Cubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two

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1 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 ubc rgonometrc B-Splne Appled to Lnear wo-pont Boundary Value Problems of Order wo Nur Nadah Abd Hamd Ahmad Abd Majd and Ahmad Izan Md Ismal Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 Abstract Lnear two-pont boundary value problems of order two are solved usng cubc trgonometrc B-splne nterpolaton method (BIM) ubc trgonometrc B-splne s a pecewse functon consstng of trgonometrc equatons hs method s tested on some problems and the results are compared wth cubc B-splne nterpolaton method (BIM) from the lterature BIM s found to approxmate the soluton slghtly more accurately than BIM f the problems are trgonometrc Keywords trgonometrc B-splne two-pont boundary value problem splne nterpolaton cubc splne I INRODUION OUNDARY value problems are abundant n the feld of Bphyscs chemstry and engneerng hs paper consders the smplest form of boundary value problems whch s lnear two-pont boundary value problems of order two as n () = x [ a b] u( a) = α u( b) = β u'' x p x u' x q x u x r x 0 [ ] and q( x ) < 0 on [ ab ] p qr ab () Moreover another smplfed form of the problems can be wrtten as n () hs form can also be transformed nto () ( p( x) u ( x) ) r( x) [ ] ' ' = x a b u a = u b = 0 [ ] and p( x ) > 0 on [ ab ] p r a b () he contnuty negatvty and postvty condtons are necessary for the exstence and unqueness of the problems [- ] hese problems are consdered because the methods proposed n ths paper are prototypes Hence t s wse to test them on the most drect form of the problems N N Abd Hamd s wth the School of Mathematcal Scences Unverst Sans Malaysa Penang 800 Malaysa (60-659; e-mal: nurnadah_abdhamd@yahoocom) A Abd Majd s wth the School of Mathematcal Scences Unverst Sans Malaysa Penang 800 Malaysa (e-mal: majd@csusmmy) A I Md Ismal s wth the School of Mathematcal Scences Unverst Sans Malaysa Penang 800 Malaysa (e-mal: zan@csusmmy) he use of monomal cubc splne n solvng these problems was frst explored by Bckley n 968 [] Hs work was further explored n [5 6] n the followng year Followng these development many other analyss and mprovements were made throughout the years as n [7 8] and the references theren However only n 006 the use of cubc B-splne a better representaton than the monomal cubc splnes was suggested by aglar et al hs method was called cubc B- splne nterpolaton method (BIM) [] ontnung wth ths work we appled the same procedure usng another type of splnes called trgonometrc B-splne rgonometrc splne functon was frst ntroduced by I J Schoenberg n 96 In ths work he proved the exstence of trgonometrc B-splne functon a locally supported trgonometrc splne functon As the name suggests trgonometrc B-splne s constructed from trgonometrc functons as opposed to polynomal functons n the case of B-splne he dervaton and propertes of ths functon are dscussed n [9-] II UBI RIGONOMERI B-SPLINE BASIS he bass of trgonometrc B-splne of order s shown n the followng formula x [ x x ) ( x) = 0 otherwse From there trgonometrc B-splne bass of order k = can be calculated usng the recursve equaton n () alculatng up to degree where k = the resultng bass x s shown n () [] k x x sn k = x x k sn x x x x k sn x x k sn k ( x) () Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 77

2 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 where ( x) = σ ( x ) x [ x x ] σ( x ) ς( x ) σ ( x ) ς( x ) σ( x ) x [ x x ] ς( x ) σ ( x ) ( x ) ( x ) σ( x ) ς( x ) x [ x x ] ς ( x ) ς( x ) σ( x ) ς ( x ) x [ x x ] θ σ ς σ x x x x ( x ) = ς( x ) = sn sn h h ( h) θ = sn sn sn () x s a pecewse trgonometrc funton of degree wth contnuty Plots of cubc trgonometrc B-splne bass together wth cubc B-splne bass s shown n Fg Fg : ubc trgonometrc B-splne bass ( x) and cubc B- splne bass B ( x ) Some propertes of cubc trgonometrc B-splne bass arelsted below: x s nonnegatve () () ( x) 0 when x [ x x ] () ( x ) = and zero otherwse III UBI RIGONOMERI B-SPLINE ubc trgonometrc B-splne bass can be manpulated to generate a pecewse functon called cubc trgonometrc B- splne ubc trgonometrc B-splne S ( x ) s a lnear combnaton of the cubc trgonometrc B-splne bass as shown n (5) n S ( x) = ( x) x [ x x ] n (5) 0 n = herefore smlar to ( x ) S x s also a pecewse trgonometrc functon of degree wth contnuty Here can be any real number and hence wll be manpulated n solvng the boundary value problems From property () of the bass functon t can be observed x x there are only four nonzero that wthn nterval [ ] bass functons namely ( x) ( x) ( x) ( x ) hus for * [ ] x x x and ( *) = ( *) ( *) ( *) ( x* ) ς ( x ) σ( x ) ς ( x ) σ( x ) ς( x ) ς ( x ) ς( x ) σ( x ) = θ σ( x ) ς( x ) σ ( x ) ς( x ) σ( x ) ς ( x ) σ ( x ) σ ( x ) S x x x x Snce ( x ) = 0 S ( x ) = ( x ) ( x ) ( x ) h sn h = h cos( h) h h sn ( h) (6) Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 78

3 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 By takng the frst and second dervatve of (5) and evaluatng at x we have (7) and (8) h = [] S ' x 0 S '' x = h ( cos) h h h h 6 cos cos h cot cos( h) he smplfcatons of S ( cos) h h h h 6 cos cos (7) x and ts dervatves at x are very useful n solvng the problems IV UBI RIGONOMERI B-SPLINE INERPOLAION MEHOD (BIM) In order to solve the problem cubc trgonometrc B-splne ab s frst presumed to be the soluton of () S ( x ) on [ ] Substtutng S x nto () we have = [ ] = α = β S '' x p x S ' x q x S x r x x a b S a S b hen evaluatng (9) at x results = S '' x p x S ' x q x S x r x = 0 n (9) (8) (0) ( ) h cos h h h 6 cos cos h cot cos( h) h ( cos( h) ) h h 6 cos cos h [] 0 h p x q x h h sn h cos( h) h h sn ( h) = r x hus (0) can be smplfed nto an expresson consstng only of and ollectng these terms we have (0) comprses of S ( x ) S '( x ) and S '' x whch are already smplfed n prevous secton herefore substtutng (6) (7) and (8) nto (0) we have Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 79

4 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 h cos h h h 6 cos cos h p( x ) h h q( x ) sn ( h) h cot cos( h) q( x ) cos( h) h cos h h h 6 cos cos h p( x ) h h q( x ) sn ( h) = r x Smlarly the boundary condtons can be smplfed nto () and () = 0 h h = sn h cos( h) h h sn ( h) S a S x = α () = h sn h = h n S b S x n = β n n cos ( h) h h sn ( h) () () and () can be arranged n a matrx equaton of the form [ ] [ ] [ ] n n n ( n ) () A = R () where the frst and last lnes of A are the boundary condtons from () and () respectvely whereas the rest are from () () s a lnear system of order ( n ) ( n ) where s the unknown vector Hence can be solved by takng - =A R (5) Lastly the obtaned values of for = n are substtuted n (5) whch becomes the approxmated analytcal soluton to () V NUMERIAL EXAMPLES AND DISUSSIONS () BIM was mplemented on Problems and 5 wth n = 0 herefore there are coeffcents that are needed to be solved he problems and ther 9 respectve exact solutons are as the followng: Problem 5 [] d x du x e = e dx dx x [ 0 ] u( 0) = 0 u( ) = 0 x [ ] u'' x u' x = e x 0 u 0 = 0 u = 0 x Exact soluton: u( x) = x( e ) Problem 5 [] ( ) = ( ) x u'' x x u' x u x x e x [ ] u u Exact soluton: u( x) = ( x ) e x 0 0 = = 0 Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 80

5 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 Problem 5 [] u'' x π u x = π sn πx x [ ] u = u = Exact soluton: u( x) = sn ( π x) Problem 5 [] u'' x u x = 0 x 0 u 0 = 0 u = snh [ ] Exact soluton: u( x) = snh ( x) Maxmum-norm and L -norm are used to gauge the accuracy of the method Suppose u( x ) and S ( x ) are the exact and approxmate solutons of () respectvely Maxmum-norm or max-norm measures the upper bound of the absolute error and s calculated usng the followng formula Max-Norm = S( x ) u( x ) = S( x ) u( x ) max hus max-norm reports the largest error of the approxmated soluton L -norm measures the dstrbuton of the absolute error and s calculated usng the followng formula L -norm = S( x ) u( x ) = [ S( x ) u( x )] able shows the max-norms and L -norms for each problem usng BIM compared to BIM ABLE NORMS FOR PROBLEMS AND 5 USING BIM AND BIM Problem Method Max-Norm L -Norm 5 BIM 8996E E-0 BIM 68895E E-0 5 BIM 08E-0 50E-0 BIM 5790E-0 898E-0 5 BIM 088E E-0 BIM 096E-0 69E-0 5 BIM 50E-05 79E-0 BIM 78E-0 80E-0 BIM dd not approxmate the soluton better for Problems 5 5 and 5 but dd so for Problem 5 hs mght be due to the trgonometrc equatons appearng n the dfferental equaton and exact soluton herefore three more problems of trgonometrc nature were tested usng BIM and BIM hey are Problems and 57 he results are shown n able Problem 55 [] π π u'' x u x = 0 x 0 u 0 = u = rue soluton: u( x) = cos( x) ( ) sn ( x) Problem 56[] π u'' ( x) u' ( x) u( x) = cos ( x) x 0 π u( 0) = 0 u = 0 rue soluton: u( x) = [ sn ( x) cos ( x) ] 0 Problem 57 [] π π u'' x u x = cos x x 0 u 0 = 0 u = 0 rue soluton: u( x) = cos( x) sn ( x) cos x 6 ABLE II MAX-NORM AND L -NORM VALUES FOR PROBLEMS AND 57 USING BIM AND BIM Problem Method Max-Norm L -Norm 5 BIM 5E-05 00E-0 BIM 56E E-05 6 BIM 88E E-0 BIM 05E-0 5E-0 7 BIM 078E E-0 BIM 8E-0 778E-0 For all the problems BIM gave slghtly better approxmatons than that of BIM Hence t s safe to say that BIM approxmates lnear two-pont boundary value problems better for problems nvolvng trgonometrc expressons VI ONLUSIONS Referrng to our prevous work n [] we used extended cubc B-splne n place of B-splne Extended cubc B-splne s an mproved verson of B-splne where one free parameter s added to the bass functon [] he results of BIM BIM and extended cubc B-splne nterpolaton method (EBIM) are shown n able From the table t s obvous that EBIM approxmates the soluton a lot better than BIM and BIM But BIM produced more accurate results compared to BIM f the Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 8

6 World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 Dgtal Open Scence Index Mathematcal and omputatonal Scences Vol: No:0 00 wasetorg/publcaton/90 problems were trgonometrc hus f cubc trgonometrc B- splne wth a free parameter s avalable further tests can be done on the problems and our hypothess would be such splne would approxmate the soluton to Problem 5 better than EBIM ABLE III ERROR FOR PROBLEMS AND 5 USING BIM BIM AND EBIM Problem Method Max-Norm L -Norm 5 BIM 8996E E-0 BIM 68895E E-0 EBIM 5E E-06 5 BIM 08E-0 50E-0 BIM 5790E-0 898E-0 EBIM 00E-06 68E-06 5 BIM 088E E-0 BIM 096E-0 69E-0 EBIM 07E E-09 5 BIM 50E-05 79E-0 BIM 78E-0 80E-0 EBIM 507E-0 795E-0 REFERENES [] R L Burden and J D Fares Numercal Analyss Belmont A: Brooks/ole th ed pp [] R P Agarwal Boundary Value Problems for Hgher Order Dfferental Equatons) Sngapore: World Scentfc Publshn o Pte Ltd 986 pp 89-0 [] H aglar N aglar and K Elfatur B-splne nterpolaton compared wth fnte dfference fnte element and fnte volume methods whch appled to two-pont boundary value problems Appled Mathematcs and omputaton vol 75 ssue pp [] W G Bckley Pecewse cubc nterpolaton and two-pont boundary problems he omputer Journal vol ssue pp [5] E L Albasny and W D Hoskns ubc splne solutons to two-pont boundary value problems he omputer Journal vol ssue pp [6] D J Fyfe he use of cubc splnes n the soluton of two-pont boundary value problems he omputer Journal vol ssue pp [7] E Al-Sad ubc splne method for solvng two-pont boundary value problems Journal of Appled Mathematcs and omputng vol 5 ssue pp [8] A Khan Parametrc cubc splne soluton of two pont boundary value problems Appled Mathematcs and omputaton vol 5 ssue pp [9] P Koch Lyche M Neamtu and L Schumaker ontrol curves and knot nserton for trgonometrc splnes Advances n omputatonal Mathematcs vol ssue pp [0] A Nkols Numercal solutons of ordnary dfferental equatons wth quadratc trgonometrc splnes Appled Mathematcs E-Notes vol pp [] G Walz Identtes for trgonometrc B-splnes wth an applcaton to curve desgn BI Numercal Mathematcs vol 7 ssue pp [] N S Asathamb Numercal Analyss heory and Practce Orlando FL: Sauders ollege Publshng 995 pp [] N N Abd Hamd A Abd Majd A I Md Ismal Extended cubc B- splne nterpolaton method appled to lnear two-pont boundary value problems World Academy of Scence Engneerng and echnology (WASE) vol 6 pp February 00 [] G Xu and G-Z Wang Extended cubc unform B-splne and [alpha]- B-splne Acta Automatca Snca vol ssue 8 pp Internatonal Scholarly and Scentfc Research & Innovaton (0) 00 8