Cubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem
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1 Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs ad Physcs, Faculty of Egeerg, Tata Uversty, Tata, Egypt Departmet of Egeerg athematcs ad Physcs, Faculty of Egeerg, Beha Uversty, Shoubra, Caro, Egypt. * Zahmed_@yahoo.com Abstract: Thrd ad fourth order coverget s based o cubc opolyomal sple fucto at mdkotes are preseted for the umercal soluto of a secod order two-pot boudary value problem wth Neuma codtos. Usg ths sple fucto a few cosstecy relatos are derved for computg approxmatos to the soluto of the problem. Covergece aalyss of these s s dscussed two umercal examples are gve to llustrate practcal usefuless of the ew s. [Joural of Amerca Scece. ;6(:97-]. (ISSN: Keywords: Cubc opolyomal sple; two-pot boudary value problem; Neuma boudary codtos.. Itroducto: I approxmato theory sple fuctos occupy a mportat posto havg a umber of applcatos, especally the umercal soluto of boudary-value problems. We shall cosder a umercal soluto of the followg lear secod order two-pot boudary value problem, see [5]. ( y + f ( x y g ( x, x [ a, b] (. Subject to Neuma boudary codtos: y ( a A y ( b A (. Where A,, are fte real costats. The fuctos f(x ad g(x are cotuous o the terval [a,b]. The aalytcal soluto of (. subjected to (. caot be obtaed for arbtrary choces of f(x ad g(x. The umercal aalyss lterature cotas lttle o the soluto of secod order two-pot boudary value problem (. subjected to Neuma boudary codtos (..whle The lear secod order two-pot boudary value problem (. subjected to Drchlet boudary codtos solved by dfferet types of sple fuctos, see [, 7, 8, 9]. Ramada et al. [5] solved the problem (. subjected to (. usg quadratc polyomal sple, cubc polyomal sple ad quadratc opolyomal sple at mdkotes. I ths paper, we develop cubc opolyomal sple at mdkotes to get smooth approxmatos for the soluto of the problem (. subjected to Neuma boudary codtos (... Dervato of the : We troduce a fte set of grd pots x by dvdg the terval [a, b] to equal parts. x a + h,,,, ba x a, x b ad h (. Let y(x be the exact soluto of the system (. ad (. ad S be a approxmato to y y(x obtaed by the sple fucto Q (x passg through the pots (x, s ad (x +, s +. Each opolyomal sple segmet Q (x has the form. Q x a s k x x + b cos k x x + c x x + d,,,... (. Where a, b, c ad d are costats ad k s the frequecy of the trgoometrc fuctos whch wll be used to rase the accuracy of the ad equato (. reduces to cubc polyomal sple fucto [a,b]whe k, Choosg the sple fucto ths form wll eable us to geeralze other exstg s by arbtrary choces of the parameters α ad β whch wll be defed later at the ed of ths secto. Thus, our cubc opolyomal sple s ow defed by the relatos: [ ] + S ( x C [ a, b] S x Q x, x x, x,,,..., ( edtor@amercascece.org
2 Joural of Amerca Scece ;6( The four coeffcets (. eed to be obtaed terms of S, D,, T a d T Q x S + + ( Where Q x D (.4 Q x + + ( v Q ( x [ T + T + ] We obta va a straghtforward calculato taθ secθ a [ T + T ], b [ T + T ] k k k h h C D + T + T d S + D T + T k k 4k [ ], [ ] (.5 Where θ kh ad,, - Now usg the cotuty codtos ( ad (., that s the cotuty of cubc opolyomal sple S(x ad ts frst ad secod dervates at the pot (x, s, where the two cubcs Q - (x ad Q (x jo, we ca have ( m ( m Q x Q x, m,, (.6 Usg Eqs. (., (.4, (.5 ad (.6 yeld the relatos: h D + D S sec cos sec S θ θ + + θ + + k k taθ h + ( T + T + T+ (.7 k 4k [ ] h h s θ s e cθ [ D ] + D (.8 k taθ secθ cosθ secθ ( T + T + T k k k ( From Eqs. (.7 (.9 we get the followg relato: ( α β α S S + S h + +,,,..., + + (. Where θ sθ θ s θ + 4sθ θ( + cosθ α β θ sθ θ sθ f S + g w th f f x ad g g x The relato (. gves (- lear algebrac equatos the ( ukows S +½,,,,. -, so we eed two more equatos, oe at each ed of the rage of tegrato for drect computato of S +½. These two equatos are deduced by Taylor seres ad the of udetermed coeffcets. These equatos are hs S + S h w + w + w + w at ( ( 5 7 ( ( 5 7 (. S S + hs h w + w + w + w, at (. Where w 's wll be determed later to get the requred order of accuracy. The local trucato errors t,,,.. assocated wth the scheme (. (. ca be obtaed as follows, we rewrte the scheme(. (. the form y + y h w y + w y + w y + w y + t, at ( ( ( ( ( 5 7 ( α β α ( ( ( + + (. y y + y + y + y + t, at,,..., ( (.4 y y + hy h w y + w y + w y + w y + t at ( ( ( ( (, 5 7 (.5 ( The terms y ad y Eq. (.4 are expaded aroud the pot x usg Taylor seres ad the expressos for t,, - ca be obtaed. Also, expressos for t ;, are obtaed by expadg Eqs. (. ad (.5 aroud 98 edtor@amercascece.org
3 Joural of Amerca Scece ;6( the pot x ad x, respectvely, usg Taylor seres ad the expressos for t ;, ca be obtaed as ( w + w + 5w + 7w ( 6 ( [ ( w+ w+ w+ w ] ( w + 9w + 5 w + 49 w 4 ( w + 7 w+ 5 w + 4 w ( w+ 8 w+ 65 w+ 4 w h 6 y (6 7 t + + O ( h ;, ( (4 5 (5 ( β ( 5 5α β 4 (4 ( α β + α α β 5 (5 9 8α β 6 ( O( h ;, The scheme (. (. gves rse to a famly of s of dfferet orders as follows: For α ad β. Thrd order For (w, w, w, w (4, -,, /d where d 4 The the local trucato errors gve by equato (.6 are t 47 5 (5 6 + O ( h,, ( 6 7 O ( h,,,..., 4 + (.7. Fourth order For (w, w, w, w (67, -98, 98, -47/d where d 576 The the local trucato errors gve by equato (.6 are 6 (6 7 + O( h,, 576 t 6 (6 7 O( h,,,..., 4 + (.8 Remark ( Whe 6 α ad β, the the 8 8 scheme (. s reduced to quadratc polyomal sple [, ]. ( Whe α ad β, the the 4 4 scheme (. s reduced to cubc polyomal sple [4] ( Whe α ad β, we get ew scheme that produces umercal results better tha both quadratc ad cubc polyomal sples [,, 4].. Sple solutos: The sple soluto of (. wth the boudary codto (. s based o the lear equatos gve by (. (., let Y y, S S, C ( C T ( t E e Y S + +, + Be -dmesoal colum vectors, the we ca wrte the stadard matrx equatos for the opolyomal sple the form. ( NY C + T ( NS C (. ( NE T We also have N N + h BF; F dag f (. + The three bad symmetrc matrx N has the form: N O The matrx B has the form: w w w w α β α α β α B O α β α α β α w w w w For the vector C, we have (. ( edtor@amercascece.org
4 Joural of Amerca Scece ;6( ha+ h w g + w g + w g 5 + w g 7, C h α g β g α g + +,,,..., + ha h w g w g w g 5 w g , (.5 Set N + J (.6 Where O J 4. Covergece aalyss o E (.7 (.8 Our ma purpose ow s to derve a boud. We ow tur back to the error equato ( (. ad rewrte t the form E N T Ths mples that ( + J + h BF T ( I+ ( J + h BF T ( E I + J + h B F T I order to derve the boud o followg two lemmas are eeded. E (4., the Lemma 4. ([]. If G s a square matrx of order ad G <, the ( I + G exsts ad ( I + G < G Lemma 4.; the matrx ( J h BF + s + osgular f h w f < where h w( β + α w ( b a + h 8 Proof. Sce, ( N + J + h BF I + J + h B F ad the matrx s osgular, so to prove N osgular t s suffcet to show ( I + ( J + h BF osgular. oreover, F f m a x a x b f ( x (4. h ( b a + h, see [ 6] (4. 8 J (4.4 Also, B α + β (4.5 ( +.( + J h B F J h B F (4.6 Therefore, substtutg F,, J a d B (4.6 we get ( α β h ( J + h BF ( b a + h + h + f 8 (4.7 Sce, h w f < (4.8 h w( β + α Therefore, Eq. (4.8 leads to J + h B F (4.9 From Lemma 4., t shows that the matrx N s osgular. Sce, ( J h B F + < so usg Lemma (4.ad Eq. (4. follow that T E (4. J + h BF From Eq. (.7 we have 47 5 ( 5 T h 5 ; 5 m ax a x b y x 576 The T E O( h J + h BF (4. Also, from Eq. (.8 we have 6 ( 6 T h 6 ; 6 m ax a x b y ( x edtor@amercascece.org
5 Joural of Amerca Scece ;6( The E J T + h BF O 4 ( h (4. We summarze the above results the ext theorem. Theorem 4. Let y(x s the exact soluto of the cotuous boudary value problem (. wth the boudary codto (. ad let,,,..., satsfes the dscrete y + boudary value problem ( (.. Further, f e y S the E O( h 4 - E O( h, for thrd order coverget, for fourth order coverget Whch are gve by (4. ad (4., eglectg all errors due to roud off. 5. Numercal examples ad dscusso: We ow cosder two umercal examples llustratg the comparatve performace of cubc opolyomal sple ( (. over quadratc opolyomal sple ad the two polyomal sple s (quadratc ad cubc. All calculatos are mplemeted by ATLAB 7 Example Cosder the boudary value problem, see [5] y + y (5. c o s ( y y s The aalytcal soluto of (5. s cos y x cos x + s ( x (5. s Example Cosder the boudary value problem, see [5] y + xy x x + x s x + 4x cos x (5. ( y, y s The aalytcal soluto of (5. s y( x ( x s ( x (5.4 The umercal results of examples ad are preseted tables ad, respectvely, for our fourth order. A comparso betwee the (. ad the exstg s Ramada et al. [5] are provded tables ad 4. Table : Approxmate, Exact Solutos ad axmum errors ( absolute value for Example usg our fourth order. S (approxmated y (Exact E (Error a a * -5 Table : Approxmate, Exact Solutos ad axmum errors ( absolute value for Example usg our fourth order. S (approxmated y (Exact E (Error Table : axmum errors ( absolute value for Example. Our fourth order Our thrd order Quadratc opoly. [5] Cubc poly. [5] Quadratc poly. [5] edtor@amercascece.org
6 Joural of Amerca Scece ;6( Table 4: axmum errors ( absolute value for Example. Our fourth order Our thrd order Quadratc opoly. [5] Cubc poly. [5] Quadratc poly. [5] Cocluso: Two ew s are preseted for solvg secod order two-pot boudary value problem wth Neuma codtos. These s are show to be optmal thrd ad optmal fourth orders whch are better tha the two polyomal sple s (quadratc ad cubc sples ad quadratc opolyomal sple. oreover, opolyomal sple has less computatoal cost over other polyomal sple s. The obtaed umercal results show that the proposed s mata a remarkable hgh accuracy whch make them are very ecouragg over other exstg s. Correspodg author Z.A. ZAk * Departmet of Egeerg athematcs ad Physcs, Faculty of Egeerg, Beha Uversty, Shoubra, Caro, Egypt. Zahmed_@yahoo.com 7. Refereces. A. Kha, Parametrc cubc sple soluto of two-pot boudary value problems, Appled athematcs ad Computatos 54 ( E.A.Al-Sad, Sple solutos for system of secod order boudary-value problems, t. J. Comput. ath 6 ( I. Hossam, S.Z. Sakr ad W.K. Zahra, Quadratc sple soluto to two pot boudary value problems, Delta Joural of Scece, 7 ( E.A. Al-Sad, The use of cubc sples s the umercal soluto of a system of secod-order boudary value problems. Comput. ath. Applcs 4 ( A. Ramada, I. F. Lashe, W.K. Zahra, Polyomal ad o-polyomal sple approaches to the umercal soluto of secod-order boudary-value problems, Appled athematcs ad Computatos 84 ( D. urray, Explct verses of Toepltz ad Assocated atrces, Azam J. 44, 85-5,. 7. W.K. Zahra, Numercal solutos for boudary value problems for ordary dfferetal equatos by usg sple fuctos,.sc. thess, Faculty of Egeerg, Tata Uversty, Egypt, J.L. Blue, Sple fucto s for olear boudary value problems, commucatos of the AC ( C.V. Raghavarao, Y.V.S.S. Sayasraju, S. Suresh, A ote o applcato of cubc sples to two-pot boudary value problems, Comput. ath. Applcs 7( ( R.A. Usma, Dscrete s for a boudary value problem wth egeerg applcatos, ath. Comput. (44 ( // edtor@amercascece.org
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