9-9 A Fd th hlf-g po of f < < < < f f f A B C 8 D 8 E F G H o of th bov %9 - p.-7
- b T b f d f d T f T b T d d 8 f 8 8 8 C /
9 - Fd th Fou tfom of f, f. w A w w C w w E w w G w w B w w D w w F w H o of th bov. %9 - I{ } w d co wd w I{ } w cow w d w I{ } D w w Fd th v plc tfom of F ω A ωt B coωt C coωt ω ω ω D ωt ω E co ωt ω F t ωt ω
- G tωt ω H o of th bov %9 7- F ω ω ω ω { F } coωt C U plc tfom to olv t t t d t d t d t A [ t ] B [ t ] C [ t ]! dt dt dt t t t d t d t d t D [ t ] E [ t ] F [ t ]! dt! dt dt d t G [ t ] H o of th bov %9 t! dt 7- T plc tfom d d [ Y ] [ Y ] [ Y ] Y d d dy dy Y Y Y d d dy dy Y d d Y dy d l Y l l lc Y c Y t! { } d t! t t dt
9 - t d t { t } Y! dt D U Mthod of Fobu to olv th gl oluto of! A c c! B c c! C c c! D c c! E c c! F c c! G c c!!!!!!! H o of th bov c d c bt cott %9 8- 令
-, 令 M!!! { }!!! 令 7!! 7! co
9-7 M { }!! 7!!!! 7! c c c co c c c!! c!! c C 7 Th R--C two how h uodl put v t ω t, d th output voltg co th cpcto dcbd b th dfftl d vo t dvo t quto: v t v t o dt dt wh th coffct dtmd b th vlu of ch pv compot. You qud to clcult th put fquc ω o tht wll cu th output v o t to hv ct 9 ph dl wth pct to th put v t, th output ch t td tt ml, th ptcul oluto of th dfftl quto. b B ug th dfftl opto D d d d d d d d 9 d d d d d -wtt D D D D, th dfftl quto o
-8 Pl dtm th coct ptto of th ptcul oluto p fo olvg, d ou do ot hv to olv th coffct t. 7%9 - d v dv v v t wt dt dt v p t wt w t D D D w { } { } D w 9D w { w t} D w w w { w t} w co wt w wt w w v t wt 9 co wt w w b m m m m m ±,,, ± c co c c c c c h 數 令 p Aco B C D E F G 7 Solv th dfftl quto co d co d %9 - co co co ot ct q.
9-9 co d I co co d co d φ φ co φ φ co φ φ co c co 8 d d Solv th dfftl quto 9 > d d 8%9 - d d 9 d d t d 令, t l >, D dt t { D D D 9} m 9 m ± t t h c c c c t 數 令 p A t A p 8 8 8 h p c c 8 t t p { } D 9 8
- Evlut th tgl d C wh : C coutclocw. %9 -!!!! d d Fd th gvlu d copodg omld gvcto om qul to fo th mt. %9 9- 令 A dt λ λ λ λi A,, λ p EV 9
9 - othooml gvcto R p EV othooml gvcto R p EV othooml gvcto R Th poto of ptcl of m m t tm t dcbd ll phcl qutt SI ut: t j t t t C :, ] [, t. t V d W dot th vg pd cl d wo do to mov th ptcl fom t to t, pctvl. Choo th coct w of V,W fom th followg:,; b,; c, ; d, ;, ;
- f, ; g,; h, ;, ; j, ;, ; l o of th bov. b If th t lctc fld E,, co co j. Wht th wo W E do b th fld E to mov th ptcl of chg q log th pcfd pth C : t, t [,]? ; b ; c ; d ; co ; f ; g ; h ; co ; j co co ; l o of th bov. 7%9 -,7- b7-8 t t d t t j j t dt t d d dt t t dt dt t d t t dt t dt t vg pd V t W F d F d m d d t t dt C C t t dt K C C C
9 - j b E 量 co co φ φ E φ co φ co, φ co φ co, φ φ co, 數 φ co c W qe d q φ d q dφ qφ C C,, co c,, co K C Th moto of tg govd b th ptl dfftl quto PDE : u c ; wh u, t th dplcmt of th ptcl t poto d tt u tm t, c l cott, th ubcpt tt, dot t,, pctvl. Th followg fgu how cto of th tg t om tt t t, pl oughl tch th foc vcto mpog o th lluttd tg cto. b t th tg h ft lgth, d th two d ld vtcll wthout fcto,.. boud codto BC :
- u, t u, t, wh th ubcpt dot. O c dv dct mod u, t X T t fucto tfg th PDE d BC b ug th mthod of pto of vbl. Pl tch th ptl pofl X fo th lowt th otvl mod. c I th pc of tl codto IC: u, f, u t, g, o uull pd th oluto tm of th mod: u, t { A } u, t, wh A } th coffct fo mod { u, t, th ubttut IC to tv A }. Although th pcpl of { uppoto wo fo th PDE of th poblm u tt c u, t could fl om oth PDE. Pl pcf tho of th followg PDE fo whch uppoto do NOT ppl. d g utt p u ; b utt p u q, t ; c u tt u ut ; u tt p u ut ; u tt uut u ; f u t p[ u ] utt utt p t ut ; h utt p[ p, t] u u 8%9 - 略 b 令 u, t X T t P.D.E X T& X T&& c X T λ X c T X λ X ; X X λ, N X,co X, X co, X co, c uppoto bf
9-9 B Pl tt TURE o FASE fo th followg ttmt. If ou ol w Tu o Fl wthout complt plto, ou gt pot. You d to pl t bfl to gt full cdt. If A d B vtbl mtc M d B ml to A, th, b t fo tg >, T A d F B ml. : R R b l tfomto. If T wh ullt T. T, th c If vcto pc V th dct um of W d W, th d {} ll dpdt t. {,, } othooml b fo P. F W W. f Th vcto gpc of l opto T gvcto of T. 念 TRUE 令 P AP B P AP P AP P AP P AP P A P B A ~ B btrue T T Nul T {} ullt T ctrue V W W 若 V W W W W {} dfase ll dpdt. %9
- FASE {,, } othogol b, but ot othooml b. ffase 零 量 If A mt, Pl fd th qud multplcto b cofcto po log th t ow to clcult t dtmt. b How m multplcto do w d to clcult f w ppl th lmt ow opto clcultg th dtmt? c Fom d b, f ou d to wt umcl pogm to clcult mt dtmt, whch mthod do ou pf? Pl jutf ou w. %9 - 令 A A A A A O O A A A 行列 類 A! b A O O
9-7 O A 行列 降 類 A! c 兩 都! 列 理 t A b mt tht ml to low tgul mt d h th dtct gvlu λ, λ, K, λ wth copodg multplct m, m, K, m. Wht ta d dta? %9 8- A ~ P P AP A PP t A t PP t P P t m λ m λ dt A dt PP dt Pdt dt P dt m λ λ λ m m m λ I R, lt S { u, u, u}, wh u,,,, u,,,, d u,,,. U th Gm-Schmdt poc d comput othooml b v, v, } fo th ubpc ps. %9 { v - Gm-Schmdt poc v u,,, v v,
-8 u, v v u v,,,,,,,,, v, v v, v u, v u, v v u v v v, v v, v,,,,,,,,,,,, v, v,,,,,,,,,,, A comp put tp of collctbl to th poduct bo, o ch bo d qul popoto. If cutom dcd to collct ll of th collctbl, wht th pctd umb of th poduct bo tht h o h hould bu? %9 率 令 X, X, X, X, X, X 數量 P X P X P X P X d ~ P X P X X Go{ P X },,,,,, 令 X 數量 X X X X X X X X E X E X E X E X E X E X E.7
9-9 How m dfft w c ou put dtcl bd to dfft bo? 8%9 率 不 9 數 H bo C C bd I tud t w dcovd tht % of th ptg of ct gll ot ogl. A collcto % of th c m mt judgg f ptg uthtc o cop. If h bu pc thg tht t ogl, wht th pobblt tht t ot? 7%9 率 令 A ogl ptg Ω 7.7. 令 B uthtc ptg A A c.8..8. B uthtc ptg B c fcttou ptg B c fcttou ptg B uthtc ptg c c P A B.. P A B.7 P B.7.8..
- 8 Suppo th cout codd b Gg cout follow Poo poc wth vg of cout p mut. Wht th pobblt tht th o cout cod tvl? b Wht th pobblt tht th ft cout occu l th cod? c Suppo th o cout th ft mut, wht th pobblt tht ft cout occu th t mut? Poo dtbuto: λ λ Pobblt m fucto P X,,,, K! m E λ. wh XNo. of cout tm tvl. %9 率 令 X Gg cout 讀 數量 X ~ Po λ 令 Y Gg cout 讀 數量 Y ~ Po λ Y P Y! b 令 Z Gg cout 讀 數量 Z ~ Po λ.. P Z! c P X Z!. X
9-9. oluto of Ug th mthod of ducto of od, fd oth l dpdt oluto. %9 - P.-8 h 令 φ φ φ φ φ φ O.D.E. φ φ φ φ φ dφ 降 O.D.E. φ d dφ d φ lφ l l c dφ d φ φ φ l l c l l l l c φ c
- φ c φ c φ c c φ c c h 令 φ φ φ φ φ O.D.E. φ φ φ φ φ φ φ φ φ dφ 降 O.D.E. φ d dφ d φ lφ l l l c φ lφ l l l l c l l c φ c c d φ c d c d
9 - d c d d c φ c c c Ug plc tfom olv th boud vlu poblm,,. %9 7- plc tfom 令 A [ Y ] [ Y ] Y Y A AY A Y { Y } A I.C. A Fd th gl oluto of th followg dfftl quto d how th
- dtl., < <. %9 9-. Bl 9 令 O.D.E. c J c Y cj cy Vf th dvgc thom b wog out th thom wth th gvg vcto fucto V d th volum V, wh V d V: th co, θ <, <,, th ut vcto th dcto of. %9 7-
9 - V V dv dv D D ddd dddθ 令 φ φ j dd da φ j dd φ coθ θ j ddθ V da ddθ V da ddθ θ dd S S V da Vdv S D U th mthod of dgolto to obt th gl oluto of th followg quto: 8- d wh pm dot. dt %9
- AX X dt λ λ λ λi A, ± λ λ λ λ 令 P D AP P 令 PY X APY Y P DY APY P Y t t t PY X
9-7 t t t u u Solv th dffuo quto α fo < <, t > t ubjct to th tl codto u, fo < <, d th boud codto u, t u hu, t fo t >, h >. %9 - 令 u, t X T t P.D.E. α X T XT& X T& X λx λ X α T T & λα T X λx ; X, X hx m λ m ± λ λ w, < w < X Acow B w X wa w wbco w B.C.: X A X hx wbcow hb w w co w h w w h t w 令 { > h t }, < < X B T& α T T t 令 u, t B α t α t
-8 I.C.: u, B d d B 7 Comput co d. %9 - co d R d 令 f, R lm f lm co R lm f lm co co d R d R { R R }
9-9 9 Fd th Fou of th followg podc fucto < < f %9 < - 令 f { co b } b f d d d f cod co d co d co f d d d f co co
- d Fd th gl oluto of th dfftl quto. d %9 - du 令 u d du du u u d d d I I u d c u c c Fd th cotou tgl d wth cotou C, d b. %9 - d d C : 令 f : C
9 - 路 lm R f d d C C R b 路 路 [ ] lm lm R lm R d d f d d f R R d d C C Rduc th followg mt to t chlo d ow-duc chlo fom 8 b Solv th tm of l quto 8 %9-8 8
- b 8 Solv th tl vlu poblm d d co,. %9 - d d co d d c t I d t c c d d I t t c t t c d c t c c t c
9 - c c t c t I.C. c c t c t Fd th gl oluto of th dfftl quto d d,. %9 d d - m 令 { m m m } m m m ± c col c l 7 Fd th gl oluto of th dfftl quto d d l, >. %9 d d - t d 令, t l >, D dt { D D D } t { D D } t m m m, t t h c ct c cl 令 p At B
- A B p t l h p c cl l 8 Fd th gl oluto of th dfftl quto d d d d. %9 - m m m, h 令 p A Bco O.D.E A, B, C p co c c c C c h p co p { } { } D D D D { } co D 9 Fd plc tfom of { t t t}, b v plc tfom. %9
9-7- } { t t t b t t co Fd gvlu d gvcto of th mt A. %9 7- dt λ λ λ λi A,, λ : λ : λ : λ
- 9 U plc tfom to olv th followg poblm fo t, t d t, dt d,. %9 dt 7- plc tfom 令 Y Y A, B Y A B Y A B Y t { Y } t A cot B t, A, B t t cot t A B Cod tm govd b th followg quto A λb, wh A, B. Fd th gvlu λ d gvcto of th bov quto. 7- A λb A λ B %9
9-7 若 零 dt A λb λ dt A λb λ, λ λ λ Th quto h th gl oluto c c. Fd th gul oluto of. [Th gl oluto c c pt fml of tght l wh ch l copod dft vlu of c. th vlop of c c wll b th gul oluto of. Not tht fml of cuv f,, c, wh c th pmt whch dtm dfft mmb of th fml, gl, vlop cuv. Th vlop of th cuv fml c b obtd b f lmtg c fom th two quto f,, c d fc,, c.] c %9-8 令 p p p pp p p p p p d p p c d c c
-8 dp d p p c p c c c d d c c c c c c c c Fd th du of cuvtu of th ght-hdd ccul hl dfd b th vcto quto t coθ θ j bθ,, b >, θ < - θ coθ θj bθ & θ θ coθj b & θ coθ θj & & b cuvtu & b b du of cuvtu ρ b %9 Vf th dtt b fdg th Fou fo f, t. %9 - P.-8 令 f co co T
9-9 d co d co f co T T olv th ht coducto poblm wth th mthod of pto of vbl, T T,, T,,,, T,. b How do ou olv th poblm, f th boud codto T T,, T,,,, T,? Not: Aw th quto b bfl. Th o d to olv th poblm. %9 - 令 T, X Y P.D.E X Y XY X Y Y λy ; Y Y λ X Y X λx
-,,, co Y λ X X B A X A X B X B T co, B T co, co d B T co, b 令,,, T T T T T T T T T T T T T
9-7- Evlut th followg tgl - t t dt. 9 t t dt t 7- d Evlut th followg tgl C wh C th ccl coutclocw compl pl. - 令 f, 路 R lm f R lm f d { R R } C 9 7- Evlut th followg tgl d. [Ht]: t mpop tgl of v fucto. 9 -
- d d 令 f, lm lm R lm lm R f f { } R R d
9-9 Cod th followg quto fo th tmptu u chmcl d u u ctg lb of mtl: λ [ ], < < d u u. d Fd u d λ fo mll mpltud potv oluto of th fom u ε u ε u hgh od tm of ε d ε λ λ. %9 - u εu ε u ε u ε λ λ d u u O.D.E λ[ ] d d u [ εu ε u ε u ] λ[ ] d d d εu ε u ε u ε u ε u ε λ [ ] d d d d ε u ε u d d ε λ εu ε u ε u d ε 數 u λu d B.C. u u λ,,,, u
- U plc tfom to olv th t : d d 9 f t,, dt dt f t t f < t < d f t l. H potv umb. %9 7- t < t < f t t[ u t u t ] l tu t tu t tu t t u t u t T plc tfom Y 9Y 9 Y Y 9 9 9 Y 9 9 9 9 t { Y } t t u t t t u t 9 9 [ co t ] u t 9 Solv th quto d dt t d,. %9 dt 7-
9 - T plc tfom Y Y Y Y } { Y t t t t t Y AX o, wh A, Y Fd th R of A. b Fd th complt b fo th ow pc of A. c Fd gl oluto fo vcto X bd o th ow pc tht ou got. %9 - A
- A b [ ] [ ] { }, A RS c[ ] Y A o oluto A wh, b A d,,, Dv po fo tm of b Th to ppoch cott wh bcom v lg. Fo th c whch d b, dtm th to. [Ht: It pobl to w th quto wthout lot of computto.] %9 8-
9-7 A A A A b 令 A dt λ λ λi A ± λ p EV gvcto R p EV gvcto R 令 P D AP P P PD A A lm