On approximations of trigonometric Fourier series as transformed to alternating series

Transcription

1 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary REEARCH ARTICLE OPE ACCE O aroximatios of trigoometric Fourier series as trasformed to alteratig series Zbigiew Płochoci* Adrew Mioduchowsi** *Istitute of Fudametal Techological Research ul.pawińsiego 5B 0-06 Warsaw Polad ** Uiversity of Alberta Det. of Mechaical Egieerig Edmoto Alberta Caada T6G G8 Abstract. Trigoometric Fourier series are trasformed ito: () alteratig series with terms of fiite sums or () fiite sums of alteratig series. Alteratig series may be aroximated by fiite sums more accurately as comared to their simle trucatio. The accuracy of these aroximatios is discussed. eywords: Fourier series alteratig series aroximatio/trucatio of series MC 00 code: B05 0A5 I. Itroductio Trigoometric Fourier series have may alicatios amog others i solvig some roblems with (artial) differetial euatios esecially cocerig heat coductio roblems (see for examle [] [] [3]). I such cases oe or two of the followig series occur: F( x) acos x F3( x) a cos x (.) ( ) F( x) bsi x F( x) b si x where: () all the series are assumed to be coverget; () the coefficiets a a b b rereset suitable Fourier trasforms of a aalyzed fuctio; (3) ad the further aalysis of the above series is limited to sice (3a) each fuctio may be reseted as a sum of its eve ad odd arts x [0] ( f ( x) [ f ( x) f ( x)] [ f ( x) f ( x)] ) (3b) cosie is a eve fuctio ad sie is a odd oe ad (3c) cosie ad sie are both eriodical fuctios (i some cases this limitatio follows immediately from the meaig of the cosidered roblem). I order to obtai detailed iformatio from the Fourier series (reresetig the solutio of a give roblem) usually it is ecessary to erform some umerical calculatios. Two roblems the arise: () aroximatio of real values of ideedet variable x by ratioal umbers ad () criteria of aroximatio of the series by fiite sums withi a give accuracy. The solutio to the first roblem is ow ad used i all umerical calculatios: each real umber ca be aroximated with arbitrarily assumed accuracy by a ratioal umber i.e. by the ratio / where ad stad for itegers. Therefore i order to determie values of the series give by Es.(.) withi a give accuracy it is sufficiet to determie their values for variable x belogig to a sufficietly dese set of ratioal umbers. I our case it is therefore sufficiet to determie values of the cosidered series for a suitable set of ratioal values of variable (.) x 0 0. Thus i further course the followig series will be cosidered (the argumet x / will ot be otified for simlicity): P a g e

2 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary (.3) a F cos F bsi with ad satisfyig Ie.(.). a F3 cos F b si ( ) The aim of this aer is to solve the secod roblem metioed above for some series of the tye (.3) i.e. to fid ossible geeral aroximatios of such series by fiite sums ad accuracy criteria for these aroximatios. II. imle trasformatio Each of the trigoometric fuctios i Es.(.3) eriodically chage the sig ad reeat its absolute value with icreasig i.e. the followig relatioshis tae lace: cos( ) ( ) cos cos[( ) ] ( ) cos( ) (.) si( ) ( ) si si[( ) ] ( ) si( ). Usig these relatioshis oe may rewrite each series (.3) i the two followig forms: (.) F ( ) a cos 0 cos ( ) a o F3 ( ) a cos 0 F ( ) b si F ( ) si ( ) b o cos ( ) a o b 0 0 si si ( ) b The first form does ot reuire ay additioal assumtios o the cosidered series sice the series is summated term by term without chages of summatio order. The secod form does ot chage the value of the series if it is absolutely covergig because summatio breas the order of summatio of the iitial series. However the trucated secod form is always eual to the trucated first oe (at the same level) because fial sum is ideedet of the order of summatio. Additioally criteria for treatig the first form as a alteratig series are more comlicated ad geerally difficult to satisfy (see Remar i ec.) whereas the secod form has simler structure ad is more coveiet i alicatios. For this reaso the further aalysis will be limited to the secod form oly. The secod form of (.) allows the iitial series to be aroximated by a fiite sum of suitable aroximated alteratig series. This aroximatio will be discussed i ec.. III. O aother ote The secod form of (.) may be esecially useful for series of the tye (3.) A A A 0. The absolute error remaider 0 of trucatio of such a series o level (the absolute value of the trucatio R A ) may be estimated as (see [] [5]) 0 o 5 P a g e

3 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary (3.) R A ad relative trucatio error as (3.3) (aroximated euality holds if A R is sufficietly small as comared to ). IV. Towards alteratig series The secod form of (.) reresets alteratig series (see Aedix) if (sufficiet coditios at a give ad ): is a odd iteger ad the terms Al (with (.) A a A3 a A b A b ) costitute a ositive decreasig seueces (with resect to ): (.) Al Al ( ) 0 l 3 resectively; or is a eve iteger ad the terms Al costitute alteratig decreasig seueces (with resect to ): Al ( ) Al Al Al l 3 (.3) ( ) resectively. Remar : Aalogous coditios may be formulated for the first form of (.). The it will be see that such coditios deed o the structure of the cosidered series ad values of ad ad it is ot always ossible to satisfy them whereas coditios for the secod form of (.) are ideedet of ad. I coclusio: If the above coditios are satisfied the the iitial series of (.3) may be trasformed ito a fiite sum of a alteratig series. V. Aroximatio of Fourier series as trasformed ito alteratig series Usually umerical iformatio from a give series is obtaied through aroximatio by a suitable fiite sum usig the trucated series. I our case these trucated series are give by the secod form of (.) with the secod sums limited to trucatio level istead of. Remar : These trucated series are eual to iitial oes (.3) trucated at level (after the -th term) i.e. with the sums limited to istead of with (5.) ( ). eries (.) may be treated as alteratig series (see Aedix) if the criteria metioed above [see Es. (.)] are satisfied. I the simle aroach such series may be estimated by fiite sums F ad F beig the alteratig series trucated after the -th ad the ( ) -th terms resectively (see Aedix). However a more accurate aroximatio of such series is roosed here (see Aedix). Alyig this roosal the series occurrig i (.) [or i comact form i (. ) (see footote )] may be aroximated as follows: 6 P a g e

4 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary (5.) where ad F F cos ( ) a ( ) a 0 F F si ( ) b ( ) b 0 F3 F3 cos ( ) a ( ) a 0 F F si ( ) b ( ) b 0 are give by Es.(.). The (relative) errors of these aroximatios (as referred to F ) may be geerally estimated as follows: cos a 3 cos a (5.3) F F si b si b F F resectively. These errors are four times smaller tha the estimated geeral trucatio errors. Remar 3: ee Remar A i Aedix. Remar : Of course level of a give aroximatio should be determied accordig to a assumed accuracy of aroximatio. VI. amle comutig As a examle we reset a few results of umerical aalysis to the series: (6.) F cos ex[ t]. After the simle trasformatio reseted i ec. with assumed to be a odd umber it may be rewritte i the form of the followig fiite sum of alteratig series [see E.(.) ]: (6.) F cos ( ) ex[ ( ) t] 0 (the secod form is adoted as more coveiet for aalysis). Remar 5: We will use trucated series therefore the reuiremet of absolutely covergig of series (6.) to be trasformed ito series (6.) is ot imortat (fial sum is ideedet of the summatio order) although series (6.) is absolutely covergig. First of all we shall illustrate the accuracy of aroximatio (5.) for t ad / /0 / / 3/ 9 /0 (with ossible cases of m / m ). We assume that the estimated * 5 relative error of this aroximatio does ot exceed O 5 0 (this corresods to the accuracy of umerical evaluatio i roud to 5 digits). I each articular case level of the aroximatio satisfyig the assumed accuracy is foud [ad corresodig level ( ) ] with corresodig estimated error. For comariso there were comuted also: the exact values of series (6.) the values of trucated series F [ad F F with ( ) ] estimated error ( F F ) / F [ ( F F ) / F with ( ) ]. I additio each alteratig series (6.) is of the tye (3.) with 7 P a g e

5 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary (6.3) 0 ex[ ( ) ] ex[ ( ) ] t t [the ratio of the ( ) -th term to the -th is the greatest oe for 0 trucatio of series (6.) o level may be estimated accordig to (3.3) as (6.) F cos ex[ ( ) t] ]. Therefore the (relative) error of The values of these estimated errors are also icluded i the Tables with the values of for orietatio. All the results of these calculatios are give i Tables ad 3. uitable umbers are give i roud to 5 digit (although calculatios were erformed with suitable higher accuracy i some cases i roud eve to 00 digits if it was ecessary). Table. Table. Table 3. Aedix. O alteratig series ad their aroximatios Alteratig series ( ) is uderstood as a (coverget) series of the tye: (A. ) ( ) c ; c c ; lim c 0 ; 0 the latter relatioshi is the coditio for covergece of the series (accordig to the Leibiz criterio). Remar A: A little more geeral (covergig) series with the coditio c c 0... istead of (A.) may be treated as the sum of a fiite sum ad a alteratig series as defied above. ice If is trucated at a eve level j (i.e. after the -th term) the it may be estimated as follows. ( ) j j j c ad ( ) j c j therefore j (A.a) j j where (A.3) ( ) c ; 0 if is trucated at a odd level j the ( ) c j j j therefore (A.b) j j. ice (A.) ( ) c therefore the absolute error of trucatio of at level caot exceed the last term (the last term of ( ) j j j c ad ). The relative error of such a trucatio (as referred to the smaller mi [ ( ) ] c ) may be estimated as follows: ˆ c. (A.5) mi c tae ito accout i.e to 8 P a g e

6 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary However may be aroximated a little more exactly (ad therefore also the aroximatio error may be estimated a little more recisely as comared to the trucatio error). I order to fid such a geeral aroximatio we divide as follows: (A.6) or i details (see Fig.A.) (A.6 ) Fig.A.. ( ) ( ) ( ) 0 R R c R R c c... j j j j j R R c c.... j j j j j3 ice of (A.) 3 the differeces of the two eighborig terms of costitute a ositive mootoically decreasig seuece ad therefore geerally (A.7) R R for each. Taig ito accout (A.) ad (A.) oe may coclude that to obtai better aroximatio of (as comared to ) it is sufficiet to add/subtract a fractio of c to/from i the case of odd/eve resectively (see Fig.A.). Thus oe has to cosider the followig aroximatio: x ( ) xc ( x)( ) c 0 x. (A.8) The (absolute) error of such a aroximatio is x x (A.9 ) R ( x) c xc where (A.8) (A.6) with (A.) ad the estimatio (A.0) ( ) m m m R c c c are used. ice of (A.7) ad (A.) the followig relatioshis tae lace: R (A.) c ad therefore the uatity (A.) x R ( x) c c has the lowest uer limitatio i the whole domai of Fig.A.. Thus the best geeral aroximatio of is (A.3) 3 ( ) c ( ) c R [(see (A.)] for x (see Fig.A.). c or i details 3 j j c j j c j (A.3 ) j 3 j j c j j c j j. The (absolute) error of this aroximatio (A.) 3 3 R c c c c 9 P a g e

7 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary [where the estimatio (A.0) is used] does ot exceed ¼ of the last term tae ito accout. The relative error (as referred to the smaller (A.5) i.e to mi [ ( ) ] c ) c mi is four times smaller tha the estimated trucatio error [cf. Ie.(A.5)]. Remar A: I articular case of alteratig series of tye (3.) the trucatio at the level may geerate smaller estimated error (3.) as comared to (A.5) if / [but if aroximatio (A.3) is alied istead of a trucatio the the estimated aroximatio error is smaller if /5!]. Literature [] H..Carslaw ad J.C.Jaeger Coductio of heat i solids d ed. Claredo Press Oxford 989; see also: A.V.Luiov Aalytical heat diffusio theory (trasl. from Russia) Academic Press ew Yor Lodo 968; J.Taler ad P.Duda olvig direct ad iverse heat coductio roblems riger- Verlag Berli-Heidelberg 006 [] Z.Płochoci ad A.Mioduchowsi A idea of thi-late thermal mirror. I. Mirror created by a heat ulse Arch.Mech. 9 o 77 0 (997); II. Mirror created by a costat heat flux 50 o 9 6 (998); A.Mioduchowsi Z.Płochoci Thermal stresses i a coatig layer. Geeral theoretical scheme Acta Mechaica (00) [3] G.I.Rudi Tiermicziesije arjazheija i deformacji w lastiie ri wozdiejstwii ocietrirowaych otoow eergii (Thermal stresses ad deformatios i a layer uder actio of cocetrated eergy fluxes) Izh. Fiz.Zh. 5 o (I Russia Eglish traslatio i Joural of Egieerig Physics) (988) [].o zeregi iesończoe PW Warszawa 956 (Polish traslatio from th Germa editio Theorie ud Awedug der uedliche Reihe; there exists also a Eglish traslatio Theory ad alicatio of ifiite series) [5] Z.Płochoci A.Mioduchowsi ufficiet accuracy criteria for trucatio of some series J.Iterolatio ad Arox. i ci.com. 03 jiasc 0007 (003) [6] M.Abramowitz ad I.A.tegu (Eds) Hadboo of mathematical fuctios with formulas grahs ad mathematical tables Ch.6 atioal Bureau of tadards Al.Math.eries 55 0d ed.; htt://eole.math.sfu.ca/~abramowitz_ad_stegu.df Table. t 0.0 / *) F / / / 3 / 3/ 6 5/0 3/ 9 / / **) ***) P a g e

8 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary *) Also F [ F with ( ) ] ad F sice all these uatities are eual withi * 5 a accuracy of O 5.0. For examle: for / / F = F = F = **) with ( ). ***) ote that i this case /5 (see Remar A). Table. t 0.05 / *) F / / 0.98 / / 6 3/ / **) *) Also F [ F with ( ) ] ad F sice all these uatities are eual i roud to 5 digits. For examle for / / F = F = F = **) with ( ). Table 3. t 0. / *) F / / / / / **) *) Also F [ F with ( ) ] ad F sice all these uatities are eual i roud to 5 digits. For examle for / / F = F = F **) with ( ) P a g e

9 Zbigiew Płochoci It. Joural of Egieerig Research ad Alicatios I: 8-96 Vol. 6 Issue (Part - ) Jauary Fig. A. Iterretatio of (A.3) for j (a) ad j (b) [see (A.6 )] Fig.A.. Otimizatio of error x 3 P a g e