Chapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
|
|
- Flora McDaniel
- 5 years ago
- Views:
Transcription
1 Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q( x) f ( x) a x b (a) and the boundar condtons (BC) are gven at both end of the doman e.g. (a) = and (b) =. The are generall fxed boundar condtons or Drchlet Boundar Condton but can also be subject to other tpes of BC e.g. Neumann BC or Robn BC. 4. LINEAR FINITE DIFFERENCE (FD) METHOD Fnte dfference method converts an ODE problem from calculus problem nto algebrac problem. In FD, and are expressed as the dfference between adjacent values, for example, (x h) (x h) (x) h () whch are derved from the Talor seres expanson, (x+h) = (x) + h (x) + (h /) (x) + () (x h) = (x) h (x) + (h /) (x) + (3) Kosash Chapter ODE Boundar Value Problems
2 If () s added to (3) and neglectng the hgher order term (O (h 3 )), we wll get (x h) (x) (x h) (x) (4) h The dfference Eqs. () and (4) can be mplemented n [x = a, x n = b] (see Fgure) f few fnte ponts n are defned and dvdng doman [a,b] nto n ntervals of h whch s defned xn x h x x (5) n - [x,x n ] doman dvded nto n ntervals. FD method gves dervatve of values at pont, as follow (x ) (x) (x-) h (6) (x ) (x ) (x ) (x ) h (7) Substtutng (6) & (7) nto (a), we get - h q(x ) (x ) h p(x ) (x ) h f(x ) h p(x ) (x ) (8) or can be smplfed to, - h q(x ) h p(x ) h f(x ) h p(x ) (9) Eq. (9) s then appled to each of the nternal nodes, =,,n. Ths wll produce a sstem of lnear equatons of tr dagonal form. The sstem of lnear equatons can then be solved usng the Thomas algorthm (but we wll solve usng sparse matrx technque). Kosash Chapter ODE Boundar Value Problems
3 Example Solve the followng nd order ODE, 7 3 for [,] wth Drchlet boundar condton, () = and () =, usng FD wth h =. (Note n ths example p(x) = 7, q(x) = 3 and f(x) = ). The dfference equaton (9) for ths problem s, ( 3.5h) ( - 3h ) ( 3.5h) h for h =., =. (C ) The dfference equaton s appled to the nternal nodes ( = ), Applng (C ) at =, we obtan the followng sstem of lnear equatons Note that nodes = & are end nodes, whch have specfed values and the do not appear n the sstem. The above sstem of lnear equatons can be wrtten n tr Kosash Chapter ODE Boundar Value Problems 3
4 dagonal matrx form, whch can be solved usng Thomas algorthm but we wll solve usng sparse technque The soluton of the above sstem s T = [, 3, 4, 5, 6, 7, 8, 9, ] = [.64,.9443,.76,.33,.84,.3,.885,.598,.99] Note that the dfference equatons at nodes = and = n = are slghtl modfed b the end nodes, () = - h q(x ) h p(x ) 3 h f(x ) - h p(x ) = h p(x ) 9 - h q(x ) h f(x ) - h p(x ) 9 () The sample MATLAB code (Example.m) I developed to solve ths problem s shown below. %*************************************************************************** % Program to solve lnear ODE boundar value problems wth FD % Both ends are subjected to Drchlet boundar condtons % nputs : % a,b are startng and endng ponts % alfa,beta are the values at the startng and endng ponts % n s the number of nterval Kosash Chapter ODE Boundar Value Problems 4
5 % outputs : % are the values of at node ponts %*************************************************************************** clear;clf; %********** INPUT ******************************************************* a = nput(ʹenter the startng x >ʹ); b = nput(ʹenter the endng x >ʹ); alfa = nput(ʹenter the value at startng x >ʹ); beta = nput(ʹenter the value at endng x >ʹ); n = nput(ʹenter the number of dvsons >ʹ); pfunc = nput(ʹenter the functon p(x) please: ʹ,ʹsʹ); qfunc = nput(ʹenter the functon q(x) please: ʹ,ʹsʹ); ffunc = nput(ʹenter the functon f(x) please: ʹ,ʹsʹ); p = nlne(pfunc); q = nlne(qfunc); f = nlne(ffunc); %*********** Constants ************ h = (b a)/n; % the nterval sze x = lnspace(a,b,n+); %************ Elements of the matrx A A(:n,:n ) =.; % ntalze matrx A for = :n % ths loop calculate element for nodes to n % n the dagram but n the % matrx t s from = to n f == A(,)= (. q(x(+)) * h*h); A(,+) = +.5 * p(x(+)) * h; ff() = f(x(+)) * h*h (. p(x(+)) *.5 * h) * alfa; elsef == n A(, ) =. p(x(+)) *.5 * h; A(,) = (. q(x(+)) * h*h); ff() = f(x(+)) * h*h ( +.5 * p(x(+)) * h) * beta; else A(, ) =. p(x(+)) *.5 * h; A(,) = (. q(x(+)) * h*h); A(,+) = +.5 * p(x(+)) * h; Kosash Chapter ODE Boundar Value Problems 5
6 ff() = f(x(+)) * h*h; end end %************ Solve the sstem of lnear equatons usng sparse matrx ASparse = sparse(a); = ASparse\ffʹ; %*********** PLOT OF RESULT ******** plot(x,[alfa;;beta],ʹb ʹ) xlabel(ʹxʹ),label(ʹʹ); legend(ʹnumercal resultsʹ) Enter the startng x > Enter the endng x > Enter the value at startng x > Enter the value at endng x > Enter the number of dvsons > Enter the functon fle of p(x) please: 7. Enter the functon fle of q(x) please: 3. Enter the functon fle of f(x) please:..4. numercal results x Now suppose the ODE Eq. () Kosash Chapter ODE Boundar Value Problems 6
7 p( x) q( x) f ( x) a x b () s now subject to boundar condtons (a) = and (b) =. The boundar at b s a dervatve boundar condton or Neumann Boundar Condton. For the problem the value of (b) must be calculated (part of the soluton) so dfference equatons must be wrtten for =,,,n. For =,,,,,n accordng to the FD equaton (9) s but for = n t s dfferent as (b) needs (x n +h) as shown below. (xn h) (xn ) (xn h) n n n- (xn ) h h () It s clear from () that (b+h) s needed, and ths s outsde the doman as shown n the fgure below. To calculate n+, we wll make use of the gven (xn ) and followng (6), n n n (3) h So, n n h n (4) If (4) s substtuted nto () hn - n n n (5) h and substtutng (5) and n- h q(x ) h f(xn ) n n n nto (), we get h h p(x n ) (6) Kosash Chapter ODE Boundar Value Problems 7
8 Eq. (6) s the FD equaton appled at node = n. Example Solve the followng nd order ODE, 5 4 For [,] and boundar condtons () = and (). Use h =., n ths problem p(x) = 5, q(x) = 4 and f(x) =.. The FD equatons are gven b (9), (.5h) ( - 4h ) (.5h) h For h =., =. (C ) (C ) s applcable for =, but for =, (6) s used (C ) The sstem of lnear equatons to be solved s In matrx form, Kosash Chapter ODE Boundar Value Problems 8
9 The soluton vector s T = [.75,.536,.393,.38,.68,.39,.55,.6,.,.] As a practce, modf the code used n Example to solve ths one. Another possble boundar condton s Robn Boundar Condton. In ths case the BC s expressed as an expresson, A n + B n = (7) Usng backward dfference to express n, A n + B n n = (8) h Or, n βh Bn (A h B ) (9) Example 3 Solve the followng nd order ODE, 5 4 Kosash Chapter ODE Boundar Value Problems 9
10 For [,] and boundar condtons: ()= and ().5ʹ()=.5. Use FD method wth h =.. For = the FD equatons are the same as prevous example, but for = the FD equaton s calculated (9) wth A =, B =.5 dan =.5. =.5 + =.5 The formed sstem of lnear equatons s The soluton s T = [.98,.3446,.465,.4543,.474,.473,.4675,.457,.4444,.435] As a practce, modf the code used n Example to solve ths one. 4. NON LINEAR FINITE DIFFERENCE (FD) METHOD The soluton of non lnear ODE usng FD method s smlar to the lnear FD except for non lnear ODE the solutons s obtaned teratvel. In non lnear problem p(x,) and q(x,) are functon of x and or dervatves of, so at each teraton k th, p(x,) and q(x,) are calculated usng the values at (k ) th teraton. In other words we lnearze the ODE nto, ( k ) (k ) p(x, ) q(x, ) f(x) () As n an teratve technques, we must start from an ntal estmate (). The values of the ntal estmate can affect the rate of convergence, so we must prudentl guess (). The smplest technque s to assume lnear (x) as shown n the followng fgure. Kosash Chapter ODE Boundar Value Problems
11 Then teratons are done untl the left sde dffer wthn a certan tolerance from the rght sde. The above method nvolves solvng lnear FD equatons agan and agan. A more explct technque can also be obtaned, we start b wrtng the ODE agan, f(x) - p(x, ) q(x, ) () After expressng and usng (6) and (7) and substtutng them nto (), h h f(x ) p(x, ) q(x, ) () Rearrangng (), can be explctl expressed as h f(x ) h p(x, ) h q(x, ) (3) Note that can be calculated teratvel f all s on the rght hand sde are the known from the prevous calculaton or ntall assumed at the begnnng of the teraton. Kosash Chapter ODE Boundar Value Problems
12 4 (k ) - h f(x ) h p(x, )( - ) h q(x, ) (4) Fausett suggested that convergence can be accelerated f we add to both sde of (3), 4( ω) 4ω h f(x ) h p(x, )( ) h q(x, ) (5) And s calculated teratvel, 4( ω) 4ω - h f(x ) hp(x, )( - ) h q(x, The teraton process s stopped when a specfed convergence crtera has been reached. An acceptable convergence crtera s (6) ) k k (7) Example 4 Solve the followng non lnear ODE, ( ) ( ) For [,] and boundar condtons () = and () =. p(x,) = q(x,) = + and f(x) =. Use the explct method wth h =. and ntal estmate for = whch assumed lnear () = [.,.,.3,.4,.5,.6,.7,.8,.9]. The explct form of (Eq. 6) for ths example s, 4( ω) 4ω - h h( )( - ) h ( ) 4( ω) 4ω -..( )( - ).( ) The result s shown n Fgure below. () Kosash Chapter ODE Boundar Value Problems
13 4. 3 Tutoral Questons. Fnd the temperature profle nsde the tube wall. Hot flud flows nsde the tube such that the nsde temperature s C. The dfferental equaton for the temperature dstrbuton s gven b d T dt dr r dr The boundar condtons are T() = C and T(5) = C. Ths s a lnear nd order lnear ODE wth Drchlet boundar condtons, so use lnear FD technque wth h =. Hot flud flowng nsde a tube.. Deflecton of prsmatc smpl supported beam shown below Kosash Chapter ODE Boundar Value Problems 3
14 s gven b the followng equaton. d qlx qx EI dx Where V s unforml dstrbuted load, L s the length of the beam, I s the moment of nerta of the beam cross secton (I = wh 3 /, w s wdth and h s heght of the beam) and E s the modulus of elastct of the beam. For the case n hand E = GPa, L =.m, w = 5.cm, h =.cm, V = 5N/m and I = 4.66x 6 m 4, Fnd (x). It s clear from the problem that the boundar condtons are Drchlet.e., () = () =., Rewrte the governng equaton Vx EI x - L.8x(x ) From the above, p(x) =, q(x) = and f(x) =.8 x (x.). Dvde the beam nto nterval gvng h =. and generate plot of the beam deflecton. 3. Use lnear shootng method and lnear Fnte Dfference Method to fnd the soluton of Kosash Chapter ODE Boundar Value Problems 4
15 ODE Range Boundar condtons Analtcal soluton h = 4 ( x) [,] () = () = (x) = e x x.5 ( e e ) x 4 e 4. Use FD method (Neumann Boundar condtons) to fnd the soluton of ODE Range Boundar condtons h = [,] () =.5 () = 5. Use FD method (Robn Boundar condtons) to fnd the soluton of ODE Range Boundar condtons h = e x [,] () =.5 ().5 () =.5 6. Use non lnear FD method to fnd the soluton of ODE Range Boundar condtons Analtcal soluton h = 3 [,] () = () = 3 (x) = x.5 Comment on the use of dfferent sa from.5.5. Adopt the followng convergence crtera k k Kosash Chapter ODE Boundar Value Problems 5
DUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationPART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationDifferentiating Gaussian Processes
Dfferentatng Gaussan Processes Andrew McHutchon Aprl 17, 013 1 Frst Order Dervatve of the Posteror Mean The posteror mean of a GP s gven by, f = x, X KX, X 1 y x, X α 1 Only the x, X term depends on the
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationTransfer 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
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationThe Expectation-Maximization Algorithm
The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.
More informationUNIVERSITY 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
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationME 501A Seminar in Engineering Analysis Page 1
umercal Solutons of oundary-value Problems n Os ovember 7, 7 umercal Solutons of oundary- Value Problems n Os Larry aretto Mechancal ngneerng 5 Semnar n ngneerng nalyss ovember 7, 7 Outlne Revew stff equaton
More informationChapter 4: Root Finding
Chapter 4: Root Fndng Startng values Closed nterval methods (roots are search wthn an nterval o Bsecton Open methods (no nterval o Fxed Pont o Newton-Raphson o Secant Method Repeated roots Zeros of Hgher-Dmensonal
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationFTCS Solution to the Heat Equation
FTCS Soluton to the Heat Equaton ME 448/548 Notes Gerald Recktenwald Portland State Unversty Department of Mechancal Engneerng gerry@pdx.edu ME 448/548: FTCS Soluton to the Heat Equaton Overvew 1. Use
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationSummary with Examples for Root finding Methods -Bisection -Newton Raphson -Secant
Summary wth Eamples or Root ndng Methods -Bsecton -Newton Raphson -Secant Nonlnear Equaton Solvers Bracketng Graphcal Open Methods Bsecton False Poston (Regula-Fals) Newton Raphson Secant All Iteratve
More informationLecture 2 Solution of Nonlinear Equations ( Root Finding Problems )
Lecture Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton o Methods Analytcal Solutons Graphcal Methods Numercal Methods Bracketng Methods Open Methods Convergence Notatons Root Fndng
More informationMAE140 - Linear Circuits - Winter 16 Midterm, February 5
Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () 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
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More informationGrid Generation around a Cylinder by Complex Potential Functions
Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationTHE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES
THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationNumerical Transient Heat Conduction Experiment
Numercal ransent Heat Conducton Experment OBJECIVE 1. o demonstrate the basc prncples of conducton heat transfer.. o show how the thermal conductvty of a sold can be measured. 3. o demonstrate the use
More informationMath1110 (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.
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationBoundary Value Problems. Lecture Objectives. Ch. 27
Boundar Vaue Probes Ch. 7 Lecture Obectves o understand the dfference between an nta vaue and boundar vaue ODE o be abe to understand when and how to app the shootng ethod and FD ethod. o understand what
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More information% & 5.3 PRACTICAL APPLICATIONS. Given system, (49) , determine the Boolean Function, , in such a way that we always have expression: " Y1 = Y2
5.3 PRACTICAL APPLICATIONS st EXAMPLE: Gven system, (49) & K K Y XvX 3 ( 2 & X ), determne the Boolean Functon, Y2 X2 & X 3 v X " X3 (X2,X)", n such a way that we always have expresson: " Y Y2 " (50).
More information: Numerical Analysis Topic 2: Solution of Nonlinear Equations Lectures 5-11:
764: Numercal Analyss Topc : Soluton o Nonlnear Equatons Lectures 5-: UIN Malang Read Chapters 5 and 6 o the tetbook 764_Topc Lecture 5 Soluton o Nonlnear Equatons Root Fndng Problems Dentons Classcaton
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationCalculus of Variations Basics
Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y
More informationReview of Taylor Series. Read Section 1.2
Revew of Taylor Seres Read Secton 1.2 1 Power Seres A power seres about c s an nfnte seres of the form k = 0 k a ( x c) = a + a ( x c) + a ( x c) + a ( x c) k 2 3 0 1 2 3 + In many cases, c = 0, and the
More informationMAE140 - Linear Circuits - Fall 10 Midterm, October 28
M140 - Lnear rcuts - Fall 10 Mdterm, October 28 nstructons () 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
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationMATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)
1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationFinite 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
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More information36.1 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
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationELE B7 Power Systems Engineering. Power Flow- Introduction
ELE B7 Power Systems Engneerng Power Flow- Introducton Introducton to Load Flow Analyss The power flow s the backbone of the power system operaton, analyss and desgn. It s necessary for plannng, operaton,
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationNewton s Method for One - Dimensional Optimization - Theory
Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton You are free to Share to copy,
More informationConsistency & Convergence
/9/007 CHE 374 Computatonal Methods n Engneerng Ordnary Dfferental Equatons Consstency, Convergence, Stablty, Stffness and Adaptve and Implct Methods ODE s n MATLAB, etc Consstency & Convergence Consstency
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More information2.29 Numerical Fluid Mechanics
REVIEW Lecture 10: Sprng 2015 Lecture 11 Classfcaton of Partal Dfferental Equatons PDEs) and eamples wth fnte dfference dscretzatons Parabolc PDEs Ellptc PDEs Hyperbolc PDEs Error Types and Dscretzaton
More information(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1
More informationMathematical Methods (10/24.539) V. General Boundary Value Problems (BVPs)
Mathematcal Methods (10/4.539) V. General Boundary Value Problems (BVPs) Introducton In the prevous secton we focused on varous schemes (both analytcal and numercal) for solvng general IVPs. We now emphasze
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationSOLVING NON-LINEAR SYSTEMS BY NEWTON s METHOD USING SPREADSHEET EXCEL Tay Kim Gaik Universiti Tun Hussein Onn Malaysia
SOLVING NON-LINEAR SYSTEMS BY NEWTON s METHOD USING SPREADSHEET EXCEL Tay Km Gak Unverst Tun Hussen Onn Malaysa Kek Se Long Unverst Tun Hussen Onn Malaysa Rosmla Abdul-Kahar
More informationThe Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method
Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationKey Words: Hamiltonian systems, canonical integrators, symplectic integrators, Runge-Kutta-Nyström methods.
CANONICAL RUNGE-KUTTA-NYSTRÖM METHODS OF ORDERS 5 AND 6 DANIEL I. OKUNBOR AND ROBERT D. SKEEL DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 304 W. SPRINGFIELD AVE. URBANA, ILLINOIS
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More information(1) The saturation vapor pressure as a function of temperature, often given by the Antoine equation:
CE304, Sprng 2004 Lecture 22 Lecture 22: Topcs n Phase Equlbra, part : For the remander of the course, we wll return to the subject of vapor/lqud equlbrum and ntroduce other phase equlbrum calculatons
More informationCOSC 4397 Parallel Computation
COSC 4397 Solvng the Laplace Equaton wth MPI Sprng Numercal dfferentaton forward dfference formula y From the defnton of dervatves f( x+ f( f ( = lm h h one can derve an approxmaton for the st dervatve
More informationSolutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1
Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton
More information