Ordinary Differential Equations
|
|
- Maud Bryan
- 5 years ago
- Views:
Transcription
1 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations MCS 507 Lecture 17 Mathematical, Statistical and Scientific Software Jan Verschelde, 4 October 2013 Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
2 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
3 a simple pendulum Imagine a sphere attached to a massless rod oscilating back and forth due to gravity: where t is time, starting at 0; θ (t) + α sin(θ(t)) = 0, θ is the angle of deviation the rod makes from its (vertical) position at rest; α = g/l, where g is the gravitational constant and L is the length of the rod. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
4 using sympy For small angles, θ 0: sin(θ) θ. >>> from sympy import * >>> L, t, g = var( L, t, g ) >>> theta = Function( theta ) >>> eq = L*Derivative(theta(t),t,2) + g*t >>> dsolve(eq,theta(t)) theta(t) == C1 + C2*t - g*t**3/(6*l) Using more terms in a series approximation for sin(θ), we obtain more accurate solutions. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
5 first order equations Introducing an auxiliary variable v(t) = θ (t), we transform θ (t) + α sin(θ(t)) = 0 into a system of first order differential equations: θ (t) = v(t) v (t) = α sin(θ(t)) θ with initial conditions: θ(0) = π/6 and v(0) = 0. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
6 forward Euler method We use the definition θ θ(t + h) θ(t) (t) = lim h 0 h to discretize the time domain with step size t. At t = t k, we approximate θ(t k ) θ k and θ(t k+1 ) θ k+1 for and t k+1 = t k + t. So θ (t) = v(t) is approximated by θ k+1 θ k t Then we compute θ k+1 = θ k + tv k. Similarly, v (t) = α sin(θ(t)) is approximated by v k+1 = v k α t sin(θ k ). = v k. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
7 plotting the evolution Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
8 programming the Euler method import scipy as sp import matplotlib.pyplot as plt def pendulum(tmax, nstep, theta0, veloc0, alpha): Return the motion (theta, v, t) of a pendulum, governed by the ODE: theta (t) + alpha*sin(theta(t)) = 0, where the parameters are tmax : time t ranges from 0 to tmax, nstep : the number of time steps, theta0 : angle at t = 0, veloc0 : velocity at t = 0, alpha : value for the parameter. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
9 the code in pendulum tstep = tmax/float(nstep) tspan = sp.linspace(0, tmax, nstep+1) veloc = sp.zeros(nstep+1) theta = sp.zeros(nstep+1) veloc[0] = veloc0 theta[0] = theta0 for k in range(nstep): theta[k+1] = theta[k] + tstep*veloc[k] veloc[k+1] = veloc[k] - alpha*tstep*sp.sin(theta[k+1]) return theta, veloc, tspan Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
10 the constants def test_values(): Returns values for the input data: tmax, nstep, theta0, veloc0, and alpha. theta0 = sp.pi/6 nstep = 1000 tmax = 10 veloc0 = 0 alpha = 5 return tmax, nstep, theta0, veloc0, alpha Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
11 the function main def main(): Defines the test values and computes the trajectory of the pendulum. tmx, dim, pos0, vel0, alp = test_values() theta, vel, tspan = pendulum(tmx, dim, pos0, vel0, alp) fig = plt.figure() fig.add_subplot(211) plt.plot(tspan, theta) plt.title( angle as function of time ) fig.add_subplot(212) plt.plot(tspan, vel, label= velocity(t) ) plt.title( velocity as function of time ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
12 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
13 calling ODEPACK do from scipy.integrate.odepack import odeint then help(odeint) shows odeint(func, y0, t,...) Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack. Solves the initial value problem for stiff or non-stiff systems of first order ode-s:: dy/dt = func(y,t0,...) where y can be a vector. ODEPACK is a FORTRAN77 library which implements Alan Hindmarsh s solvers for ordinary differential equations, available at Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
14 defining the right hand side import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def fun(y, t): Returns the right hand side of the ODE dy/dt = fun(y,t). rhs = sp.array([0, 0], float) rhs[0] = y[1] rhs[1] = -5.0*sp.sin(y[0]) return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
15 the function main def main(): Computes the motion of a pendulum, governed by the ODE: theta (t) + alpha*sin(theta(t)) = 0, tmax = 10 nsteps = 1000 theta0 = sp.pi/6 veloc0 = 0 tspan = sp.linspace(0, tmax, nsteps+1) initc = sp.array([theta0, veloc0]) sol = odeint(fun, initc, tspan) theta = sol[:, 0] veloc = sol[:, 1] then plot (tspan,theta) and (tspan,veloc). Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
16 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
17 differential equations Consider three bodies with masses m 1, m 2, m 3 with positions (x 1 (t), y 1 (t)), (x 2 (t), y 2 (t)), (x 3 (t), y 3 (t)). For the first body: d 2 x 1 (t) dt 2 m = 2 (x 1 (t) x 2 (t)) ( (x1 (t) x 2 (t)) 2 + (y 1 (t) y 2 (t)) 2) 3/2 m 3 (x 1 (t) x 3 (t)) ( (x1 (t) x 3 (t)) 2 + (y 1 (t) y 3 (t)) 2) 3/2 d 2 y 1 (t) dt 2 =... Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
18 setting up the problem With u 1 (t) and v 1 (t), the velocities of x 1 (t) and y 1 (t): dx 1 (t) dt du 1 (t) dt dy 1 (t) dt dv 1 (t) dt = u 1 (t) =... = v 1 (t) =... Adding the equations for the other two bodies, we get 12 first order differential equations. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
19 the figure eight Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
20 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
21 defining the system import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def fun(zvr, time): zvar is a vector with 12 entries arranged in blocks of 4: x[k](time), x [k](time), y[k](time), y [k](time) for the three masses k = 1, 2, 3. zeros = [0 for k in xrange(12)] rhs = sp.array(zeros, float) # take three equal masses mass = [1, 1, 1] Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
22 relabeling # relabel input vector zvr (x1p, u1x, y1p, v1y) = (zvr[0], zvr[1], zvr[2], zvr[3]) (x2p, u2x, y2p, v2y) = (zvr[4], zvr[5], zvr[6], zvr[7]) (x3p, u3x, y3p, v3y) = (zvr[8], zvr[9], zvr[10], zvr[11]) # u and v are first derivatives of x and y (rhs[0], rhs[2]) = (u1x, v1y) (rhs[4], rhs[6]) = (u2x, v2y) (rhs[8], rhs[10]) = (u3x, v3y) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
23 computing distances # compute squared distances dx12 = x1p - x2p sdx12 = dx12**2 dx13 = x1p - x3p sdx13 = dx13**2 dx23 = x2p - x3p sdx23 = dx23**2 dy12 = y1p - y2p sdy12 = dy12**2 dy13 = y1p - y3p sdy13 = dy13**2 dy23 = y2p - y3p sdy23 = dy23**2 # denominators d12 = (sdx12 + sdy12)**1.5 d13 = (sdx13 + sdy13)**1.5 d23 = (sdx23 + sdy23)**1.5 Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
24 assigning the result # righthandside for second order rhs[1] = - mass[1]*dx12/d12 - mass[2]*dx13/d13 rhs[3] = - mass[1]*dy12/d12 - mass[2]*dy13/d13 rhs[5] = - mass[0]*(-dx12)/d12 - mass[2]*dx23/d23 rhs[7] = - mass[0]*(-dy12)/d12 - mass[2]*dy23/d23 rhs[9] = - mass[0]*(-dx13)/d13 - mass[1]*(-dx23)/d23 rhs[11] = - mass[0]*(-dy13)/d13 - mass[1]*(-dy23)/d23 return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
25 the function main def main(): Plots the trajectories of 3 equal masses forming a figure 8. # initial positions ip1 = [ , ] ip2 = [-ip1[0], -ip1[1]] ip3 = [0, 0] # initial velocities iv3 = [ , ] iv2 = [-iv3[0]/2, -iv3[1]/2] iv1 = iv2 # input for initial righthandside vector initz = [ip1[0], iv1[0], ip1[1], iv1[1], \ ip2[0], iv2[0], ip2[1], iv2[1], \ ip3[0], iv3[0], ip3[1], iv3[1]] Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
26 solving the problem # solving the IVP period = 2*sp.pi/3 dim = 1000 tspan = sp.linspace(0, period, dim+1) path = odeint(fun, initz, tspan) # extracing the trajectories (x1p, y1p) = (path[:, 0], path[:, 2]) (x2p, y2p) = (path[:, 4], path[:, 6]) (x3p, y3p) = (path[:, 8], path[:, 10]) # plotting the trajectories plt.figure() plt.plot(x1p, y1p, r, x2p, y2p, g, x3p, y3p, b ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
27 an animation def animate(zvr, nbframes): Makes an animation of the trajectories computed by odeint in zvr, using nbframes. (x1p, x2p, x3p) = (zvr[:, 0], zvr[:, 4], zvr[:, 8]) (y1p, y2p, y3p) = (zvr[:, 2], zvr[:, 6], zvr[:, 10]) dim = len(x1p) deltaframe = dim/nbframes frame = deltaframe plt.ion() fig = plt.figure() axs = fig.add_subplot(111) axs.set_xlim(-1.5, 1.5) axs.set_ylim(-0.5, 0.5) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
28 making the frames of the animation for i in range(nbframes): (s1x, s1y) = (x1p[1:frame], y1p[1:frame]) (s2x, s2y) = (x2p[1:frame], y2p[1:frame]) (s3x, s3y) = (x3p[1:frame], y3p[1:frame]) axs.plot(s1x, s1y, r, \ s2x, s2y, g, \ s3x, s3y, b ) fig.canvas.draw() plt.pause(1) frame = frame + deltaframe raw_input( hit enter to quit\n ) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
29 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
30 the tractrix problem A tractor is connected to a trailer by a rigid bar of unit length. The tractor moves in a circle. The path of the trailer is the solution of a system of ordinary differential equations. Generalization: trailer is predator, tractor is prey. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
31 tractor, trailer, and bar Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
32 the model Given are (x 1 (t), x 2 (t)) defining the path of the tractor and L is the length of the rigid bar. Wanted: (y 1 (t), y 2 (t)), the path of the trailer. The velocity vector of the trailer is parallel to the direction of the bar: ( y 1 y 2 ) ( y1 x = λ 1 y 2 x 2 ), λ > 0. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
33 the ODE system The velocity vector of the trailer is the projection of the velocity vector of the tractor onto the direction of the bar: u = (y 1 x 1, y 2 x 2 ) (y 1 x 1, y 2 x 2 ) and v = (x 1, x 2 ), we compute the projection of the velocity vector as (v T u)u. Then the system of first-order equation that defines the path of the trailer is given by ( ) y 1 y 2 = (v T u)u. with u the normalized vector of the direction of the bar and v the velocity vector of the tractor. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
34 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
35 the tractor import scipy as sp import numpy as np import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def tractor(time): Returns a 4-tuple xp, yp, xv, yv the coordinates xp, yp of the position and the components xv, yv of the velocity vector for the tractor at the given time. xpos = sp.cos(time) ypos = sp.sin(time) xvel = -sp.sin(time) yvel = sp.cos(time) return xpos, ypos, xvel, yvel Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
36 the trailer def trailer(yval, time): Defines the right hand side of the system for the trailer. rhs = np.array([0, 0], float) x1p, x2p, x1v, x2v = tractor(time) rhs[0] = yval[0] - x1p rhs[1] = yval[1] - x2p rhs = rhs/np.linalg.norm(rhs) vtu = x1v*rhs[0] + x2v*rhs[1] rhs = vtu*rhs return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
37 solving the IVP def main(): Defines the setup for the system for the trailer and solves it. endtime = 20 dim = 100 tspan = sp.linspace(0, endtime, dim+1) initc = sp.array([2, 0]) path = odeint(trailer, initc, tspan) xtrail = path[:, 0] ytrail = path[:, 1] xtrac, ytrac, x1v, x2v = tractor(tspan) plt.figure() plt.plot(xtrac, ytrac, r, xtrail, ytrail, g ) for i in xrange(0, dim, 6): plt.plot(sp.array([xtrac[i], xtrail[i]]), \ sp.array([ytrac[i], ytrail[i]]), b ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
38 Summary + Exercises Appendices C, D, and E of the book are on differential equations. SciPy.integrate exports odeint() of odepack. A.C. Hindmarsh: ODEPACK, A Systematized Collection of ODE Solvers. In IMACS Transactions on Scientific Computation, Volume 1, pages 55-64, 1983, edited by R.S. Stepleman. 1 Extend the model for the n-body problem so it works for any number of bodies. 2 Instead of odeint() for the planar 3-body problem, write code for the forward Euler method. For which value(s) of the step size do you get the figure eight? 3 Make an animation of the trajectories of tractor, trailer, and the moving bar. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
39 more exercises 4 Solving y = s(t)w where s(t) is the speed and w the normalized direction vector of a predator chasing a prey gives in y the coordinates of the predator. Set up a model for a prey to move in a straight line and let s(t) be a large enough constant so the prey gets caught. Plot the trajectories of prey and predator. 5 Read the documentation about solving ODEs in Sage and use Sage to plot the trajectories either for the figure eight planar 3-body or the tractrix problem. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39
Ordinary Differential Equations
Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations
More informationOrdinary Differential Equations
Ordinary Differential Equations 1 2 3 MCS 507 Lecture 30 Mathematical, Statistical and Scientific Software Jan Verschelde, 31 October 2011 Ordinary Differential Equations 1 2 3 a simple pendulum Imagine
More informationUniform and constant electromagnetic fields
Fundamentals of Plasma Physics, Nuclear Fusion and Lasers Single Particle Motion Uniform and constant electromagnetic fields Nuno R. Pinhão 2015, March In this notebook we analyse the movement of individual
More informationThe Inverted Pendulum
Lab 1 The Inverted Pendulum Lab Objective: We will set up the LQR optimal control problem for the inverted pendulum and compute the solution numerically. Think back to your childhood days when, for entertainment
More informationMAS212 Scientific Computing and Simulation
MAS212 Scientific Computing and Simulation Dr. Sam Dolan School of Mathematics and Statistics, University of Sheffield Autumn 2017 http://sam-dolan.staff.shef.ac.uk/mas212/ G18 Hicks Building s.dolan@sheffield.ac.uk
More informationThe Shooting Method for Boundary Value Problems
1 The Shooting Method for Boundary Value Problems Consider a boundary value problem of the form y = f(x, y, y ), a x b, y(a) = α, y(b) = β. (1.1) One natural way to approach this problem is to study the
More informationAPPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS
LECTURE 10 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems For Initial Value problems (IVP s), conditions are specified
More informationLecture 19: Calculus of Variations II - Lagrangian
Lecture 19: Calculus of Variations II - Lagrangian 1. Key points Lagrangian Euler-Lagrange equation Canonical momentum Variable transformation Maple VariationalCalculus package EulerLagrange 2. Newton's
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More information3 Space curvilinear motion, motion in non-inertial frames
3 Space curvilinear motion, motion in non-inertial frames 3.1 In-class problem A rocket of initial mass m i is fired vertically up from earth and accelerates until its fuel is exhausted. The residual mass
More informationOrdinary Differential Equations (ODE)
Ordinary Differential Equations (ODE) Why study Differential Equations? Many physical phenomena are best formulated mathematically in terms of their rate of change. Motion of a swinging pendulum Bungee-jumper
More informationChapter 3: 2D Kinematics Tuesday January 20th
Chapter 3: 2D Kinematics Tuesday January 20th Chapter 3: Vectors Review: Properties of vectors Review: Unit vectors Position and displacement Velocity and acceleration vectors Relative motion Constant
More informationMotion in Two or Three Dimensions
Chapter 3 Motion in Two or Three Dimensions PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 3 To use vectors
More informationThe Pendulum. The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations.
The Pendulum Introduction: The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations. Equipment: Simple pendulum made from string
More information1 The pendulum equation
Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating
More informationExperiment 9: Compound Pendulum
COMSATS nstitute of nformation Technology, slamabad Campus PHYS - 108 Experiment 9: Compound Pendulum A compound pendulum (also known as a physical pendulum) consists of a rigid body oscillating about
More informationResearch Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations
Research Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations Ramses van Zon SciNet HPC Consortium November 25, 2014 Ramses van Zon (SciNet HPC Consortium)Research
More informationSolving Systems of ODEs in Python: From scalar to TOV. MP 10/2011
Solving Systems of ODEs in Python: From scalar to TOV. MP 10/2011 Homework: Integrate a scalar ODE in python y (t) =f(t, y) = 5ty 2 +5/t 1/t 2 y(1) = 1 Integrate the ODE using using the explicit Euler
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationExercise_set7_programming_exercises
Exercise_set7_programming_exercises May 13, 2018 1 Part 1(d) We start by defining some function that will be useful: The functions defining the system, and the Hamiltonian and angular momentum. In [1]:
More informationHere we consider soliton solutions of the Korteweg-de Vries (KdV) equation. This equation is given by u t + u u x + 3 u
Lab 3 Solitons Lab Objective: We study traveling wave solutions of the Korteweg-de Vries (KdV) equation, using a pseudospectral discretization in space and a Runge-Kutta integration scheme in time. Here
More informationOn Schemes for Exponential Decay
On Schemes for Exponential Decay Hans Petter Langtangen 1,2 Center for Biomedical Computing, Simula Research Laboratory 1 Department of Informatics, University of Oslo 2 Sep 24, 2015 1.0 0.8 numerical
More informationMath 266: Ordinary Differential Equations
Math 266: Ordinary Differential Equations Long Jin Purdue University, Spring 2018 Basic information Lectures: MWF 8:30-9:20(111)/9:30-10:20(121), UNIV 103 Instructor: Long Jin (long249@purdue.edu) Office
More informationOld Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall
More informationModeling with First-Order Equations
Modeling with First-Order Equations MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Radioactive Decay Radioactive decay takes place continuously. The number
More informationChapter 9b: Numerical Methods for Calculus and Differential Equations. Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers
Chapter 9b: Numerical Methods for Calculus and Differential Equations Initial-Value Problems Euler Method Time-Step Independence MATLAB ODE Solvers Acceleration Initial-Value Problems Consider a skydiver
More informationMath 104: l Hospital s rule, Differential Equations and Integration
Math 104: l Hospital s rule, and Integration Ryan Blair University of Pennsylvania Tuesday January 22, 2013 Math 104:l Hospital s rule, andtuesday Integration January 22, 2013 1 / 8 Outline 1 l Hospital
More informationWeight Change and Predator-Prey Models
Lab 2 Weight Change and Predator-Prey Models Lab Objective: We use IVP methods to study two dynamical systems. The first system is a weight change model based on thermodynamics and kinematics. The second
More informationProblem Set 1 solutions
University of Alabama Department of Physics and Astronomy PH 301 / LeClair Fall 2018 Problem Set 1 solutions 1. If a(t), b(t), and c(t) are functions of t, verify the following results: ( d [a (b c)] =
More informationInverse Problems. Lab 19
Lab 19 Inverse Problems An important concept in mathematics is the idea of a well posed problem. The concept initially came from Jacques Hadamard. A mathematical problem is well posed if 1. a solution
More informationDriven, damped, pendulum
Driven, damped, pendulum Note: The notation and graphs in this notebook parallel those in Chaotic Dynamics by Baker and Gollub. (There's a copy in the department office.) For the driven, damped, pendulum,
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 informationHomework 2 Computational Chemistry (CBE 60553)
Homework 2 Computational Chemistry (CBE 60553) Prof. William F. Schneider Due: 1 Lectures 1-2: Review of quantum mechanics An electron is trapped in a one-dimensional box described by
More informationDynamics Final Report
Dynamics Final Report Sophie Li and Hannah Wilk 1 Abstract We set out to develop a n-rigid body solver in MatLab. We wanted a user to define how many rigid bodies there are, and our system creates the
More informationFigure 2.1 The Inclined Plane
PHYS-101 LAB-02 One and Two Dimensional Motion 1. Objectives The objectives of this experiment are: to measure the acceleration due to gravity using one-dimensional motion, i.e. the motion of an object
More informationModeling with First-Order Equations
Modeling with First-Order Equations MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Radioactive Decay Radioactive decay takes place continuously. The number
More informationOrdinary Differential equations
Chapter 1 Ordinary Differential equations 1.1 Introduction to Ordinary Differential equation In mathematics, an ordinary differential equation (ODE) is an equation which contains functions of only one
More informationAM205: Assignment 3 (due 5 PM, October 20)
AM25: Assignment 3 (due 5 PM, October 2) For this assignment, first complete problems 1, 2, 3, and 4, and then complete either problem 5 (on theory) or problem 6 (on an application). If you submit answers
More informationVolumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
More informationFirst Order Systems of Linear Equations. or ODEs of Arbitrary Order
First Order Systems of Linear Equations or ODEs of Arbitrary Order Systems of Equations Relate Quantities Examples Predator-Prey Relationships r 0 = r (100 f) f 0 = f (r 50) (Lokta-Volterra Model) Systems
More informationUNIVERSITETET I OSLO
(Continued on page 2.) UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: INF1100 Introduction to programming with scientific applications Day of examination: Tuesday, December
More informationLab I. 2D Motion. 1 Introduction. 2 Theory. 2.1 scalars and vectors LAB I. 2D MOTION 15
LAB I. 2D MOTION 15 Lab I 2D Motion 1 Introduction In this lab we will examine simple two-dimensional motion without acceleration. Motion in two dimensions can often be broken up into two separate one-dimensional
More informationAIMS Exercise Set # 1
AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest
More informationLeast squares and Eigenvalues
Lab 1 Least squares and Eigenvalues Lab Objective: Use least squares to fit curves to data and use QR decomposition to find eigenvalues. Least Squares A linear system Ax = b is overdetermined if it has
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationLab I. 2D Motion. 1 Introduction. 2 Theory. 2.1 scalars and vectors LAB I. 2D MOTION 15
LAB I. 2D MOTION 15 Lab I 2D Motion 1 Introduction In this lab we will examine simple two-dimensional motion without acceleration. Motion in two dimensions can often be broken up into two separate one-dimensional
More informationMTH_256: Differential Equations Examination II Part 1 of 2 Technology Allowed Examination. Student Name:
01 March 2018 Technology Allowed Examination Kidoguchi\m256_x2.1soln.docx MTH_256: Differential Equations Examination II Part 1 of 2 Technology Allowed Examination Student Name: STUDNAME Instructions:
More informationChapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
More informationProblem 1. Mathematics of rotations
Problem 1. Mathematics of rotations (a) Show by algebraic means (i.e. no pictures) that the relationship between ω and is: φ, ψ, θ Feel free to use computer algebra. ω X = φ sin θ sin ψ + θ cos ψ (1) ω
More informationNoether s Theorem. 4.1 Ignorable Coordinates
4 Noether s Theorem 4.1 Ignorable Coordinates A central recurring theme in mathematical physics is the connection between symmetries and conservation laws, in particular the connection between the symmetries
More informationOrdinary Differential Equations (ODEs) Background. Video 17
Ordinary Differential Equations (ODEs) Background Video 17 Daniel J. Bodony Department of Aerospace Engineering University of Illinois at Urbana-Champaign In this video you will learn... 1 What ODEs are
More informationA645/A445: Exercise #1. The Kepler Problem
A645/A445: Exercise #1 The Kepler Problem Due: 2017 Sep 26 1 Numerical solution to the one-degree-of-freedom implicit equation The one-dimensional implicit solution is: t = t o + x x o 2(E U(x)). (1) The
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationOrdinary Differential Equations II: Runge-Kutta and Advanced Methods
Ordinary Differential Equations II: Runge-Kutta and Advanced Methods Sam Sinayoko Numerical Methods 3 Contents 1 Learning Outcomes 2 2 Introduction 2 2.1 Note................................ 4 2.2 Limitations
More informationStudy guide: Generalizations of exponential decay models
Study guide: Generalizations of exponential decay models Hans Petter Langtangen 1,2 Center for Biomedical Computing, Simula Research Laboratory 1 Department of Informatics, University of Oslo 2 Sep 13,
More informationDM534 - Introduction to Computer Science
Department of Mathematics and Computer Science University of Southern Denmark, Odense October 21, 2016 Marco Chiarandini DM534 - Introduction to Computer Science Training Session, Week 41-43, Autumn 2016
More informationn-dimensional LCE code
n-dimensional LCE code Dale L Peterson Department of Mechanical and Aeronautical Engineering University of California, Davis dlpeterson@ucdavisedu June 10, 2007 Abstract The Lyapunov characteristic exponents
More information1 Motion of a single particle - Linear momentum, work and energy principle
1 Motion of a single particle - Linear momentum, work and energy principle 1.1 In-class problem A block of mass m slides down a frictionless incline (see Fig.). The block is released at height h above
More information{ } is an asymptotic sequence.
AMS B Perturbation Methods Lecture 3 Copyright by Hongyun Wang, UCSC Recap Iterative method for finding asymptotic series requirement on the iteration formula to make it work Singular perturbation use
More informationOrdinary Differential Equations. Monday, October 10, 11
Ordinary Differential Equations Monday, October 10, 11 Problems involving ODEs can always be reduced to a set of first order differential equations. For example, By introducing a new variable z, this can
More informationLecture4- Projectile Motion Chapter 4
1 / 32 Lecture4- Projectile Motion Chapter 4 Instructor: Prof. Noronha-Hostler Course Administrator: Prof. Roy Montalvo PHY-123 ANALYTICAL PHYSICS IA Phys- 123 Sep. 28 th, 2018 2 / 32 Objectives Vector
More informationThe Principle of Least Action
The Principle of Least Action In their never-ending search for general principles, from which various laws of Physics could be derived, physicists, and most notably theoretical physicists, have often made
More informationConditioning and Stability
Lab 17 Conditioning and Stability Lab Objective: Explore the condition of problems and the stability of algorithms. The condition number of a function measures how sensitive that function is to changes
More informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
More informationMAT 275 Laboratory 4 MATLAB solvers for First-Order IVP
MAT 275 Laboratory 4 MATLAB solvers for First-Order IVP In this laboratory session we will learn how to. Use MATLAB solvers for solving scalar IVP 2. Use MATLAB solvers for solving higher order ODEs and
More informationSOLUTIONS, PROBLEM SET 11
SOLUTIONS, PROBLEM SET 11 1 In this problem we investigate the Lagrangian formulation of dynamics in a rotating frame. Consider a frame of reference which we will consider to be inertial. Suppose that
More informationOrdinary Differential Equations
Ordinary Differential Equations Swaroop Nandan Bora swaroop@iitg.ernet.in Department of Mathematics Indian Institute of Technology Guwahati Guwahati-781039 Modelling a situation We study a model, a sort
More informationChapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:
Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion
More informationClassical Mechanics III (8.09) Fall 2014 Assignment 3
Classical Mechanics III (8.09) Fall 2014 Assignment 3 Massachusetts Institute of Technology Physics Department Due September 29, 2014 September 22, 2014 6:00pm Announcements This week we continue our discussion
More informationSchool of Engineering Faculty of Built Environment, Engineering, Technology & Design
Module Name and Code : ENG60803 Real Time Instrumentation Semester and Year : Semester 5/6, Year 3 Lecture Number/ Week : Lecture 3, Week 3 Learning Outcome (s) : LO5 Module Co-ordinator/Tutor : Dr. Phang
More informationStudy guide for Exam 1. by William H. Meeks III October 26, 2012
Study guide for Exam 1. by William H. Meeks III October 2, 2012 1 Basics. First we cover the basic definitions and then we go over related problems. Note that the material for the actual midterm may include
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Transient Response of a Chemical Reactor Concentration of a substance in a chemical reactor
More informationSimulated Data Sets and a Demonstration of Central Limit Theorem
Simulated Data Sets and a Demonstration of Central Limit Theorem Material to accompany coverage in Hughes and Hase. Introductory section complements Section 3.5, and generates graphs like those in Figs.3.6,
More informationAlgorithms for Uncertainty Quantification
Technische Universität München SS 2017 Lehrstuhl für Informatik V Dr. Tobias Neckel M. Sc. Ionuț Farcaș April 26, 2017 Algorithms for Uncertainty Quantification Tutorial 1: Python overview In this worksheet,
More informationNumerical Methods for the Solution of Differential Equations
Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predator - Prey Model Trajectories and the nonlinear conservation law James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline
More informationEECS 700: Exam # 1 Tuesday, October 21, 2014
EECS 700: Exam # 1 Tuesday, October 21, 2014 Print Name and Signature The rules for this exam are as follows: Write your name on the front page of the exam booklet. Initial each of the remaining pages
More informationVector Fields and Solutions to Ordinary Differential Equations using Octave
Vector Fields and Solutions to Ordinary Differential Equations using Andreas Stahel 6th December 29 Contents Vector fields. Vector field for the logistic equation...............................2 Solutions
More informationLecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25
Lecture 2 Introduction to Differential Equations Roman Kitsela October 1, 2018 Roman Kitsela Lecture 2 October 1, 2018 1 / 25 Quick announcements URL for the class website: http://www.math.ucsd.edu/~rkitsela/20d/
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Two Dimensions; Vectors Vectors and Scalars Addition of Vectors Graphical Methods (One and Two- Dimension) Multiplication of a Vector by a Scalar Subtraction of Vectors Graphical
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationParticles and Cylindrical Polar Coordinates
Chapter 2 Particles and Cylindrical Polar Coordinates TOPICS Here, we discuss the cylindrical polar coordinate system and how it is used in particle mechanics. This coordinate system and its associated
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More informationPHYSICS 210 SOLUTION OF THE NONLINEAR PENDULUM EQUATION USING FDAS
PHYSICS 210 SOLUTION OF THE NONLINEAR PENDULUM EQUATION USING FDAS 1. PHYSICAL & MATHEMATICAL FORMULATION O θ L r T m W 1 1.1 Derivation of the equation of motion O Consider idealized pendulum: Mass of
More information10.34: Numerical Methods Applied to Chemical Engineering. Lecture 19: Differential Algebraic Equations
10.34: Numerical Methods Applied to Chemical Engineering Lecture 19: Differential Algebraic Equations 1 Recap Differential algebraic equations Semi-explicit Fully implicit Simulation via backward difference
More informationOrdinary differential equations. Phys 420/580 Lecture 8
Ordinary differential equations Phys 420/580 Lecture 8 Most physical laws are expressed as differential equations These come in three flavours: initial-value problems boundary-value problems eigenvalue
More informationAP Physics Free Response Practice Oscillations
AP Physics Free Response Practice Oscillations 1975B7. A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is drawn aside through
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationMatlab Course. Anna Kristine Wåhlin. Department of Geophysics, University of Oslo. January Matlab Course p.1/??
Matlab Course Anna Kristine Wåhlin Department of Geophysics, University of Oslo January 2003 Matlab Course p.1/?? Numerical estimate of the derivative An estimate of the time derivative of dataset at time
More informationLecture 1. Scott Pauls 1 3/28/07. Dartmouth College. Math 23, Spring Scott Pauls. Administrivia. Today s material.
Lecture 1 1 1 Department of Mathematics Dartmouth College 3/28/07 Outline Course Overview http://www.math.dartmouth.edu/~m23s07 Matlab Ordinary differential equations Definition An ordinary differential
More informationMath 215/255 Final Exam, December 2013
Math 215/255 Final Exam, December 2013 Last Name: Student Number: First Name: Signature: Instructions. The exam lasts 2.5 hours. No calculators or electronic devices of any kind are permitted. A formula
More informationOcean Modeling - EAS Chapter 2. We model the ocean on a rotating planet
Ocean Modeling - EAS 8803 We model the ocean on a rotating planet Rotation effects are considered through the Coriolis and Centrifugal Force The Coriolis Force arises because our reference frame (the Earth)
More informationBoyce/DiPrima/Meade 11 th ed, Ch 1.1: Basic Mathematical Models; Direction Fields
Boyce/DiPrima/Meade 11 th ed, Ch 1.1: Basic Mathematical Models; Direction Fields Elementary Differential Equations and Boundary Value Problems, 11 th edition, by William E. Boyce, Richard C. DiPrima,
More informationME 274 Spring 2017 Examination No. 2 PROBLEM No. 2 (20 pts.) Given:
PROBLEM No. 2 (20 pts.) Given: Blocks A and B (having masses of 2m and m, respectively) are connected by an inextensible cable, with the cable being pulled over a small pulley of negligible mass. Block
More informationClassical Mechanics and Statistical/Thermodynamics. August 2018
Classical Mechanics and Statistical/Thermodynamics August 2018 1 Handy Integrals: Possibly Useful Information 0 x n e αx dx = n! α n+1 π α 0 0 e αx2 dx = 1 2 x e αx2 dx = 1 2α 0 x 2 e αx2 dx = 1 4 π α
More informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More informationDifferential Equations Spring 2007 Assignments
Differential Equations Spring 2007 Assignments Homework 1, due 1/10/7 Read the first two chapters of the book up to the end of section 2.4. Prepare for the first quiz on Friday 10th January (material up
More informationSheet 5 solutions. August 15, 2017
Sheet 5 solutions August 15, 2017 Exercise 1: Sampling Implement three functions in Python which generate samples of a normal distribution N (µ, σ 2 ). The input parameters of these functions should be
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More information