n = 0 Our second-order method Our fourth-order method Our sxth-order method Our eght-order method Our tenth-order method Our twelfth-order method Table 4. Observed maxmum absolute errors n Examples and 2 Example n = Example 2 n = Example n = 22 Example 2 n = 22 λ λ λ λ λ λ Concluson The new methods of orders and 2 based on non-polynomal splne are developed for the soluton of specal lnear tenth-order boundary-value problems. Tabs. and 2 shows that our methods produced better n comparson wth the method developed n [2]. References [] K. Atknson. An ntroducton to numercal analyss second edton. Johan Wley and Sons Inc 989. [2] S. Chandrasekhar. Hydrodynamc and Hydromagnetc Stablty. Clarendon press New York 98. Oxford 96. [3] M. Daele G. Berghe H. Meyer. A smooth approxmaton for the soluton of a fourth-order boundary value problem based on non-polynomal splnes. Journal of Computatonal and Appled Mathematcs : [4] K. Djdejel E. Twzell. Numercal methods for specal nonlnear boundary value problems of order 2m. Journal of Computatonal and Appled Mathematcs : [5] P. Henrc. Dscrete Varable Methods n Ordnary Dfferental Equatons. Wley New York 96. [6] A. Lamn H. Mraou et al. Splne soluton of some lnear boundary value problems. Appled Mathematcs E-Notes : [7] M. Ramadan I. Lashen W. Zahra. Hgh order accuracy nonpolynomal splne solutons for 2µth order two pont boundary value problems. Appled Mathematcs and Computaton : [8] J. Rashdna R. Jallan K. Farajeyan. Splne approxmate soluton of eghth-order boundary-value problems. Internatonal Journal of Computer Mathematcs : [9] M. Scott H. Watts. A Systematzed collecton of codes for solvng two-pont BVPs Numercal Methods for Dfferental Systems. Academc press 976. [0] S. Sddq G. Akram. Soluton of 0th-order boundary value problems usng non-polynomal splne technque. Appled Mathematcs and Computaton : [] S. Sddq G. Akram. Soluton of tenth-order boundary value problems usng eleventh degree splne. Appled Mathematcs and Computaton : [2] S. Sddq E. Twzell. Splne solutons of lner tenth-order boundary value problems. Internatonal Journal of Computer Internatonal Journal of Computer : [3] E. Twzell A. Boutayeb K. Djdjel. Numercal methods for eghth tenth and twelfth-order egenvalue problems arsng n thermal nstablty. Advances n Computatonal Mathematcs 994 2: WJMS emal for subscrpton: nfo@wjms.org.uk