COMPUTING FOLRIER AND LAPLACE TRANSFORMS. Sven-Ake Gustafson. be a real-valued func'cion, defined for nonnegative arguments.

Transcription

1 77 COMPUTNG FOLRER AND LAPLACE TRANSFORMS BY MEANS OF PmER SERES EVALU\TON Sv-Ak Gustafso 1. NOTATONS AND ASSUMPTONS Lt f b a ral-valud fuc'cio, dfid for ogativ argumts. W shall discuss som aspcts of th umrical valuatio of th Laplac trasform (1.1) (Lf) (\) -At f(t)dt, ad th Fourir trasform (1. 2) (Ff) (w) iwtf(t)dt. t will tur out to b advatagous to trat (1.1) ad (1.2) sparatly, v if (1.2) is obtaid by sttig A= -iw i (1.1). W shall fi our discussio to th cass A ad w ral. W obsrv that th twosidd Fourir trasfol~t ca b cast o th form of (1.2) sic ( f -iwt f(-t)dt. Thrfor, th ivrs Laplac trasform may b calculatd by mas of valuatig itg-rals of th typ of (1.1) ad (1.2). (S.g. [1] ad [4].) our tratmt w shall assum that f(t) may b calculatd for a arbitrary argumt t with kow, fiit accuracy. ordr to assss th accuracy of th calculatd valus of (1.1) ad (1.2) w must kow that f

2 78 blogs to a class of fuctios with crtai qualitativ charactristics. For xampl, mrly to assum that f is boudd ad tiuous dos o'c suffic to driv rror bouds, v if a larg umbr of fuctioal valus f (t) ar calcula-td. 2. TRA.l\SFORM NTEGRALS AND POWER SERES Lt h > b a fixd umbr. W fid (+l)h f -At -At f(t)dt "!., -f(t)dt =O h =O h -Ah -\t f(h+t)dt Thus (2.1) -Atf(t)dt =O -A.h x a, X a h -At f(h+t)dt R>vri tig (L2) i th sam way w obtai (2. 2) f iwt f(t)dt Z~:: =O ' z iwh iwt - f (h+'c) clt A obvious stratgy 'P/Ould b first to calculat a ad by mas of umrical itgratio ad th valuat th powr sris, prfrably usig a vrgc acclratio schm; such as thos discussd i [2] ad [4]" W obsrv that th rat of vrgc of th powr sris is iflucd by th choic of h particular, if w tak h = T/w i (2.2) w gt (2. 3) b T/W iwt f(h+t)dt This altratig sris may, for a fairly larg class of fuctios f,

3 79 b valuatd usig th Eulr trasformatio which i this cas is quivalt to rpatd avragig of th partial sums of th altratig sris i (2. 3). This schm ca b sho m to vrg, if f is a polyomial i t, or if f admits a rprstatio as th Stiltjs' itg-ral (2.4) f(t) -tx da(x), 'N'hr a is of boudd varia tio o [, ] ad idpdt of t Compu tatioal xprimts hav b carrid out ad it turd out that fairly may fuctioal valus wr rquird to dtrmi a ad b i (2.1) ad (2.2) >vith high accuracy. 'rhis fact has giv a ictiv to try othr approachs, sic a ad b dpd o A ad w ad must hc b valuatd for ach valu of ths paramtrs. Prformig a chag of variabls i (2.1) ad (2.2) w gt th xprssios (2.5) (Lf) (A) 1 A f -tf(t/a)dt =O x a -h 1 X a ), h -tf th+t)d _.A t (2.6) (Ff) (wl 1 () -itf(t/a)dt zb =O z = ihw h l b itf(h+t)dt Assu.m that N trms ar rquird to valuat th powr sris i (2.5) ad (2.6) with dsird accuracy for all A ad w. Ths N trms ar mpltly dtrmid by th valus of f i th boudd itrvals

4 8 (2. 7) (2.8) [ Nh], l Assum ow that w kow a fuctio g togthr with its trasforms Lg ad Fg. Th w fid (2. 9) (Fg) (l) - (Ff) (W) =O f g approximats f wll o th i trval (2.8) ad thr is a vrgc acclratio schm which dlivrs a. g ood stimat of th sris i (2.9) usig N trms, th (Fg) (W) is a good approximatio of (Ff) (w) A rlatd statmt holds for (Lf) (A) W ot that if w is larg, th (2.8) dfis a short itrval but if w is small, th th itrval (2.8) is log. Thrfor a gral s tra tgy is to approximat f wll clos to, if w is larg, whil good approximatios for t larg ar rquird for w small A similar rul holds for th calculatio of {Lf) (A) 3. CALCULATON OF Lf AND Ff FOR LARGE PA.Rl\..METERS )t AND W xampl: W illustrat th gral idas put forward abov by a umrical EXAMPLE Calculat Lf ad Ff for f (t) =.Q. (l+t). Clos to th origi w may approxima-t f by th first fw trms of its Taylor xpasio,.il,(l+t) E trig this xprssio ito (1.1) w gt (3.1) -At.t(l+t)dt = A-2 -!c c-4-6;\-s + 24;\-6-121c-7 +..,

5 81 which is th asymp totic xpasio dlivrd by Wa.tso's lmma. For :\ = 1 o would trucat th xpasio (3.1) aftr 6 trms gttig th stimat this particular cas th xpasio (3.1) uld also b drivd by mas of itgratio by parts ad th a simpl xprssio is also ob'caid for th rmaidr trt. Ff is dfid by mas of aalytic tiuatio. Takig th ral part of (Ff) (a) w fid (3.2) swt }i, (1 +t) d'c -2 -w stad of approximatig f by a polyomial w try a xpot.ial fil Thus w pu'c f (t) i (t) whr (3. 3) * f (t) \!.. r=l - (r-1) t v "'"r Vi!hr yr ar dtaid such that itrpolats f at th quidistat grid (i-l)~t, i = l,...,. ~tis a stp-width. Thus (3.4) (Lf*) (:\) L Y rr/ (:\+r -1) r=l r:l'his foula prmits asy tabula tio of L'' ()c) giv blow: ad som sampl valus a.r

6 8:< TABLE 1 Estimats of : -At i(l+t)dt basd o (3.3) ad (3.4) with 11t =.1 A = 5 A = W fid immdiatly (Ff*) (W) r=l y (r-1-iw), r givig th approximatio r (3.5) swt i ( l+t) dt ~ W giv som sampl valus: r=l 2 2 (r-l)y /(w +(r-1) ) r TABLE 2 Estimats of 11t =.1 oo swt i(l+t)dt basd o (3.3) ad (3.4) with A = 1 A =

7 83 1\T ot tha-t 'ch accuracy of th approximatios basd o (3, 3) mpars favourably vith thos basd o powr xpasio of (l+t) Aothr stratgy ' d-trmi xpo-tial approximatios of f is dscribd i [3] o Th mthods giv thr rquir that f admits a rprstatio (2.4)o Udr this ditio it is fairly asy to driv a rror boud by applyig o of th vrgc schms i [2] to (2.9). 4. REMARKS ON THE CASE OF SMALL PARAME'l'ER VALUES As statd at th d of Sctio 2, (2. 9) rquirs that f is wll approximatd for t larg by a fuctio f*, whos trasform (1.1) or (1. 2) is kow. Put u = 1/t a.d g(u) = f(l/u) f w ca struct a polyomial P which approximats g accura-tly for whr T > is a kow umbr, th w gt (4ol) T iut p(l/t)dt. To valuat th itgral at th right had sid of (4.1) w d to dtrmi th umbrs (W) T iwt -r t dt r,1,.. 'cgratig by parts w fid th s tabl rcurrc rlatio iut r rt + iw c r r (u) Th star-tig valu c 1 (W) must b calcula.td umrically. W fid c 1 (w) ( iut it dt dt ) t t ' T TW

8 84 ad th lattr itgral is asily calculatd usig umrical itgratio i juctio with vrgc acclratio as dscribd arlir. ordr to valuat (1.2) w also d to dtrmi T iwtf(t)dt, o which ca b achivd by mas of stadard umrical mthods. Th Laplac trasforms ar tratd i a similar way. REFERENCES [1] B. Davis ad B. Marti, Numrical ivrsio of th Laplac trasform: a survy ad mpariso of mthods,. Compo Phys. 33 (1979), [2] S.-A. Gustafso, Covrgc acclratio of powr sris, Computig 21 (1978), [3] S.-A. Gustafso ad G. Dahlquist, O th mputatio of slowly vrgt itgrals, Mthod ud Vrfahr dr Mathmatisch Physik, 6 (1972), [4] F.R. d Hoog,.H. Kight ad A.N. Stoks, A improvd mthod for umrical ivrsio of Laplac t'asforms, SAM. Sci. Stat. Comput. 3 (1982), Ctr for Mathmatical Aalysis Australia Natioal Uivrsity Cabrra ACT 261 AUSTRALA