Exact Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics

Size: px
Start display at page:

Download "Exact Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics"

Transcription

1 ISSN Englnd K Journl of Informion nd Compuing Sin Vol. 6 No. pp. 9- E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis A.R. Shh + E.M.E.Zd. * nd K.A.prl Mhmis Dprmn Ful of Sin Tif nirsi El-Tif El- Hih P.O.Bo Kingdom of Sudi Ari Mhmis DprmnFul of Sin El-Mini nirsi El-Mini Egp. Mhmis Dprmn Ful of Sin Zgzig nirsi Zgzig Egp. Rid Dmr pd Dmr Asr. In his ril inroduing n gnrl nsz h improd / - pnsion - mhod is proposd o onsru soluions of som nonlinr pril diffrnil quions in mhmil phsis i h gnrlizd Zhro quions h oupld Mris quions h +- dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion in rms of h hproli funions rigonomri funions nd rionl funion hr sisfis sond ordr linr ordinr diffrnil quion. Whn h prmrs r n spil lus h solir r drid from h rling s. This mhod is rlil simpl nd gis mn n soluions. / Kords: Th improd - pnsion mhod Trling soluions Th gnrlizd Zhro quions Th oupld Mris quions Th + dimnsionl Fornrg Whihm quion Th +-dimnsionl Wu-Zhng quions.. Inroduion Nonlinr pril diffrnil quions r non o dsri id ri of phnomn no onl in phsis hr ppliions nd or mgno fluid dnmis r surf gri s lromgni rdiion rions nd ion ousi s in plsm jus o nm f u lso in iolog nd hmisr nd srl ohr filds. I is on of h imporn ss in h sud of h nonlinr pril diffrnil quions o s nd plii soluions. In h ps srl dds oh mhmiins nd phsiiss h md mn mps in his dirion. Vrious mhods for oining soluions o nonlinr pril diffrnil quions hd n proposd. Among hs r h inrs sring mhod [] Hiro s ilinr mhod [] Blund rnsformion [] Pinlépnsion [5] sin osin mhod [6] homognous ln mhod [7] homoop prurion mhod [ ] riion mhod [] Adomin domposiion mhod [5] nh - funion mhod [6 ] Joi llipi funion pnsion mhod [9 ] F-pnsion mhod [ 5] nd Ep-funion mhod [6 ]. Wng l [9] proposd n mhod lld h / pnsion mhod o loo for h rling soluions for nonlinr pril diffrnil quions NPDEs. B using h / pnsion mhod Zd l [] nd h modifid / pnsion mhod Shh [] h sussfull oind mor rling soluions for som imporn NPDEs. Rnl uo l [] hd dlopd h / pnsion mhod for soling h NPDEs. In his ppr us h impromn / pnsion mhod o find h rling soluions for h gnrlizd Zhro quions h oupld Mris quions h +-dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion.. Dsripion of h impromn / pnsion mhod for NPDEs + Corrsponding uhor. E-mil ddrss: shh@hoo.om * E-mil ddrss: gprl@hoo.om Pulishd World Admi Prss World Admi nion

2 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis In his sion gi h dild dsripion of our mhod. Suppos h nonlinr oluion quion s in o indpndn rils nd is gin P u u u u u u.. hr u u is n unnon funion P is polnomil in u u nd is rious pril driis in hih h highs ordr driis nd nonlinr rms r inold. To drmin u pliil h folloing fi sps []: Sp : W us h folloing rlling rnsformion: u hr is onsn o drmind lr. Th NPDE is rdud o nonlinr ordinr diffrnil quion NODE in : Sp. W suppos h folloing sris pnsion s soluion of Eq. : P.... i n m i n i im n n hr i i... m r onsns o drmind lr m is posii ingr nd sisfis sond ordr linr ordinr diffrnil quion 5 hr is rl onsns. Th gnrl soluions of Eq. 5 n lisd s follos. Whn oin h hproli funion soluion of Eq.5 C osh C sinh. 6 Whn oin h rigonomri funion soluion of Eq.5 C sin C os. 7 Whn oin h rionl funion soluion of Eq.5 hr C nd C r rirr onsns. C C. Sp. Drmin h posii ingr m lning h highs ordr nonlinr rms nd h highs ordr drii in Eqs. or. Sp. Susiuing Eq. long ih 5 ino lning h dnominor nd hn sing ll h offiins of / i i.. o zro ild s of lgri quions for hih h onsns i... m nd. i Sp 5. Assuming h h onsns i i... m nd n oind soling h lgri quions in Sp hn susiuing hs onsns nd h non gnrl soluions of Eq. 5 ino n oin h plii soluions of Eq. immdil.. Appliions of h improd / pnsion mhod for NPDEs In his sion ppl h improd / - pnsion mhod o onsru h rling JIC mil for onriuion: dior@ji.org.u

3 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- JIC mil for susripion: pulishing@wa.org.u soluions for som nonlinr PDEs i h gnrlizd Zhro quions h oupld Mris quions h +-dimnsionl Wu-Zhng quions nd h + dimnsionl Fornrg Whihm quion hih r r imporn in h mhmil phsis nd h n pid nion mn rsrhrs... Empl. Th gnrlizd- Zhro quions In his sion h gnrlizd- Zhro quions for h ompl nlop [] rds: i 9 hr is nonzro onsn. L us ssum h rling soluion of Eqs 9 in h form: V i hr V r rl funions nd r onsns o drmind lr. Susiuing ino Eqs.9 h:. V V B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions in h folloing forms: V hr sisfis Eq.5 nd r onsns o drmind lr. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for nd. Soling h s of lgri quions using Mpl or Mhmi h Cs. hr nd r rirr onsns.

4 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis Cs. hr nd r rirr onsns. No h hr r ohr ss hih r omid hr. W jus lis som soluions orrsponding o ss o illusr h ffinss of h improd pnsion mhod. / sing s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h gnrlizd- Zhro quions 9. Whn oin h hproli funion soluions of Eq.9 i C osh C sinh C sinh C osh nd V [ C {osh [ C {osh sinh sinh [ C [ C sinh } C sinh } C C C {sinh {sinh osh osh ] ] osh }] osh }] 5 hr. In priulr sing C C h folloing solir soluions of gnrlizd- Zhro quions 9 r disord i nh nd V nh. 6 Sing gin C C C hn h solir soluions of gnrlizd- Zhro quions 9 h folloing form: i oh[ ] JIC mil for onriuion: dior@ji.org.u

5 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- nd V oh [ ] 7 C hr nh. I is s o s h if C C C nd r n s ohr spil lus in propr mor solir soluions of Eq. 9 n oind hr omi hm for simplii. Whn g h rigonomri funion soluions of Eq.9 sin os i C C C os C sin nd V [ C {sin os [ C } C os {os C sin }] sin ] [ C{sin os } C{os sin }] [ C os C sin ] In priulr sing C C h folloing solir soluions of gnrlizd- Zhro quions 9 r disord i o nd V o 9 Sing gin C C C hn h solir soluions of gnrlizd- Zhro quions 9 h folloing form: i n nd V n C hr n. C Whn g h soluions of Eq.9 h folloing form: i C C C JIC mil for susripion: pulishing@wa.org.u

6 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis nd C C V. C In h s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h gnrlizd- Zhro quions 9. Whn oin h hproli funion soluions of Eq.9 [ C sinh C osh ] [ C{osh sinh } C{sinh i[ ] osh }] nd V [ C {osh [ C {osh sinh sinh [ C sinh [ C sinh C } C C } C osh {sinh osh ] {sinh ] osh }] osh }] Whn g h rigonomri funion soluions of Eq.9 [ C os C sin ] [ C {sin os } C {os i[ ] sin }] nd V [ C {sin os [ C {sin os [ C os [ C os C } C C } C sin ] {os sin ] {os sin sin }] }]. hr. I is s o s h if C C nd r n s ohr spil lus in propr mor solir soluions of Eq. 9 n oind hr omi hm for simplii... Empl. Th oupld Mris quions In his susion sud h oupld Mris quions []. iq Q QR R R Q. In ordr o s h soluions of Eqs. suppos JIC mil for onriuion: dior@ji.org.u

7 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9-5 Q u [ i l] 5 hr nd l r onsns o drmind lr. Susiuing Eq.5 ino Eqs. h : i u R u R u uu. u ur 6 W us h folloing rling rnsformions u R V 7 hr nd r onsns o drmind lr is n onsn Eqs. 6 om h folloing NODEs: V V B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for l nd. Soling h s of lgri quions using Mpl or Mhmi h Cs. 9 hr nd r rirr onsns. Cs. hr nd r rirr onsns. sing s 9 nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h oupld Mris quions. Whn oin h hproli funion soluions of Eq. JIC mil for susripion: pulishing@wa.org.u

8 6 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis Q [ C {osh [ i l] [ C sinh C osh ] sinh } C{sinh osh }] nd R [ C {osh [ C{osh [ C sinh C osh ] sinh } C {sinh osh [ C ] sinh C osh sinh } C{sinh osh }] }] hr [ ]. In priulr sing C C h folloing solir soluions of h oupld Mris quions r disord [oh ] Q [ oh }] [ i l] nd R oh oh oh [ oh ] Whn oin h hproli funion soluions of Eq. Q [ C{sin [ C os C sin ] os } C{os sin [ i l] }] 5 nd R [ C os C sin ] [ C {sin os } C {os sin [ C ] os C sin [ C{sin os } C{os sin }] }] 6 In priulr sing C C h folloing solir soluions of h oupld Mris quions r disord n ] Q [ i l] 7 [ n ] JIC mil for onriuion: dior@ji.org.u

9 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- JIC mil for susripion: pulishing@wa.org.u 7 nd ] n [ n ] n [ n R hr ] [... Empl. Th +-dimnsionl Wu-Zhng quions In his susion sud h +-dimnsionl Wu-Zhng quions [56].. u u u u u u uu u 9 L us ssum h rling soluions of Eqs 9 in h folloing forms: W V u hr is n rirr onsn. Susiuing ino Eqs. 9 h: L V W V W W VV V V W V hr L is h ingrion onsn. B lning h highs ordr drii rms nd nonlinr rms in Eqs. suppos h Eqs. on h soluions in h folloing forms:. W V hr nd r onsns o drmind lr. Susiuing Eqs. long ih 5 ino Eqs. nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eqs. r onrd ino polnomils in /. Sing h offiin of hs polnomils o zro dri s of lgri quions for L nd. Soling h s of lgri quions using Mpl or Mhmi h

10 Cs. A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis L hr nd r rirr onsns. Cs L sing s nd h gnrl soluions of Eq.5 n find h folloing rling soluions of h +-dimnsionl Wu-Zhng quions 9. Whn oin h hproli funion soluions of Eqs.9 [ C sinh C osh ] [ C{osh sinh } C{sinh osh }] 5 [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] V 6 [ C{osh [ C sinh C osh ] [ C{osh sinh } C{sinh osh sinh } C{sinh [ C sinh C osh osh }] ] }] 6 JIC mil for onriuion: dior@ji.org.u

11 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9-9 nd 6 W [ ][ C sinh C osh ] [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] [ C{osh sinh } C{sinh osh }] [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] 7 [ C{osh sinh } C{sinh osh }] [ C sinh C osh ] hr. In priulr sing C C h folloing solir soluions of h +-dimnsionl Wu-Zhng quions r disord oh ] [nh ] oh oh V 6 [nh ] 9 [ oh }] nd 6 W [ ] oh [ oh }] oh [nh [ oh }] ] [nh ] 5 Whn g h rigonomri funion soluions of Eqs.9 [ C{sin os [ C{sin os [ C os [ C os } C{os } C{os C sin ] C sin ] sin sin }] }] 5 V 6 [ C{sin os [ C{sin os } C{os } C{os [ C os C sin ] [ C os C sin ] sin }] sin }] 5 nd JIC mil for susripion: pulishing@wa.org.u

12 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis 6 W [ ][ C os C sin ] [ C{sin os } C{os sin }] [ C os C sin ] [ C{sin os } C{os sin }] [ ][ C{sin os } C{os sin }] [ C os C sin ] [ C{sin os } C{os sin }]. [ C os C sin ] 5 In priulr sing C C h folloing solir soluions of h +-dimnsionl Wu- Zhng quions r disord n ] [o ] 5 [ n }] n V 6 [o ] 55 [ n }] nd 6 W [ ]n [ n ] n [ [ n ] ][o [o ]. B h similr mnnr n oin h soluions for h s. W omid h rsuls of s. for onnin... Empl Th + dimnsionl Fornrg Whihm quion In his susion sud h + dimnsionl Fornrg Whihm quion [7]. u u u uu uu u u. 57 Th rling rnsformion prmis us onring Eq.57 o h folloing ODE:. 5 B lning h highs ordr drii rms nd nonlinr rms in Eqs. 5 g 59 On susiuing Eq.59 long ih 5 ino Eqs. 5 nd lning h dnominor nd olling ll rms ih h sm ordr of / oghr h lf hnd sid of Eq. 5 r onrd ino ] 56 JIC mil for onriuion: dior@ji.org.u

13 Journl of Informion nd Compuing Sin Vol. 6 No. pp 9- polnomil in /. Sing h offiin of his polnomil o zro dri s of lgri quions for nd. Soling h s of lgri quions using Mpl or Mhmi h 6 6 hr nd r rirr onsns. Consqunl h rling soluion s h folloing form: [ ]{ Asinh B osh } 6 B A osh A B sinh hr A B r rirr onsns nd.. Conlusion Th proposd mhod in his ppr is mor gnrl hn h nsäz in '/-pnsion mhod [] nd modifid '/- pnsion mhod []. If s h prmrs in. nd.5 o spil lus h o o mhods n rord our proposd mhod. Thrfor h n mhod is mor porful hn h '/-pnsion mhod nd modifid '/- pnsion mhod nd som n ps of rlling soluions nd solir soluions ould pd for som PDEs. This mhod is onis ffi nd n pplid o ohr nonlinr oluion quions in mhmil phsis. 5. Anoldgmn This or is prill suppord Tif nirsi Sudi Ari undr h rn No Th uhors ish o hn Tif nirsi for his suppor. 6. Rfrns [] M.J. Aloiz nd P.A. Clrson. Solions nonlinr Eoluion Equions nd Inrs Sring Trnsform. Cmridg: Cmridg ni. Prss 99. [] R.Hiro. E soluion of h KdV quion for mulipl ollisions of soluions. Phs. R. Lrs. 97 7: 9-9. [] M.R.Miur. Blund Trnsformion. Brlin: Springr-Vrlg 97. [] C.Rogrs nd W.F.Shdi. Blund Trnsformions. N Yor: Admi Prss 9. [5] J.Wiss M.Tor nd.rnll. Th Pinl propr for pril diffrnil quions. J.Mh.Phs. 9 : 5-56 [6] D.S.Wng Y.J.Rn nd H.Q.Zhng. Furhr ndd sinh-osh nd sin-os mhods nd n non rling soluions of h +-dimnsionl disprsi long quions. Appl. Mh.E-Nos. 5 5: [7] M.L. Wng. E soluions for ompound KdV-Burgrs quion. Phs. L. 996 A : [] J.H. H. Th homoop prurion mhod for nonlinr osillors ih disoninuiis. Appl. Mh. Compu. 5: 7-9. [9] J.H. H. Comprison of homoop prurion mhod nd homoop nlsis mhod. Appl. Mh. Compu. 56: [] J.H. H. Homoop prurion mhod for ifurion of nonlinr quions. In. J. Nonlinr Si. Numr. Simul. 5 6: 7-. [] E.M.E.Zd T.A. Nofl nd K.A.prl. Th homoop prurion mhod for soling nonlinr Burgrs nd n oupld MKdV quions. Zishrif fur Nurforshung. 6: [] H.M. Liu. nrlizd riionl prinipls for ion ousi plsm s H s smi-inrs mhod. Chos Solions & Frls. 5 : [] H.M. Liu. Vriionl pproh o nonlinr Elrohmil ssm. In. J. Nonlinr Si. Numr. Simul. 5: [] E. Bolin J. Bizr nd A.R.Vhidi. A n ompuionl mhod for Lpl rnsforms domposiion mhod. Appl. Mh. Compu. 5: -6. JIC mil for susripion: pulishing@wa.org.u

14 A.R. Shh l: E Soluions for Som Nonlinr Pril Diffrnil Equions in Mhmil Phsis [5] E.Bolin J.Bizr nd A.R.Vhidi. Soluion of ssm of nonlinr quions Adomin domposiion mhod. Appl. Mh. Compu. 5: 7-5. [6] H.A. Aduslm. On n improd ompl nh -funion mhod. In. J. Nonlinr Si.Numr. Simul. 5 6: [7] E.M.E.Zd Hssn A.Zdn nd Khld A. prl. roup nlsis nd modifid ndd Tnh- funion o find h inrin soluions nd solion soluions for nonlinr Eulr quions. In. J. Nonlinr Si. Numr. Simul. 5: -. [] L. D-Shng. Fng nd Z. Hong-Qing. Soling h + dimnsionl highr ordr Bror-Kup ssm i rnsformion nd nh- funion mhod. Chos Solions & Frls. : -5. [9] Y. Chn nd Q. Wng. Endd Joi llipi funion rionl pnsion mhod nd undn fmilis of Joi llipi funions soluions o + dimnsionl disprsi long quion. Chos Solions nd Frls. 5 : [] S.Liu Z. Fu S.D. Liu nd Q. Zho. Joi llipi funion pnsion mhod nd priodi soluions of nonlinr quions. Phs. Lrs A. 9: [] D. Lu. Joi llipi funion soluions for o rin Boussinsq quions. Chos Solions nd Frls. 5 : 7-5. [] E.M.E. Zd H.A. Zdn nd K.A. prl. On h solir soluions for nonlinr Hiro Ssum oupld KdV of quions. Chos Solions Frls. : 5. [] M.A. Adou. Th ndd F-pnsion mhod nd is ppliions for lss of nonlinr oluion quion. Chos Solions nd Frls. 7 : [] M.Wng nd X. Li. Appliions of F-pnsion o priodi soluions for n Hmilonin mpliud quion. Chos Solions nd Frls. 5 : [5] S. Zhng nd T.C. Xi. A gnrlizd F-pnsion mhod nd n soluions of Konoplhno- Duros quions. Appl. Mh. Compu. 6 : 9-. [6] J.H. H nd X.H. Wu. Ep-funion mhod for nonlinr quions. Chos Solions nd Frls. 6 : 7-7. [7] S. Zhng. Appliion of Ep-funion mhod o highr dimnsionl nonlinr oluion quion. Chos Solions nd Frls. : [] S. Zhng. Appliion of Ep-funion mhod o Rii quion nd n soluions ih hr rirr funions of Bror- Kup- Kuprshmid quions. Phs. Lrs A. 7: 7-. [9] M.Wng X.Li nd J.Zhng. Th / - pnsion mhod nd rling soluions of nonlinr oluion quions in mhmil phsis. Phs.Lrs A. 7: 7-. [] E.M.E.Zd nd K.A.prl. Th / - pnsion mhod for finding rling soluions of nonlinr PDEs in mhmil phsis. J. Mh. Phs. 9 5: 5-5. [] E.M.E.Zd nd Khld A.prl. Som ppliions of h / pnsion mhod o non-linr pril diffrnil quions. Appl. Mh. nd Compu. 9 : -. [] A.R. Shh. Th rling soluions of h prurd nonlinr Shrödingr quion nd h ui quini inzurg Lndu quion using h modifid /-pnsion mhod. Applid Mh. Compuion. 7: -. [] S. uo Y. Zhou nd C. Zho. Th improd '/- pnsion mhod nd is ppliions o h Bror Kup quions nd pproim long r quions. Applid Mh. Compuion. 6: [] M. A. Adou. An ndd Rii quion rionl pnsion mhod nd is ppliion. In. Journl Nonlinr Sin. 9 7: [5] Z.Y.M. Homoop prurion mhod for h Wu-Zhng quion in fluid dnmis. J. Phs.: Conf. Sr. 96: -. [6] X. Ji C. Chn J.E. Zhng nd Y. Li. Li smmr nlsis nd som n soluions of h Wu-Zhng quion. Journl Mh. Phs. pp.-6. [7] L. Tin nd Y. o. Th glol ror of h isous Fornrg - Whihm quion. Nonlinr Anlsis. 9 7: JIC mil for onriuion: dior@ji.org.u

A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique

A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique Inrnionl hmil orum no. 667-67 Sud of h Soluions of h o Volrr r rdor Ssm Using rurion Thniqu D.Vnu ol Ro * D. of lid hmis IT Collg of Sin IT Univrsi Vishnm.. Indi Y... Thorni D. of lid hmis IT Collg of

More information

3.4 Repeated Roots; Reduction of Order

3.4 Repeated Roots; Reduction of Order 3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r &

More information

Global Solutions of the SKT Model in Population Dynamics

Global Solutions of the SKT Model in Population Dynamics Volm 7 No 7 499-5 ISSN: 3-88 rin rion; ISSN: 34-3395 on-lin rion rl: h://ijm ijm Glol Solion of h SK Mol in Polion Dnmi Rizg Hor n Mo Soilh USH El li Ezzor lgir lgri rizg@gmilom USH El li Ezzor lgir lgri

More information

Single Correct Type. cos z + k, then the value of k equals. dx = 2 dz. (a) 1 (b) 0 (c)1 (d) 2 (code-v2t3paq10) l (c) ( l ) x.

Single Correct Type. cos z + k, then the value of k equals. dx = 2 dz. (a) 1 (b) 0 (c)1 (d) 2 (code-v2t3paq10) l (c) ( l ) x. IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 www.kolsss.om Qusion. & Soluion. In. Cl. Pg: of 6 TOPIC = INTEGRAL CALCULUS Singl Corr Typ 3 3 3 Qu.. L f () = sin + sin + + sin + hn h primiiv of f()

More information

Mathcad Lecture #4 In-class Worksheet Vectors and Matrices 1 (Basics)

Mathcad Lecture #4 In-class Worksheet Vectors and Matrices 1 (Basics) Mh Lr # In-l Workh Vor n Mri (Bi) h n o hi lr, o hol l o: r mri n or in Mh i mri prorm i mri mh oprion ol m o linr qion ing mri mh. Cring Mri Thr r rl o r mri. Th "Inr Mri" Wino (M) B K Poin Rr o

More information

UNSTEADY HEAT TRANSFER

UNSTEADY HEAT TRANSFER UNSADY HA RANSFR Mny h rnsfr problms rquir h undrsnding of h ompl im hisory of h mprur vriion. For mpl, in mllurgy, h h ring pross n b onrolld o dirly ff h hrrisis of h prossd mrils. Annling (slo ool)

More information

HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution

More information

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm

More information

Analytical Solution of A Differential Equation that Predicts the Weather Condition by Lorenz Equations Using Homotopy Perturbation Method

Analytical Solution of A Differential Equation that Predicts the Weather Condition by Lorenz Equations Using Homotopy Perturbation Method Globl Journl of Pur nd Applid Mhmis. ISSN 0973-768 Volum 3, Numbr 207, pp. 8065-8074 Rsrh Indi Publiions hp://www.ripubliion.om Anlyil Soluion of A Diffrnil Equion h Prdis h Whr Condiion by Lornz Equions

More information

Lecture 21 : Graphene Bandstructure

Lecture 21 : Graphene Bandstructure Fundmnls of Nnolcronics Prof. Suprio D C 45 Purdu Univrsi Lcur : Grpn Bndsrucur Rf. Cpr 6. Nwor for Compuionl Nnocnolog Rviw of Rciprocl Lic :5 In ls clss w lrnd ow o consruc rciprocl lic. For D w v: Rl-Spc:

More information

Chapter 3. The Fourier Series

Chapter 3. The Fourier Series Chpr 3 h Fourir Sris Signls in h im nd Frquny Domin INC Signls nd Sysms Chpr 3 h Fourir Sris Eponnil Funion r j ros jsin ) INC Signls nd Sysms Chpr 3 h Fourir Sris Odd nd Evn Evn funion : Odd funion :

More information

Fourier Series and Parseval s Relation Çağatay Candan Dec. 22, 2013

Fourier Series and Parseval s Relation Çağatay Candan Dec. 22, 2013 Fourir Sris nd Prsvl s Rlion Çğy Cndn Dc., 3 W sudy h m problm EE 3 M, Fll3- in som dil o illusr som conncions bwn Fourir sris, Prsvl s rlion nd RMS vlus. Q. ps h signl sin is h inpu o hlf-wv rcifir circui

More information

Equations and Boundary Value Problems

Equations and Boundary Value Problems Elmn Diffnil Equions nd Bound Vlu Poblms Bo. & DiPim, 9 h Ediion Chp : Sond Od Diffnil Equions 6 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ กด วล ยร ชต Topis Homognous

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

Adrian Sfarti University of California, 387 Soda Hall, UC Berkeley, California, USA

Adrian Sfarti University of California, 387 Soda Hall, UC Berkeley, California, USA Innionl Jonl of Phoonis n Oil Thnolo Vol. 3 Iss. : 36-4 Jn 7 Rliisi Dnis n lonis in Unifol l n in Unifol Roin s-th Gnl ssions fo h loni 4-Vo Ponil in Sfi Unisi of Clifoni 387 So Hll UC Bkl Clifoni US s@ll.n

More information

Math 266, Practice Midterm Exam 2

Math 266, Practice Midterm Exam 2 Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.

More information

THE EXTENDED TANH METHOD FOR SOLVING THE -DIMENSION NONLINEAR DISPERSIVE LONG WAVE EQUATION

THE EXTENDED TANH METHOD FOR SOLVING THE -DIMENSION NONLINEAR DISPERSIVE LONG WAVE EQUATION Jornl of Mhemil Sienes: Adnes nd Appliions Volme Nmer 8 Pes 99- THE EXTENDED TANH METHOD FOR SOLVING THE ( ) -DIMENSION NONLINEAR DISPERSIVE LONG WAVE EQUATION SHENGQIANG TANG KELEI ZHANG nd JIHONG RONG

More information

( ) ( ) + = ( ) + ( )

( ) ( ) + = ( ) + ( ) Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( +

More information

LINEAR 2 nd ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

LINEAR 2 nd ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS Diol Bgyoko (0) I.INTRODUCTION LINEAR d ORDER DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS I. Dfiiio All suh diffril quios s i h sdrd or oil form: y + y + y Q( x) dy d y wih y d y d dx dx whr,, d

More information

Systems of First Order Linear Differential Equations

Systems of First Order Linear Differential Equations Sysms of Firs Ordr Linr Diffrnil Equions W will now urn our nion o solving sysms of simulnous homognous firs ordr linr diffrnil quions Th soluions of such sysms rquir much linr lgbr (Mh Bu sinc i is no

More information

On the Existence and uniqueness for solution of system Fractional Differential Equations

On the Existence and uniqueness for solution of system Fractional Differential Equations OSR Jourl o Mhms OSR-JM SSN: 78-578. Volum 4 ssu 3 Nov. - D. PP -5 www.osrjourls.org O h Es d uquss or soluo o ssm rol Drl Equos Mh Ad Al-Wh Dprm o Appld S Uvrs o holog Bghdd- rq Asr: hs ppr w d horm o

More information

Stability and Optimal Harvesting of Modified Leslie-Gower Predator-Prey Model

Stability and Optimal Harvesting of Modified Leslie-Gower Predator-Prey Model Journl of Phsis: Confrn Sris PAPR OPN ACCSS Sbili nd Oiml rvsing of Modifid Lsli-Gowr Prdor-Pr Modl To i his ril: S Toh nd M I Azis 08 J. Phs.: Conf. Sr. 979 0069 Viw h ril onlin for uds nd nhnmns. This

More information

Midterm. Answer Key. 1. Give a short explanation of the following terms.

Midterm. Answer Key. 1. Give a short explanation of the following terms. ECO 33-00: on nd Bnking Souhrn hodis Univrsi Spring 008 Tol Poins 00 0 poins for h pr idrm Answr K. Giv shor xplnion of h following rms. Fi mon Fi mon is nrl oslssl produd ommodi h n oslssl sord, oslssl

More information

Inventory Model with Quadratic Demand under the Two Warehouse Management System

Inventory Model with Quadratic Demand under the Two Warehouse Management System Prin : - nlin : - A K Mlik l. / nrnionl Jornl of Enginring nd hnology JE nnory Modl wih Qdri Dmnd ndr h wo Wrhos Mngmn ysm A K Mlik Dipk Chkrory Kpil Kmr Bnsl nd * ish Kmr Assoi Profssor Dprmn of Mhmis

More information

Systems of First Order Linear Differential Equations

Systems of First Order Linear Differential Equations Sysms of Firs Ordr Linr Diffrnil Equions W will now urn our nion o solving sysms of simulnous homognous firs ordr linr diffrnil quions Th soluions of such sysms rquir much linr lgbr (Mh Bu sinc i is no

More information

Fourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t

Fourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih

More information

Relation between Fourier Series and Transform

Relation between Fourier Series and Transform EE 37-3 8 Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Rlion bwn ourir Sris n Trnsform Th ourir Trnsform T is riv from h finiion of h ourir Sris S. Consir, for xmpl, h prioic complx sinl To wih prio

More information

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2

Math 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2 Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary

More information

M.A.K.Azad, L.S. Andallah

M.A.K.Azad, L.S. Andallah Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 878 ISSN 9-558 Anlil Solions of D Inomrssil Nir- Soks Eqions for Tim Dnn Prssr Grin MAKAz LS Anllh Asr- In his r w rsn nlil solions of wo imnsionl

More information

UNSTEADY STATE HEAT CONDUCTION

UNSTEADY STATE HEAT CONDUCTION MODUL 5 UNADY A HA CONDUCION 5. Inroduion o his poin, hv onsidrd onduiv h rnsfr problms in hih h mprurs r indpndn of im. In mny ppliions, hovr, h mprurs r vrying ih im, nd rquir h undrsnding of h ompl

More information

Special Curves of 4D Galilean Space

Special Curves of 4D Galilean Space Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Spl Curvs of D ll Sp Mhm Bkş Mhmu Ergü Alpr Osm Öğrmş Fır Uvrsy Fuly of S Dprm of Mhms 9 Elzığ Türky

More information

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7

CIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7 CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

The Laplace Transform

The Laplace Transform Th Lplc Trnform Dfiniion nd propri of Lplc Trnform, picwi coninuou funcion, h Lplc Trnform mhod of olving iniil vlu problm Th mhod of Lplc rnform i ym h rli on lgbr rhr hn clculu-bd mhod o olv linr diffrnil

More information

Binomials and Pascal s Triangle

Binomials and Pascal s Triangle Binomils n Psl s Tringl Binomils n Psl s Tringl Curriulum R AC: 0, 0, 08 ACS: 00 www.mthltis.om Binomils n Psl s Tringl Bsis 0. Intif th prts of th polnomil: 8. (i) Th gr. Th gr is. (Sin is th highst

More information

CS 541 Algorithms and Programs. Exam 2 Solutions. Jonathan Turner 11/8/01

CS 541 Algorithms and Programs. Exam 2 Solutions. Jonathan Turner 11/8/01 CS 1 Algorim nd Progrm Exm Soluion Jonn Turnr 11/8/01 B n nd oni, u ompl. 1. (10 poin). Conidr vrion of or p prolm wi mulipliiv o. In i form of prolm, lng of p i produ of dg lng, rr n um. Explin ow or

More information

Lecture 4: Laplace Transforms

Lecture 4: Laplace Transforms Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions

More information

16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Ideal Nozzle Fluid Mechanics

16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Ideal Nozzle Fluid Mechanics 6.5, Rok ropulsion rof. nul rinz-snhz Lur 3: Idl Nozzl luid hnis Idl Nozzl low wih No Sprion (-D) - Qusi -D (slndr) pproximion - Idl gs ssumd ( ) mu + Opimum xpnsion: - or lss, >, ould driv mor forwrd

More information

A L A BA M A L A W R E V IE W

A L A BA M A L A W R E V IE W A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N

More information

Inventory Management Model with Quadratic Demand, Variable Holding Cost with Salvage value

Inventory Management Model with Quadratic Demand, Variable Holding Cost with Salvage value Asr Rsr Journl of Mngmn Sins ISSN 9 7 Vol. 8- Jnury Rs. J. Mngmn Si. Invnory Mngmn Modl wi udri Dmnd Vril Holding Cos wi Slvg vlu Mon R. nd Vnkswrlu R. F-Civil Dp of Mmis Collg of Miliry Enginring Pun

More information

Quality Improvement of Unbalanced Three-phase Voltages Rectification

Quality Improvement of Unbalanced Three-phase Voltages Rectification SEI 9 5 h Inrnionl Confrn: Ss of Elroni, hnologis of Inforion nd louniions Mrh -6, 9 UNISIA Quliy Ipron of Unlnd hr-phs ols Rifiion Fi Zhr AMAOUL *, Musph.RAOUFI * nd Mouly hr LAMCHICH * * Dprn of physis,

More information

More on FT. Lecture 10 4CT.5 3CT.3-5,7,8. BME 333 Biomedical Signals and Systems - J.Schesser

More on FT. Lecture 10 4CT.5 3CT.3-5,7,8. BME 333 Biomedical Signals and Systems - J.Schesser Mr n FT Lcur 4CT.5 3CT.3-5,7,8 BME 333 Bimdicl Signls nd Sysms - J.Schssr 43 Highr Ordr Diffrniin d y d x, m b Y b X N n M m N M n n n m m n m n d m d n m Y n d f n [ n ] F d M m bm m X N n n n n n m p

More information

ON A METHOD FOR FINDING THE NUMERICAL SOLUTION OF CAUCHY PROBLEM FOR 2D BURGERS EQUATION

ON A METHOD FOR FINDING THE NUMERICAL SOLUTION OF CAUCHY PROBLEM FOR 2D BURGERS EQUATION Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 ON A MEHOD FOR FINDING HE NUMERICAL SOLUION OF CAUCHY PROBLEM FOR D BURGERS EQUAION Mir Rsulo Prof. Been Uniersi Deprmen

More information

Solutions for Nonlinear Partial Differential Equations By Tan-Cot Method

Solutions for Nonlinear Partial Differential Equations By Tan-Cot Method IOSR Journl of Mhemics (IOSR-JM) e-issn: 78-578. Volume 5, Issue 3 (Jn. - Feb. 13), PP 6-11 Soluions for Nonliner Pril Differenil Equions By Tn-Co Mehod Mhmood Jwd Abdul Rsool Abu Al-Sheer Al -Rfidin Universiy

More information

Inverse Fourier Transform. Properties of Continuous time Fourier Transform. Review. Linearity. Reading Assignment Oppenheim Sec pp.289.

Inverse Fourier Transform. Properties of Continuous time Fourier Transform. Review. Linearity. Reading Assignment Oppenheim Sec pp.289. Convrgnc of ourir Trnsform Rding Assignmn Oppnhim Sc 42 pp289 Propris of Coninuous im ourir Trnsform Rviw Rviw or coninuous-im priodic signl x, j x j d Invrs ourir Trnsform 2 j j x d ourir Trnsform Linriy

More information

Life Science Journal 2014;11(9) An Investigation of the longitudinal fluctuations of viscoelastic cores

Life Science Journal 2014;11(9)   An Investigation of the longitudinal fluctuations of viscoelastic cores Lif Sin Journl (9) h://wwwlifiniom n Invigion of h longiuinl fluuion of violi or Kurnov Ni yg, Bjnov Vul Gmz Drmn of Gnrl Mh, Sumgi S Univriy, Sumgi, ZE 5, zrijn vul@gmilom r: I i nry o l rolm from ynmi

More information

The Procedure Abstraction Part II: Symbol Tables and Activation Records

The Procedure Abstraction Part II: Symbol Tables and Activation Records Th Produr Absrion Pr II: Symbol Tbls nd Aivion Rords Th Produr s Nm Sp Why inrodu lxil soping? Provids ompil-im mhnism for binding vribls Ls h progrmmr inrodu lol nms How n h ompilr kp rk of ll hos nms?

More information

1 Finite Automata and Regular Expressions

1 Finite Automata and Regular Expressions 1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o

More information

Generalized Projective Synchronization Using Nonlinear Control Method

Generalized Projective Synchronization Using Nonlinear Control Method ISSN 79-3889 (prin), 79-3897 (online) Inernionl Journl of Nonliner Siene Vol.8(9) No.,pp.79-85 Generlized Projeive Synhronizion Using Nonliner Conrol Mehod Xin Li Deprmen of Mhemis, Chngshu Insiue of Tehnology

More information

A modified hyperbolic secant distribution

A modified hyperbolic secant distribution Songklnkrin J Sci Tchnol 39 (1 11-18 Jn - Fb 2017 hp://wwwsjspsuch Originl Aricl A modifid hyprbolic scn disribuion Pnu Thongchn nd Wini Bodhisuwn * Dprmn of Sisics Fculy of Scinc Kssr Univrsiy Chuchk

More information

Section 2: The Z-Transform

Section 2: The Z-Transform Scion : h -rnsform Digil Conrol Scion : h -rnsform In linr discr-im conrol sysm linr diffrnc quion chrcriss h dynmics of h sysm. In ordr o drmin h sysm s rspons o givn inpu, such diffrnc quion mus b solvd.

More information

a b v a v b v c v = a d + bd +c d +ae r = p + a 0 s = r + b 0 4 ac + ad + bc + bd + e 5 = a + b = q 0 c + qc 0 + qc (a) s v (b)

a b v a v b v c v = a d + bd +c d +ae r = p + a 0 s = r + b 0 4 ac + ad + bc + bd + e 5 = a + b = q 0 c + qc 0 + qc (a) s v (b) Outlin MULTIPLE-LEVEL LOGIC OPTIMIZATION Gionni D Mihli Stnfor Unirsit Rprsnttions. Tonom of optimition mthos: { Gols: r/l. { Algorithms: lgri/booln. { Rul-s mthos. Empls of trnsformtions. Booln n lgri

More information

An Optimal Ordering Policy for Inventory Model with. Non-Instantaneous Deteriorating Items and. Stock-Dependent Demand

An Optimal Ordering Policy for Inventory Model with. Non-Instantaneous Deteriorating Items and. Stock-Dependent Demand Applid Mhmicl Scincs, Vol. 7, 0, no. 8, 407-4080 KA Ld, www.m-hikri.com hp://dx.doi.org/0.988/ms.0.56 An piml rdring Policy for nvnory Modl wih Non-nsnnous rioring ms nd Sock-pndn mnd Jsvindr Kur, jndr

More information

ELECTRIC VELOCITY SERVO REGULATION

ELECTRIC VELOCITY SERVO REGULATION ELECIC VELOCIY SEVO EGULAION Gorg W. Younkin, P.E. Lif FELLOW IEEE Indusril Conrols Consuling, Di. Bulls Ey Mrking, Inc. Fond du Lc, Wisconsin h prformnc of n lcricl lociy sro is msur of how wll h sro

More information

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]

S.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15] S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:

More information

P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

More information

Jonathan Turner Exam 2-10/28/03

Jonathan Turner Exam 2-10/28/03 CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm

More information

KITAGAWA NC TILTING ROTARY TABLES

KITAGAWA NC TILTING ROTARY TABLES MOS 101 (S)120 140 (S)181 182 (S)251 (S)321 OIMM IIN XIS SIN ROI N OMON N N IMININ IION S SIONR IIN N ROR MOOR SS OM SIN MINIMIS INR- RN ROM IR R MOOR OR IS IR-R O RN MOISR N ORIN MRIS ROM NRIN OR O IS

More information

CSE 421 Algorithms. Warmup. Dijkstra s Algorithm. Single Source Shortest Path Problem. Construct Shortest Path Tree from s

CSE 421 Algorithms. Warmup. Dijkstra s Algorithm. Single Source Shortest Path Problem. Construct Shortest Path Tree from s CSE Alorihm Rihr Anron Dijkr lorihm Sinl Sor Shor Ph Prolm Gin rph n r r Drmin in o ry r rom Iniy hor ph o h r Epr onily hor ph r Eh r h poinr o pror on hor ph Conr Shor Ph Tr rom Wrmp - - I P i hor ph

More information

Air Filter 90-AF30 to 90-AF60

Air Filter 90-AF30 to 90-AF60 Ai il -A o -A6 Ho o Od A /Smi-sndd: Sl on h fo o. /Smi-sndd symol: Whn mo hn on spifiion is uid, indi in lphnumi od. Exmpl) -A-- Sis ompil ih sondy is Mil siion Smi-sndd Thd yp Po siz Mouning lo diion

More information

PHA Second Exam. Fall 2007

PHA Second Exam. Fall 2007 PHA 527 Scond Exm Fll 2007 On my honor, I hv nihr givn nor rcivd unuhorizd id in doing his ssignmn. Nm Pu ll nswrs on h bubbl sh OAL /30 ps Qusion S I (ru or Fls) (5 poins) ru (A) or Fls (B). On h bubbl

More information

Supporting Online Materials for

Supporting Online Materials for Suppoing Onlin Mils o Flxibl Schbl nspn Mgn- Cbon Nnoub hin Film Loudspks Lin Xio*, Zhuo Chn*, Chn Fng, Ling Liu, Zi-Qio Bi, Yng Wng, Li Qin, Yuying Zhng, Qunqing Li, Kili Jing**, nd Shoushn Fn** Dpmn

More information

Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011

Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011 plc Trnorm Nionl Chio Tung Univriy Chun-Jn Ti /9/ Trnorm o Funcion Som opror rnorm uncion ino nohr uncion: d Dirniion: x x, or Dx x dx x Indini Ingrion: x dx c Dini Ingrion: x dx 9 A uncion my hv nicr

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

Global analysis of a delay virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response

Global analysis of a delay virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response Glol nlsis of del irus dnmis model wih Beddingon-DeAngelis inidene re nd CTL immune response Lish Ling Shool of Mhemis nd Phsis Uniersi of Siene nd Tehnolog Beijing Beijing Chin 369558953@63om Yongmei

More information

T h e C S E T I P r o j e c t

T h e C S E T I P r o j e c t T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T

More information

h : sh +i F J a n W i m +i F D eh, 1 ; 5 i A cl m i n i sh» si N «q a : 1? ek ser P t r \. e a & im a n alaa p ( M Scanned by CamScanner

h : sh +i F J a n W i m +i F D eh, 1 ; 5 i A cl m i n i sh» si N «q a : 1? ek ser P t r \. e a & im a n alaa p ( M Scanned by CamScanner m m i s t r * j i ega>x I Bi 5 n ì r s w «s m I L nk r n A F o n n l 5 o 5 i n l D eh 1 ; 5 i A cl m i n i sh» si N «q a : 1? { D v i H R o s c q \ l o o m ( t 9 8 6) im a n alaa p ( M n h k Em l A ma

More information

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l

More information

o C *$ go ! b», S AT? g (i * ^ fc fa fa U - S 8 += C fl o.2h 2 fl 'fl O ' 0> fl l-h cvo *, &! 5 a o3 a; O g 02 QJ 01 fls g! r«'-fl O fl s- ccco

o C *$ go ! b», S AT? g (i * ^ fc fa fa U - S 8 += C fl o.2h 2 fl 'fl O ' 0> fl l-h cvo *, &! 5 a o3 a; O g 02 QJ 01 fls g! r«'-fl O fl s- ccco > p >>>> ft^. 2 Tble f Generl rdnes. t^-t - +«0 -P k*ph? -- i t t i S i-h l -H i-h -d. *- e Stf H2 t s - ^ d - 'Ct? "fi p= + V t r & ^ C d Si d n. M. s - W ^ m» H ft ^.2. S'Sll-pl e Cl h /~v S s, -P s'l

More information

Last 4 Digits of USC ID:

Last 4 Digits of USC ID: Chemistry 05 B Practice Exam Dr. Jessica Parr First Letter of last Name PLEASE PRINT YOUR NAME IN BLOCK LETTERS Name: Last 4 Digits of USC ID: Lab TA s Name: Question Points Score Grader 8 2 4 3 9 4 0

More information

8. Relax and do well.

8. Relax and do well. CHEM 1314 3;30 pm Theory Exam III John III. Gelder November 13, 2002 Name TA's Name Lab Section INSTRUCTIONS: 1. This examination consists of a total of 8 different pages. The last page include a periodic

More information

02/05/09 Last 4 Digits of USC ID: Dr. Jessica Parr

02/05/09 Last 4 Digits of USC ID: Dr. Jessica Parr Chemistry 05 B First Letter of PLEASE PRINT YOUR NAME IN BLOCK LETTERS Exam last Name Name: 02/05/09 Last 4 Digits of USC ID: Dr. Jessica Parr Lab TA s Name: Question Points Score Grader 2 2 9 3 9 4 2

More information

PHA Second Exam. Fall On my honor, I have neither given nor received unauthorized aid in doing this assignment.

PHA Second Exam. Fall On my honor, I have neither given nor received unauthorized aid in doing this assignment. Nm: UFI #: PHA 527 Scond Exm Fll 20 On my honor, I hv nihr givn nor rcivd unuhorizd id in doing his ssignmn. Nm Pu ll nswrs on h bubbl sh OAL /200 ps Nm: UFI #: Qusion S I (ru or Fls) (5 poins) ru (A)

More information

Elsayed M. E. Zayed 1 + (Received April 4, 2012, accepted December 2, 2012)

Elsayed M. E. Zayed 1 + (Received April 4, 2012, accepted December 2, 2012) ISSN 746-7659, England, UK Journal of Information and Computing Science Vol. 8, No., 03, pp. 003-0 A modified (G'/G)- expansion method and its application for finding hyperbolic, trigonometric and rational

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

New Exact Traveling Wave Solutions of Nonlinear Evolution Equations with Variable Coefficients

New Exact Traveling Wave Solutions of Nonlinear Evolution Equations with Variable Coefficients Studies in Nonlinear Sciences (: 33-39, ISSN -39 IDOSI Publications, New Exact Traveling Wave Solutions of Nonlinear Evolution Equations with Variable Coefficients M.A. Abdou, E.K. El-Shewy and H.G. Abdelwahed

More information

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS

ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h

More information

Pupil / Class Record We can assume a word has been learned when it has been either tested or used correctly at least three times.

Pupil / Class Record We can assume a word has been learned when it has been either tested or used correctly at least three times. 2 Pupi / Css Rr W ssum wr hs b r wh i hs b ihr s r us rry s hr ims. Nm: D Bu: fr i bus brhr u firs hf hp hm s uh i iv iv my my mr muh m w ih w Tik r pp push pu sh shu sisr s sm h h hir hr hs im k w vry

More information

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm

More information

The model proposed by Vasicek in 1977 is a yield-based one-factor equilibrium model given by the dynamic

The model proposed by Vasicek in 1977 is a yield-based one-factor equilibrium model given by the dynamic h Vsick modl h modl roosd by Vsick in 977 is yild-bsd on-fcor quilibrium modl givn by h dynmic dr = b r d + dw his modl ssums h h shor r is norml nd hs so-clld "mn rvring rocss" (undr Q. If w u r = b/,

More information

Revisiting what you have learned in Advanced Mathematical Analysis

Revisiting what you have learned in Advanced Mathematical Analysis Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr

More information

Week 06 Discussion Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 8

Week 06 Discussion Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 8 STAT W 6 Discussion Fll 7..-.- If h momn-gnring funcion of X is M X ( ), Find h mn, vrinc, nd pmf of X.. Suppos discr rndom vribl X hs h following probbiliy disribuion: f ( ) 8 7, f ( ),,, 6, 8,. ( possibl

More information

POLYTECHNIC OF NAMIBIA

POLYTECHNIC OF NAMIBIA POLYTECHNIC OF NAMIBIA DEPARTMENT OF HEALTH SCIENCES BACHELOR OF ENVIRONMENTAL HEALTH SCIENCES HEALTH SCIENCE CHEMISTRY (HSC 511S) NQF level 5 SECOND OPPORTUNITY EXAMINATION November 2014 TIME: MARKS:

More information

Faculty of Natural and Agricultural Sciences Chemistry Department. Semester Test 1. Analytical Chemistry CMY 283. Time: 120 min Marks: 100 Pages: 6

Faculty of Natural and Agricultural Sciences Chemistry Department. Semester Test 1. Analytical Chemistry CMY 283. Time: 120 min Marks: 100 Pages: 6 Faculty of Natural and Agricultural Sciences Chemistry Department Semester Test 1 Analytical Chemistry CMY 283 Date: 5 September 2016 Lecturers : Prof P Forbes, Dr Laurens, Mr SA Nsibande Time: 120 min

More information

Faculty of Natural and Agricultural Sciences Chemistry Department. Semester Test 1 MEMO. Analytical Chemistry CMY 283

Faculty of Natural and Agricultural Sciences Chemistry Department. Semester Test 1 MEMO. Analytical Chemistry CMY 283 Faculty of Natural and Agricultural Sciences Chemistry Department Semester Test 1 MEMO Analytical Chemistry CMY 283 Date: 5 September 2016 Lecturers : Prof P Forbes, Dr Laurens, Mr SA Nsibande Time: 90

More information

Chapter 4 Multifield Surface Bone Remodeling

Chapter 4 Multifield Surface Bone Remodeling hr Mulifild Surf on Rmodling In hr, h horil nd numril rul of inrnl on rmodling wr rnd. Exnion o mulifild urf on rmodling i diud in hi hr. horil rdiion of urf on rmodling in h dihyi of h long on undr vriou

More information

Computer Aided Geometric Design

Computer Aided Geometric Design Copue Aided Geoei Design Geshon Ele, Tehnion sed on ook Cohen, Riesenfeld, & Ele Geshon Ele, Tehnion Definiion 3. The Cile Given poin C in plne nd nue R 0, he ile ih ene C nd dius R is defined s he se

More information

APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS

APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS Intrntionl Journl o Sintii nd Rrh Publition Volum, Iu 5, M ISSN 5-353 APPLICATIONS OF THE LAPLACE-MELLIN INTEGRAL TRANSFORM TO DIFFERNTIAL EQUATIONS S.M.Khirnr, R.M.Pi*, J.N.Slun** Dprtmnt o Mthmti Mhrhtr

More information

-Z ONGRE::IONAL ACTION ON FY 1987 SUPPLEMENTAL 1/1

-Z ONGRE::IONAL ACTION ON FY 1987 SUPPLEMENTAL 1/1 -Z-433 6 --OGRE::OA ATO O FY 987 SUPPEMETA / APPR)PRATO RfQUEST PAY AD PROGRAM(U) DE ARTMET OF DEES AS O' D 9J8,:A:SF ED DEFS! WA-H ODM U 7 / A 25 MRGOPf RESOUTO TEST HART / / AD-A 83 96 (~Go w - %A uj

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

P a g e 3 6 of R e p o r t P B 4 / 0 9

P a g e 3 6 of R e p o r t P B 4 / 0 9 P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J

More information

Applications of these ideas. CS514: Intermediate Course in Operating Systems. Problem: Pictorial version. 2PC is a good match! Issues?

Applications of these ideas. CS514: Intermediate Course in Operating Systems. Problem: Pictorial version. 2PC is a good match! Issues? CS514: Inmi Co in Oing Sm Poo Kn imn K Ooki: T liion o h i O h h k h o Goi oool Dii monioing, h, n noiiion gmn oool, h 2PC n 3PC Unling hm: om hing n ong om o onin, om n mng ih k oi To, l look n liion

More information

Cylindrically Symmetric Marder Universe and Its Proper Teleparallel Homothetic Motions

Cylindrically Symmetric Marder Universe and Its Proper Teleparallel Homothetic Motions J. Bsi. Appl. i. Res. 4-5 4 4 TeRod Publiion IN 9-44 Journl of Bsi nd Applied ienifi Reserh www.erod.om Clindrill mmeri Mrder Universe nd Is Proper Teleprllel Homohei Moions Amjd Ali * Anwr Ali uhil Khn

More information

Shortest Paths. CSE 421 Algorithms. Bottleneck Shortest Path. Negative Cost Edge Preview. Compute the bottleneck shortest paths

Shortest Paths. CSE 421 Algorithms. Bottleneck Shortest Path. Negative Cost Edge Preview. Compute the bottleneck shortest paths Shor Ph CSE Alorihm Rihr Anron Lr 0- Minimm Spnnin Tr Ni Co E Dijkr lorihm m poii o For om ppliion, ni o mk n Shor ph no wll in i rph h ni o yl - - - Ni Co E Priw Topoloil Sor n or olin h hor ph prolm

More information

Solutions and Ions. Pure Substances

Solutions and Ions. Pure Substances Class #4 Solutions and Ions CHEM 107 L.S. Brown Texas A&M University Pure Substances Pure substance: described completely by a single chemical formula Fixed composition 1 Mixtures Combination of 2 or more

More information

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%

More information

Chapter 4 Circular and Curvilinear Motions

Chapter 4 Circular and Curvilinear Motions Chp 4 Cicul n Cuilin Moions H w consi picls moing no long sigh lin h cuilin moion. W fis su h cicul moion, spcil cs of cuilin moion. Anoh mpl w h l sui li is h pojcil..1 Cicul Moion Unifom Cicul Moion

More information

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps

More information