Chapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations
|
|
|
- Iris Townsend
- 6 years ago
- Views:
Transcription
1 Cter. Runge-Kutt nd Order Metod or Ordnr Derentl Eutons Ater redng ts cter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl eutons nd ow to use t to solve roblems. Wt s te Runge-Kutt nd order metod? Te Runge-Kutt nd order metod s numercl tecnue used to solve n ordnr derentl euton o te orm d ( ( d Onl rst order ordnr derentl eutons cn be solved b usng te Runge-Kutt nd order metod. In oter sectons we wll dscuss ow te Euler nd Runge-Kutt metods re used to solve ger order ordnr derentl eutons or couled (smultneous derentl eutons. How does one wrte rst order derentl euton n te bove orm? Emle Rewrte n Soluton d d d d d d d ( 5.e ( ( orm. ( 5.e ( 5.e d In ts cse..
2 .. Cter. Emle Rewrte n e Soluton (.e d d e d d sn( ( ( orm. d d sn( d sn( d e In ts cse sn( ( e ( 5 ( 5 ( 5 Runge-Kutt nd order metod Euler s metod s gven b ( ( were ( To understnd te Runge-Kutt nd order metod we need to derve Euler s metod rom te Tlor seres. d d d ( ( (... d! d! d ( ( '( ( ''( (... (!! As ou cn see te rst two terms o te Tlor seres ( re Euler s metod nd ence cn be consdered to be te Runge-Kutt st order metod. Te true error n te romton s gven b ( ( Et... (!! So wt would nd order metod ormul loo le. It would nclude one more term o te Tlor seres s ollows.
3 Runge-Kutt nd Order Metod.. (! Let us te generc emle o rst order ordnr derentl euton d e ( 5 d ( e Now snce s uncton o ( ( d (5 d ( e [( e ]( e e ( ( e 5e 9 Te nd order ormul or te bove emle would be! ( ( 5 e e 9! However we lred see te dcult o vng to nd ( n te bove metod. Wt Runge nd Kutt dd ws wrte te nd order metod s ( (6 were ( ( (7 Ts orm llows one to te dvntge o te nd order metod wtout vng to clculte (. So ow do we nd te unnowns nd. Wtout roo (see Aend or roo eutng Euton ( nd (6 gves tree eutons. Snce we ve eutons nd unnowns we cn ssume te vlue o one o te unnowns. Te oter tree wll ten be determned rom te tree eutons. Generll te vlue o s cosen to evlute te oter tree constnts. Te tree vlues generll used or re nd nd re nown s Heun s Metod te mdont metod nd Rlston s metod resectvel.
4 .. Cter. Heun s Metod Here s cosen gvng resultng n ( were ( (9 ( (9b Ts metod s grcll elned n Fgure. Sloe ( Sloe redcted Averge Sloe [ ( ( ] Fgure Runge-Kutt nd order metod (Heun s metod. Mdont Metod Here s cosen gvng resultng n ( were
5 Runge-Kutt nd Order Metod..5 ( (b Rlston s Metod Here s cosen gvng resultng n ( were ( ( (b Emle A bll t K s llowed to cool down n r t n mbent temerture o K. Assumng et s lost onl due to rdton te derentl euton or te temerture o te bll s gven b dθ -.67 ( θ dt were θ s n K nd t n seconds. Fnd te temerture t t seconds usng Runge- Kutt nd order metod. Assume ste sze o seconds. Soluton.67 ( θ ( t θ.67 ( θ dθ dt Per Heun s metod gven b Eutons ( nd (9 θ θ θ ( t ( t θ t θ θ ( ( t θ o
6 ..6 Cter. (.67 (.5579 ( t θ ( (.5579 ( 6.9 ( θ θ (.5579 (.7595 ( K t t θ 655.6K θ ( t ( ( ( t θ ( ( θ θ (.69 ( ( K θ θ ( 5. 7 K Te results rom Heun s metod re comred wt ect results n Fgure. Te ect soluton o te ordnr derentl euton s gven b te soluton o nonlner euton s θ.959ln.59 tn θ Te soluton to ts nonlner euton t t s s θ ( K (.θ.67 t. 9
7 Runge-Kutt nd Order Metod..7 Temerture θ (K - Ect 5 Tme t(sec Fgure Heun s metod results or derent ste szes. Usng smller ste sze would ncrese te ccurc o te result s gven n Tble nd Fgure below. Tble Eect o ste sze or Heun s metod E % Ste sze θ ( t t Temerture θ ( Ste sze Fgure Eect o ste sze n Heun s metod.
8 .. Cter. In Tble Euler s metod nd te Runge-Kutt nd order metod results re sown s uncton o ste sze Tble Comrson o Euler nd te Runge-Kutt metods Ste sze θ ( Euler Heun Mdont Rlston wle n Fgure te comrson s sown over te rnge o tme. θ(k Temerture 9 7 Anltcl Mdont Rlston Heun 6 Euler Tme t (sec Fgure Comrson o Euler nd Runge Kutt metods wt ect results over tme. How do tese tree metods comre wt results obtned we ound ( drectl? O course we now tt snce we re ncludng te rst tree terms n te seres te soluton s olnoml o order two or less (tt s udrtc lner or constnt n o te tree metods re ect. But or n oter cse te results wll be derent. Let us te te emle o d e ( 5. d te rst tree terms o te Tlor seres gves I we drectl nd
9 Runge-Kutt nd Order Metod..9 were! ( ( ( e ( 5e 9 For ste sze o. usng Heun s metod we nd (.6. 9 Te ect soluton e e gves (.6 (.6 (.6 e e.969 Ten te bsolute reltve true error s t % For te sme roblem te results rom Euler s metod nd te tree Runge-Kutt metods re gven n Tble. Tble Comrson o Euler s nd Runge-Kutt nd order metods (.6 Ect Euler Drect nd Heun Mdont Rlston Vlue % t Aend A How do we get te nd order Runge-Kutt metod eutons? We wrote te nd order Runge-Kutt eutons wtout roo to solve d ( ( (A. d s ( (A. were ( (A. nd ( (A.b
10 .. Cter. (A. Te dvntge o usng nd order Runge-Kutt metod eutons s bsed on not vng to nd te dervtve o smbolcll n te ordnr derentl euton So ow do we get te bove tree Eutons (A.? Ts s te ueston tt s nswered n ts Aend. Wrtng out te rst tree terms o Tlor seres re! O d d d d (A.5 were Snce d d we cn rewrte te Tlor seres s! O (A.6 Now d d. (A.7 Hence! O d d O (A. Now te term used n te Runge-Kutt nd order metod or cn be wrtten s Tlor seres o two vrbles wt te rst tree terms s O (A.9 Hence O O
11 Runge-Kutt nd Order Metod.. Eutng te terms n Euton (A. nd Euton (A. we get (A. ORDINARY DIFFERENTIAL EQUATIONS Toc Runge nd Order Metod or Ordnr Derentl Eutons Summr Tetboo notes on Runge nd order metod or ODE Mjor Generl Engneerng Autors Autr Kw Lst Revsed October Web Ste tt://numerclmetods.eng.us.edu
ORDINARY DIFFERENTIAL EQUATIONS
6 ORDINARY DIFFERENTIAL EQUATIONS Introducton Runge-Kutt Metods Mult-step Metods Sstem o Equtons Boundr Vlue Problems Crcterstc Vlue Problems Cpter 6 Ordnr Derentl Equtons / 6. Introducton In mn engneerng
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
ORDINARY DIFFERENTIAL EQUATIONS EULER S METHOD
Numercal Analss or Engneers German Jordanan Unverst ORDINARY DIFFERENTIAL EQUATIONS We wll eplore several metods o solvng rst order ordnar derental equatons (ODEs and we wll sow ow tese metods can be appled
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
Computation of Fifth Degree of Spline Function Model by Using C++ Programming
www.ijci.org 89 Computton o Ft Degree o plne Functon Model b Usng C Progrmmng Frdun K. Hml, Aln A. Abdull nd Knd M. Qdr Mtemtcs Dept, Unverst o ulmn, ulmn, IRAQ Mtemtcs Dept, Unverst o ulmn, ulmn, IRAQ
Chapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
Model Fitting and Robust Regression Methods
Dertment o Comuter Engneerng Unverst o Clorn t Snt Cruz Model Fttng nd Robust Regresson Methods CMPE 64: Imge Anlss nd Comuter Vson H o Fttng lnes nd ellses to mge dt Dertment o Comuter Engneerng Unverst
Topic 6b Finite Difference Approximations
/8/8 Course Instructor Dr. Rymond C. Rump Oice: A 7 Pone: (95) 747 6958 E Mil: rcrump@utep.edu Topic 6b Finite Dierence Approximtions EE 486/5 Computtionl Metods in EE Outline Wt re inite dierence pproximtions?
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
FORMULAE FOR FINITE DIFFERENCE APPROXIMATIONS, QUADRATURE AND LINEAR MULTISTEP METHODS
Jourl o Mtemtcl Sceces: Advces d Alctos Volume Number - Pges -9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS QUADRATURE AND LINEAR MULTISTEP METHDS RAMESH KUMAR MUTHUMALAI Dertmet o Mtemtcs D G Vsv College
DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
Some Unbiased Classes of Estimators of Finite Population Mean
Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: 3 4767 P-IN: 3-4759 Www.Ijms.Org Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs,
Chapter Direct Method of Interpolation More Examples Chemical Engineering
hter 5. Direct Method of Interoltion More Exmles hemicl Engineering Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t
CISE 301: Numerical Methods Lecture 5, Topic 4 Least Squares, Curve Fitting
CISE 3: umercl Methods Lecture 5 Topc 4 Lest Squres Curve Fttng Dr. Amr Khouh Term Red Chpter 7 of the tetoo c Khouh CISE3_Topc4_Lest Squre Motvton Gven set of epermentl dt 3 5. 5.9 6.3 The reltonshp etween
Introduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
GAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
Lecture 26 Finite Differences and Boundary Value Problems
4//3 Leture 6 Fnte erenes and Boundar Value Problems Numeral derentaton A nte derene s an appromaton o a dervatve - eample erved rom Talor seres 3 O! Negletng all terms ger tan rst order O O Tat s te orward
PLK VICWOOD K.T. CHONG SIXTH FORM COLLEGE Form Six AL Physics Optical instruments
AL Pysics/pticl instruments/p.1 PLK VICW K.T. CHNG SIXTH FRM CLLEGE Form Six AL Pysics pticl Instruments pticl instruments Mgniying glss Microscope Rercting telescope Grting spectrometer Qulittive understnding
Remember: Project Proposals are due April 11.
Bonformtcs ecture Notes Announcements Remember: Project Proposls re due Aprl. Clss 22 Aprl 4, 2002 A. Hdden Mrov Models. Defntons Emple - Consder the emple we tled bout n clss lst tme wth the cons. However,
Chapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
ME 501A Seminar in Engineering Analysis Page 1
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 More oundr-vlue Prolems nd genvlue Prolems n Os Lrr retto Menl ngneerng 5 Semnr n ngneerng nlss ovemer 9, 7 Outlne Revew oundr-vlue prolems Soot
OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
The proof is a straightforward probabilistic extension of Palfrey and Rosenthal (1983). Note that in our case we assume N
PROOFS OF APPDIX A Proo o PROPOSITIO A pure strtey Byesn-s equlbr n te PUprtcpton me wtout lled oters: Te proo s strtorwrd blstc etenson o Plrey nd Rosentl 983 ote tt n our cse we ssume A B : It s esy
Lesson 16: Basic Control Modes
0/8/05 Lesson 6: Basc Control Modes ET 438a Automatc Control Systems Technology lesson6et438a.tx Learnng Objectves Ater ths resentaton you wll be able to: Descrbe the common control modes used n analog
The Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
Chapter Direct Method of Interpolation More Examples Electrical Engineering
Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o
Mathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
Shuai Dong. Isaac Newton. Gottfried Leibniz
Computatonal pyscs Sua Dong Isaac Newton Gottred Lebnz Numercal calculus poston dervatve ntegral v velocty dervatve ntegral a acceleraton Numercal calculus Numercal derentaton Numercal ntegraton Roots
CENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
Quiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
Section 4.7 Inverse Trigonometric Functions
Section 7 Inverse Trigonometric Functions 89 9 Domin: 0, q Rnge: -q, q Zeros t n, n nonnegtive integer 9 Domin: -q, 0 0, q Rnge: -q, q Zeros t, n non-zero integer Note: te gr lso suggests n te end-bevior
Numerical Differentiation
Part 5 Capter 19 Numercal Derentaton PowerPonts organzed by Dr. Mcael R. Gustason II, Duke Unversty Revsed by Pro. Jang, CAU All mages copyrgt Te McGraw-Hll Companes, Inc. Permsson requred or reproducton
36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to
ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to
Jens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
, rst we solve te PDE's L ad L ad n g g (x) = ; = ; ; ; n () (x) = () Ten, we nd te uncton (x), te lnearzng eedbac and coordnates transormaton are gve
Freedom n Coordnates Transormaton or Exact Lnearzaton and ts Applcaton to Transent Beavor Improvement Kenj Fujmoto and Tosaru Suge Dvson o Appled Systems Scence, Kyoto Unversty, Uj, Kyoto, Japan suge@robotuassyoto-uacjp
Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas
Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous
consider in the case of 1) internal resonance ω 2ω and 2) external resonance Ω ω and small damping
consder n the cse o nternl resonnce nd externl resonnce Ω nd smll dmpng recll rom "Two_Degs_Frdm_.ppt" tht θ + μ θ + θ = θφ + cos Ω t + τ where = k α α nd φ + μ φ + φ = θ + cos Ω t where = α τ s constnt
Physics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
: 5: ) A
Revew 1 004.11.11 Chapter 1: 1. Elements, Varable, and Observatons:. Type o Data: Qualtatve Data and Quanttatve Data (a) Qualtatve data may be nonnumerc or numerc. (b) Quanttatve data are always numerc.
General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation
General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton 1 1 1 + = d d to solve or d o you should rst rewrte t as 1 1 1 = d
Outline. Review Numerical Approach. Schedule for April and May. Review Simple Methods. Review Notation and Order
Sstes of Ordnar Dfferental Equatons Aprl, Solvng Sstes of Ordnar Dfferental Equatons Larr Caretto Mecancal Engneerng 9 Nuercal Analss of Engneerng Sstes Aprl, Outlne Revew bascs of nuercal solutons of
4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
Homework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given:
Homework 9 Solutons PROBLEM ONE 1 (Exercses from the book, th edton,, 1-) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE Soluton: 5! (b) SCHOOL Soluton:! (c) SALESPERSONS Soluton:
The Karush-Kuhn-Tucker. Nuno Vasconcelos ECE Department, UCSD
e Karus-Kun-ucker condtons and dualt Nuno Vasconcelos ECE Department, UCSD Optmzaton goal: nd mamum or mnmum o a uncton Denton: gven unctons, g, 1,...,k and, 1,...m dened on some doman Ω R n mn w, w Ω
Endogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract
Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous
Math Week 5 concepts and homework, due Friday February 10
Mt 2280-00 Week 5 concepts nd omework, due Fridy Februry 0 Recll tt ll problems re good for seeing if you cn work wit te underlying concepts; tt te underlined problems re to be nded in; nd tt te Fridy
Substitution Matrices and Alignment Statistics. Substitution Matrices
Susttuton Mtrces nd Algnment Sttstcs BMI/CS 776 www.ostt.wsc.edu/~crven/776.html Mrk Crven crven@ostt.wsc.edu Ferur 2002 Susttuton Mtrces two oulr sets of mtrces for roten seuences PAM mtrces [Dhoff et
Effects of polarization on the reflected wave
Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton
UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 2017/2018 FINITE ELEMENT AND DIFFERENCE SOLUTIONS
OCD0 UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 07/08 FINITE ELEMENT AND DIFFERENCE SOLUTIONS MODULE NO. AME6006 Date: Wednesda 0 Ma 08 Tme: 0:00
Multivariate Ratio Estimator of the Population Total under Stratified Random Sampling
Open Journal of Statstcs, 0,, 300-304 ttp://dx.do.org/0.436/ojs.0.3036 Publsed Onlne July 0 (ttp://www.scrp.org/journal/ojs) Multvarate Rato Estmator of te Populaton Total under Stratfed Random Samplng
Complex Numbers, Signals, and Circuits
Complex Numbers, Sgnals, and Crcuts 3 August, 009 Complex Numbers: a Revew Suppose we have a complex number z = x jy. To convert to polar form, we need to know the magntude of z and the phase of z. z =
INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING Dundgl, Hyderbd - 5 3 FRESHMAN ENGINEERING TUTORIAL QUESTION BANK Nme : MATHEMATICS II Code : A6 Clss : II B. Te II Semester Brn : FRESHMAN ENGINEERING Yer : 5 Fulty
Applied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
Fundamental Theorem of Calculus
Funmentl Teorem of Clculus Liming Png 1 Sttement of te Teorem Te funmentl Teorem of Clculus is one of te most importnt teorems in te istory of mtemtics, wic ws first iscovere by Newton n Leibniz inepenently.
Chapter 6. Operational Amplifier. inputs can be defined as the average of the sum of the two signals.
6 Operatonal mpler Chapter 6 Operatonal mpler CC Symbol: nput nput Output EE () Non-nvertng termnal, () nvertng termnal nput mpedance : Few mega (ery hgh), Output mpedance : Less than (ery low) Derental
Section 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
Numbers Related to Bernoulli-Goss Numbers
ursh Journl of Anlyss n Nuber heory, 4, Vol., No., -8 Avlble onlne t htt://ubs.sceub.co/tnt///4 Scence n Eucton Publshng OI:.69/tnt---4 Nubers Relte to Bernoull-Goss Nubers Mohe Oul ouh Benough * érteent
12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
Numerical Solution of Ordinary Differential Equations
College of Engneerng and Computer Scence Mecancal Engneerng Department otes on Engneerng Analss Larr Caretto ovember 9, 7 Goal of tese notes umercal Soluton of Ordnar Dfferental Equatons ese notes were
Solubilities and Thermodynamic Properties of SO 2 in Ionic
Solubltes nd Therodync Propertes of SO n Ionc Lquds Men Jn, Yucu Hou, b Weze Wu, *, Shuhng Ren nd Shdong Tn, L Xo, nd Zhgng Le Stte Key Lbortory of Checl Resource Engneerng, Beng Unversty of Checl Technology,
MATH1013 Tutorial 12. Indefinite Integrals
MATH Tutoril Indefinite Integrls The indefinite integrl f() d is to look for fmily of functions F () + C, where C is n rbitrry constnt, with the sme derivtive f(). Tble of Indefinite Integrls cf() d c
Outline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III
Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order
4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
Finite Difference Method
7/0/07 Instructor r. Ramond Rump (9) 747 698 rcrump@utep.edu EE 337 Computatonal Electromagnetcs (CEM) Lecture #0 Fnte erence Method Lecture 0 These notes ma contan coprghted materal obtaned under ar use
LAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION IN A TWO-LAYERED SLAB
Journl of Appled Mthemtcs nd Computtonl Mechncs 5, 4(4), 5-3 www.mcm.pcz.pl p-issn 99-9965 DOI:.75/jmcm.5.4. e-issn 353-588 LAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION
Advanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)
Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed
Math1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure
Stratified Extreme Ranked Set Sample With Application To Ratio Estimators
Journl of Modern Appled Sttstcl Metods Volume 3 Issue Artcle 5--004 Strtfed Extreme Rned Set Smple Wt Applcton To Rto Estmtors Hn M. Smw Sultn Qboos Unversty, smw@squ.edu.om t J. Sed Sultn Qboos Unversty
MAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
Chapter Direct Method of Interpolation More Examples Civil Engineering
Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this
Force = F Piston area = A
CHAPTER III Ths chapter s an mportant transton between the propertes o pure substances and the most mportant chapter whch s: the rst law o thermodynamcs In ths chapter, we wll ntroduce the notons o heat,
Lecture 36. Finite Element Methods
CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht. In the lst clss, we dscussed on the ppromte methods
Module 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
Chapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
CHAPTER 4d. ROOTS OF EQUATIONS
CHAPTER 4d. ROOTS OF EQUATIONS A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng by Dr. Ibrahm A. Assakka Sprng 00 ENCE 03 - Computaton Methods n Cvl Engneerng II Department o
Chemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W
ES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
Dennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1
Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:
A new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
5 The Laplace Equation in a convex polygon
5 Te Laplace Equaton n a convex polygon Te most mportant ellptc PDEs are te Laplace, te modfed Helmoltz and te Helmoltz equatons. Te Laplace equaton s u xx + u yy =. (5.) Te real and magnary parts of an
Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
Math 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
Module 11 Design of Joints for Special Loading. Version 2 ME, IIT Kharagpur
Module 11 Desgn o Jonts or Specal Loadng Verson ME, IIT Kharagpur Lesson 1 Desgn o Eccentrcally Loaded Bolted/Rveted Jonts Verson ME, IIT Kharagpur Instructonal Objectves: At the end o ths lesson, the
Chemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F E F E + Q! 0
DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica
demo8.nb 1 DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA Obectves: - defne matrces n Mathematca - format the output of matrces - appl lnear algebra to solve a real problem - Use Mathematca to perform
PHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014
PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 Work-Knetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o
A Simple Research of Divisor Graphs
The 29th Workshop on Combnatoral Mathematcs and Computaton Theory A Smple Research o Dvsor Graphs Yu-png Tsao General Educaton Center Chna Unversty o Technology Tape Tawan yp-tsao@cuteedutw Tape Tawan
Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
Lecture Note 4: Numerical differentiation and integration. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 4: Numericl differentition nd integrtion Xioqun Zng Sngi Jio Tong University Lst updted: November, 0 Numericl Anlysis. Numericl differentition.. Introduction Find n pproximtion of f (x 0 ),
3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4
// Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples
International Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl
Many-Body Calculations of the Isotope Shift
Mny-Body Clcultons of the Isotope Shft W. R. Johnson Mrch 11, 1 1 Introducton Atomc energy levels re commonly evluted ssumng tht the nucler mss s nfnte. In ths report, we consder correctons to tomc levels
Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
![The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).](/thumbs/78/77467690.jpg "The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).")
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples