Chapter 4 The Wave Equation
|
|
- Milton Simmons
- 6 years ago
- Views:
Transcription
1 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 sound or water waves. It arses n dfferent felds such as acoustcs, electromagnetcs, or flud dynamcs. In ts smplest form, the wave equaton refers to a scalar functon u = u(r,t), r R n that satsfes: u t = c u. (4.) Here denotes the Laplacan n R n and c s a constant speed of the wave propagaton. An even more compact form of Eq. (4.) s gven by u =, where = c t s the d Alembertan. 4. The Wave Equaton n D The wave equaton for the scalar u n the one dmensonal case reads u t = u c x. (4.) The one-dmensonal wave equaton (4.) can be solved exactly by d Alembert s method, usng a Fourer transform method, or va separaton of varables. To llustrate the dea of the d Alembert method, let us ntroduce new coordnates (ξ, η) by use of the transformaton ξ = x ct, η = x+ct. (4.3) 35
2 In the new coordnate system one can wrte and Eq. (4.) becomes u xx = u ξ ξ + u ξ η + u ηη, c u tt = u ξ ξ u ξ η + u ηη, u =. (4.4) ξ η That s, the functon u remans constant along the curves (4.3),.e., Eq. (4.3) descrbes characterstc curves of the wave equaton (4.) (see App. B). Moreover, one can see that the dervatve u/ ξ does not depends on η,.e., η ( u ξ After ntegraton wth respect to ξ one obtans ) = u ξ = f(ξ). u(ξ,η) = F(ξ)+G(η), where F s the prmtve functon of f and G s the constant of ntegraton, n general the functon of η. Turnng back to the coordnates (x, t) one obtans the general soluton of Eq. (4.) u(x,t) = F(x ct)+g(x+ct). (4.5) 4.. Soluton of the IVP Now let us consder an ntal value problem for Eq. (4.): u tt = c u xx, t, u(x,) = f(x), (4.6) u t (x,) = g(x). To wrte down the general soluton of the IVP for Eq. (4.), one needs to exspress the arbtrary functon F and G n terms of ntal data f and g. Usng the relaton one becomes: t F(x ct) = cf (x ct), where F (x ct) := ξ F(ξ) u(x,) = F(x)+G(x) = f(x); u t (x,) = c( F (x)+g (x)) = g(x).
3 After dfferentaton of the frst equaton wth respect to x one can solve the system n terms of F (x) and G (x),.e., F (x) = ( f (x) ) c g(x), G (x) = ( f (x)+ ). c g(x) Hence F(x) = f(x) c x g(y)dy+c, G(x) = f(x)+ c x g(y)dy C, where the ntegraton constant C s chosen n such a way that the ntal condton F(x) + G(x) = f(x) s fullfeld. Alltogether one obtans: u(x,t) = ( ) f(x ct)+ f(x+ct) + x+ct g(y) dy. (4.7) c x ct 4.. Numercal Treatment 4... A Smple Explct Method The frst dea s just to use central dfferences for both tme and space dervatves,.e., u j+ u j + u j t = c u j + u j + u j x, (4.8) or, wth α = c t/ x u j+ = u j + ( α )u j + α (u j + + u j ). (4.9) Schematcal representaton of the scheme (4.9) s shown on Fg. 4.. Note that one should also mplement ntal condtons (4.6). In order to mplement the second ntal condton one needs the vrtual pont u, u t (x,) = g(x ) = u u t +O( t ). Fg. 4. Schematcal vsualzaton of the numercal scheme (4.9) for (4.). x x x + t j+ t j t j
4 Wth g := g(x ) one can rewrte the last expresson as u = u tg +O( t ), and the second tme row can be calculated as u = tg +( α ) f + α ( f + f + ), (4.) where u(x,) = u = f(x ) = f. von Neumann Stablty Analyss In order to nvestgate the stablty of the explct scheme (4.9) we start wth the usual ansatz (.) ε j+ = g j e kx, whch leads to the followng expresson for the amplfcaton factor g(k) g = ( α )g +α gcos(k x). After several transformatons the last expresson becomes just a quadratc equaton for g, namely g β g+ =, (4.) where β = α sn ( k x). Solutons of the equaton for g(k) read g, = β ± β. Notce that f β > then at least one of absolute value of g, s bgger that one. Therefor one should desre for β <,.e., and g, = β ± β g = β + β =. That s, the scheme (4.9) s condtonal stable. The stablty condton reads ( ) k x α sn, what s equvalent to the standart CFL condton (.7)
5 t j+ Fg. 4. Schematcal vsualzaton of the mplct numercal scheme (4.) for (4.). x x x + t j t j α = c t x An Implct Method One can try to overcome the problems wth condtonal stablty by ntroducng an mplct scheme. The smplest way to do t s just to replace all terms on the rght hand sde of (4.8) by an average from the values to the tme steps j + and j,.e, u j+ u j + u j ( t = c x u j + j u +u j +uj+ + ) j+ u +u j+. (4.) Schematcal dagramm of the numercal scheme (4.) s shown on Fg. (4.). Let us check the stablty of the mplct scheme (4.). To ths am we use the standart ansatz leadng to the equaton for g(k) wth ε j+ = g j e kx β g g+β = ( ) k x β = +α sn. One can see that β for all k. Hence the solutons g, take the form and g, = ± β β g = ( β ) β =. That s, the mplct scheme (4.) s absolute stable. Now, the queston s, whether the mplct scheme (4.) s better than the explct scheme (4.9) form numercal pont of vew. To answer ths queston, let us analyse dsperson relaton for the wave equaton (4.) as well as for both schemes (4.9) and
6 Fg. 4.3 Dsperson relaton for the one-dmensonal wave equaton (4.), calculated usng the explct (blue curves) and mplct (red curves) methods (4.9) and (4.). ω t/α α= α=.8 α=. α= α=.8 α= k x/π (4.). The exact dsperson relaton s ω = ±ck,.e, all Fourer modes propagate wthout dsperson wth the same phase velocty ω/k = ±c. Usng the ansatz u j ekx ωt j for the explct method (4.9) one obtans: whle for the mplct method (4.) cos(ω t) = α ( cos(k x)), (4.3) cos(ω t) = +α ( cos(k x)). (4.4) One can see that for α both methods provde the same result, otherwse the explct scheme (4.9) always exceeds the mplct one (see Fg. (4.3)). For α = the scheme (4.9) becomes exact, whle (4.) devates more and more from the exact value of ω for ncreasng α. Hence, for Eq. (4.) there are no motvaton to use mplct scheme nstead of the explct one Examples Example. Use the explct method (4.9) to solve the one-dmansonal wave equaton (4.): u tt = 4u xx for x [, L] and t [,T] (4.5) wth boundary condtons u(, t) = u(l, t) =.
7 u 5 5 t Fg. 4.4 Space-tme evoluton of Eq. (4.5) wth the ntal 5 dstrbuton u(x, ) = sn(π x), u t (x,) =. 4 x 6 8 Assume that the ntal poston and velocty are Other parameters are: u(x,) = f(x) = sn(πx), and u t (x,) = g(x) =. Space nterval L= Space dscretzaton step x =. Tme dscretzaton step t =.5 Amount of tme steps T = Frst one can fnd the d Alambert soluton. In the case of zero ntal velocty Eq. (4.7) becomes u(x,t) = f(x t)+ f(x+t) = snπ(x t)+snπ(x+t) = sn(πx) cos(πt),.e., the soluton s just a sum of a travellng waves wth ntal form, gven by f(x). Numercal soluton of (4.5) s shown on Fg. (4.4). Example. Solve Eq. (4.5) wth the same boundary condtons. Assume now, that ntal dstrbutons of poston and velocty are, x [, x ]; u(x,) = f(x) = and u t (x,) = g(x) = g, x [x, x ];, x [x, L]. Other parameters are:
8 Fg. 4.5 Space-tme evoluton of Eq. (4.5) wth the ntal dstrbuton u(x, ) =, u t (x,) = g(x). Intal nonzero velocty g =.5 Intal space ntervals x = L/4, x = 3L/4 Space nterval L= Space dscretzaton step x =. Tme dscretzaton step t =.5 Amount of tme steps T = 4 Numercal soluton of the problem s shown on Fg. (4.5). Example 3. Vbratng Strng Use the explct method (4.9) to solve the wave equaton for a vbratng strng: u tt = c u xx for x [, L] and t [,T], (4.6) where c = wth the boundary condtons u(,t) = u(l,t) =. Assume that the ntal poston and velocty are u(x,) = f(x) = sn(nπx/l), and u t (x,) = g(x) =, n =,,3,.... Other parameters are: Space nterval L= Space dscretzaton step x =. Tme dscretzaton step t =.5 Amount of tme steps T = Usually a vbratng strng produces a sound whose frequency s constant. Therefore, snce frequency characterzes the ptch, the sound produced s a constant note. Vbratng strngs are the bass of any strng nstrument lke gutar or cello. If the speed of propagaton c s known, one can calculate the frequency of the sound pro-
9 duced by the strng. The speed of propagaton of a wave c s equal to the wavelength multpled by the frequency f : c = λ f If the length of the strng s L, the fundamental harmonc s the one produced by the vbraton whose nodes are the two ends of the strng, so L s half of the wavelength of the fundamental harmonc, so f = c L Solutons of the equaton n queston are gven n form of standng waves. The standng wave s a wave that remans n a constant poston. Ths phenomenon can occur because the medum s movng n the opposte drecton to the wave, or t can arse n a statonary medum as a result of nterference between two waves travelng n opposte drectons (see Fg. (4.6)) n = n = n = n = 4 n = 5 n = Fg. 4.6 Standng waves n a strng. The fundamental mode and the frst fve overtones are shown. The red dots represent the wave nodes.
10 4. The Wave Equaton n D 4.. Examples 4... Example. Use the standart fve-pont explct method (4.9) to solve a two-dmansonal wave equaton u tt = c (u xx + u yy ), u = u(x,y,t) on the rectangular doman [, L] [, L] wth Drchlet boundary condtons. Other parameters are: Space nterval L= Space dscretzaton step x = y =. Tme dscretzaton step t =.5 Amount of tme steps T = Intal condton u(x,y,) = 4x y( x)( y) Numercal soluton of the problem for two dfferent tme moments t = and t = 5 can be seen on Fg. (4.7) t = t = 5 Fg. 4.7 Numercal soluton of the two-dmensonal wave equaton, shown for t = and t = 5.
The equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
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 informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
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 informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
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 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 informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12
REVIEW Lecture 11: 2.29 Numercal Flud Mechancs Fall 2011 Lecture 12 End of (Lnear) Algebrac Systems Gradent Methods Krylov Subspace Methods Precondtonng of Ax=b FINITE DIFFERENCES Classfcaton of Partal
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 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 information12. 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
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 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 informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
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 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 informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
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 informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationModule 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
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationSUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationSolutions to Exercises in Astrophysical Gas Dynamics
1 Solutons to Exercses n Astrophyscal Gas Dynamcs 1. (a). Snce u 1, v are vectors then, under an orthogonal transformaton, u = a j u j v = a k u k Therefore, u v = a j a k u j v k = δ jk u j v k = u j
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
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(
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
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 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 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 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 information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationLab session: numerical simulations of sponateous polarization
Lab sesson: numercal smulatons of sponateous polarzaton Emerc Boun & Vncent Calvez CNRS, ENS Lyon, France CIMPA, Hammamet, March 2012 Spontaneous cell polarzaton: the 1D case The Hawkns-Voturez model for
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
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 informationACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
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 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 information829. An adaptive method for inertia force identification in cantilever under moving mass
89. An adaptve method for nerta force dentfcaton n cantlever under movng mass Qang Chen 1, Mnzhuo Wang, Hao Yan 3, Haonan Ye 4, Guola Yang 5 1,, 3, 4 Department of Control and System Engneerng, Nanng Unversty,
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 informationGroup Analysis of Ordinary Differential Equations of the Order n>2
Symmetry n Nonlnear Mathematcal Physcs 997, V., 64 7. Group Analyss of Ordnary Dfferental Equatons of the Order n> L.M. BERKOVICH and S.Y. POPOV Samara State Unversty, 4430, Samara, Russa E-mal: berk@nfo.ssu.samara.ru
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationSection 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
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 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 informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
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 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 informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationMAE140 - 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
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 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 informationTensor Smooth Length for SPH Modelling of High Speed Impact
Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru
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 informationDigital Signal Processing
Dgtal Sgnal Processng Dscrete-tme System Analyss Manar Mohasen Offce: F8 Emal: manar.subh@ut.ac.r School of IT Engneerng Revew of Precedent Class Contnuous Sgnal The value of the sgnal s avalable over
More informationQueens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Scence Numercal Methods CSCI 361 / 761 Sprng 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 16 Lecture 16a May 3, 2018 Numercal soluton of systems
More informationPhysics 4B. A positive value is obtained, so the current is counterclockwise around the circuit.
Physcs 4B Solutons to Chapter 7 HW Chapter 7: Questons:, 8, 0 Problems:,,, 45, 48,,, 7, 9 Queston 7- (a) no (b) yes (c) all te Queston 7-8 0 μc Queston 7-0, c;, a;, d; 4, b Problem 7- (a) Let be the current
More informationA constant recursive convolution technique for frequency dependent scalar wave equation based FDTD algorithm
J Comput Electron (213) 12:752 756 DOI 1.17/s1825-13-479-2 A constant recursve convoluton technque for frequency dependent scalar wave equaton bed FDTD algorthm M. Burak Özakın Serkan Aksoy Publshed onlne:
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
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 information1-Dimensional Advection-Diffusion Finite Difference Model Due to a Flow under Propagating Solitary Wave
014 4th Internatonal Conference on Future nvronment and nergy IPCB vol.61 (014) (014) IACSIT Press, Sngapore I: 10.776/IPCB. 014. V61. 6 1-mensonal Advecton-ffuson Fnte fference Model ue to a Flow under
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
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 informationIrregular vibrations in multi-mass discrete-continuous systems torsionally deformed
(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationNumerical Simulation of Wave Propagation Using the Shallow Water Equations
umercal Smulaton of Wave Propagaton Usng the Shallow Water Equatons Junbo Par Harve udd College 6th Aprl 007 Abstract The shallow water equatons SWE were used to model water wave propagaton n one dmenson
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 16
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 218 LECTURE 16 1 why teratve methods f we have a lnear system Ax = b where A s very, very large but s ether sparse or structured (eg, banded, Toepltz, banded plus
More informationIntegrals and Invariants of
Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationΔ x. u(x,t) Fig. Schematic view of elastic bar undergoing axial motions
ME67 - Handout 4 Vbratons of Contnuous Systems Axal vbratons of elastc bars The fgure shows a unform elastc bar of length and cross secton A. The bar materal propertes are ts densty ρ and elastc modulus
More informationarxiv: v1 [physics.flu-dyn] 16 Sep 2013
Three-Dmensonal Smoothed Partcle Hydrodynamcs Method for Smulatng Free Surface Flows Rzal Dw Prayogo a,b, Chrstan Fredy Naa a a Faculty of Mathematcs and Natural Scences, Insttut Teknolog Bandung, Jl.
More informationPerfect Fluid Cosmological Model in the Frame Work Lyra s Manifold
Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra
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 informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More information10.34 Fall 2015 Metropolis Monte Carlo Algorithm
10.34 Fall 2015 Metropols Monte Carlo Algorthm The Metropols Monte Carlo method s very useful for calculatng manydmensonal ntegraton. For e.g. n statstcal mechancs n order to calculate the prospertes of
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
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 informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
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 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 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 informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
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 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 informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationResearch Article Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations
Appled Mathematcs Volume 22, Artcle ID 4587, 8 pages do:.55/22/4587 Research Artcle Cubc B-Splne Collocaton Method for One-Dmensonal Heat and Advecton-Dffuson Equatons Joan Goh, Ahmad Abd. Majd, and Ahmad
More informationNormally, in one phase reservoir simulation we would deal with one of the following fluid systems:
TPG4160 Reservor Smulaton 2017 page 1 of 9 ONE-DIMENSIONAL, ONE-PHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases
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 informationOPTIMISATION. 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
More information