Comparison between Fourier and Corrected Fourier Series Methods

Transcription

1 Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1 Nor Hafizah bii Zaial ad 1,* Adem Kilicma 1 Deparme of Mahemaics, Faculy of Sciece, Uiversii Pura Malaysia, 434 UPM Serdag, Selagor, Malaysia Isiue for Mahemaical Research, Uiversiy Pura Malaysia, 434 UPM Serdag, Selagor, Malaysia orzaial_89@yahoo.com ad akilicma@pura.upm.edu.my *Correspodig auhor ABSTRACT Fourier series is a mehod ha ca solve for may problems especially for solvig various differeial equaios of ieres i sciece ad egieerig. However, Gibbs oscillaios will be occurs whe usig a rucaed Fourier series. Thus, we solve he problems by usig he correced Fourier series. Here, we wa o compare he resuls bewee he soluios ha we ge from Fourier series mehod ad Correced Fourier series mehod. The compariso bewee hese wo mehods will be our fidig. Keywords: Fourier series, correced fourier series, Gibbs oscillaio. 1. INTRODUCTION A ordiary differeial equaio (ODE) ha coais oe or more derivaives of a ukow fucio, which we call y( ) for is he idepede variable. A ODE is said o be of order if he h derivaive of he ukow fucio y is he highes derivaive of y i he equaio. We ca wrie hem as F( y, y ) = (1) A parial differeial equaio (PDE) is a mahemaical equaio havig parial derivaives wih respec o more ha oe variable. Cosider a geeral form of he secod-order liear parial differeial equaio as

2 Nor Hafizah bii Zaial & Adem Kilicma u u u u u L a( y) b( y) c( y) d( y) e( y) f ( y) u g( y) = y y y () The we ofe eed o specify some supplemeary codiios, which called boudary value codiios or iiial value codiios o solve he problem. Bu, by applied mehod of Galerki wih correced Fourier series as is basis fucio, we do o eed o ofe specify ay supplemeary codiios. The fucio approimaio is eeded o elimiae he Gibbs pheomeo ha occurs whe usig a rucaed Fourier series or oher eigefucio series a a simple discoiuiy.. FOURIER AND CORRECTED FOURIER SERIES METHOD Fourier series decomposes a periodic fucio io a sum of simple oscillaig fucios, amely sies ad cosies. Fourier series are very impora o he egieer ad physicis because hey allow he soluio of ODEs i coecio wih forced oscillaios ad he approimaio of periodic fucios, (Kreyszig (11)). Meawhile, he correced Fourier series is a combiaio of he uiformly coverge Fourier series ad he correcio fucio cosiss of algebraic polyomials ad Heaviside sep fucios ad is required by he aperiodiciy a he edpois (i.e., f () f ( ) ) ad he fiie discoiuiies i bewee, (Zhag (7))..1 Fourier series To defie Fourier series, we iroduce a fucio f () as a periodic fucio. Suppose ha f () is a give fucio of period π, he = 1 ( a cos b ) f ( ) = a si where he Fourier coefficies of f () give by he Euler formulas (3) 74 Malaysia Joural of Mahemaical Scieces

3 Compariso bewee Fourier ad Correced Fourier Series Mehods 1 π 1 π a = π f ( ) d, a = π π f ( ) cos d π 1 π b = π f ( ) si d π,. Correced Fourier series Suppose y () is ay mh quasi-smooh fucio, ad has possible discoiuiies a ( = 1,,, J ). The heorem of he correced Fourier series i (Zhag (7)) saes ha a mh quasi-smooh fucio ca be approimaed uiformly by a correced Fourier series cosisig of hree pars, which are a mh uiformly coverge Fourier series, a o-moreha (m1) h-order polyomial, ad a mh iegral of he Heaviside sep fucios a he discoiuiies. Therefore, y() is approimaed by he followig correced Fourier series, y( ) = ( ) m 1 l m iα Ae al b H ( ), < l= 1 l! m! π α = (4) where is A is he Fourier proecio of y( ) o he basis fucio i he ierval, ]. [ A = 1 y( ) e iα d i e α, ha (5) 3. CORRECTED FOURIER SERIES METHOD FOR SOLVING TWO UNKNOWNS PROBLEM A firs, we cosider he geeral form of he secod order liear PDEs i he regio [, ] [, ], Malaysia Joural of Mahemaical Scieces 75

4 Nor Hafizah bii Zaial & Adem Kilicma u( u( u( p1 ( p( p3( L { u( } = f ( u( u( p4( p5( p6( u( (6) where p l ( wih l = 1,,, 6 ad f ( are quasi-smooh fucios wih wo ukows havig he discoiuiies a ( = 1,,, J ) ad ( = 1,,, J ). Defiiio 1: Ay fucio, φ( o he basis fucio regio [, ] [, ], geerally wrie as ( β m ) i he i e α 1 i( α βm F φ( φ( e dd m = (7) π mπ is called he Fourier proecio where α = ad β m =. Lemma 1. Assume uɶ ( is he soluio of equaio (6). The ecessary ad sufficie codiios for equaio (3) wih uɶ ( o be equivale o is Fourier proecios { } 1 1 m F L u( = F f ( m (8) ha is, uɶ ( saisfies he cosisecy codiios of he edpois ad discoiuiies L{ uɶ ( } = f ( L uɶ ( = f ( wih = 1,,, J, ad { } = 1,,, J where ad eiher pl ( ) ad f ( ) ad deoe he fiie discoiuiies i 76 Malaysia Joural of Mahemaical Scieces

5 Compariso bewee Fourier ad Correced Fourier Series Mehods f ( f (, f (, ( f (, ) f (, )) ( f (,) f (,)) f (, ) f (,) f (, ) f (,). (9) L ~ is a periodic, quasismooh coiuous fucio whose Fourier series is uiformly coverge L uɶ ( f ( is equivale o Saisfyig he cosisecy codiios { u( } f ( ) wihou Gibbs oscillaio. This meas ha { } is Fourier series. Thus Equaio (6) Equaio (8). I order u( ) o be a soluio of equaio (6): i. u( ad is firs derivaive, ad ii. The secod derivaive of u(, u( ad cao have ay discoiuiies oher ha u( mus be coiuous, u( u(, ad. ad u( Therefore, u( ) mus be secod quasi-smooh fucio wih wo variables ( ad ) ad has possible discoiuiies a ( = 1,,, J ) ad ( = 1,,, J ). 3.1 Derivaive of he correced Fourier series Accordig o (Zhag (7)), Theorem. sae ha ay m -h quasismooh coiuous fucio ca be approimaed uiformly by he sum of a m -h uiformly coverge Fourier series ad a polyomial o more ha ( m 1) -h order. This heorem has bee proof for oe ukow. Now, we eed his heorem o he case wih wo ukows. Theorem 1 Ay m -h d quasi-smooh coiuous fucio um ( Sm ([, ],[, ]) ca be approimaed uiformly by he sum of a m -h uiformly coverge Fourier series ad a polyomial o more ha ( m 1) -h order Malaysia Joural of Mahemaical Scieces 77

6 Nor Hafizah bii Zaial & Adem Kilicma u( = A e a a a e 3 i( α βm ) iβm m 1m m 3m < m < m <! 3! l l iα b1 b b3 e dll <! 3! l = 1 l = 1 l! l! (1) Proof A firs, he fucio u( ) is a secod quasi-smooh coiuous fucio wih respec o. Therefore, for ay [, ], u( = A ( e a ( a ( a ( 3 i α (11) 1 3 <! 3! I he above equaio, A ( ) ad al ( ) are secod quasi-smooh coiuous fucio wih respec o. I equaio (11), A ( ad al ( where ( l = 1,,3) ca be furher epaded io he followig A ( = A m < al ( = a lm m < π mπ where α = ad β m =.! 3 3! iβ m me b1 b b, 3 3 iβ m e dl1 dl d, ( l = 1,,3 ); (1) l3! 3! The, by subsiuig A ( ) ad al ( ) io equaio (8), we have u( = A e a a a e 3 i( α βm ) iβm m 1m m 3m < m < m <! 3! l l iα b1 b b3 e dll <! 3! l = 1 l = 1 l! l!. (13) As i he ay oher Galerki mehods, he correced Fourier series will be rucaed so ha N ad m M here afer. I Equaio (13), ie 78 Malaysia Joural of Mahemaical Scieces

7 Compariso bewee Fourier ad Correced Fourier Series Mehods ukows d ll where ( l, l = 1,,3) are obaied by solvig he followig liear equaios l 3 3 l u( d H ( l l, ) = (14) ll l= 1 l = 1 ( l )! ( l )! o where (, =,1,) ad i depeds o he boudary values of u( ) ad i firs ad secod derivaives oly. We ca say here ha he firs hree erms o he righ-had side of he equaio (13) are ideically zero caused by he i periodiciy of eiher e α i m or e β. The we arrage he ie ukows d ll io a vecor ordered as ( d11, d1, d13, d1, d, d3, d31, d3, d33) ad he equaios ordered as whe =, he =,1,, whe = 1, he =,1,, ad =, =,1,, so ha he coefficie mari of he liear equaios is up-riagular ad ca be easily ivered. 3. Compuaio of he coefficies Noe ha whe alm, b l ad d ll are suiable chose, he Gibbs oscillaios which ofe rouble he regular Fourier series mehod a edpois ad discoiuiies are elimiaed. The coefficies A m are readily obaied by he followig Fourier proecio: 3 l 3 l alm F1 bl F l = 1 l! l 1! = l m Am = F u(. m 3 3 l l dll F 1 F l= 1 l = 1 l! l! m (15) Beig looked a i he aoher way, wih coefficies alm ( l = 1,,3), a lm ( l = 1,,3), b ( 1,,3) l l = ad dll ( l, l = 1,,3) ye o be deermied, Equaio (13) represes all possible correced Fourier series soluios of Equaio (3), which are uiformly coverge uil heir secod derivaives. Now we have N 1 ad M 1 orhogoal codiios by applyig Fourier proecio Equaio (7) o Equaio (3) ( N 1 ad M 1) imes, ad J 1, J 1 cosisecy codiios (Lemma), (Zhag (5)). The, u( ) is formally epressed as follows Malaysia Joural of Mahemaical Scieces 79

8 Nor Hafizah bii Zaial & Adem Kilicma u( = c u ( c u ( u ( (16) 1 1 where c 1 ad c are wo cosas. Viewig u 1 ( ad u ( as wo liearly idepede soluios ad u (, ) as he specific soluio, Equaio (16) is a geeral soluios of Equaio (3). No boudary codiios are eplicily iroduced whe Equaio (16) is obaied. I is worhwhile oig ha u (, u ( ad 1 u (, ) are Galerki approimaed soluios by usig he correced Fourier series, (Zhag (7)). 4. NUMERICAL PROCEDURE We solve he problem by usig boh mehod, Fourier ad correced Fourier series mehod. We show he differece bewee hese wo mehods by graph. I problem 1, we cosider he fucio i oe variable. I problem, we are solvig he PDEs problem which is hea equaio. Problem 1 Fid he Fourier series of he fucio f ( ) =. Figure 1 Figure Figure 1 shows he soluio of problem 1 by usig Fourier series while Figure shows he soluio by usig Correced Fourier series. The gree color refers o he fucio of f ( ) =, blue color refers o he fucio series afer we rucaed he series io 3 erms, yellow refers o 5 erms of he series ad red color refers o 1 erms of he series. 8 Malaysia Joural of Mahemaical Scieces

9 Compariso bewee Fourier ad Correced Fourier Series Mehods Problem (PDEs Hea Problem) The Figure 3 shows he soluio of u( ) for f ( ) =, c = 4cm /sec, = 1 cm ad several values of. The Figure 4 shows he soluio of π u( ) for f ( ) = 1si, c = 1.158cm /sec, 8 cm ad several values 8 of. We ca see here ha, by usig Fourier series, he differece bewee graphs wih differece value of is very big raher ha by usig correced Fourier. Problem 3 Figure 3 Figure 4 u u = i regio [,8] [,1] Figure 5 Malaysia Joural of Mahemaical Scieces 81

10 Nor Hafizah bii Zaial & Adem Kilicma I he Figure 5, we se he value of equals o 5. By usig correced Fourier series, ad we rucaed he series for = 1, m = 1, =, m = ad = 3, m = 3. From he graph, we ca see ha here is o differece for he soluio. Thus, o Gibbs oscillaios appear for his soluio. 5. CONCLUSIONS The correced Fourier series (CFS) is free of he Gibbs pheomeo, alhough he quasi-smooh fucio ca be aperiodic ad have discoiuiies i geeral. CFS are used o solve he problem ha have osigular coefficies whe heir eac soluio do o always eis. We have such soluios ha are beig uiformly coverge uil is m-h derivaive i he eire regio of he equaios by usig CFS. The soluios of he problem ha solve by usig correced Fourier series are depeds o he value of uɶ ( ha we assume i he begiig of he calculaios. So, he differece value of uɶ ( will give he differece fial soluio for each problem. REFERENCES Joaha, V.. Gibbs Pheomeo i Specral Models, AT 745. Kreyszig, E. 11. Advaced Egieerig Mahemaics. 1 h Wiley & Sos, Ic. Ed. Joh Zhag, Q. H., Che, S. ad Qu, Y. 5. Correced Fourier series ad is applicaio o fucio approimaio. Ieraioal J. Mah. Mah. Sci. 1: Zhag, Q. H., Che, S., Ma, J. ad Qu, Y. 7. Soluios of liear ordiary differeial equaios wih o-sigular varyig coefficies by usig correced Fourier series. App.Mah. Comp. 187: Malaysia Joural of Mahemaical Scieces