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

Size: px
Start display at page:

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

Transcription

1 Inrnionl hmil orum no 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 Sin IT Univrsi Vishnm.. Indi sr This r dvlos roxim nlil soluions of h gnrl form of o Volrr r rdor ssm using h rurion hniqu. Hr i is shown h rsuls of h rvious wor [] om riulr s of h rsn wor. hmis suj lssifiions: D 799 Kwords: rurion hniqu o Volrr r rdor * orrsonding uhor -mil ddrss: dvgrgim@gmil.om. Inroduion Th mhmil sud of r rdor ssms in oulion dnmis hs n h suj of svrl rn rs sring wih h wor of o Volrr. In h sud of non linr ssm of diffrnil quions suh s in h o Volrr quion nlil soluions r usull unnown. In his s

2 668 D.Vnu ol Ro nd Y... Thorni in ordr o nlz h hviour of h ssm w usull rsor o numrill ingrd hniqus suh s rurion hniqu. Th rurion hniqu dnds on h xisn of smll or lrg rmrs in h non linr rolms. Th gnrl form of quions whih w onsidrd hr is x& x x x. x& x x x hr x is r oulion nd x is rdor oulion whr r ll osiiv onsns wih ngiv fd i.. >. Vrm [] oind x soluions of h quions. x& x x x & x x. undr h ssumion. Wilson [6] gv h form of x soluions of. wih h ssumion nml whr r funions of im whr is onsn. urnsid [] gv h form of x soluions wih n ssumion & & K..urh nd D.V..Ro [] oind roxim nlil soluions of h quions. x& x x x. x & x x Ths r dvlos roxim nlil soluions of h gnrl o Volrr quions. using rurion hniqu. Hr i is shown h h soluions of. om riulr s of... rurion hod Th rurion mhod [] hs n usd widl in non linr mhnis. This mhod n lid o ir of firs ordr diffrnil quions of h. x & f x μ g x. & f x μ g x wih h iniil ondiions x nd. Th linr rms r wrin in funions f nd f. Th non linr rms r wrin in funions g nd g. Th rmr μ idll smll rmr is ssoid wih g nd g. Th soluion is found s owr sris in μ. This sris will onvrg ridl if μ is smll. In ri μ is inrodud rifiill. Th sris soluion sough is of h form

3 Soluions of h o Volrr r rdor ssm 669 x x μ x μ x.... μ μ... Th rms x nd r lld gnring rms nd r h x soluions of h linr quions x & f x & f x. Th rms x x r lld orrion rms. n imorn fur of h rurion mhod is h oh gnring nd orrion rms in h soluions r oind solving linr diffrnil quions onl.. liion o gnrl r rdor ssm of quions Th o Volrr quions. hv h quilirium oin. sids h quilirium oins Dfining h vrils x x. w n oin h diffrnil quions in nd s givn low. Th vrils nd rrsn r nd rdor oulion dviions from h quilirium vlu givn in quion. & &. Inroduing h rmr µ ino h nonlinr rms of h ov quions ling h rurion hniqu. & μ μ μ μ &. Soluions r sough in h form μ μ.... μ μ... Th rmr μ will s qul o uni fr h soluions r drmind. susiuing. ino. nd quing qul owrs of μ on oh sids w oin &.6 & & &.7

4 67 D.Vnu ol Ro nd Y... Thorni Th iniil ondiions r lid o h gnring soluions nd. Th diffrnil quions for orrsonding rms hn hv zro iniil ondiions. Th soluions of h linr quions.6 r oind s [ ].8 [ ].9 whr whr wih whr nd r iniil ondiions of nd rsivl. Hving drmind nd quions.7 w nx solvd for nd o ild.. whr Q R R

5 Soluions of h o Volrr r rdor ssm 67 Q Q R Q R R Q Q R nd Q R

6 67 D.Vnu ol Ro nd Y... Thorni Using hs firs orrion rms n rori soluion o. is. In rms of h originl r nd rdor oulions x nd x. h soluions r x. x. Susiuing in.. h quilirium oin of. is wih Thn w g roxim nlil soluions of. of h form x x whr i i onsns oind ing in. nd..

7 Soluions of h o Volrr r rdor ssm 67 Hr i is shown h h soluions of h o volrr quions. m riulr s of h gnrl o Volrr quions.. us of h ddiionl orrion rms h ov soluions n xd o mor ur nd vlid for lrgr dviions from h quilirium oin. Th ur of h soluions rurion mhod n imrovd furhr drmin sond nd highr ordr orrion rms.. Conlusions Th roxim nlil soluions of h gnrl o Volrr quions hv n drmind using rurion mhod. Th rurion mhod n lid o ohr non linr mhmil modls of oulion dnmis. Rfrns [] R.R.urnsid no on x soluions of h r-rdor quions ull. h iol [] W.J. CunningHm Inroduion o nonlinr nlsis rw Hill w Yor 98. [] K..ur nd D.V..Ro roxim nlil soluions of nrl o Volrr quions Journl of hmil nlsis nd liions w Yor Vol o [].. ilov n Inroduion of hmil olog Wil Inr sin w Yor 969. [] V.S.Vrm x Soluions of Sil r-rdor or Coming Sis Ssm ull h. iol [6].J.Wilson Vrm s r-rdor rolm ull h iol Rivd: ril

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

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

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

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

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

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

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

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

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

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

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

Exact Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics

Exact Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics ISSN 76-7659 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

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

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

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

Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.

Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline. Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.

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

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

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

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

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

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

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

International Journal on Recent and Innovation Trends in Computing and Communication ISSN: Volume: 5 Issue:

International Journal on Recent and Innovation Trends in Computing and Communication ISSN: Volume: 5 Issue: Inrnionl Journl on Rn nd Innovion rnds in Compuing nd Communiion ISSN: -869 Volum: Issu: 78 97 Dvlopmn of n EPQ Modl for Drioring Produ wih Sok nd Dmnd Dpndn Produion r undr Vril Crrying Cos nd Pril Bklogging

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

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

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

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

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

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

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

Price Dependent Quadratic Demand Inventory Models with Variable Holding Cost and Inflation Rate

Price Dependent Quadratic Demand Inventory Models with Variable Holding Cost and Inflation Rate Pric Dndn Qudric Dmnd nvnory Modls wi Vril Holding os nd nlion R SBN: 978-8-97-8-9 R. Vnswrlu Gim Univrsiy rngvjl_v@yoo.co.in M. S. Rddy BVSR Enginring ollg nvnsrinu@gmil.com n m is md o dvlo n invnory

More information

Right Angle Trigonometry

Right Angle Trigonometry Righ gl Trigoomry I. si Fs d Dfiiios. Righ gl gl msurig 90. Srigh gl gl msurig 80. u gl gl msurig w 0 d 90 4. omplmry gls wo gls whos sum is 90 5. Supplmry gls wo gls whos sum is 80 6. Righ rigl rigl wih

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

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

2. The Laplace Transform

2. The Laplace Transform Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin

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

Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =

Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = = L's rvs codol rol whr h v M s rssd rs o h rdo vrl. L { M } rrr v such h { M } Assu. { } { A M} { A { } } M < { } { } A u { } { } { A} { A} ( A) ( A) { A} A A { A } hs llows us o cosdr h cs wh M { } [ (

More information

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:

a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system: Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor.

More information

Charging of capacitor through inductor and resistor

Charging of capacitor through inductor and resistor cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.

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

Derivation of the differential equation of motion

Derivation of the differential equation of motion Divion of h iffnil quion of oion Fis h noions fin h will us fo h ivion of h iffnil quion of oion. Rollo is hough o -insionl isk. xnl ius of h ll isnc cn of ll (O) - IDU s cn of gviy (M) θ ngl of inclinion

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

ROBOTIC BACKHOE WITH HAPTIC DISPLAY

ROBOTIC BACKHOE WITH HAPTIC DISPLAY Dni Modling nd Conrol Dsign of ROBOTIC BACOE WIT APTIC DISPLAY Jo Frnl orgi Insiu of Thnolog Aril 4, 3 I. Inroduion A. Bground Th rdiionl hod o onrol hdruli quin hs n olishd wih h us of nul roorionl vlvs.

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

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

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

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

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

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

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if. Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[

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

Construction 11: Book I, Proposition 42

Construction 11: Book I, Proposition 42 Th Visul Construtions of Euli Constrution #11 73 Constrution 11: Book I, Proposition 42 To onstrut, in givn rtilinl ngl, prlllogrm qul to givn tringl. Not: Equl hr mns qul in r. 74 Constrution # 11 Th

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

Ratio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute

Ratio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute Raio-Produc T Exonnial Esimaor For Esimaing Fini Poulaion Man Using Informaion On Auxiliar Aribu Rajsh Singh, Pankaj hauhan, and Nirmala Sawan, School of Saisics, DAVV, Indor (M.P., India (rsinghsa@ahoo.com

More information

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1 Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

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

Major: All Engineering Majors. Authors: Autar Kaw, Luke Snyder

Major: All Engineering Majors. Authors: Autar Kaw, Luke Snyder Nolr Rgrsso Mjor: All Egrg Mjors Auhors: Aur Kw, Luk Sydr hp://urclhodsgusfdu Trsforg Nurcl Mhods Educo for STEM Udrgrdus 3/9/5 hp://urclhodsgusfdu Nolr Rgrsso hp://urclhodsgusfdu Nolr Rgrsso So populr

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

1 Introduction to Modulo 7 Arithmetic

1 Introduction to Modulo 7 Arithmetic 1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w

More information

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture: Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin

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

Ch 1.2: Solutions of Some Differential Equations

Ch 1.2: Solutions of Some Differential Equations Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of

More information

NUPOC STUDY GUIDE ANSWER KEY. Navy Recruiting Command

NUPOC STUDY GUIDE ANSWER KEY. Navy Recruiting Command NUPOC STUDY GUIDE ANSWE KEY N cruiing Commnd SOLUTIONS TO MATHEMATICS POBLEMS. Susiuing gin soluions sos,i is corrc nsr.. On - ln, coos o foci Fc, nd F -c, nd dno consn disnc s. Tn, using disnc formul

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

CHAPTER 9. Compressible Flow. Btu ft-lb lbm ft-lb c p = = ft-lb slug- R. slug- R. 1 k. p p. p v p v. = ρ ρ

CHAPTER 9. Compressible Flow. Btu ft-lb lbm ft-lb c p = = ft-lb slug- R. slug- R. 1 k. p p. p v p v. = ρ ρ CHPTER 9 Cmrssibl Flw 9 Bu f-lb lbm f-lb c 778 6 lbm- R Bu slug slug- R f-lb cv c R 6 76 96 96 slug- R Bu 7 lbm R f-lb slug- R Bu 778 f - lb slug lbm c 9 c cv + R c cv c + R r c R c R / ( ) 9 If s, Eq

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

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

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

CHAPTER 7. X and 2 = X

CHAPTER 7. X and 2 = X CHATR 7 Sco 7-7-. d r usd smors o. Th vrcs r d ; comr h S vrc hs cs / / S S Θ Θ Sc oh smors r usd mo o h vrcs would coclud h s h r smor wh h smllr vrc. 7-. [ ] Θ 7 7 7 7 7 7 [ ] Θ ] [ 7 6 Boh d r usd sms

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

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

Formal Concept Analysis

Formal Concept Analysis Forml Conpt Anlysis Conpt intnts s losd sts Closur Systms nd Implitions 4 Closur Systms 0.06.005 Nxt-Closur ws dvlopd y B. Gntr (984). Lt M = {,..., n}. A M is ltilly smllr thn B M, if B A if th smllst

More information

Reliability Analysis of a Bridge and Parallel Series Networks with Critical and Non- Critical Human Errors: A Block Diagram Approach.

Reliability Analysis of a Bridge and Parallel Series Networks with Critical and Non- Critical Human Errors: A Block Diagram Approach. Inrnaional Journal of Compuaional Sin and Mahmais. ISSN 97-3189 Volum 3, Numr 3 11, pp. 351-3 Inrnaional Rsarh Puliaion Hous hp://www.irphous.om Rliailiy Analysis of a Bridg and Paralll Sris Nworks wih

More information

CS 688 Pattern Recognition. Linear Models for Classification

CS 688 Pattern Recognition. Linear Models for Classification //6 S 688 Pr Rcogiio Lir Modls for lssificio Ø Probbilisic griv modls Ø Probbilisic discrimiiv modls Probbilisic Griv Modls Ø W o ur o robbilisic roch o clssificio Ø W ll s ho modls ih lir dcisio boudris

More information

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton Journl of Modrn hysics, 014, 5, 154-157 ublishd Onlin August 014 in SciRs. htt://www.scir.org/journl/jm htt://dx.doi.org/.436/jm.014.51415 Th Angulr Momnt Diol Momnts nd Gyromgntic Rtios of th Elctron

More information

Library Support. Netlist Conditioning. Observe Point Assessment. Vector Generation/Simulation. Vector Compression. Vector Writing

Library Support. Netlist Conditioning. Observe Point Assessment. Vector Generation/Simulation. Vector Compression. Vector Writing hpr 2 uomi T Prn Gnrion Fundmnl hpr 2 uomi T Prn Gnrion Fundmnl Lirry uppor Nli ondiioning Orv Poin mn Vor Gnrion/imulion Vor omprion Vor Wriing Figur 2- Th Ovrll Prn Gnrion Pro Dign-or-T or Digil I nd

More information

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review

CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht

More information

Mundell-Fleming I: Setup

Mundell-Fleming I: Setup Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)

More information

Lecture 2: Current in RC circuit D.K.Pandey

Lecture 2: Current in RC circuit D.K.Pandey Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging

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

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

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 ) AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc

More information

CHAPTER 9 Compressible Flow

CHAPTER 9 Compressible Flow CHPTER 9 Comrssibl Flow Char 9 / Comrssibl Flow Inroducion 9. c c cv + R. c kcv. c + R or c R k k Rk c k Sd of Sound 9.4 Subsiu Eq. 4.5.8 ino Eq. 4.5.7 and nglc onial nrgy chang: Q WS + + u~ u~. m ρ ρ

More information

Problem 1. Solution: = show that for a constant number of particles: c and V. a) Using the definitions of P

Problem 1. Solution: = show that for a constant number of particles: c and V. a) Using the definitions of P rol. Using t dfinitions of nd nd t first lw of trodynis nd t driv t gnrl rltion: wr nd r t sifi t itis t onstnt rssur nd volu rstivly nd nd r t intrnl nrgy nd volu of ol. first lw rlts d dq d t onstnt

More information

K x,y f x dx is called the integral transform of f(x). The function

K x,y f x dx is called the integral transform of f(x). The function APACE TRANSFORMS Ingrl rnform i priculr kind of mhmicl opror which ri in h nlyi of om boundry vlu nd iniil vlu problm of clicl Phyic. A funcion g dfind by b rlion of h form gy) = K x,y f x dx i clld h

More information

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9 Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function:

More information

Option Pricing When Changes of the Underlying Asset Prices Are Restricted

Option Pricing When Changes of the Underlying Asset Prices Are Restricted Journal of Mahmaial Finan 8-33 doi:.436/jmf..4 Publishd Onlin Augus (hp://www.sirp.org/journal/jmf) Opion Priing Whn Changs of h Undrling Ass Pris Ar Rsrid Absra Gorg J. Jiang Guanzhong Pan Li Shi 3 Univrsi

More information

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

PHA First Exam Fall On my honor, I have neither given nor received unauthorized aid in doing this assignment. PHA 527 Firs Exm Fll 20 On my honor, I hv nihr givn nor rcivd unuhorizd id in doing his ssignmn. Nm Qusion S/Poins I. 30 ps II. III. IV 20 ps 5 ps 5 ps V. 25 ps VI. VII. VIII. IX. 0 ps 0 ps 0 ps 35 ps

More information

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely . DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

More information

A Simple Method for Determining the Manoeuvring Indices K and T from Zigzag Trial Data

A Simple Method for Determining the Manoeuvring Indices K and T from Zigzag Trial Data Rind 8-- Wbsi: wwwshimoionsnl Ro 67, Jun 97, Dlf Univsiy of chnoloy, Shi Hydomchnics Lbooy, Mklw, 68 CD Dlf, h Nhlnds A Siml Mhod fo Dminin h Mnouvin Indics K nd fom Ziz il D JMJ Jouné Dlf Univsiy of chnoloy

More information

INTERQUARTILE RANGE. I can calculate variabilityinterquartile Range and Mean. Absolute Deviation

INTERQUARTILE RANGE. I can calculate variabilityinterquartile Range and Mean. Absolute Deviation INTERQUARTILE RANGE I cn clcul vribiliyinrquril Rng nd Mn Absolu Dviion 1. Wh is h grs common fcor of 27 nd 36?. b. c. d. 9 3 6 4. b. c. d.! 3. Us h grs common fcor o simplify h frcion!".!". b. c. d.

More information

VOLUME 13, ARTICLE 3, PAGES PUBLISHED 19 AUGUST DOI: /DemRes

VOLUME 13, ARTICLE 3, PAGES PUBLISHED 19 AUGUST DOI: /DemRes Dmogrhi Rsrh fr xdid onlin journl of r-rviwd rsrh nd ommnry in h oulion sins ublishd by h Mx Plnk Insiu for Dmogrhi Rsrh Konrd-Zus Sr. 1 D-18057 Rosok GERMANY www.dmogrhi-rsrh.org DEMOGRAPHIC RESEARCH

More information

Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example

Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)

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

EEE 303: Signals and Linear Systems

EEE 303: Signals and Linear Systems 33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =

More information

Chapter 3 Linear Equations of Higher Order (Page # 144)

Chapter 3 Linear Equations of Higher Order (Page # 144) Ma Modr Dirial Equaios Lcur wk 4 Jul 4-8 Dr Firozzama Darm o Mahmaics ad Saisics Arizoa Sa Uivrsi This wk s lcur will covr har ad har 4 Scios 4 har Liar Equaios o Highr Ordr Pag # 44 Scio Iroducio: Scod

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information