MULTIPLE TIME-SCALE APPROXIMATIONS FOR THE TIME DEPENDENT P 1 AND P 3 EQUATIONS

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1 Mh Copuo Supropug Ror Phy Nulr ologl pplo Pl Pp vgo Fr Spr -5 5 o C-ROM r Nulr Soy LGrg Pr IL (5) MULTIPLE TIME-SCLE PPROXIMTIONS FOR THE TIME EPENENT P N P EQUTIONS ruo Mr Iu ür Rorhrh Forhugru rlruh Poh rlruh r@r.. G. Cu Iu ür rh u Rorhrh Uvrä rlruh H-u-Nu-Sr. 7 Poo Nr rlruh u@r.u-rlruh. STRCT Th wor pr h rvo o lo or pro wll or hr -l pproo o h po quo h o-group -p P - P - quo or hoogou ulplyg u plr gory. Th rul prou y h hr-l pproo or h po quo r how o prlly ur h url rul prou y h gov-yp lgorh u prouo o y gly l o opuol rour. For h P quo lo-or hr-l pproo hv rv or h uro lu hghrorr phrl hro wll or h pruror or wo group o ly uro. Th vlop o h hr-l pproo o rly o pog pro o p ; urhror hy or h poly o r vgo o h r w h P - P - uo quo rpvly. I prulr g r r orv rgrg h -hvor o h vrou pproo or h uro urr llr r r orv rgrg h r h -hvor o h vrou pro or h uro lu whl h o w h vrou pproo or h pruror ruo r glgl. Furhror h hr-l pproo or h P - P -quo hv how o upror o h rul or y h prov qu ho r h lrur prulrly or h hor l h pr o lrg rvy ro. I ury h ho o ulpl-l po or w wy or vlopg vry ur v pproo or olvg ypl -y uh h p uro rpor /or uo quo wh ly uro. EYWORS: Mulpl-Sl Epo Ror P N -Equo

2 . INTROUCTION wll ow h r hvor o ulr y govr y h l o h prop ly uro prouo. Thu rll opuo o h r lol powr ruo wh h ror or woul rqur h oluo o h p- - rgy- gl-p uro rpor quo wh ly uro. I pr howvr o l o olv ly h -p uro rpor quo or rl ror or gor u o h rou rqur opur powr ory. Populr pproo lu h yh p pro ho [] r ho [] ol po ho [] h qu- ho [4] h po ror ol [5] or h [67] or ol ho [8]. Th r ho or ulg h uro hvor o lgh wr ror (LWR) rly o olvg h p uro uo quo whou p- pro ug or h or ol ho. For ror y howvr h uro gro ou llr h or lgh wr ror. Coquly r ro o h rvv r h rpor or uo quo ( u or LWR ) woul l o prohvly log opuol ory u h opuol p y h uro gro. Th oy r or ror lulo r h qu- ho [9] or h po ho []. Howvr h pro o p wh h ho o uly ur. Th l or gro o h prop ly uro u h y o quo prou y y o h ov pproo o ply hvor. For uh y rgulr pruro ho l o pr urly h ror log- hvor u o h vl uulo o h ovrll rror whh grow v h lol rror vry p r ll. O h ohr h h o lrly pr l ugg h u o ulpl-l po [] or rvg oluo h r hr ypo vly or lrg vlu. Th -l or uh po r gv y h l o h prop ly uro prouo. Th ov o h wor o pply h ho o ulpl-l po o vlop lyl pproo or rg h hvor o h uro populo h pruror oro ulplyg y wh ly uro. wll how h -p h pproo uo p o h uro gro. Thror lrg -p u whou vrly g h oluo ury. Th prpl o pplyg h ulpl-l po ho r or So y ug h po quo. Th uly ury y o h ulpl l quo o or opug h hvor o h uro populo r u So. lo pr h So r rul ro vrl llurv opuo ug h hr-l pproo or h P P quo or l hoogou l ror. Sg r r orv rgrg h -hvor o h vrou pproo or h uro urr llr r r orv rgrg h r h -hvor o h vrou pro or h uro lu whl h o or h pruror ruo r glgl. Furhror h hr-l pproo or h P - P -quo r how o upror o h prov qu oluo r h lrur prulrly or h hor l h pr o lrg rvy ro. Th hr-l pproo (wh wo group o ly uro) or -p P vlop h pp. r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

3 . THREE TIME-SCLES EXPNSIONS FOR THE POINT INETICS EQUTION Th op h prl pplo o h ho o ulpl l po wll llur h o y ug h po quo or ulplyg y wh ly uro prouo. lhough h po quo rpr r plo o h ol quo hy o po h ypl hvor rou y h ly uro. Th ulply o -l wll r y ug o h ulpl l po ho. Th vlop o h ulpl -l quo wll o y org h po quo wh group o ly uro whh wr r or : ρ λ () Λ λ. () Λ I h ov quo o h vrl; () o h -p populo o uro wh v l- Λ; () rpr h -p oro o h h - pruror wh y o λ ro ; ρ rpr h rvy o h y; ; o h ol ur o group o pruror. To llur h pro o -l h po quo Eq. Eq. r r h ollowg o-ol or: (). (4) Th ollowg o hv u h ov quo: ρ (5) Λ (6) λ Λ. (7) Th l volv plly Eq. Eq. r o h ol prr h ll prr y pg h ol prr o h hr l ollow: (8) orgly h p vrl r or o uo o h hr l ly: ( ) (9) (). r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

4 Th rvv opror lo p h hr l : () I vw o h ov po ollow h h uro populo h pruror oro p orlly r o h or (). () Ev hough h ury o h pproo how Eq. Eq. r grl or or r r r h rpv po por o o h whl h ury o h gl-l po uully prov y rg hghrorr r h rpv po h ury o ulpl-l po prov oly ol ul-l oro o vll. Th ulpl-l quo r o y rg Eq. Eq. Eq. o h ol po quo Eq. Eq. 4 uquly ollg h r h o h powr o. Thu ollg h r orrpog o o h roh- (low-) powr yl h ollowg quo or h uro populo h pruror oro : (4) ;. (5) Th y o r-orr oupl orry rl quo rpr y Eq. 4 Eq. 5 prov oro rgrg h y hvor o h l o h orr o h prop uro gro. Th y rly olv o o h ollowg lo-or oluo: (6) (7) (8) Th ov oluo o -p pol uo h y ur o whh oly r wh hlp o h oluo o h hghr-orr quo o. Th o prov oro rgrg h lu o h ly uro prouo o h hor- l o h prop uro gro. Collg h o o h r orr o h po yl h ollowg quo or uro ruo h pruror oro : (9). () r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 4/

5 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 5/ Th ov y o hr oupl quo hrr y h logr l o wh h prouo o ly uro. To olv h quo w u Eq. 6 hrough Eq. 8 Eq. 9 Eq.. Th rul h ollowg quo or h uro ruo : () Th ov quo u p o y r h lr r ro-orr r. Rqurg hror h vhg o ll r l o h ollowg quo or h quy : () Th ov quo rly olv o o () Th quy rou h lu o h ly uro prouo o h quo or h hor horo o h prop uro prouo. No h h o o o whh woul r o r lr y ug h l oo. Irg ow Eq. o Eq. pl h lr orly h rulg r orr rl quo or olv o o (4). (5) I h ov quo h o u p o. Thr pro r y ug Eq. 6 rough Eq. 8 Eq. 4 Eq. 5 Eq.. Th rg quo or h pruror oro gv h rqur or h o. Th rqur ulll y pog h oo: (6) (7) No h h o (8) h u Eq. 6 Eq. 7 orr o ply h oo. Th ov y o prl rl quo olv ly o o: [ ] [ ] r r (9)

6 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 6/ [ ] [ ] r r r r () whr r () I h ov pro h o r o o r ro h l oo or h ruo o uro pruror. I por o o h Eq. 9 Eq. volv olly h low -l whh h h y o quo o logr. Th vlop o h ulpl-l po opp h po h pproo o h po u o o r orr oluo. Collg h rul o yl h ollowg r-orr pro or h oro o uro pruror: () () (4) No h wh h ror prop rl (.. wh ρ ) h whl o whh h rovl gulry Eq. hrough Eq. 4 u url l. Th prol ruv y rplg h pol uo wh h pol rulg ro oluo or o group o ly uro ly ρ q whh quvl o ug r-orr orro o h roorr rul. Th pl h h pro o h quy o q ρ (5) whr 4 q (6) Λ. λ (7) I Eq. 7 h uo ru ovr ll h vll group o pruror. Furhror o vo ovrg h prouo o pruror h pro o h qu o r. (8) h g h o whh ppr Eq. hrough Eq. 4 r r o h l oo or h l oo o o:

7 ( ) q (9) q ( ) ( ) r (4) ( ) ( ) r ( ( ) ( ) ) r (4) r. Th y o quo oprg Eq. Eq. 4 wh h rpv o gv y Eq. 5 hrough Eq. 4 wll ll urhr o h hr-l pproo. Th l ruo o uro pruror hr urg h y lulo (y h vlu h o h prvou opuol -p) or ro r y.g.. Th y o quo whh opr h hr-l λλ pproo or h P or P -quo o y ollowg rly lr ph or h po quo ov [6 7 8]. Howvr h orrpog hl rvo o o rg y w op r orly or volv h ho or h po quo; hror h rvo or h P -quo r rlg o h pp. Th rvo or h P -quo ou h rr.. ILLUSTRTIVE COMPUTTIONS USING THE THREE-SCLE PPROXIMTIONS Th llurv opuol rul pr h o vgg h y ury o h hr-l pproo h w vlop h prvou o. I o pl r o h ury o h hr-l pproo y opro o h r po quo wh group o ly uro. O our uh opro u lo o ou h h h r -group po quo o olv rvly o hr wll r-o w oluo ury opuol rqur. Th r ho or olvg h po quo ly o or ror rv url ho o h gov lgorh [4]. Fgur llur h lo gr w h rul prou y h rv gov lgorh or ror wh 6 group o ly uro urgog rog pov rvy ro (ρ.9 $) h rul o y ug h hr-l pproo. For opro h rul prou y h plr wo-l pproo (orrpog wh h ruo o h ur o ly uro group) whh o rgly wor r lo p. Th hr-l pproo prou vry ur rul prlly guhl ro h gov rul. r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 7/

8 5 uro populo [-] ur (gov) wo-l ppro. hr-l ppro... [] Fgur : Nuro populo ollowg pov rvy ro (r.9 $): opro o wo- hr-l ppro. wh h rv gov lgorh Th hvor o h r w h r rv gov lgorh wh 6 group o ly uro h hr-l pproo p Fg. h h olu vlu o h r r lrly u r qu ll olu vlu. Th rlv r y low r [-] r.. [] Fgur : Evoluo o h r w h hr-l pproo oluo h rv gov lgorh (6 group) Hvg h ury pol o h hr-l pproo h rg quo o r h o vlug h opuol y o h pproo. S h rpv pproo ply volv h opuol vluo o lo or pro url y r upror o y ho h woul rqur h r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 8/

9 url oluo o h po quo. Thu h ruo o prop ly uro pr y h hr-l pproo vlu ly y prulr po whl h pro o h ruo y ug h gov ho rly rqur rv pro y ug p hr y h lo o h uro gro /or h ovrg rro l y h ur or h gov lgorh. To llur h pol y o h hr-l pproo hr hv vg. Th r ( C ) volv ll pw rvy pruro (ρ. $) h u oly vry ll hg h uro populo. Th o ( C ) volv lrg pruro (ρ.9 $) whh u v r h uro populo. Th hr ( C ) volv rvy ro -w h r wo (ρ.75 $). No h h rv opuo ug h gov lgorh (wh 6 ly group) or C C wr opp y h lo rro (. ) whl C w opp y h powr lo rro (p 5 l vlu). Th y o h ul l pproo oluo llur y opug h ruo or g y ug h hr-l pproo o h gov lgorh or h lvl o ury. Th rul or h ruo or wll horh ll or hor g r pr Tl or h hr r h orgog uo o h ovrg rro ploy or h gov lgorh. For h ll pruro (ρ. $) h lg p o. h g vr w ( or o.).6 ( or o.) 66. Th g r or rlly or lrg pruro (ρ.9 $) v wh h rhg o h lg or h uro populo u ruo o h ovrll p o.. Th lrg g o or rlv g pruro (ρ.75 $) oo wh g ovr ll p h h lg or uro populo r rh rly ulouly. Tl : T g or h hr-l pproo vru h rv gov lgorh (6 group) (gov) C g C g C g > > > 4 > 5 llur y h rul how Tl ov h g or po lulo p vry rogly o h ovrg rro ; h rog p g h quo o h g o h vlu or o u h o-ll prouo ( oppo o rrh ) o wh lg wh hgg rvy vlu. For pl opur o h ovrg rro o ul vlu p o h rvy hg; orovr h ul vlu vry ovr lrg rg pg o h p o. Thu h SS4 ly o [] whh u h po ho h ul vlu or 5* -6. O h ohr h h ly o SIMMER-III [9] whh u h qu- ho h ul vlu or.. Th rul how Tl h h plo o h hr-l pproo o ( or uur) prouo o oul vo h prol. r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 9/

10 Rul ro vrl llurv opuo or h p- p prol ug h hr-l pproo vlop h pp wll pr ollowg ug l o hoogou l o u o vhg lu oo or oh h prop ly uro o h rpol l our. Th rl -6 propr hrrg h l r ollow: ( vν Σ ). ; ;. ; ;. ;. ; - - Λ ; λ.96 ; λ.6 ;. Th rl propr (.g. prop ly uro propr) ho ov r h or h po lulo orr o l h uo o h ulpl -l ruur o h P - P -quo y opro o h po quo. Th ollowg ol qu r u or h hr-l pproo o h P -quo:. 4 4 Th llurv opuo o pr ( Fg. hrough 9) r or (or) rvy ro o ρ. $. Rll lo h h oluo o h -p uo quo or h pl prg gory o o h po- quo ulpl y /o uol p p. Th ul or o y h hr-l pproo o h prop uro lu or h uo P P -quo r rpvly: ( ) o(. ) ( ) o. 7 5 ( ) ( 55 ) ( ) (.. ) 9.87 ( ) 9.58 (. ) φ 464 φ 464 φ P P ( ( 55 ) (.67 ) ( ) ) o(.4668 )).4 u o h yry hr o h l-gory o h prg prol h rul o pr h So r ply oly or (h gv) hl o h -. Fgur p h p- ruo o h uro lu h hr-l pproo pr y Eq. 46 or h P -quo. (45) (46) (47) φ Fgur : Nuro lu ruo opu wh h hr-l pproo o h P quo or rvy ro o r.$ Fg. lrly how oly h hr log-lv r lu h hvor o h prop uro lu o h -l rlv o ulr ror. Th horr lv r [.. h r r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

r Eq. 46 h r hr r Eq. 47] vh r oo quly o vly lu h r o h prop uro lu o h ror-yp l or h wor. Fgur 4: ruo o ly uro pr. or h r group opu wh h hr-l ppro. o h P quo or rvy ro o r.$. Fgur 5: ruo o ly uro pr.

11 r Eq. 46 h r hr r Eq. 47] vh r oo quly o vly lu h r o h prop uro lu o h ror-yp l or h wor. Fgur 4: ruo o ly uro pr. or h r group opu wh h hr-l ppro. o h P quo or rvy ro o r.$. Fgur 5: ruo o ly uro pr. or h o group opu wh h hr-l ppro. o h P quo or rvy ro o r.$. Fgur 4 5 p rpvly h p- ruo o h r h o group o ly uro h hr-l pproo. Th o-ruo o oh h prop ly uro h pl ro lo ppr Fg o 5 p vw o h hoogou l gory or h llurv pl. Th uro lu gv y Eq whh pr h hr-l pproo orrpog o h uo P -quo rpvly how lr p- hvor h p Fgur. Th r w h lu opu ug Eq rg ll opuo ro l l oo g h rvy ro o ρ.$ r p Fg oh o h gur ply pov p h rvl low. r whh h r vh rply. Th ro or h hp o h p o ppr y g Eq Th hr-l pproo or h P or h P -quo o o rpvly hr ol pol uo wh lrg gv po; urhror h rpv r (whh r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

I ohr wor pog o pro o p or olvg h uro rpor quo woul u h hvor o h P /or P pproo o h h o h uo pproo; h r how Eq.

12 ul h prop uro prouo) r o l. Th ol pol r ppr u o h -rvv o h hghr-orr phrl hro uro lu. Th rvv r r r wh rvg h uo pproo ro h uro rpor quo. Howvr h p oor r pr o ( pr wh h prov qu- ho []) h ol pol uo woul o ppr hr wh h orrpog uo or h P or P -quo. I ohr wor pog o pro o p or olvg h uro rpor quo woul u h hvor o h P /or P pproo o h h o h uo pproo; h r how Eq or h rpv hr-l pro or h uro lu r ro h h h rg quo h rpv P - P - pproo wr o whou hvg pog p- pro o h ull-lg uro rpor quo. To ur h r h rpv prop r h r whh ppr olly Eq (u o o ppr h uo pproo r y Eq. 45) rpr h rpor o h -hvor o h uro ruo wh h prg l ror. Th olu vlu o h p p Fg. 6 7 p o ohr y prr; h p u h gg o h r ou % o h olu hg h uro lu or h pl (ρ.$). rwr h pol wh h lrg gv po y vry rply. For logr vlu h r p Fg. 7 8 vh h log-r hvor o Eq govr y l r. φ Fgur 6: r [ φ φ P ] ug Eq. (85) (84). φ Fgur 7: r [ φ φ P ] ug Eq. (86) (84) r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

Coprg Fgur 6 7 h h olu vlu h uro o h p or h hr-l pproo o h P -quo r lo o h rul o ug h hr-l pproo o h P -quo. Th -p r whh r u o φ r rlv oly o l whh h ol wo pol uo P horr h h l or or lulo ulr ror.

13 Coprg Fgur 6 7 h h olu vlu h uro o h p or h hr-l pproo o h P -quo r lo o h rul o ug h hr-l pproo o h P -quo. Th -p r whh r u o φ r rlv oly o l whh h ol wo pol uo P horr h h l or or lulo ulr ror. O h ohr h h pl-p r r ro h lghly r rgu o h o uo h ppr Eq rpvly. Th orgog orvo h h orl ol wor volv ug h hr-l pproo or h P - quo [7 8] rg ll prov o h quly o h rul r hr hvor or opr o ug h hr-l pproo or h P -quo o h hr-l pproo or h uo quo. H h hr-l pproo or h P -quo y vgou o u or or ur olg o h pl ruo o h uro lu (plly h ol r wh h o 4 r ) u h hr-l pproo or h P -quo woul u or olg o r o r or ulr ror. Fgur 8: r [ P ] opu wh h hr l pproo or h P h uo pproo rpvly. Fgur 8 p h ruo p o h r [ P ] opu y ug h hr l pproo or h P - h uo quo rpvly. Th r w h wo pproo g rhg p o. % or vry hor rvl ly r g h r opuo. Th plo o h r prov y g h rpv quo: o h o h h hr- l pproo or h uro urr opu ro h uo quo o y ulplyg h r po oluo y h rpv ory pl ruo; o h ohr h ll o h hor- p r rop ou urg h y Eq..7. Th orrpog rul or h hr-l vlop o P pproo o h uro urr orrpog o h P -quo o h ruur P. hvor p- vry lr o h o r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

Th hvor o h (p- p) pruror ruo rly l or h hr-l pproo o h uo h P - h P -quo h og pol uo r l h rpv pproo. Th r o h rvy ro gu o h hr-l pproo or h uo h P -quo rpvly r llur Fg. 9 or prop rl rvy ro.

14 Th hvor o h (p- p) pruror ruo rly l or h hr-l pproo o h uo h P - h P -quo h og pol uo r l h rpv pproo. Th r o h rvy ro gu o h hr-l pproo or h uo h P -quo rpvly r llur Fg. 9 or prop rl rvy ro. Th lrg rvy ro hghlgh o oly h hor- hvor o φ φ ] u lo uror h h or lrg pruro h h r [ P r o o vh progr w h or h llr rvy ro φ φ ] r (o ρ.$) or Fg. 6. how Fg. 9 h r [ ovr h rvl or h opuo wh h r o r og ou.8 % rhg ypo vlu o ou. % o h olu hg uro lu urg h r opuo or h prg hoogou l ror. Th φ φ ] or h po h h lu o h hvor o h r [ P hr-l pproo or h P -quo or proou or rog r whh h prop uro prouo o h ovrll voluo o h uro ruo. Th r h orrpog hr-l pproo or h pruror oro r o v o h gu o rvy ro or ro lry u rlo o Fg P φ Fgur : r [ φ φ P ] or prop rl rvy ro.. CONCLUSIONS Th ho o ulpl l po or w wy or h vlop o vry ur v ul l pproo uo or r pproo o h p ol quo wh ly uro. Th vlop ul l ly pproo uo r powrul v ool or h ly o h hvor o ulplyg y wh ly uro prouo. Th ly o h ury o h ul l pproo oluo h how vry goo gr h opro wh h r rv oluo wh 6 group o ly uro o h o h gov lgorh. lhough g hv or vry oguro h ul ou o h g p rogly o h y prr h ovrg rro o h url lgorh. Th g vr w or o up o 5 or h lulo. r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 4/

15 I lo por o o h lhough h hr-l pproo o h wor l o wo group o ly uro; h lo oul rly rov whou lo o oy h ur l wr o r o our or v l. I uh h ul-l pproo woul vlop y ug h lry r rgy llur So II lo h pp; or v l or pl h orrpog vro o Eq. 6 Eq. 7 woul ou y o ourh orr o h y o h o orr. Th ulpl l po ho h u o vlop lo-or hr-l pproo or h p P -quo or hoogou ulplyg rl plr gory wh wo group o ly uro. Clo-or hr-l pproo hv o or h uro lu hghr-orr phrl hro wll or h pruror or wo group o ly uro. Th vlop o h hr-l pproo o rly o pog pro o p ; urhror hy or h poly o r vgo o h r w h P - P - uo quo rpvly. Thu g r r orv rgrg h -hvor o h vrou pproo or h uro urr llr r r orv rgrg h r h -hvor o h vrou pro or h uro lu whl h o w h vrou pproo or h pruror ruo r glgl. Furhror h hr-l pproo or h P - P -quo hv how o upror o h rul or y h prov qu ho r h lrur prulrly or h hor l h pr o lrg rvy ro. O wy or plg ulpl -l pproo or rg w oluo rg or h -p uro rpor quo wh ly uro y plg h r o uro rpor o y ug h lo-or pro vlop h wor or h hr-l pproo o h P -quo. Suh plg o h r o uro rpor o woul l o vrl ur h o h orrpog o h rpv l urlyg h ulpl l po. Th gl u-r woul orly llr h h r oul hror vr rly u h u-r woul o logr. I prpl uh prour woul ou o vro o h r y lo prog (h lo orrpog o p -l); uh vro prour uully or h h r rv o. Thu h g y or y plg lo-or pro rv ro ulpl- po o o ro pol u o h lrgr pol p u o h lyl pproo oluo u ro h or hl ho. Furhror lo-or pro rv ro ulpl- po lo u or wh uory lro ho. To ur h ho o ulpl l po or w wy or vlopg vry ur v pproo or olvg ypl -y uh h p uro rpor /or uo quo wh ly uro. Furhr o yo h prg pl prov h wor woul h oro o or h hr -l o po h woul pr ulpl-l o oly u lo p. r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 5/

16 REFERENCES.. O. O R. J. Nuhol: Irouory Nulr Ror y r Nulr Soy L Grg Pr Il (985).. J. rly rw II. F. H: Nurl Soluo o h T-p Mulgroup uo Equo Nulr S Egrg Vol. pp. 4- (968).. J.. Yy: Th Soluo o h Sp-T Nuro Group uo Equo y T-ouou Syh Mho Nulr S Egrg Vol. 9 pp. 8 (967). 4.. O J. Mll: Qu Tr o Spl Pho Ror y Nulr S Egrg Vol. 6 pp (966). 5. G. R. p: Phy o Nulr o Wly Rg (965). 6. S. Lguh W. Murr W. Wrr: Cor-Mh Flu-Epo Mho or h ly o Sp-T E Lrg Lgh Wr Ror Cor Nulr S Egrg Vol. 6 pp (977). 7. J. J. org: Mor Cor Mh Mho Nw vlop o h 7 Topl. Mg. Copuol Mho Nul. Eg. Wllurg V (979). Vol S. Sh: ly Nol Mho or Solvg h Two-Group Mulol S Tr Nuro uo Equo ro Mhu Iu o Thology (979). 9. S. oo l.: SIMMER-III: v Copur Progr or LMFR Svr ly NP'9 Toyo Jp (99).. H. U. Wr l.: Coprv ly o Hypohl Lo-o-Flow Irr LMFR Cor Ug r Copur Mol or Coo hr Prol EUR 95 EN (989).. Th SS4 LMFR ly Co Sy NL/RS 8-8 rvo (988).. J. vor J.. Col: Mulpl Sl Sgulr Pruro Mho ppl Mhl S Vol. 4 Sprgr Nw Yor (996).. G. Sr: Nw Mho o Epo Mhl Phy Il Nuovo Co Vol. 6 pp (965). 4. J. J. gov: Nurl Soluo o h O-Group Sp-Ip Ror Equo or Nuro y Gv h E Rvy rgo Nol L. NL- 6 (96). 5. SIMMER-III: Copur Progr or LMFR Cor rupv ly (Vro.G) JNC TN 94 - (). 6.. R. Mr. G. Cu: Mulpl T-Sl Epo or Nuro : II. Illurv pplo o P P Equo p y Nulr S Egrg (5). 7.. R. Mr. G. Cu: Mulpl T-Sl Epo or Nuro : I. Illurv pplo o h Po Mol p y Nulr S Egrg (5). 8.. R. Mr: E hrlg Nähruglöug ür hägg ol- Trporglhug FZ 696 rlruh () 9. J. ur L. Hlo: Nulr Ror ly Joh Wly & So I. Nw Yor (976). r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 6/

17 . J. J. ur W. R. Mr: Trpor Thory Joh Wly & So Nw Yor (979).. M. Clr Jr.. F. H: Nurl Mho o Ror ly Nulr S Thology Pr Nw Yor (964). S. Goluoglu H. L. o: T-p Thr-ol Nuro Trpor Mhoology Nulr S Egrg Vol. 9 pp () PPENIX Thr-Sl Epo pproo or h P -Equo Ug r oo h -p P -pproo o h uro rpor quo wh group o ly uro o-ol hoogou l gory opr h ollowg oupl prl rl quo: φ( ) ( ) Σ φ( ) ( ) νσ φ( ) λc ( ) (.) v ( ) φ( ) Σr ( ) (.) v C ( ) νσ φ( ) λc ( ).... (.) Pror o pplyg ulpl-l po ho h ov quo o ror ol or y g h ollowg qu: v( νσ ) ( νσ ) vc Σr ν Σ νσ Σ νσ (.4) (.5) (.6) (.7) (.8) whr h rly vlu o h y o y νσ Σ νσ (.9) whl h r rvy hg h y o y Σ νσ νσ. (.) Ug h ov o Eq... lg h rulg quo o wo group o ly uro yl h ollowg or or h o-ol P-quo: φ ( ) φ (.) (.) r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 7/

18 φ. (.) So II w rou o Eq... h ol prr (.4) p h rvv opror h r (.5) I ollow h h uro lu φ h uro urr h pruror oro rpr r o h or φ (.6) (.7). (.8) Iroug Eq..4.8 o Eq... qug h o o l powr o rpvly wll yl r o y o quo or h rpv uo Eq Thu h ro-orr r (whh r p o o our) yl h ollowg quo or h uro lu h uro urr h pruror oro : ( ) (.9) (.). (.) Equo.9. r h hvor o uro populo o h l o prop uro prouo. Thr oluo o lylly wll how o o pol uo op y hr lr oo o wo pol uo p h rvy gv (.. < ) or y lr oo o rgoor uo p h rvy gv (.. >). Th o gv rvy ( < ) I h h ul oluo o Eq..9. wr h ollowg or: (.) ( ) ; ( ) C ; ( ) ( ) ; ( ) ( ). Th ollowg oo hv u Eq...5: (.) (.4) (.5) (.6) r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 8/

19 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 9/ (.7) (.8) ; Z (.9). Z (.) Th vrl Z ( ) whh ppr ov h pro o o h wo roo o h qur quo. Z Z (.) Th qu C whh ppr Eq...5 r p o u y uo o. Thror h qu u r y ug h r orr quo whh r (.) (.). (.4) N h qu r l ro pprg Eq...4 y ug Eq..4.5 oghr wh h r or o Eq... hu og y o our quo or h qu (whh rpr h uro lu urr h l o h ly uro). Th rulg quo r h rqur o p o (whh orrpo o h roh-orr r ); h rqur yl h ollowg quo or rg C : ; (.5). C C (.6) Solvg h ov quo gv h ollowg pro or C uo o h lowr -l ; (.7). C C (.8) y h ov pro h uo C rou h lu o h ly uro prouo o Eq...5 whh r h hor l o wh h prouo o prop uro. Equo.7.8 r ow u o ply Eq.

20 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /..4 whh pol o olv h lr lylly o o h ollowg pro or h uro lu h uro urr :.. (.9).. (.4).. (.4).. (.4) Th ollowg o hv u h ov pro: (.4). (.44) Th qu Eq..9.4 u p o h l u y p o h lowr l. Th lr p r y ollowg h prour or h qu C ; h prour ully l o h oo (.45) (.46) whr. (.47) Th lo-or oluo o Eq wr h or [ ] [ ] r r (.48) [ ] [ ] r r r r (.49) whr r (.5) r rg vrou o h rul o hu r oll o o h ollowg hr-l wo-group pproo o h -p P -quo: ; φ (.5) (.5)

21 ( ) ( ) ; ( ) ( ). (.5) (.54) Th o pov rvy ( > ) I h h oluo o Eq..9. wr h ollowg or: o [ ( ) ( )] ; [ o( ) ( )] C ; ( ) o( ) Th ollowg o wr u h ov pro: [ ( ) o( )] ( ) ; [ ]. (.55) (.56) (.57) (.58) (.59) Z ; (.6) Z ; (.6) ( ) [ ( ) o( )] (.6) ( ) [ ( ) o( )]. (.6) I Eq..6.6 h qu Z ( ) o h wo roo o h qur quo: Z ( ) Z. (.64) Furhror h r-orr uro lu uro urr o h or [ ( ) o( )] [ ( ) o( )] ; (.65) [ ( ) o( )] [ ( ) o( )] ; (.66) [ o( ) ( )] [ o( ) ( )] ; [ o( ) ( )] [ o( ) ( )] ; whr ; (.67) (.68) (.69) (.7). r rg vrou o h rul oll o o h ollowg hrl wo-group pproo o h -p P -quo: r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 /

22 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 / [ ] [ ] [ ] ; o o o φ (.7) [ ] [ ] o o (.7) [ ] [ ] ; o o (.7) [ ] [ ]. o o (.74) Ju h how or h po quo pr So II h ury o url opuo ug h hr-l pproo gly prov y rplg h pol uo or h prop uro prouo r (.. ) wh h oluo or h quo rg o-group o ly uro ly ρ q whh quvl o g h rpl q ρ (.75) Eq or h pro o. Th ov rpl pl ug r-orr orro o h ro-orr rul. For oy wh h rpl h vrl q o r y logy wh h o-group o ly uro ollow: Λ λ (.76) (.77) q 4. (.78). (.79) Th o r r y ug h l oo o o (.8)

23 r Nulr Soy Topl Mg Mh & Copuo vgo Fr 5 / (.8) r r φ (.8) r r. (.8) whr h qu r h roo o h u quo 6 (.84) whl h qu r. (.85) Th l oo hr y y vlu y vlu opu urg h y lulo ro h l opuol -p or y ug h ollowg r vlu: φ φ ; ; ; ; Λ λ / or ; ; or < or or >.

G. Ayyappan 1, J. Udayageetha 2

G. Ayyappan 1, J. Udayageetha 2 rol Jourl o S ovv hl rh JS olu 5 u 7 8-7 SS 47-7 r & SS 47-4 Ol O: hp://oor/4/47-45 wwwrjourlor Tr Soluo o / / rl uu y wh rory rv o roull o roull Fk v rrvl rkow ly rpr Sup lk Ayypp J Uyh pr o h ohrry Er