Math 124B January 31, 2012
|
|
- Laureen Brown
- 6 years ago
- Views:
Transcription
1 Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat and wave equations with homogeneous boundary conditions, we woud ike to turn to inhomogeneous probems, and use the Fourier series in our search for soutions. We start with the foowing boundary vaue probem for the inhomogeneous heat equation with homogeneous Dirichet conditions. ut ku xx = fx, t, for < x <, t >, ux, = φx, 1 u, t = u, t =. A naïve separation of variabes wi not amount to much, since the source function fx, t may not be separabe. Instead, we can use the eigenfunctions of the corresponding homogeneous eigenvaue probem, and ook for a soution in the series form ux, t = u n t sin nπx. 2 This is, of course, the Fourier sine series of ux, t for each fixed t >, the coefficients of which wi vary with the variabe t. Such an expansion aways exists due to competeness of the set of eigenfunctions sinnπx/}. The coefficients wi then be given by the Fourier sine coefficients formua u n t = 2 We can simiary expand the source function, fx, t = ˆ ux, t sin nπx dx. f n t sin nπx, where f n t = 2 ˆ fx, t sin nπx dx. 3 Now, since we are ooking for a twice differentiabe function ux, t that satisfies the homogeneous Dirichet boundary conditions, we can differentiate the Fourier series 2 term by term to obtain u xx x, t = u n t nπ nπx sin. 4 Notice that to be abe to differentiate twice, we need to guarantee that u x satisfies homogeneous Neumann conditions, which can be arranged by taking the odd extension of u to the interva,, and then the periodic extension to the rest of the rea ine. We can aso differentiate the series 2 with respect to t to obtain since the Fourier coefficients of u t x, t are u t x, t = ˆ 2 u t x, t sin nπx dx = t u nt sin nπx, 5 ˆ 2 ux, t sin nπx dx = u nt. Differentiation under the above integra is aowed since the resuting integrand is continuous. 1
2 Substituting 5 and 4 into the equation, and using 3, we have u nt + k But then, due to the competeness, u n + k nπ un t sin nπx = f n t sin nπx. nπ un = f n t, for n = 1, 2,..., which are ODEs for the coefficients u n t of the series 2. Using the integrating factor e knπ/2t, we can rewrite the ODE as e knπ/2t u n = e knπ/t f n t. Hence, we get But the initia condition impies thus, u n t = ue knπ/2t + e knπ/2 t ux, = φx = u n = φ n = 2 So the soution can be written in the series form as ux, t = φ n e knπ/2t + ˆ e knπ/2s f n s ds. u n sin nπx, φx sin nπx dx. 6 e knπ/2 s t f n s ds sin nπx, 7 where φ n, and f n are the Fourier coefficients of the initia data φx and the source term fx, t, and can be found from 6 and 3 respectivey. Notice that the first coefficient term in the above series comes from the homogeneous heat equation, whie the second term can be thought of as coming from the variation of parameters for the inhomogeneous equation. The case of the Neumann boundary conditions for the inhomogeneous heat equation is simiar, with the ony difference that one ooks for a series soution in terms of cosines, rather than the sine series 2. The boundary vaue probem for the inhomogeneous wave equation, utt c 2 u xx = fx, t, for < x <, ux, = φx, u t x, = ψx, 8 u, t = u, t =, can aso be soved by expanding the soution into the series 2. In this case the ODEs for the coefficients wi be nπ u n + c 2 un = f n t, for n = 1, 2,..., with the initia conditions u n = φ n, u n = ψ n. These ODEs can be soved expicity using variation of parameters to obtain the coefficients u n t, with which one can form the series soution 2. 2
3 7.1 Inhomogeneous boundary conditions We next study the case of inhomogeneous boundary conditions, and consider the foowing boundary vaue probem for the inhomogeneous heat equation. ut ku xx = fx, t, for < x <, t >, ux, = φx, 9 u, t = ht, u, t = jt. We wi use the method of shifting the data to reduce this probem to one with homogeneous boundary conditions, which wi be equivaent to 1. The idea of this method is to subtract a function satisfying the inhomogeneous boundary conditions from the soution to the above probem. Let us define the function Ux, t = 1 x ht + x jt, for which triviay U, t = ht, and U, t = jt. But then for the new quantity vx, t = ux, t Ux, t, we have v t kv xx = u t ku xx U t ku xx = fx, t 1 x h t x j t, vx, = ux, Ux, = φx 1 x v, t = u, t U, t = ht ht =, v, t = u, t U, t = jt jt =. h x j, So vx, t satisfies the foowing boundary vaue probem with homogeneous boundary conditions, v t kv xx = fx, t, for < x <, t >, vx, = φx, v, t = v, t =, 1 where fx, t = fx, t 1 x h t x j t, φx = φx 1 x h x 11 j. Probem 1 is equivaent to 1, so the function vx, t can be found by formua 7, in which φ n and f n s must be repaced by respectivey φ n and f n s. Ceary, φ n and f n t can be found in terms of the Fourier coefficients of φx and fx, t using 11. Having found the function vx, t in the series form, vx, t = φ n e knπ/2t + we can write the soution to probem 9 as e knπ/2 s t fn s ds sin nπx, ux, t = vx, t + Ux, t = 1 x ht + x jt + φ n e knπ/2t + In the case of Neumann boundary conditions, e knπ/2 s t fn s ds sin nπx. 12 u x, t = ht, u x, t = jt, 3
4 one shoud use the function ˆ V x, t = Ux, t dx = x x2 ht + x2 2 2 jt to shift the data, since V x, t = U, t = ht, and V x, t = U, t = jt. For the inhomogeneous boundary vaue probem for the wave equation, utt c 2 u xx = fx, t, for < x <, ux, = φx, u t x, = ψx, u, t = ht, u, t = jt, 13 one can shift the data in the same way to reduce the probem to one equivaent to 8. An aternative method to shifting the data, is to use expansion 2 to find the soution of probem 9 directy. The difference between this case and the case of probem 1 is in that the soution does not satisfy the homogeneous boundary conditions, so the series 2 can not be differentiated term by term when substituting into the equation. The way out is to separatey expand u xx x, t as But then using Green s second identity, we can compute w n t = 2 ˆ ˆ b a u xx x, t = u xx x, t sin nπx dx = 2 ˆ nπ nπx ux, t sin w n t sin nπx. 14 f g gf dx = f g fg b a, dx + 2 u x sin nπx nπ u cos nπx Noticing that the first boundary term vanishes, and using the boundary conditions for u from 9 in the second term, we have nπ w n t = un t 2nπ 2 1n jt + 2nπ ht Then substituting 14 and 5 into the equation, and using 15, we get nπ u nt k un t 2nπ 2 1n jt + 2nπ ht sin nπx 2 Using the competeness, we obtain the foowing ODEs for the coefficients u n t. u n + k nπ un = f n t 2nπ 2 1n jt ht, with u n = φ n. =. f n t sin nπx. 4
5 One can sove these ODEs using the same integrating factor as before to get u n t = φ n e knπ/2t + e knπ/2 s t f n s 2nπ 2 1n js hs ds. The soution to 9 wi then be given by the series 2. It is eft as an exercise to convince yourseves that this soution is equivaent to 12. For the inhomogeneous wave probem 13 this method wi yied the foowing ODEs for the coefficients with the initia conditions u n + c 2 nπ un = f n t 2nπ 2 1n jt ht, u n = φ n, u n = ψ n. Soving these ODEs by variation of parameters, one can find the soution to 13 as the series Concusion Using the eigenfunctions corresponding to the associated homogeneous boundary vaue probems as buiding bocks, we searched for a soution to the inhomogeneous heat and wave equations in the form of a series in terms of these eigenfunctions. This ead to ODEs for the coefficients of the series, which is akin to the method of variation of parameters, where one varies the coefficients in the inear combination of soutions to the homogeneous equation to obtain a soution of the inhomogeneous one. We studied two methods of soving probems with inhomogeneous boundary conditions, the first of which amounted to merey shifting the boundary data to reduce the probem to one with homogeneous boundary conditions. The second method was to directy ook for a soution in the form of a series in terms of the eigenfunctions of the associated homogeneous probem. In this case one can no onger directy substitute the series into the equation due to the issue of boundary terms when differentiating Fourier series. However, after taking care of these boundary terms, we arrived at ODEs for the coefficients of this expansion as we, soving which wi give the series soution 2. 5
Math 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationHomework #04 Answers and Hints (MATH4052 Partial Differential Equations)
Homework #4 Answers and Hints (MATH452 Partia Differentia Equations) Probem 1 (Page 89, Q2) Consider a meta rod ( < x < ), insuated aong its sides but not at its ends, which is initiay at temperature =
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More informationMA 201: Partial Differential Equations Lecture - 10
MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationMA 201: Partial Differential Equations Lecture - 11
MA 201: Partia Differentia Equations Lecture - 11 Heat Equation Heat conduction in a thin rod The IBVP under consideration consists of: The governing equation: u t = αu xx, (1) where α is the therma diffusivity.
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More information1 Heat Equation Dirichlet Boundary Conditions
Chapter 3 Heat Exampes in Rectanges Heat Equation Dirichet Boundary Conditions u t (x, t) = ku xx (x, t), < x (.) u(, t) =, u(, t) = u(x, ) = f(x). Separate Variabes Look for simpe soutions in the
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 7
Strauss PDEs 2e: Section 4.3 - Exercise 1 Page 1 of 7 Exercise 1 Find the eigenvaues graphicay for the boundary conditions X(0) = 0, X () + ax() = 0. Assume that a 0. Soution The aim here is to determine
More informationWeek 5 Lectures, Math 6451, Tanveer
Week 5 Lectures, Math 651, Tanveer 1 Separation of variabe method The method of separation of variabe is a suitabe technique for determining soutions to inear PDEs, usuay with constant coefficients, when
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationEXISTENCE OF SOLUTIONS FOR ONE-DIMENSIONAL WAVE EQUATIONS WITH NONLOCAL CONDITIONS
Eectronic Journa of Differentia Equations, Vo. 21(21), No. 76, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (ogin: ftp) EXISTENCE OF SOLUTIONS
More informationMath 124B January 24, 2012
Mth 24B Jnury 24, 22 Viktor Grigoryn 5 Convergence of Fourier series Strting from the method of seprtion of vribes for the homogeneous Dirichet nd Neumnn boundry vue probems, we studied the eigenvue probem
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 6. Solve the completely inhomogeneous diffusion problem on the half-line
Strauss PDEs 2e: Section 3.3 - Exercise 2 Page of 6 Exercise 2 Solve the completely inhomogeneous diffusion problem on the half-line v t kv xx = f(x, t) for < x
More informationPhysics 505 Fall 2007 Homework Assignment #5 Solutions. Textbook problems: Ch. 3: 3.13, 3.17, 3.26, 3.27
Physics 55 Fa 7 Homework Assignment #5 Soutions Textook proems: Ch. 3: 3.3, 3.7, 3.6, 3.7 3.3 Sove for the potentia in Proem 3., using the appropriate Green function otained in the text, and verify that
More informationVibrations of Structures
Vibrations of Structures Modue I: Vibrations of Strings and Bars Lesson : The Initia Vaue Probem Contents:. Introduction. Moda Expansion Theorem 3. Initia Vaue Probem: Exampes 4. Lapace Transform Method
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationHaar Decomposition and Reconstruction Algorithms
Jim Lambers MAT 773 Fa Semester 018-19 Lecture 15 and 16 Notes These notes correspond to Sections 4.3 and 4.4 in the text. Haar Decomposition and Reconstruction Agorithms Decomposition Suppose we approximate
More informationHigher dimensional PDEs and multidimensional eigenvalue problems
Higher dimensiona PEs and mutidimensiona eigenvaue probems 1 Probems with three independent variabes Consider the prototypica equations u t = u (iffusion) u tt = u (W ave) u zz = u (Lapace) where u = u
More informationPhysics 505 Fall Homework Assignment #4 Solutions
Physics 505 Fa 2005 Homework Assignment #4 Soutions Textbook probems: Ch. 3: 3.4, 3.6, 3.9, 3.0 3.4 The surface of a hoow conducting sphere of inner radius a is divided into an even number of equa segments
More informationSuggested Solution to Assignment 5
MATH 4 (5-6) prti diferenti equtions Suggested Soution to Assignment 5 Exercise 5.. () (b) A m = A m = = ( )m+ mπ x sin mπx dx = x mπ cos mπx + + 4( )m 4 m π. 4x cos mπx dx mπ x cos mπxdx = x mπ sin mπx
More informationChapter 5. Wave equation. 5.1 Physical derivation
Chapter 5 Wave equation In this chapter, we discuss the wave equation u tt a 2 u = f, (5.1) where a > is a constant. We wi discover that soutions of the wave equation behave in a different way comparing
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationApplications of the Maximum Principle
Jim Lambers MAT 606 Spring Semester 2015-16 Lecture 26 Notes These notes correspond to Sections 7.4-7.6 in the text. Applications of the Maximum Principle The maximum principle for Laplace s equation is
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationLecture notes. May 2, 2017
Lecture notes May, 7 Preface These are ecture notes for MATH 47: Partia differentia equations with the soe purpose of providing reading materia for topics covered in the ectures of the cass. Severa exampes
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 15 Heat with a source So far we considered homogeneous wave and heat equations and the associated initial value problems on the whole line, as
More informationSCHOOL OF MATHEMATICS AND STATISTICS. Mathematics II (Materials) Section A. Find the general solution of the equation
Data provided: Formua Sheet MAS250 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics II (Materias Autumn Semester 204 5 2 hours Marks wi be awarded for answers to a questions in Section A, and for your
More informationWave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationLECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL
More informationMAS 315 Waves 1 of 8 Answers to Examples Sheet 1. To solve the three problems, we use the methods of 1.3 (with necessary changes in notation).
MAS 35 Waves of 8 Answers to Exampes Sheet. From.) and.5), the genera soution of φ xx = c φ yy is φ = fx cy) + gx + cy). Put c = : the genera soution of φ xx = φ yy is therefore φ = fx y) + gx + y) ) To
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationAn approximate method for solving the inverse scattering problem with fixed-energy data
J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationA proof for the full Fourier series on [ π, π] is given here.
niform convergence of Fourier series A smooth function on an interval [a, b] may be represented by a full, sine, or cosine Fourier series, and pointwise convergence can be achieved, except possibly at
More informationModule 22: Simple Harmonic Oscillation and Torque
Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More informationMath 124A October 11, 2011
Math 14A October 11, 11 Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This corresponds to a string of infinite length. Although
More informationDiffusion on the half-line. The Dirichlet problem
Diffusion on the half-line The Dirichlet problem Consider the initial boundary value problem (IBVP) on the half line (, ): v t kv xx = v(x, ) = φ(x) v(, t) =. The solution will be obtained by the reflection
More informationWavelet Galerkin Solution for Boundary Value Problems
Internationa Journa of Engineering Research and Deveopment e-issn: 2278-67X, p-issn: 2278-8X, www.ijerd.com Voume, Issue 5 (May 24), PP.2-3 Waveet Gaerkin Soution for Boundary Vaue Probems D. Pate, M.K.
More informationSrednicki Chapter 51
Srednici Chapter 51 QFT Probems & Soutions A. George September 7, 13 Srednici 51.1. Derive the fermion-oop correction to the scaar proagator by woring through equation 5., and show that it has an extra
More informationWMS. MA250 Introduction to Partial Differential Equations. Revision Guide. Written by Matthew Hutton and David McCormick
WMS MA25 Introduction to Partia Differentia Equations Revision Guide Written by Matthew Hutton and David McCormick WMS ii MA25 Introduction to Partia Differentia Equations Contents Introduction 1 1 First-Order
More informationb n n=1 a n cos nx (3) n=1
Fourier Anaysis The Fourier series First some terminoogy: a function f(x) is periodic if f(x ) = f(x) for a x for some, if is the smaest such number, it is caed the period of f(x). It is even if f( x)
More information17 Source Problems for Heat and Wave IB- VPs
17 Source Problems for Heat and Wave IB- VPs We have mostly dealt with homogeneous equations, homogeneous b.c.s in this course so far. Recall that if we have non-homogeneous b.c.s, then we want to first
More informationHomework 5 Solutions
Stat 310B/Math 230B Theory of Probabiity Homework 5 Soutions Andrea Montanari Due on 2/19/2014 Exercise [5.3.20] 1. We caim that n 2 [ E[h F n ] = 2 n i=1 A i,n h(u)du ] I Ai,n (t). (1) Indeed, integrabiity
More informationMA 201: Method of Separation of Variables Finite Vibrating String Problem Lecture - 11 MA201(2016): PDE
MA 201: Method of Separation of Variables Finite Vibrating String Problem ecture - 11 IBVP for Vibrating string with no external forces We consider the problem in a computational domain (x,t) [0,] [0,
More informationLecture 11. Fourier transform
Lecture. Fourier transform Definition and main resuts Let f L 2 (R). The Fourier transform of a function f is a function f(α) = f(x)t iαx dx () The normaized Fourier transform of f is a function R ˆf =
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationMethod of Separation of Variables
MODUE 5: HEAT EQUATION 11 ecture 3 Method of Separation of Variables Separation of variables is one of the oldest technique for solving initial-boundary value problems (IBVP) and applies to problems, where
More informationMath 201 Assignment #11
Math 21 Assignment #11 Problem 1 (1.5 2) Find a formal solution to the given initial-boundary value problem. = 2 u x, < x < π, t > 2 u(, t) = u(π, t) =, t > u(x, ) = x 2, < x < π Problem 2 (1.5 5) Find
More information# Points Score Total 100
Name: PennID: Math 241 Make-Up Final Exam January 19, 2016 Instructions: Turn off and put away your cell phone. Please write your Name and PennID on the top of this page. Please sign and date the pledge
More informationTransforms and Boundary Value Problems
Transforms and Boundary Vaue Probems (For B.Tech Students (Third/Fourth/Fifth Semester Soved University Questions Papers Prepared by Dr. V. SUVITHA Department of Mathematics, SRMIST Kattankuathur 6. CONTENTS
More information11.1 One-dimensional Helmholtz Equation
Chapter Green s Functions. One-dimensiona Hemhotz Equation Suppose we have a string driven by an externa force, periodic with frequency ω. The differentia equation here f is some prescribed function) 2
More informationSeparation of variables in two dimensions. Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 )
Separation of variables in two dimensions Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 ) Separation of variables in two dimensions Overview of method: Consider linear, homogeneous
More informationLECTURE 19: SEPARATION OF VARIABLES, HEAT CONDUCTION IN A ROD
ECTURE 19: SEPARATION OF VARIABES, HEAT CONDUCTION IN A ROD The idea of separation of variables is simple: in order to solve a partial differential equation in u(x, t), we ask, is it possible to find a
More informationNOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs
More informationFourier series. Part - A
Fourier series Part - A 1.Define Dirichet s conditions Ans: A function defined in c x c + can be expanded as an infinite trigonometric series of the form a + a n cos nx n 1 + b n sin nx, provided i) f
More informationA Two-Parameter Trigonometric Series
A Two-Parameter Trigonometric Series Xiang-Qian Chang Xiang.Chang@bos.mcphs.edu Massachusetts Coege of Pharmacy and Heath Sciences Boston MA 25 Let us consider the foowing dua questions. A. Givenafunction
More informationStrauss PDEs 2e: Section Exercise 4 Page 1 of 6
Strauss PDEs 2e: Section 5.3 - Exercise 4 Page of 6 Exercise 4 Consider the problem u t = ku xx for < x < l, with the boundary conditions u(, t) = U, u x (l, t) =, and the initial condition u(x, ) =, where
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationJackson 4.10 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 4.10 Homework Probem Soution Dr. Christopher S. Baird University of Massachusetts Lowe PROBLEM: Two concentric conducting spheres of inner and outer radii a and b, respectivey, carry charges ±.
More information= 1 u 6x 2 4 2x 3 4x + 5. d dv (3v2 + 9v) = 6v v + 9 3v 2 + 9v dx = ln 3v2 + 9v + C. dy dx ay = eax.
Math 220- Mock Eam Soutions. Fin the erivative of n(2 3 4 + 5). To fin the erivative of n(2 3 4+5) we are going to have to use the chain rue. Let u = 2 3 4+5, then u = 62 4. (n(2 3 4 + 5) = (nu) u u (
More informationMAT 167: Advanced Linear Algebra
< Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationMidterm 2: Sample solutions Math 118A, Fall 2013
Midterm 2: Sample solutions Math 118A, Fall 213 1. Find all separated solutions u(r,t = F(rG(t of the radially symmetric heat equation u t = k ( r u. r r r Solve for G(t explicitly. Write down an ODE for
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More information12.2. Maxima and Minima. Introduction. Prerequisites. Learning Outcomes
Maima and Minima 1. Introduction In this Section we anayse curves in the oca neighbourhood of a stationary point and, from this anaysis, deduce necessary conditions satisfied by oca maima and oca minima.
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationMATH 425, HOMEWORK 5, SOLUTIONS
MATH 425, HOMEWORK 5, SOLUTIONS Exercise (Uniqueness for the heat equation on R) Suppose that the functions u, u 2 : R x R t R solve: t u k 2 xu = 0, x R, t > 0 u (x, 0) = φ(x), x R and t u 2 k 2 xu 2
More informationLecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling
Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest
More information17 Lecture 17: Recombination and Dark Matter Production
PYS 652: Astrophysics 88 17 Lecture 17: Recombination and Dark Matter Production New ideas pass through three periods: It can t be done. It probaby can be done, but it s not worth doing. I knew it was
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationThe Wave Equation. Units. Recall that [z] means the units of z. Suppose [t] = s, [x] = m. Then u = c 2 2 u. [u]
The Wave Equation UBC M57/316 Lecture Notes c 014 by Phiip D. Loewen Many PDE s describe wave motion. The simpest is u tt = c u xx. Here c > 0 is a given constant, and u = u(x, t is some quantity of interest
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationu(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0
Bocconi University PhD in Economics - Microeconomics I Prof M Messner Probem Set 4 - Soution Probem : If an individua has an endowment instead of a monetary income his weath depends on price eves In particuar,
More informationCryptanalysis of PKP: A New Approach
Cryptanaysis of PKP: A New Approach Éiane Jaumes and Antoine Joux DCSSI 18, rue du Dr. Zamenhoff F-92131 Issy-es-Mx Cedex France eiane.jaumes@wanadoo.fr Antoine.Joux@ens.fr Abstract. Quite recenty, in
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationMath Assignment 14
Math 2280 - Assignment 14 Dylan Zwick Spring 2014 Section 9.5-1, 3, 5, 7, 9 Section 9.6-1, 3, 5, 7, 14 Section 9.7-1, 2, 3, 4 1 Section 9.5 - Heat Conduction and Separation of Variables 9.5.1 - Solve the
More information