N 1. (q k+1 q k ) 2 + α 3. k=0


 MargaretMargaret Betty Grant
 1 years ago
 Views:
Transcription
1 Teoretisk Fysik Handin problem B, SI1142, Spring 2010 In 1955 Fermi, Pasta and Ulam 1 numerically studied a simple model for a one dimensional chain of nonlinear oscillators to see how the energy distribution is randomized in a system with many degrees of freedom. Instead of the rapid transition to a uniform energy distribution one would naively expect, they observed an extremely slow randomization. In subsequent work by Zabusky and Kruskal 2 the relation of the FermiPastaUlam model to the Kortewegde Vries (KdV) equation was noted, and in a numeric solution of the KdV equation the appearance of solitary wave pulses was observed which they named solitons. This work motivated the search for a analytic solution of the KdV equation which was achieved two years later. 3 This latter work started a period of very intensive research on a class of nonlinear partial differential equations which today are known as soliton equations. The FermiPastaUlam model describes a weakly nonlinear, fixedend onedimensional chain of (N 1) moving mass points and is given by the Hamiltonian H = k=1 1 2m p2 k + K 2 k=0 (q k+1 q k ) 2 + α 3 k=0 (q k+1 q k ) 3 (1) where q 0 q N 0 and q k and p k are the coordinate and momentum for the kth particle, and the parameter α is a small nonlinear coupling; the positive parameters m and K are the particle mass and spring constant, respectively. As was realized by Zabusky and Kruskal (ZK), the equations of motion following from this Hamiltonian (the dots mean differentiation with respect to time t), m q k = K(q k+1 2q k + q k 1 ) + α[(q k+1 q k ) 2 (q k q k 1 ) 2 ] (2) in the lowestorder continuum limit take the following form, q tt = (c ɛq x )q xx (3) with some constants c 0 and ɛ. Eq. (3) can be viewed as ordinary wave equation whose wave speed c depends on the spatial derivative q x, c 2 = (c ɛq x). Typical solutions of this equations for positive ɛ describe pulses moving to the left or right while changing shape and steepening until their leading edge develops a vertical shock front (like water waves in the sea before they break close to the beach), at which point Eq. (3) looses validity. Nonetheless, prior to the formation of the shock, it was found that Eq. (3) provides a quite reasonable description of the Fermi PastaUlam model behavior. ZK found that, to avoid shock wave behavior, one has to take into account dissipation, which can be done by including the nextorder spatial derivative term, q tt = c 2 0q xx + ɛq x q xx + βq xxxx (4) for some constant β. For both convenience and simplicity, ZK now insist on periodic boundary conditions, restrict their attention to waves traveling on one direction only, and elect a moving frame. After replacing x by x = x c 1 t, t by t = c 2 t, q x (x, t) by c 3 u( x, t) in Eq. (4) with particular constants c j and neglecting terms proportional to ɛ 2 they obtained the KdV equation in the following form, u t + uu x + δ 2 u x x x = 0, with δ a (small) constant. In the following we will write this equation as u t + uu x + δ 2 u xxx = 0, u = u(x, t) (5) 1 Los Alamos Scientific Report, unpublished 2 Zabusky and Kruskal, Phys. Rev. Lett. 15, 240 (1965) 3 Gardner, Greene, Kruskal and Miura, Phys. Rev. Lett. 19, 1095 (1967) 1
2 with u = u(x, t), to simplify notation. In their short paper referred to above, ZK first recalled the welllocalized traveling wave solutions of the the KdV equation, with U(x) = u(x, t) = U(x ct) (6) 3c cosh 2 ( c(x x 0 )/2δ), (7) which they describe as solitary wave pulses. The amplitude of such a pulse is proportional to its velocity c > 0, and the constant x 0 equal to the location of its centerofmass at t = 0. ZK then discuss the equation following from (3) without dissipative term, together with the initial condition Using the fact that u t + uu x = 0, (8) u(0, x) = cos πx. (9) u(t, x) = f(x u(t, x)t) (10) is an implicit solution of (8) for any differentiable function f, they find that the solution of Eqs. (8) and (9) tends to become discontinuous at x = 1/2 and t = T B = 1/π. They then describe their results of a numeric solution of the KdV equation with small but nonzero dissipative term, δ = (11) They find that for times smaller than the breakdown time T B, the solution is nearly the same as for δ = 0, but then the dissipative term becomes important. For nonzero δ there is no breakdown any more, but instead solitary wave pulses appear: after a while the solution looks very much like a superposition of special solutions in Eqs. (6) and (7), with different values of c and x 0, 4 and the localized pulses are very stable, preserving shape and speed always except when two of them meet, in which case there is a certain interaction period in which the pulses merge. After such interactions the individual pulses reappear which the same shapes and velocities as before the interaction. ZK also observe that, during the interaction period, the joint amplitude of the interacting solitons decreases, in contradistinction to what would happen if the pulses overlapped linearly. In this problem you are asked to fill in the details of the derivations outlined above (part A). You are then asked to solve the KdV equation numerically and, in particular, reproduce the ZK result (part B). More specific instructions together with further hints will be given below. 4 Note that this is very remarkable since the KdV equation is nonlinear, and the superposition principle therefore does not hold. 2
3 Part A: To do: Derive Eq. (2) from Eq. (1). Hint: Use the following variational principle (= Hamiltonian principle), ( t2 N 1 ) δ dt p k (t) q k (t) H(p 1 (t),..., p N 1 (t), q 1 (t),..., q N 1 (t)) t 1 k=1 = 0. (12) From this obtain 2(N 1) first order differential equations of the form q k =..., ṗ k =... (= Hamilton equations), which, after eliminating half of the variables, yield (2). Derive in detail the continuum limit of Eq. (2) as described above. Hint: To obtain the continuum limit in Eq. (3), introduce x = ka where a > 0 is the lattice constant (i.e. the distance of adjacent mass points when the chain is not moving), and write q(t, x) = q k (t) where x = ka. Approximate q k±1 (t) = q(t, x ± a) by a Taylor series, keeping only certain lower order derivative terms. Do not forget to give the relation between the parameters (i.e. write down the formulas for the parameters c 0 and ɛ in Eq. (3) in terms of m, K, a, α, etc.). Generalize the argument to get Eq. (4). Give then all the details leading to Eq. (5). Verify in detail the shock wave behavior of Eq. (8), as described above. Hint: Verify that Eq. (10) is an implicit solution of Eq. (8). Use (10) to sketch the solution of Eqs. (8) and (9) for a few different times between 0 and T B = 1/π, to show the steepening of the wave. Verify that the wave breaks at the time T B. (If you cannot show this analytically, you can also hand in numeric verifications of these facts produced with the help of MATLAB.) Derive the solutions in Eqs. (6) and (7) from the KdV equation (5). Hint: This is easy if you read Chapter 9 in FMM, of course. Part (B): To do: Solve Eq. (5) for δ = numerically by using the finite difference method (FDM; see Section 8.1 in FMM), for 0 x 2 and periodic boundary conditions, u(x + 2, t) = u(x, t). Test your program by comparing the numeric solution with the analytical solution given in Eqs. (6) and (7) for x 0 = 0.4 and some value of c for times 0 t 0.6/c. Hand in a plot showing your result together with a printout of your MATLAB program. Hint: We suggest you use the following discretization of the KdV equation, with u i,j+1 = u i,j k 6h (u i+1,j + u i,j + u i 1,j ) (u i+1,j u i 1,j ) δ 2 k 2h 3 (u i+2,j 2u i+1,j + 2u i 1,j u i 2,j ) (13) x i = ih, h = 2, i = 0, 1, 2... N 1 N t j = jk, k = t max, j = 0, 1, 2... M (14) M 3
4 and u i+n,j = u i,j (periodic boundary conditions). The initial condition is u i,0 = u(x i, 0), of course. Show that Eq. (13) indeed is a discretization of Eq. (5). To solve these equations on a computer, we suggest you use MATLAB (you can also use other program tools if you prefer). A similar MATLAB program which you can use as a starting point can be found on the course homepage. You will find that not all values of c, h and k are numerically stable, and you need to experiment to find good values. Values that are numerically stable are e.g. c = 0.1, h = 0.06, k = 0.005, (15) but you can get better results with smaller values for h and k. 5 program by either decreasing the ratio k/h 3 or/and decreasing c. We could stabilize our Find a simpler FDM discretization of the KdV equation. Implement your discretization in MATLAB (e.g.) and test your program. Hand in your program and one representative plot showing your test results. 6 Hand in plots showing your numerical solution of the KdV equation with the following initial conditions, (i) u(x, 0) = A cos(πx) 0 t 2/(Aπ) (ii) u(x, 0) = 3U(x) for x 0 = 0.4, 0 t 0.6/c (iii) u(x, 0) = U(x) for x 0 = 0.4, 0 t 0.6/c (16) with U(x) in Eq. (7), and some value of c > 0 and A > 0 which you choose, showing the solution in a six equidistant times (including initial and final time) in the given time interval. Discuss your results. Hint: ZK give the result for (i) with A = 1 which therefore would be the preferred choice, but to get reasonable results for that you need much smaller h and k values than the ones given above. If you computer cannot handle that choose a smaller values for A like A = 0.1. The initial condition u(x, 0) = 1 2n(n + 1)U(x) for n = 1, 2,... is known to be a nsoliton solution at the moment when all solitons are at the same spot (nsoliton collision). For t > 0 the solitons move to the right and separate. For negative initial condition one does not have any solitons but only dispersive waves moving to the left. Try our different initial conditions and different values for k and h in your program. Hand in one plot (iv) with an initial condition of your own choice which you find interesting. Hint: It is fine if you choose u(x, 0) = AU(x) for some other parameters of c, x 0 and A > 0, but perhaps you find something more interesting. Results from an unstable program are not interesting, of course. 5 If your computer is slow and cannot handle such small values of h and k it is OK to have larger values. 6 A negative result (program based on your discretization cannot be made stable) is OK if described properly. 4
5 GENERAL INSTRUCTIONS: A MAIN GOAL IN THIS PROJECT IS TO WRITE A GOOD SCIENTIFIC REPORT. Your solution should be selfcontained and preferably hand written (except for the program code, of course). Give a clear description of your reasoning. Your plots should have a figure caption explaining what they display, and the axes should be labeled. (The plot produced by the MATLAB program provided by us is not OK as it comes out of the program! It is OK to write the labels and figure captions by hand.) Please put your papers together (no loose pages)! Collaborations are allowed but only up to two students in one team, and each of the students should hand in his/her own report. If you collaborate and/or hand in a report in LaTex or the like you should be prepared to be asked to explain your solution in an oral presentation. Do not forget to write name, ADDRESS and personal number on your papers. 5
The Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationCollege of the Holy Cross, Spring 2009 Math 373, Partial Differential Equations Midterm 1 Practice Questions
College of the Holy Cross, Spring 29 Math 373, Partial Differential Equations Midterm 1 Practice Questions 1. (a) Find a solution of u x + u y + u = xy. Hint: Try a polynomial of degree 2. Solution. Use
More informationDynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate
Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a timevarying response termed the dynamics. The dynamics of neurons are an inevitable constraint
More informationTeoretisk Fysik KTH. Advanced QM (SI2380), test questions 1
Teoretisk Fysik KTH Advanced QM (SI238), test questions NOTE THAT I TYPED THIS IN A HURRY AND TYPOS ARE POSSIBLE: PLEASE LET ME KNOW BY EMAIL IF YOU FIND ANY (I will try to correct typos asap  if you
More informationEVOLUTION OF A WHITHAM ZONE IN KORTEWEG DE VRIES THEORY
EVOLUTION OF A WHITHAM ZONE IN KORTEWEG DE VRIES THEORY V. V. AVILOV AND ACADEMICIAN S. P. NOVIKOV We consider the analogy of shock waves in KdV theory. It is well known (see [1], pp. 261 263) that the
More informationScientific Programming
1 The wave equation Scientific Programming Wave Equation The wave equation describes how waves propagate: light waves, sound waves, oscillating strings, wave in a pond,... Suppose that the function h(x,t)
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationAccuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the onsite potential
Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the onsite potential I. Avgin Department of Electrical and Electronics Engineering,
More information5.2 Accuracy and Stability for u t = c u x
c 006 Gilbert Strang 5. Accuracy and Stability for u t = c u x This section begins a major topic in scientific computing: Initialvalue problems for partial differential equations. Naturally we start with
More informationFourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions
FourthOrder Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of MichiganDearborn,
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver. Finite Difference Methods for Partial Differential Equations As you are well aware, most differential equations are much too complicated to be solved by
More informationChapter 15, example problems:
Chapter, example problems: (.0) Ultrasound imaging. (Frequenc > 0,000 Hz) v = 00 m/s. λ 00 m/s /.0 mm =.0 0 6 Hz. (Smaller wave length implies larger frequenc, since their product,
More informationMODULE VII LARGE BODY WAVE DIFFRACTION
MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wavestructure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction.
More informationOscillations. Vern Lindberg. June 10, 2010
Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1
More informationOn computer algebraaided stability analysis of dierence schemes generated by means of Gr obner bases
On computer algebraaided stability analysis of dierence schemes generated by means of Gr obner bases Vladimir Gerdt 1 Yuri Blinkov 2 1 Laboratory of Information Technologies Joint Institute for Nuclear
More informationFinite Difference Approach to Option Pricing
Finite Difference Approach to Option Pricing February 998 CS5 Lab Note. Ordinary differential equation An ordinary differential equation, or ODE, is an equation of the form du = fut ( (), t) (.) dt where
More information20 Applications of Fourier transform to differential equations
20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use here
More informationarxiv:physics/0004029v1 [physics.edph] 14 Apr 2000
arxiv:physics/0004029v1 [physics.edph] 14 Apr 2000 Lagrangians and Hamiltonians for High School Students John W. Norbury Physics Department and Center for Science Education, University of WisconsinMilwaukee,
More informationPHY 140A: Solid State Physics. Solution to Homework #4
PHY 140A: Solid State Physics Solution to Homework #4 TA: Xun Jia 1 November 5, 006 1 Email: jiaxun@physics.ucla.edu Fall 006 Physics 140A c Xun Jia (November 5, 006 Problem #1 Problem. in Ashcroft and
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationFrom Fourier Series to Fourier Integral
From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this
More informationUpdated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum
Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi Module No. # 02 Simple Solutions of the 1 Dimensional Schrodinger Equation Lecture No. # 7. The Free
More information1.4 Using computers to solve differential equations
1.4. USING COMPUTERS TO SOLVE DIFFERENTIAL EQUATIONS67 1.4 Using computers to solve differential equations We have been looking so far at differential equations whose solutions can be constructed from
More informationChapter 15 Collision Theory
Chapter 15 Collision Theory 151 Introduction 1 15 Reference Frames Relative and Velocities 1 151 Center of Mass Reference Frame 15 Relative Velocities 3 153 Characterizing Collisions 5 154 OneDimensional
More informationPHY411. PROBLEM SET 3
PHY411. PROBLEM SET 3 1. Conserved Quantities; the RungeLenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationOn a class of Hill s equations having explicit solutions. Email: m.bartuccelli@surrey.ac.uk, j.wright@surrey.ac.uk, gentile@mat.uniroma3.
On a class of Hill s equations having explicit solutions Michele Bartuccelli 1 Guido Gentile 2 James A. Wright 1 1 Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK. 2 Dipartimento
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationLecture 31: Second order homogeneous equations II
Lecture 31: Second order homogeneous equations II Nathan Pflueger 21 November 2011 1 Introduction This lecture gives a complete description of all the solutions to any differential equation of the form
More informationInstability, dispersion management, and pattern formation in the superfluid flow of a BEC in a cylindrical waveguide
Instability, dispersion management, and pattern formation in the superfluid flow of a BEC in a cylindrical waveguide Michele Modugno LENS & Dipartimento di Fisica, Università di Firenze, Italy Workshop
More informationSecond Order Linear Partial Differential Equations. Part I
Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables;  point boundary value problems; Eigenvalues and Eigenfunctions Introduction
More informationAcceleration levels of dropped objects
Acceleration levels of dropped objects cmyk Acceleration levels of dropped objects Introduction his paper is intended to provide an overview of drop shock testing, which is defined as the acceleration
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More information particle with kinetic energy E strikes a barrier with height U 0 > E and width L.  classically the particle cannot overcome the barrier
Tunnel Effect:  particle with kinetic energy E strikes a barrier with height U 0 > E and width L  classically the particle cannot overcome the barrier  quantum mechanically the particle can penetrated
More information) and mass of each particle is m. We make an extremely small
Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Thermodynamics, 6, kl 9.5. Hjälpmedel: Students may use any book including the textbook Thermal physics. Present your solutions in details: it will
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More information3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas.
Tentamen i Statistisk Fysik I den tjugosjunde februari 2009, under tiden 9.0015.00. Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 03 = F, 46
More informationWave equation examples
Wave equation examples The wave equation is discussed in detail in the Dawkins online text, http://tutorial.math.lamar.edu/classes/de/intropde.aspx, starting on page 13. The function u(x, t) is a solution
More informationSIMPLE HARMONIC MOTION
SIMPLE HARMONIC MOTION PURPOSE The purpose of this experiment is to investigate one of the fundamental types of motion that exists in nature  simple harmonic motion. The importance of this kind of motion
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationSimple harmonic motion
PH122 Dynamics Page 1 Simple harmonic motion 02 February 2011 10:10 Force opposes the displacement in A We assume the spring is linear k is the spring constant. Sometimes called stiffness constant Newton's
More informationLab M1: The Simple Pendulum
Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of
More information5.4 The Heat Equation and ConvectionDiffusion
5.4. THE HEAT EQUATION AND CONVECTIONDIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and ConvectionDiffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The
More informationSIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I
Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged
More informationProblem Set 4: Covariance functions and Gaussian random fields
Problem Set : Covariance functions and Gaussian random fields GEOS 7: Inverse Problems and Parameter Estimation, Carl Tape Assigned: February 9, 15 Due: February 1, 15 Last compiled: February 5, 15 Overview
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion
More informationHeat equation examples
Heat equation examples The Heat equation is discussed in depth in http://tutorial.math.lamar.edu/classes/de/intropde.aspx, starting on page 6. You may recall Newton s Law of Cooling from Calculus. Just
More informationLINEAR SYSTEMS. Consider the following example of a linear system:
LINEAR SYSTEMS Consider the following example of a linear system: Its unique solution is x +2x 2 +3x 3 = 5 x + x 3 = 3 3x + x 2 +3x 3 = 3 x =, x 2 =0, x 3 = 2 In general we want to solve n equations in
More informationPhysics. Essential Question How can one explain and predict interactions between objects and within systems of objects?
Physics Special Note for the 201415 School Year: In 2013, the Maryland State Board of Education adopted the Next Generation Science Standards (NGSS) that set forth a vision for science education where
More information1 One Dimensional Horizontal Motion Position vs. time Velocity vs. time
PHY132 Experiment 1 One Dimensional Horizontal Motion Position vs. time Velocity vs. time One of the most effective methods of describing motion is to plot graphs of distance, velocity, and acceleration
More informationIntroduction to Green s Functions: Lecture notes 1
October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE16 91 Stockholm, Sweden Abstract In the present notes I try to give a better
More informationSpring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations
Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring
More informationThe one dimensional heat equation: Neumann and Robin boundary conditions
The one dimensional heat equation: Neumann and Robin boundary conditions Ryan C. Trinity University Partial Differential Equations February 28, 2012 with Neumann boundary conditions Our goal is to solve:
More information1 of 7 9/5/2009 6:12 PM
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
More informationDiscrete mechanics, optimal control and formation flying spacecraft
Discrete mechanics, optimal control and formation flying spacecraft Oliver Junge Center for Mathematics Munich University of Technology joint work with Jerrold E. Marsden and Sina OberBlöbaum partially
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationPARTICLE SIMULATION ON MULTIPLE DUST LAYERS OF COULOMB CLOUD IN CATHODE SHEATH EDGE
PARTICLE SIMULATION ON MULTIPLE DUST LAYERS OF COULOMB CLOUD IN CATHODE SHEATH EDGE K. ASANO, S. NUNOMURA, T. MISAWA, N. OHNO and S. TAKAMURA Department of Energy Engineering and Science, Graduate School
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More information3. Diffusion of an Instantaneous Point Source
3. Diffusion of an Instantaneous Point Source The equation of conservation of mass is also known as the transport equation, because it describes the transport of scalar species in a fluid systems. In this
More informationLecture L20  Energy Methods: Lagrange s Equations
S. Widnall 6.07 Dynamics Fall 009 Version 3.0 Lecture L0  Energy Methods: Lagrange s Equations The motion of particles and rigid bodies is governed by ewton s law. In this section, we will derive an alternate
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationsince by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations
More informationWaves & Oscillations
Physics 42200 Waves & Oscillations Lecture 3 French, Chapter 1 Spring 2013 Semester Matthew Jones Simple Harmonic Motion The time dependence of a single dynamical variable that satisfies the differential
More informationStochastic Doppler shift and encountered wave period distributions in Gaussian waves
Ocean Engineering 26 (1999) 507 518 Stochastic Doppler shift and encountered wave period distributions in Gaussian waves G. Lindgren a,*, I. Rychlik a, M. Prevosto b a Department of Mathematical Statistics,
More informationAn Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig
An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationModule 43: Vibrations of a Rigid Beam
Module 43: Vibrations of a Rigid Beam David L. Powers in his book, Boundary Value Problems, Third Edition (Published by Harcourt Brace Jovanovich Publishers) gives a problem concerning the vibrations of
More informationPHY 192 Absorption of Radiation Spring
PHY 192 Absorption of Radiation Spring 2010 1 Radioactivity II: Absorption of Radiation Introduction In this experiment you will use the equipment of the previous experiment to learn how radiation intensity
More information1 Variational calculation of a 1D bound state
TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationASEN 3112  Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A TwoDOF MassSpringDashpot Dynamic System Consider the lumpedparameter, massspringdashpot dynamic system shown in the Figure. It has two point
More information3. Electronic Spectroscopy of Molecules I  Absorption Spectroscopy
3. Electronic Spectroscopy of Molecules I  Absorption Spectroscopy 3.1. Vibrational coarse structure of electronic spectra. The Born Oppenheimer Approximation introduced in the last chapter can be extended
More informationExact Solutions to a Generalized Sharma Tasso Olver Equation
Applied Mathematical Sciences, Vol. 5, 2011, no. 46, 22892295 Exact Solutions to a Generalized Sharma Tasso Olver Equation Alvaro H. Salas Universidad de Caldas, Manizales, Colombia Universidad Nacional
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationPart II: Finite Difference/Volume Discretisation for CFD
Part II: Finite Difference/Volume Discretisation for CFD Finite Volume Metod of te AdvectionDiffusion Equation A Finite Difference/Volume Metod for te Incompressible NavierStokes Equations MarkerandCell
More informationGasses at High Temperatures
Gasses at High Temperatures The Maxwell Speed Distribution for Relativistic Speeds Marcel Haas, 333 Summary In this article we consider the Maxwell Speed Distribution (MSD) for gasses at very high temperatures.
More informationExamination paper for Solutions to Matematikk 4M and 4N
Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination
More information2 Background: Fourier Series Analysis and Synthesis
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2001 Lab #11: Design with Fourier Series Date: 3 6 April 2001 This is the official Lab #11 description. The
More informationApplication of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semiinfinite domain)
Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semiinfinite domain) The Fourier Sine Transform pair are F. T. : U = 2/ u x sin x dx, denoted as U
More informationGravity waves on water
Phys374, Spring 2006, Prof. Ted Jacobson Department of Physics, University of Maryland Gravity waves on water Waves on the surface of water can arise from the restoring force of gravity or of surface tension,
More informationFourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +
Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.
More informationPhysics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER
1 P a g e Work Physics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More information3.7 Nonautonomous linear systems of ODE. General theory
3.7 Nonautonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some
More informationF ij = Gm im j r i r j 3 ( r j r i ).
Physics 3550, Fall 2012 Newton s Third Law. Multiparticle systems. Relevant Sections in Text: 1.5, 3.1, 3.2, 3.3 Newton s Third Law. You ve all heard this one. Actioni contrariam semper et qualem esse
More informationLab 5: Projectile Motion
Description Lab 5: Projectile Motion In this lab, you will examine the motion of a projectile as it free falls through the air. This will involve looking at motion under constant velocity, as well as motion
More informationExperiment 4: Harmonic Motion Analysis
Experiment 4: Harmonic Motion Analysis Background In this experiment you will investigate the influence of damping on a driven harmonic oscillator and study resonant conditions. The following theoretical
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationThe 1D Wave Equation
The D Wave Equation 8.303 Linear Partial Differential Equations Matthew J. Hancock Fall 006 D Wave Equation : Physical derivation Reference: Guenther & Lee., MyintU & Debnath..4 [Oct. 3, 006] We consider
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationCHAPTER 5 THE HARMONIC OSCILLATOR
CHAPTER 5 THE HARMONIC OSCILLATOR The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment
More informationMotion of a Leaky Tank Car
1 Problem Motion of a Leaky Tank Car Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 8544 (December 4, 1989; updated October 1, 214) Describe the motion of a tank car initially
More informationSchool of Biotechnology
Physics reference slides Donatello Dolce Università di Camerino a.y. 2014/2015 mail: donatello.dolce@unicam.it School of Biotechnology Program and Aim Introduction to Physics Kinematics and Dynamics; Position
More informationand from (14.23), we find the differential cross section db do R cos (0 / 2) R sin(0/2) R 2
574 Chapter 14 Collision Theory Figure 14.10 A point projectile bouncing off a fixed rigid sphere obeys the law of reflection, that the two adjacent angles labelled a are equal. The impact parameter is
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More information