Finite Fourier Transform

Size: px

Start display at page:

Download "Finite Fourier Transform"

Audra Russell
6 years ago
Views:

Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 755.3 Fii Foi Tsom.3. odcio - Fii gl Tsom 756 Tbl Fii Foi Tsom 76.

1 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Fii Foi Tsom.3. odcio - Fii gl Tsom 756 Tbl Fii Foi Tsom H Eqio i h Fii y Codcio d Advcio H Eqio i h Sph Empls plg low ov hig ibs 78

2 756 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb NRODUCTON Th igl soms w sdid bo (Foi d plc) c b symboliclly dod s pi o opos T : T : did wih h hlp o h igl som l ( ) ( ) ( ) d D d D Ths soms c b pplid o solio o BVP o clssicl PDE s. gl h pocd cosisd o h ollowig sps: ) pplicio o h ppopi igl som limis h cospodig diviv om h qio; ) h somd qio is solvd o h somd ow cio; 3) h solio o BVP is obid by h ivs igl som. gl Tsom ov Fii vl Th igl soms w pplid i h iii d smi-iii domis. Now w w o discov how o solv BVP o PDE s i h ii domi ξ [ ] wih h hlp o h igl som. To do his w w igl soms o posss h sm popy h biliy o limi h cospodig diviv om h PDE. s ivsig his poblm. Th gl om o igl som i h Csi coodi c b wi s sysm ov h ii domi [ ] ( ) ( ) d wh ( ) is h l o h igl som. Apply his som o h scod diviv wih spc o d s (pply igio by ps wic): how o choos ( ) ( ) d ( ) d ( ) ( ) ( ) d ( ) ( ) d ( ) ( ) ( ) ( ) d ( ) ( ) d ( )

3 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Obviosly h igl m s io h somd cio i i gl h l o som hs h popy ( ) c( ) ( ) Th codiio mids h igvl poblm. Ailiy Sm-iovill Poblm s cosid h cospodig iliy Sm-iovill Poblm (SP) wih h pm i h Csi coodi sysm: ( ) h ( ) (SP) h which ccodig o h Sm-iovill Thom posssss iiily my vls o h pm...(igvls) o which SP hs o-ivil solios ( ) (igcios). Tho igcios ( ) sisy body codiios: ( ) h ( ) ( ) h ( ) Solio o h SP o ll... picl h gl solio o h giv qio o SP is giv by ξ c cos c si ddiio h s o ll igcios { }... is compl s o ohogol cios o which w c iodc h om by ( ) d h l o h igl som b ( ) l ( ) ( ) ( ) Diio o h l ( ) wih spc o ξ wic yilds ( ) ( ξ ) ( ) ( ) Th popy which is wd. Fom his ollows h h ls m i h qio ( ) bcoms ( ) d ( ) d ( ) Now cosid homogos body codiios which obviosly bcs i is coscd om h solio sisid by h l o SP: i h ( ) h ( ) ( ) ( ) ( ) h ( ) ( ) ( ) h h

4 758 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Sbsi h l d h diviv o h l io h is ms o qio ( ) h ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( ) h h h h ( ) ( ) ( ) ( ) Tho i w choos igcios ( ) l o h igl som ( ) (d h cospodig ) sch h hy sisy h sm homogos body codiios s o h cio ( ) i h BVP o PDE s h boh h is wo ms i h pvios pssio bcom zo. So w c m h coclsio h i his cs pplicio o h igl som o h scod diviv yilds ( ) d i h l is coscd om h solio o h cospodig SP wih h sm homogos body codiios s o h cio o homogos d ( ) i BVP o PDE. codiios o boh o Robi yp h h ( ) h ( ) d h diviv is somd o ( ) d ( ) ( )... h h h ( ) h ( ) ( ) ( ) ( ) h ( ) ( ) ( ) h h ( ) ( ) ( ) h h

5 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Fii Foi Tsom ov [ ] Th igl som wih h dsid opiol popy ( ) ( ) d ( ) ( ) d d Th ohogol s o igcios { } o h glizd Foi sis psio o h cio ( ) wh h Foi coicis ( ) ( ) d... c b pplid Th psio c b wi s ( ) ( ) d ( ) d d ( ) which c b d s h ivs som i h om o h sis wih ( ) h sm l ( ). Tho w hv h ollowig igl som pis: Fii Foi Tsom ov [ ] ξ ξ dξ igl som ( ) ( ) ivs som Fii Foi Tsom ov [ ] ξ ξ dξ igl som ( ) ( ) ivs som Applicio o h Fii Foi Tsom ov [ ] o d diviv: ( ) ( ) d ( ) ( ) ( ) d ( ) ( ) ( ) ( )

6 76 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Fii Foi Tsom ( ) d igl som ( ) ( ) ivs som Body Codiios h h H [ ] h h H [ ] l ( ) ( ) ( ) ( ) Wh [ ] [ ] DN o R DN o R h igcios o h cospodig Sm-iovill Poblm: Opiol Popy d D [ ] D [ ] N [ ] D [ ] D [ ] N [ ] N [ ] N [ ] π... π si π si π π cos π... π cos π cos π π si π π si... π si π π cos π... π cos π cos... π

7 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb D [ ] R [ ] h N [ ] h R [ ] R [ ] h D [ ] R [ ] h N [ ] R [ ] h h R [ ] posiiv oos o h qio: cos H si ( ) si si ( ) ( ) si 4 cos ( ) ( ) si 4 posiiv oos o h qio: si H cos ( ) cos cos ( ) ( ) si 4 posiiv oos o h qio: cos H si si ( ) ( ) si si 4 ( ) ( ) cos si 4 si ( ) posiiv oos o h qio: si H cos ( ) ( ) cos cos si 4 posiiv oos o h qio: ( ) ( ) ( ) ( ) ( ) H H si H H cos cos H si o ( ) cos H si H H H H

8 76 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb H Eqio i h Fii y Cosid h codcio i h -dimsiol slb wih h gio l d is posd o covciv viom wih im dpd mp h h ( ) ( ) [ ] ( ) igh d is ollowig h pscibd im-dpd mp ( ) Eqio: S( ) α [ ] > iil codiio: ( ) ( ) Body codiios: h h ( ) ( ) [ ] ( ) Robi Diichl ) gl som Accodig o h bl FFT h l o h igl som cospodig o Robi-Diichl body codiios is: ( ) si ( ) wh si 4 ( ) d igvls h posiiv oos o h qio cos H si ) Tsomd qio Accodig o h bl FFT (R-D) h scod diviv o ( ) is somd o d ( ) ( ) Tsom h soc cio d iiil codiio: [ S( ) ] ( ) d S ( ) [ ( ) ] ( ) d Th h somd qio hs h om S α ( ) α Q w...

9 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb ( ) α( ) Q( ) ( ) Q ( ) αs( ) Th somd solio is dmid by viio o pm: wh α α α α τ τ Q dτ 4) ( ) Q Q dos o dpd o im h h solio dcs o Q Q ( ) h i ddiio h iiil codiio is zo Q α α 3) Solio o BVP Th solio c b obid by h ivs som ( ) ( ) ( ) Q d α α α τ τ τ 4) Sdy s solio h cs o cos body codiios d wih iom S q h sdy s solio is volmic h gio [Joh Soddd Scio ]: FT-4.mws 5) Empl (FT-.mws) S ( ) Wiho h gio: h s h ( ) ( ) cos cos Q α

10 764 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 ( ) ( ) α α Q Q α α α Obviosly h sdy s solio ( ) lim ( ) s Q iss oly wh h cio Q is im idpd. Th povidd h om h solio o somd cio i 3) w hv Q s ( ) Ad h by ivs som ( ) s( ) Q α sdy s si solio solio This is h Foi sis psio o h sdy s solio (4). FT-.mws H Eqio i h ii ly Robi - Diichl body codiios > s;wih(plos): > Digis:5; Digis : 5 > :3;:;h:3;H:h/;:; : 3 > :;:; > :; > S:; chcisic qio: > w():*cos(*)h*si(*); > plo(w()..6); : h : 3 3 H : : : : : S : w( ) : cos( 3 ) 3 si( 3 )

11 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Eigvls: > lmbd:y(..); : y (.. [ ]) > :: o m om o 5 do y:solv(w()m/..(m)/): i yp(ylo) h lmbd[]:y: : i od: > o i o 3 do lmbd[i] od; > N:-; N : 4 > :'':i:'i':m:'m':y:'y'::'': Eigcios: > []:si(lmbd[]*(-)); : si( ( 3 )) > N[]:sq(/-si(*lmbd[]*)/4/lmbd[]); si( 6 ) N : 6 l: > []():[]/N[]; ( ) : si( ( 3) ) 6 si 6 > []():sbs([]()); : si( 3 ) si( 6 ) 6 Tsomd iil Codiio d Soc cio: > []:i(*[]()..): > S[]:i(S*[]()..): > Q[]:[]()*/-lmbd[]*/N[]S[]: > QQ[]():sbs(Q[]): > []:[]*p(-^*lmbd[]^*)p(- ^*lmbd[]^*)*i(p(^*lmbd[]^*)*qq[]()..): Solio - vs Tsom: > ():sm([]*[]()..n): > plo3d(()...sbod);

12 766 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 ( ) Sdy S Solio: > s:(h*-)*(-)/(h*); 5 7 s : Tmp poils o di moms o im: > z:sbs(.()):z:sbs(.()):z3:sbs(()): > z4:sbs(3.()):z5:sbs(.()): > plo({szzz3z4z5}..coloblc); ( ). sdy s solio 3....

13 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb ) Tim-dpd body codiio h h ( ) [ ] h si Robi Diichl α α α τ τ Q dτ Assm S Q ( ) α si( ) αh si( ) h si α α τ αh τ dτ α si ατ αh τ dτ α α α cos si α ( ) αh 4 α αh cos α si 4 α α αh 4 α 4 cos( ω δ) α ( ) ( ) ( ) α

768 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7.3.3. Codcio d Advcio M Cosid h sdy s h s i h lid lowig hogh h cgl dc wih cos vlociy v. Egy is sd joily by codcio d bl moio (dvcio).

14 768 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Codcio d Advcio M Cosid h sdy s h s i h lid lowig hogh h cgl dc wih cos vlociy v. Egy is sd joily by codcio d bl moio (dvcio). Flid s h dc wih iiil mp T. Th wlls o h dc hmosd wih h mp T w. Th is iomly disibd gy soc o sgh g. H Eqio: T g ρc p T v α T T T T v y z g ( ) y ( M ) z ( ) By h chg o dpd vibl ( y z) T ( y z) Tw poblm is dcd o h ollowig BVP: h gl Tsoms: g v y z wih body codiios: y ( M ) z ( ) T Tw Diichl y z Diichl y M z z > > ( ) > ( M ) > ( M ) z Diichl y Diichl y Diichl ) gl soms All h spc vibls ivolvd i h poblm. W d limiio o h pil divivs wih spc o wo vibls o dc h qio om pil o odiy. Two o h vibls y d z hv h ii domi. Accodig o bl SP h l o igl som cospodig o Diichl-Diichl body codiios is: Tsom i z-vibl: Th is som o vibl z i h ivl z [ ] : Eigvls π Eigcios si z l ( z) si z gl som pi: ( y) ( yz) ( z)dz ( y z) ( y) ( z) Tsom i y-vibl: Th scod som o vibl y i h ivl [ M ] Eigvls y : mπ m Eigcios m si µ m y M m m z si µ m M l o igl som y gl som pi: m M ( ) ( y) m( y)dy ( y) m ( ) m ( y) m m Rcoscio o h wic somd cio is giv by ( yz) m ( ) m ( y) ( y) wh m

15 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb M M m ( ) ( y z) ( z) dzm( y) dy ( y z) m( y) dy( z) dz Tsomd soc d iiil codiios: g m ( ) M g g M ( y z) m m ( y) ( z) ( y) ( z) dzdy dzdy g M ( ) mπ ( ) g cos ) M m m( y) ( z) m dzdy M ( T T ) ( ) ( ) mπ m w (ssm Tsomd qio: ) Tsomd qio All body codiios i h somd domi homogos ho h qio is somd o m m v ( µ m ) m gm m m Gl solio o h qio cosiss o h gl solio o h homogos p pls picl solio o h o-homogos qio: v ( v) 4( µ ) v ( v) 4( µ ) g m m ( ) c c µ Fo h solio o b bodd w hv o choos c h wih oio β m v ( v ) 4 ( µ ) Th solio bcoms g m m m c β µ m Th coici c c b od om h body codiio h yilds solio o h somd qio m g β g µ m µ m 3) Solio o BVP ow c od by h dobl ivs som m m m solio o somd qio m ( ) m ( yz) m ( ) m ( y ) ( z) m g g m β m m m m y z m µ m µ m Th chg o h vibl yilds Solio: T ( yz) Tw ( yz) T w m g g m β m m m µ m µ m m ( y) ( z)

16 77 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 dc-3.mws H Ts i h 3-D cgl dc wih dvcio > s;wih(plos): > Tw:4;T:;g:;:.;M:;:;v:;:5;dT:T-Tw; Tw : 4 T : g : :. M : : v : : 5 dt : - Eigvls: > lmbd[]:*pi/;m[m]:m*pi/m; l: : π µ m : m π > [m](y):si(m[m]*y)*sq(/m);[](z):si(lmbd[]*z)*sq(/); m ( y ) : si( m π y) ( z ) : si π z > [m]:dt*i(i([m](y)y..m)*[](z)z..); m : 4 ( cos( m π) ) ( cos( π) ) m π > [m]:sbs({cos(m*pi)(-)^mcos(*pi)(-)^}[m]); m : 4 ((-) m ) ((-) ) m π > g[m]:g/*i(i([](y)y..)*[m](z)z..m); g m :. > b[m]:(^*v-sq((^*v)^4*(lmbd[]^m[m]^)))/; β m : 5 65 π 4 m π > [m]:(([m]-g[m]//(lmbd[]^m[m]^))*p(b[m]*) g[m]//(lmbd[]^m[m]^))*[m](y)*[](z); m 8 ((-) m ) ((-) ) : 5/ 65 π 4 m π m π si( m π y) si π z

17 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb SOUTON: > T(yz):Twsm(sm([m]..5)m..5): > ():sbs({z/ym/}t(yz)): > plo(()..); mp poil log h cl li T w T T > y(y):sbs({z/5}t(yz)): > plo3d(y(y)..5y..msbod); T w mp poil i h pl z 5 T > yz(yz):sbs({}t(yz)): > yz(yz):sbs({}t(yz)): > yz(yz):sbs({}t(yz)): > plo3d({yz(yz)yz(yz)5yz(yz)3}y..mz..); mp disibio wih h chg o T > im3d(t(yz)y..mz...4ms); mp disibio wih h chg o o imio s wb si

18 77 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7

19 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb

774 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7.3.4. H Eqio i h Sph covciv viom h dcio o -d Csi omlio by chg o vibl ( ) U wo body codiios U ( ) U body codiio h h U HU Cosid h codcio i h solid sph wih gl symmy.

W c wi h body codiio i h sdd om h h ) odc h w dpd vibl ( ) ( ) U h h HE bcoms U U [ ] > which ow omlly is h -d H Eqio i i h ii ivl [ ] which qis wo body codiios.

20 774 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb H Eqio i h Sph covciv viom h dcio o -d Csi omlio by chg o vibl ( ) U wo body codiios U ( ) U body codiio h h U HU Cosid h codcio i h solid sph wih gl symmy. Th spilly o-sioy mp ild ( ) dpds oly o h dil vibl. Th H Eqio: ( ) wih iiil codiio: ( ) ( ) d covciv body codiio: h ( ) [ ] > wh is mp o h sodigs (glly cio o im). W c wi h body codiio i h sdd om h h ) odc h w dpd vibl ( ) ( ) U h h HE bcoms U U [ ] > which ow omlly is h -d H Eqio i i h ii ivl [ ] which qis wo body codiios. Th is codiio is obid dicly om h qio sd o chg o vibl: U Diichl Cosid h scod body codiio : h U h U h h U U h U U h U h U h h h U h H ( ) h U HU ( ) Robi ) gl som Cosid h ii Foi som cospodig o h cs o Diichl-Robi body codiios: Eigvls posiiv oos o h qio: cos H si Eigcios si Th l ( ) si 4 ( ) si si 4 ( )

21 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb gl som pi: d U U U U Accodig o h bl FT h diviv is somd o d U U U Tsomio o h iiil codiio: d d U 3) Tsomd qio Applyig h igl som o gs U U U U U U Solio by viio o pm: U d U τ τ U d τ τ τ 4) Cs cos i is cos h U d U τ τ U τ U U cos U Solio o BVP c b od by h ivs som i h om o iii sis U U U

22 776 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 5) Empl (y-.mws) Rosig o y (y-3.mws smooh solio) Th y is ssmd o b sph o dis o C. is posd o h covciv viom. m wih h iom iiil mp o 5 C wih h covciv W coici h 5. Th y is cosidd o b do wh is miimm mp chs m o do 75 C. Thmophysicl popis o y m sd o clclio om h bl (Scio V..5 p.58). > s;wih(plos): > :.;:.6;h:5;:;i:5;:774; :. > H[]:h/-/; > :h*i*/; :.6 h : 5 : i : 5 : 774 H : : 5. Chcisic qio: > w():*cos(*)h[]*si(*); > plo(w()..5); w( ) : cos(. ) si(. ) Eigvls: > lmbd:y(..); : y (.. [ ]) > :: o m om o 5 do y:solv(w()*m..*(m)): i yp(ylo) h lmbd[]:y: : i od: > o i o 4 do lmbd[i] od; > N:-; N : 6 > :'':i:'i':m:'m':y:'y'::'': Eigcios d l (S bl (5) Diichl-Robi): > []:si(lmbd[]*); : si( )

23 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb > N[]:sq(/-si(*lmbd[]*)/4/lmbd[]); N : si(. ).5 4 > []():[]/N[]; ( ) : si( ) si(. ).5 4 > []():sbs([]()): > U[]:*i(*[]()..); U :. (. si(. ). cos(. )). ( si(. ) ) > U[]:*[]()/lmbd[]^(U[]-*[]()/lmbd[]^)*p(- lmbd[]^*/^): Solio: > ():sm(u[]*[]()..n)/: > ():sbs(-()): > :*6*::*6*6::3*6*6:3:5*6*6:4:7*6*6: z:sbs(()):z:sbs(()):z3:sbs(3() ):z4:sbs(4()): > plo({zzz3z4z5}-..coloblcsbod); ( ) 7 hos 5 hos ho mi > wih(plos): > im(()...36ms5);

24 778 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7

25 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb

sid is piodic si ( ) y ( y) igh sid is isld y M y Eqio: α y ( y ) ( )

26 78 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Empls. Cosid h codcio i h -dimsiol coss-scio o h log colm: y M y M h mp o h l sid is piodic si ( ) y ( y) igh sid is isld y M y Eqio: α y ( y ) ( ) () M) > iil codiio:. Poi h soc movig o sioy. mpls poi soc. 6-3 o movig soc - Copy.mws

27 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb [Adm] Codcio d dvcio (plg low) M y y M y M is li o symmy (isld) plg low ( y) v y y y y q s y y y y ( y ) < Eqio: ρcv p y ( y ) ( ) ( M ) N y Body codiios: ( ) q H( ) H( ) y ( ) M s Noio ρcv > p

28 78 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 ) sp M y y M y M is li o symmy (isld) plg low ( y) v y y y y q s y y ( y ) < Eqio: ρcv p y ( y ) ( ) ( M ) y Body codiios: ( ) q H( ) H( ) s (Aio: qs < o hig) y ( ) M ) Fii Foi Tsom i y : ( ) ( y) hig N [ ] qs y islio N [ ] M y M M y dy igl som ( y ) ( ) ( y) ivs som N [ ] y N [ ] M y M π... M Y π Y cos y M M M π cos y M M M... Tsomd qio: ρcv p ( )

29 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Cosid cojg poblms: < ρcv p ρcv q p < < s > ρcv p Plo o M

30 784 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Gl Cs Abiy mb N o hig ibs Fo b pomc -vibl hs o b omlizd o? m plg low ( y) v y y M y y M qs y y y y y y y M is li o symmy (isld) y y q s y y y y ( y ) <... N N N g g... N... N N N g Eqio: ρcv p y ( y ) ( ) ( M ) M y Body codiios: -- ( ) < < y ( ) < < y ( ) F( ) N ( ) q H( ) H( ) s < < M y ( ) < <

31 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb ) Fii Foi Tsom i y : ( ) ( y) M y dy igl som ( ) ( y) M ( y ) ( ) ( y) y dy ivs som N [ ] y N [ ] M y M π... M Y π Y cos y M π cos y M M M... Opiol popy: F y M M Tsomd qio: ρcv p ( ) g ( ) g q F s g q F s Cosid cojg poblms:... < ρcv p p < < ρcv > ρcv p

32 786 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 m < < m > Cojg codiios: c c c c > m m c c c m < < c m m g ( ) c c s s 3 4 ( ) 4 c s s s W W g g W g g W g W W g W g ( ) qs qs ( τ) τ s d F d F g ( ) qs τ qs ( τ) τ s d F d F

33 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb > g ( ) s m m m m c c s s m m m m m m c m c m s s s m s m m m W ( m m ) m m m m m m m W g ( ) m g m m W g ( ) m m g m m W g W m m ( ) m W g W m m ( ) m g ( ) m qs m τ qs ( τ) τ s d F d F m m m m m m g ( ) m qs m τ qs ( τ) τ s d F d F m m m m m m s q s F s m m q s F m m s m q s F m m s m s s s s < < c c 5 6 ( ) c 5 c c m m 5 6 c m m 5

34 788 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Sysm o qios o coicis: c c3 c4 c c4 c c s ( ) s ( ) c c s c c3 c4 c m c m c m 3 4 c c s ( ) s ( ) c m m m m m m m m m m c m c m s m s m c m Coicis: > c s ( ) c s ( ) c 3 3 c c s ( ) 4 4 c s ( ) c s ( ) 5 5 c s ( )

35 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb Solio o h somd qio: < c c > m < < c c s s 3 4 m m m m > c c s s < < 5 c > 5 c m Solio: < ( y ) ( ) ( y) < < ( y ) ( ) ( y) < < ( y ) ( ) ( y)

36 79 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Mpl cod: ADAM-9 -- zos N ibs.mw

37 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb

38 79 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 Homwo: M y y M y M is li o symmy (isld) plg low ( y) v y y y y q s y y y y ( y ) < Eqio: ρcv p y ( y ) ( ) ( M ) o y Body codiios: ( ) q H( ) H( ) y ( ) M s ρcv p Tsomd qio:

39 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb m < m > < c c c m m c c c m c c m m < < q s q s q s c3 c4 m m s 3 4 q c c q s c4 m c m c m m 3 4 < < V V c c 5 6 c c m m 5 6

40 794 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 c 6 m c m c m m 5 6 V 3 < < V V q s 3 3 q s q V s c7 c8 V m m s 7 8 q c c V 8 c s q V m c m c m m 7 8 V V 3 3 V > V V 4 4 c c V 9 c c V m m 9 V V 4 4 V c V m c m c m m 9 V < < V V q s 5 5 q s q V s c c V m m s q c c V c s q V m c m c m m V V 5 5

41 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb V > V < > 5 c c V 3 4 c c V m m 3 4 c c V 3 4 c c V m 3 4 m c V 3 c V 3 m V ( ) V c m m 3 c ( ) c s c3 c4 q s c3 c4 q s c4 q c5 c6 q s c4 ( ) c6 c c ( ) V c6 V s c7 c8 V q V s c7 c8 8 c q V c9 c q V s ( ) c8 V c c V ( ) V c V s c c c s q q V s c c c q V c 3 s q

42 796 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 q V s ( ) c V ( ) c c3 c4 c q s c4 q s c3 c4 c5 c6 q s c4 c 6 c c c 6 q c c s 7 8 q s c8 q 3 s c7 c8 3 c9 c c q 3 s c c c 4 c 3 3 q c c 4 s 4 q 4 s c 5 q 5 s c c 5 c 5 s q c 3

43 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb > c m m m s q c c c m m m m s q m c c c c m 5 6 m m ( ) c m c m m c m c m m 5 6 c c m m 5 6 V m m s 7 8 q c c m m ( ) c5 m V c6 m m c m c m m V m m s 7 8 q V m c c c9 c m V m m ( ) c7 m V c8 m m c m c m m 9 c c V m m 4 9 V m m s q c c V m m ( ) c9 m V c m m c m c m m 5 V m m s q V c c c m 3 V m m ( ) c m V c m c m m 3

44 798 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 > c s 3 4 q c c c m c3 m c4 m m m s 3 4 q m c c c c 5 m 6 m m m m 3 4 c5 m c6 m c m c m c c m m 5 6 c m c m m m s 7 8 q c c m m m m 5 6 c7 m c8 m 3 m 3 m 3 s 7 8 q m c c c9 c 3 m 3 c m c m c m c m m 3 m 3 m 3 m c c m 4 m m 4 m 4 s q c c c m c m c m c m m 4 m 4 m 4 m V m 5 m 5 s q c c c3 m 5 c m c m c m m 5 m 5 3 m 5

45 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb m < m > < c c c m m c c c m c c m m < < q s q s q s c3 c4 m m s 3 4 q c c q s c4 m c m c m m 3 4 < < V V c c 5 6 c c m m 5 6

46 8 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7 c 6 m c m c m m 5 6 V 3 < < V V q s 3 3 q s q V s c7 c8 V m m s 7 8 q c c V 8 c s q V m c m c m m 7 8 V V 3 3 V > V V 4 4 c c V 9 c c V m m 9 V V 4 4 V c V m c m c m m 9 V < < V V q s 5 5 q s q V s c c V m m s q c c V c s q V m c m c m m V V 5 5

47 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb V > V < > 5 c c V 3 4 c c V m m 3 4 c c V 3 4 c c V m 3 4 m c V 3 c V V ( ) 3 m V c m m 3 c ( ) c s c3 c4 q s c3 c4 q s c4 q c5 c6 q s c4 ( ) c6 c c ( ) V c6 V s c7 c8 V 4 9 q V s c7 c8 8 c q V c9 c q V s ( ) c8 V c c V ( ) V c c s q q V s c c c s q 5 q V c V s c c q 3 V s ( ) c V ( )

48 8 Chp Th gl Tsom Mhods.3 Fii Foi Tsom Novmb 6 7

Department of Mathematics. Birla Institute of Technology, Mesra, Ranchi MA 2201(Advanced Engg. Mathematics) Session: Tutorial Sheet No.

Department of Mathematics. Birla Institute of Technology, Mesra, Ranchi MA 2201(Advanced Engg. Mathematics) Session: Tutorial Sheet No. Dpm o Mhmics Bi Isi o Tchoog Ms Rchi MA Advcd gg. Mhmics Sssio: 7---- MODUL IV Toi Sh No. --. Rdc h oowig i homogos dii qios io h Sm Liovi om: i. ii. iii. iv. Fid h ig-vs d ig-cios o h oowig Sm Liovi bod