A Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions

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1 Appled Matheatcs, 1, 4, Publshed Ole Aprl 1 ( A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple Fuctos Parcha Kalya 1*, Patbada S. Raa Chadra Rao 1, Araju Sowbhagya Madhusudha Rao 1 Kakatya Isttute of Techology ad Sceces, Waragal, Ida Varadha Reddy College of Egeerg, Waragal, Ida Eal: * kk.parcha@yahoo.co Receved Deceber, 1; revsed February, 1; accepted March, 1 Copyrght 1 Parcha Kalya et al. Ths s a ope access artcle dstrbuted uder the Creatve Coos Attrbuto Lcese, whch perts urestrcted use, dstrbuto, ad reproducto ay edu, provded the orgal work s properly cted. ABSTRACT I ths coucato we have used Bckley s ethod for the costructo of a sth order sple fucto ad apply t v to solve the lear ffth order dfferetal equatos of the for y g y r where g ad r are gve fuctos wth the two dfferet probles of dfferet boudary codtos. The ethod s llustrated by applyg t to solve soe probles to deostrate the applcato of the ethods dscussed. Keywords: Cubc Sple; Trdagoal; Covetoal Approach 1. Itroducto I the recet past, several authors have cosdered the applcato of cubc sple fuctos for the soluto of two pot boudary value probles. Bckley [1] has cosdered the use of cubc sple for solvg secod order two pot boudary value probles. The essetal feature of hs aalyss s that t leads to the soluto of a set of lear equatos whose atr coeffcets are of upper Heseberg for. Bckley uses a specal otato other tha the covetoal oe for the represetato of the cubc sple, for a detaled dscusso oe ay refer to E. A. Boquez ad J. D. A. Walker [], M. M. Chawla [], ad P. S. Raachadra Rao [4-7]. We used Bckley s ethod for the costructo of a sth degree sple ad apply t to the lear ffth order dfferetal equato wth two dfferet probles wth dfferet boudary codtos. The work has bee llustrated through eaples wth h =. ad h =... Cubc Sple-Bckley s Method Suppose the terval, s dvded to subtervals wth kots, 1,, Λ, startg at, the fucto u the terval, s represeted by a cubc sple the for * Correspodg author. j j j j j j j j S a b c d (1) Proceedg to the et terval 1,, we add a ter d1 1 ; proceedg to the et terval,, we add aother ter d ad so utl we reach. Thus the fucto S j s represeted the for for j j S a b c j j j j j 1 d 1 j j j j (1.) S b c d 1 j j (1.1) S c d (1.).1. The Two-Pot Secod Order Boudary Value Proble Frst, we cosder the lear dfferetal equato P u q u r u v (1.4) Wth the boudary codtos uu at, u u at (1.) Copyrght 1 ScRes.

2 84 P. KALYANI ET AL. The uber of coeffcets (1.1) s ( + ). The satsfacto of the dfferetal equato by the sple fucto at the ( + 1) odes gves ( + 1) equatos the ( + ) ukows. Also the ed codtos (1.) gve us two ore equatos the ukows. Thus we get ( + ) equatos ( + ) ukows a, b, c, d, d 1, Λ, d 1. after deterg these ukows we substtute the (1.1) ad thus we get the cubc sple approato of u. Puttg, 1,, Λ, the sple fucto thus detered, we get the soluto at the odes. The syste of equatos to be satsfed by the coeffcets a, b, c, d, d1, Λ, d 1 are derved below. Substtutg (1.1), (1.), (1.) (1.4), at we get 1 ar b r q cr q p d r q p S,,1,,, (1.) where p p ad so o. Applyg boudary codtos (1.), we get ab a b c (1.7) 1 d If these equatos are take the order (1.7), (1.) wth, 1, Λ,, the atr of the coeffcets of the ukows d 1, d,, d1, d, c, b, a s of the Heseberg for, aely a upper tragle wth a sgle lower sub-dagoal. The forward elato s the sple, wth oly oe ultpler at each step ad the back substtuto s correspodgly easy... Costructo of the Sth Degree Sple Suppose the terval, s dvded to subtervals wth kots, 1,, Λ,. Startg at, the fucto y the terval, 1 s represeted by a sth degree sple e g h y a b c d Proceedg to the et terval 1,, we add a ter h1 1, Proceedg to the et terval, we add aother ter h ad so utl we reach. Thus the fucto y s represeted the for 1 e g h (1.8) y a b c d It ca be see that are cotuous across odes. y ad ts frst fve dervatves. Ffth Order Boudary Value Proble We cosder the lear ffth order dfferetal equato y f y r (1.9) Wth the boudary codtos y, y, y, y, y (1.1) We get ( + ) equatos ( + ) ukows abc,,, degh,,,, h1, h,, h 1. After deterg these ukows we substtute the (1.8) ad thus we get the sth degree sple approato of y. Puttg 1,,, Λ, the sple fucto thus detered, we get the soluto at the odes. The syste of equatos to be satsfed by the coeffcets abcde,,,,, g, h, h1, h,, h 1 are derved below. Fro (1.8) we get y 1g7h 7h 1 1 7h 7h h (1.11) usg (1.8) & (1.11) the dfferetal Equato (1.9) at the odes takes of the for 1 1 af bf cf df ef g f 7 h f r,1,,, (1.1) To these equatos we add those obtaed fro the boudary codtos (1.1), we get a (1.1) 1 ab c d (1.14) e g h b (1.1) 1 bc d 4e g h (1.1) Copyrght 1 ScRes.

3 P. KALYANI ET AL. 8 c (1.17) If these equatos are take the order (1.14), (1.1), (1.1) wth, 1, Λ,, (1.17), (1.1) & (1.1) the atr of the coeffcets of the ukows, h, 1, h,, h1 h, g, e, d, c, b, a s a upper tragular atr wth two lower sub dagoals. The forward elato s the sple wth oly two ultplers at each step, ad the back substtuto s correspodgly easy..1. Eaple 1 Cosder the followg ffth order lear boudary value proble y y 1 1 e, 1 (1.18) Wth the boudary codtos y y 1, y 1, y 1 e, y () by takg equal subtervals wth h =. ad h =. 1) Soluto wth h =. The sth order sple y s gve by h 1 1 s whch approates s ab c d e g h where, 1., 1. We have eght ukows abc,,, degh,,,, h1 ad eght codtos to be satsfed by these ukows are s, s, s 1, se, s 1 1 e for,1, s, s 1, s () (4) s s () Sce a, b 1, c t follows that Equato () reduces to the for 4 g h h11 s, s s d e also sce = & reduces to s 1 1 e s 1 1 e () ad equatos of () for ad It follows that we have to detere the fve ukows degh,,,, h Equato (), subject to the fve codtos e, 1 1 e s s e s,, 1 1 e, s s s (7) fro () 1 4 h h s d e g 1 1 ad s 1g7h7h1 1 (9) Substtutg (), (8), (9) (7) we get the syste of equatos d egh.1 h 1, h.47441, g h h d 4egh.187 h , 8g 1, 1 d e g Solvg these we get d.148, e.1491, g.1, h.4878, h Substtutg these values () we get s 1 4 (8) (1) (11) h.4878h y s h h h where h =. Therefore y 1 y The aalytcal soluto of the dfferetal equato (1.18) subject to the codtos s gve by y 1 e (11.1) The eact value of y It follows that the Absolute error of the uercal value of y., coputed fro the sple approato s.84 whch s very sall. ) Soluto wth h =. The terval [,1] s dvded to 4 equal subtervals we deote the kots by, 1,,, 4 where, 1.,.,.7, 4 1. The sth order sple s whch approate y s gve by e g h h h h s a b c d 1 1 (1) Copyrght 1 ScRes.

4 8 P. KALYANI ET AL. s whch are to be de- There are 1 ukows tered fro 1 codtos 4 4 s, s, s 1, s e, s, s s 1 1 e for,1,,,4 (1) I vew of the codtos s, s 1 ad s t follows that a, b 1, c hece The sple s reduces to the for 4 s d e 1 1 g h h h h Fro (14) s 1d 4e g h h 1 1 h h h 7h s g h h 1 4 (14) (1) (1) Substtutg (14), (1), (1) (1) take the order, s4, s4e, 1 1 e s s for 4,,,1, we get the followg syste of equatos.4414h.1h h1h ged 1,.897h.187h h1 h g4ed , h 7h 18h1144h 4g , h h1 9.81h g.14e.4187 d 4.887, h h g.e.1d , h g.9e.1d.44479, 1g 1. (17) Fro the above syste of equatos, we otce that the coeffcet atr s a upper tragular atr wth two lower sub dagoals. solvg the above equatos we get d., e.879, g.1, h.41, h1.1781, (18) h.89, h.744 However t ay be otced that fro the Equato (17) g 1 1 whch whe substtuted the reag equatos wll gve us a syste of equatos whch ay be solved. Substtutg (18) (14) we get the sple Approato s of y. The values of s, ad The correspodg absolute errors at 1,, tabulated Table 1. The aalytcal soluto of the dfferetal equato (1.18) wth the codtos s gve by (11.1) s syetrc about the cetral value. The sae aspect s also satsfed by the uercal approatos as s evdet fro the above table. We foud that the approate values are rearkably accurate... Eaple Cosder the followg ffth order lear boudary value proble Subject to y y 19 cos cos 41s y y y s,1 1, y 1 y1cos1, 1y 1 4cos 1 s 1cos18s1. 1) Soluto wth h 1 The sth order sple (19) 1, () s whch approates y s gve by (). The equatos to be satsfed by the coeffcets of the sple fucto are s cos1.4, s cos1.4, s 4cos 1 s , s 4 cos 1 s , (1) s cos 1 8s , 19 cos cos 41s s, s s For,1, We observe that a.4, b , c.4481 Copyrght 1 ScRes.

5 P. KALYANI ET AL. 87 Table 1. Approate solutos ad absolute errors for Eaple 1 wth h =.. X s y Absolute error also sce s s.4,.4 ad 1 the equatos of (1) for,1, reduces to 1, s s s , It follows that we have to detere the ukows de,. g, h, h 1 Equato (), subject to the fve codtos s.4, s , 1, s s s , Fro () s bc d 4e g h h () () s 1g7h 7h (4) Substtutg (), (), (4) () We get the syste of equatos d e4g8h.1 h , 1d e8g19h h , 1g144h 7h , 7h 1g, 1g Solvg these we get d.7947, e.77881, g.8844, h.887, h.1 1 also we have 1 () a.4, b , c.4481 Substtutg all these values Equato () we get the sple approato for y whch s gve by s h.7947h.77881h.844h.887h y s h where h y y () 1, The aalytcal soluto of (19) wth the codtos () s gve by 1 cos y (7) 1 t follows that the ab- s s The eact value of y solute error the uercal approato foud to be.1177 whch s very sall. ) Soluto wth h. The terval 1,1 s dvded to 4 equal subtervals we deote the kots by, 1,,, 4 where 1, 1.,,., 4 1 We assue the sple fucto s whch approates y the for s gve by (1) Fro (1) we have 4 g h h11 h h s are s cos1, s4 cos1, s 4cos 1 s 1, s4 4cos 1 s 1, s cos 1 8s 1, s s 19cos cos 41s s s b c d e The codtos to be satsfed by for,1,,, 4 fro (9) we fd that (8) (9) Copyrght 1 ScRes.

6 88 P. KALYANI ET AL. Table. Approate solutos ad absolute errors for Eaple wth h =.. X s y Absolute error a.4, b , c.4481 usg the reag codtos of (9) the order, s4 cos1, s 4cos 1 s cos cos 41s s, s s for 4,,,1, that s takg the Equatos (1), (1), (8) (9) & by substtutg the values of, 1,,, 4 We get the followg syste of equatos d e4g8h h1.1h.191h , e4g8h h1.1h.191h , 1d e8g19h 4.h1 h.187h , 1g144h 18h17h h 4.48, () 1.87d.1e1.7987g h 7.h1.781h 1.779, 1g7h h1,.d.1e g h 8.71, 1g Solvg (9) we get d.4719, e.14, g.8, h.48, h1.44, h, h.44 (1) Also we have a.4, b , c.4481 Substtutg these values (1) we get the approato s. The values of s, y ad the correspodg absolute errors at,, are etoed Table. 4. Cocluso 1 Nuercal values obtaed by the sple approato have hgh accuracy. It has bee otced that the uercal solutos obtaed are rearkably accurate ad have eglgble percetage errors eve for values of h as large as., 1.. REFERENCES [1] W. G. Bckley, Pecewse Cubc Iterpolato ad Two- Pot Boudary Value Probles, Coputer Joural, Vol. 11, No., 198, pp. -8. do:1.19/cojl/11.. [] E. A. Boquez ad J. D. A. Walker, Fourth Order Fte Dfferece Methods for Two Pot Boudary Value Probles, IMA Joural of Nuercal Aalyss, Vol. 4, No. 1, pp do:1.19/au/4.1.9 [] M. M. A. Chawla, Fourth Order Trdagoal Fte Dfferece Method for Geeral Two Pot Boudary Value Probles wth Med Boudary Codtos, Joural of the Isttute of Matheatcs ad Its Applcatos, Vol. 1, No. 1, 1978, pp do:1.19/aat/1.1.8 [4] P. S. Raa Chadra Rao, Soluto of Fourth Order Boudary Value Probles Usg Sple Fuctos, Ida Joural of Matheatcs ad Matheatcal Sceces, Vol., No. 1,, pp [] P. S. Raa Chadra Rao, Specal Multstep Methods Based o Nuercal Dfferetato for Solvg the Ital Value Proble, Appled Matheatcs ad Coputato, Vol. 181, No. 1,, pp. -1. do:1.11/j.ac..1. [] P. S. Raa Chadra Rao, Soluto of a Class of Bodary Value Probles Usg Nuercal Itegrato, Ida Joural of Matheatcs ad Matheatcal Sceces, Vol., No.,, pp [7] P. S. Raa Chadra Rao, Soluto of Ital Value Probles by Spectal Multstep Methods, Ida Joural Matheatcs ad Matheatcal Sceces, Vol.. No.,, pp Copyrght 1 ScRes.