The Wave Equation on a Disk
|
|
- Clyde Henderson
- 6 years ago
- Views:
Transcription
1 R. C. Trinity University Partial Differential Equations April 3, 214
2 The vibrating circular membrane Goal: Model the motion of an elastic membrane stretched over a circular frame of radius a. Set-up: Center the membrane at the origin in the xy-plane and let u(r,θ,t) = deflection of membrane from equilibrium at polar position (r,θ) and time t. Under ideal assumptions: ( u tt = c 2 2 u = c 2 u rr + 1 r u r + 1 ) r 2u θθ, < r < a, < θ < 2π, t >, u(a,θ,t) =, θ 2π, t >. y u= 2 Δ 2 u = c u tt a x
3 Separation of variables Setting u(r,θ,t) = R(r)Θ(θ)T(t) leads to the separated boundary value problems r 2 R +rr + ( λ 2 r 2 µ 2) R =, R() finite, R(a) =, Θ +µ 2 Θ =, Θ 2π-periodic, T +c 2 λ 2 T =. We have already seen that the second yields Θ(θ) = Θ m (θ) = Acos(mθ)+Bsin(mθ), µ = m N. For each such µ, we have also seen that the solution to the first is R(r) = J m (λr), where J m is the Bessel function of the first kind of order m.
4 We still have one more boundary condition: R(a) = J m (λa) = λa = α mn, where α mn is the nth positive zero of J m. This means that λ = λ mn = α mn a, and hence R(r) = R mn (r) = J m (λ mn r), m N, n N. Returning to T, we finally find that T(t) = T mn (t) = C cos(cλ mn t)+dsin(cλ mn t).
5 Normal modes Multiplying our results together gives the separated solutions J m (λ mn r)(acos(mθ)+bsin(mθ))(c cos(cλ mn t)+dsin(cλ mn t)). For convenience we split these in two and write u mn (r,θ,t) = J m (λ mn r)(a mn cos(mθ)+b mn sin(mθ))cos(cλ mn t), u mn(r,θ,t) = J m (λ mn r)(a mncos(mθ)+b mnsin(mθ))sin(cλ mn t). Note that, up to scaling, rotation and a phase shift in time, these have the form u(r,θ,t) = J m (λ mn r) cos(mθ) cos(cλ mn t).
6 Remarks Let s compare with the normal modes of the rectangular membrane problem: u(x,y,t) = sin(µ m x)sin(ν n y)cos(λ mn t) The functions R mn (r) are the polar analogs of X m (x) = sin(µ m x), Y n (x) = sin(ν n y). The numbers λ mn = αmn a are analogous to µ m = mπ a and ν n = nπ b. Moral: We have (essentially) replaced sine by J m and the zeros of sine by those of J m.
7 Initial conditions and superposition In order to completely determine the shape of the membrane at any time we must specify the initial conditions u(r,θ,) = f(r,θ), r a, θ 2π (shape), u t (r,θ,) = g(r,θ), r a, θ 2π (velocity). In order to meet these conditions we use superposition to build the general solution u(r,θ,t) = m= n=1 + J m (λ mn r)(a mn cos(mθ)+b mn sin(mθ))cos(cλ mn t) }{{} u mn(r,θ,t) m= n=1 J m (λ mn r)(amn cos(mθ)+b mn }{{ sin(mθ))sin(cλ mnt). } umn (r,θ,t)
8 Othogonality of Bessel functions We will see later that the functions R mn (r) = J m (λ mn r) are orthogonal relative to the weighted inner product That is, f,g = a f(r)g(r)r dr. R mn,r mk = a In addition, it can also be shown that R mn,r mn = J m (λ mn r)j m (λ mk r)r dr = if n k. a J 2 m(λ mn r)r dr = a2 2 J2 m+1(α mn ).
9 Using the orthogonality relations for Bessel and trigonometric functions, one obtains: Theorem The functions φ mn (r,θ) = J m (λ mn r)cos(mθ), ψ mn (r,θ) = J m (λ mn r)sin(mθ), (m N, n N) form a (complete) orthogonal set of functions relative to the inner product f,g = 2π a f(r,θ)g(r,θ)r dr dθ. That is, φ mn,φ jk = ψ mn,ψ jk = for (m,n) (j,k) and φ mn,ψ jk = for all (m,n) and (j,k).
10 Imposing the initial conditions Setting t = in the general solution gives f(r,θ) = u(r,θ,) = J m (λ mn r)(a mn cos(mθ)+b mn sin(mθ)) = g(r,θ) = u t (r,θ,) = = m= n=1 m= n=1 m= n=1 m= n=1 (a mn φ mn (r,θ)+b mn ψ mn (r,θ)), which are called Fourier-Bessel expansions. cλ mn J m (λ mn r)(amn cos(mθ)+b mn sin(mθ)) (cλ mn amn φ mn(r,θ)+cλ mn bmn ψ mn(r,θ)),
11 Integral formulae for a mn and b mn Using orthogonality, the usual argument gives a mn = f,φ mn φ mn,φ mn = 2π a 2π a f(r,θ)j m (λ mn r)cos(mθ)r dr dθ. Jm 2 (λ mnr) cos 2 (mθ)r dr dθ for m, n 1. Using the complementary orthogonality relation, the integral in the denominator is equal to 2π cos 2 (mθ)dθ a πa 2 J1 2(α n) if m =, Jm 2 (λ mnr)r dr = πa 2 2 J2 m+1(α mn ) if m 1.
12 We finally find that a n = a mn = 1 πa 2 J 2 1 (α n) 2π a 2 πa 2 J 2 m+1 (α mn) 2π a and likewise (using ψ mn instead) b mn = for m,n N. 2 πa 2 J 2 m+1 (α mn) f(r,θ)j (λ n r)r dr dθ, 2π a f(r,θ)j m (λ mn r) cos(mθ)r dr dθ, f(r,θ)j m (λ mn r) sin(mθ)r dr dθ,
13 Integral formulae for a mn and b mn The same reasoning using the Fourier-Bessel expansion of g(x,y) yields a n = a mn = b mn = for m,n N. 1 πcα n aj 2 1 (α n) 2 πcα mn aj 2 m+1 (α mn) 2 πcα mn aj 2 m+1 (α mn) 2π a 2π a 2π a g(r,θ)j (λ n r)r dr dθ, g(r,θ)j m (λ mn r) cos(mθ)r dr dθ, g(r,θ)j m (λ mn r) sin(mθ)r dr dθ, This (almost) completes the statement of the general solution to the vibrating circular membrane problem!
14 Remark Since cos = 1 and sin = we have m= n=1 J m (λ mn r)(a mn cos(mθ)+b mn sin(mθ))cos(cλ mn t) = a n J (λ n r)cos(cλ n t) + (as above) n=1 } {{ } m= m=1 n=1 Note that there are really no b n coefficients. This is the true form of the first series in the solution. Analogous comments hold for the second series.
15 Remark If f(r,θ) = f(r) (i.e. f is radially symmetric), then for m a mn = ( ) = ( ) 2π a a dr f(r)j m (λ mn r) cos(mθ)r dr dθ 2π cos(mθ) dθ =, } {{ } and b mn =, too. That is, there are only a n terms. Likewise, if g is radially symmetric, then for m and there are only a n terms. a mn = b mn =,
16 Example Solve the vibrating membrane problem with a = c = 1 and initial conditions f(r,θ) = 1 r 4, g(r,θ) =. Because g(r,θ) =, we immediately find that a mn = b mn = for all m and n. Because f is radially symmetric, we only need to compute a n. Since a = 1, λ mn = α mn, so 1 a n = πj1 2(α n) = 2 J 2 1 (α n) 2π 1 1 f(r)j (α n r)r drdθ (1 r 4 )J (α n r)r dr } {{ } substitute x=α n r
17 2 αn = (1 x4 α 2 n J2 1 (α n) = 2 α 2 n J2 1 (α n) αn α 4 n xj (x)dx } {{ } A ) J (x)x dx 1 αn α 4 n x 5 J (x)dx. } {{ } B According to earlier results αn α n A = xj (x)dx = xj 1 (x) = α n J 1 (α n ), αn B = x 5 J (x)dx = x 5 J 1 (x) 4x 4 J 2 (x)+8x 3 α n J 3 (x) = α 5 nj 1 (α n ) 4α 4 nj 2 (α n )+8α 3 nj 3 (α n ).
18 It follows that a n = so that finally u(r,θ,t) = 2 α 2 n J2 1 (α n) n=1 ( A 1 ) α 4 B n = 8(α nj 2 (α n ) 2J 3 (α n )) α 3 n J2 1 (α, n) 8(α n J 2 (α n ) 2J 3 (α n )) α 3 n J2 1 (α J (α n r) cos(α n t). n) Remark: This solution can easily be implemented in Maple, since the command BesselJZeros(m,n) will compute α mn numerically.
An Introduction to Bessel Functions
An Introduction to R. C. Trinity University Partial Differential Equations Lecture 17 Bessel s equation Given p 0, the ordinary differential equation x 2 y + xy + (x 2 p 2 )y = 0, x > 0 is known as Bessel
More informationd Wave Equation. Rectangular membrane.
1 ecture1 1.1 2-d Wave Equation. Rectangular membrane. The first problem is for the wave equation on a rectangular domain. You can interpret this as a problem for determining the displacement of a flexible
More informationIn what follows, we examine the two-dimensional wave equation, since it leads to some interesting and quite visualizable solutions.
ecture 22 igher-dimensional PDEs Relevant section of text: Chapter 7 We now examine some PDEs in higher dimensions, i.e., R 2 and R 3. In general, the heat and wave equations in higher dimensions are given
More informationPartial Differential Equations
Partial Differential Equations Spring Exam 3 Review Solutions Exercise. We utilize the general solution to the Dirichlet problem in rectangle given in the textbook on page 68. In the notation used there
More information1. Partial differential equations. Chapter 12: Partial Differential Equations. Examples. 2. The one-dimensional wave equation
1. Partial differential equations Definitions Examples A partial differential equation PDE is an equation giving a relation between a function of two or more variables u and its partial derivatives. The
More informationBoundary Value Problems in Cylindrical Coordinates
Boundary Value Problems in Cylindrical Coordinates 29 Outline Differential Operators in Various Coordinate Systems Laplace Equation in Cylindrical Coordinates Systems Bessel Functions Wave Equation the
More informationThe Laplacian in Polar Coordinates
The Laplacian in Polar Coordinates R. C. Trinity University Partial Differential Equations March 17, 15 To solve boundary value problems on circular regions, it is convenient to switch from rectangular
More informationMath 2930 Worksheet Final Exam Review
Math 293 Worksheet Final Exam Review Week 14 November 3th, 217 Question 1. (* Solve the initial value problem y y = 2xe x, y( = 1 Question 2. (* Consider the differential equation: y = y y 3. (a Find the
More informationIntroduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series
CHAPTER 5 Introduction to Sturm-Liouville Theory and the Theory of Generalized Fourier Series We start with some introductory examples. 5.. Cauchy s equation The homogeneous Euler-Cauchy equation (Leonhard
More informationMATH 241 Practice Second Midterm Exam - Fall 2012
MATH 41 Practice Second Midterm Exam - Fall 1 1. Let f(x = { 1 x for x 1 for 1 x (a Compute the Fourier sine series of f(x. The Fourier sine series is b n sin where b n = f(x sin dx = 1 = (1 x cos = 4
More informationBoundary value problems for partial differential equations
Boundary value problems for partial differential equations Henrik Schlichtkrull March 11, 213 1 Boundary value problem 2 1 Introduction This note contains a brief introduction to linear partial differential
More informationThe two-dimensional heat equation
The two-dimensional heat equation Ryan C. Trinity University Partial Differential Equations March 5, 015 Physical motivation Goal: Model heat flow in a two-dimensional object (thin plate. Set up: Represent
More informationBOUNDARY PROBLEMS IN HIGHER DIMENSIONS. kt = X X = λ, and the series solutions have the form (for λ n 0):
BOUNDARY PROBLEMS IN HIGHER DIMENSIONS Time-space separation To solve the wave and diffusion equations u tt = c u or u t = k u in a bounded domain D with one of the three classical BCs (Dirichlet, Neumann,
More information24 Solving planar heat and wave equations in polar coordinates
24 Solving planar heat and wave equations in polar coordinates Now that all the preparations are done, I can return to solving the planar heat and wave equations in domains with rotational symmetry. 24.1
More informationBessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy
More informationStriking a Beat. Ashley Martin PHY 495. Spring Striking a Beat. Ashley Martin PHY 495. Introduction. Outline. Cartesian Coordinates
Spring 2012 Polar Where is it optimal to strike a circular drum? Polar Daniel Bernoulli (1700-1782) - introduced concept of Bessel functions Leonhard Euler (1707-1783) - used Bessel funtions of both zero
More informationBessel functions. The drum problem: Consider the wave equation (with c = 1) in a disc with homogeneous Dirichlet boundary conditions:
Bessel functions The drum problem: Consider the wave equation (with c = 1) in a disc with homogeneous Dirichlet boundary conditions: u = u t, u(r 0,θ,t) = 0, u u(r,θ,0) = f(r,θ), (r,θ,0) = g(r,θ). t (Note
More informationMa 530. Partial Differential Equations - Separation of Variables in Multi-Dimensions
Ma 530 Partial Differential Equations - Separation of ariables in Multi-Dimensions Temperature in an Infinite Cylinder Consider an infinitely long, solid, circular cylinder of radius c with its axis coinciding
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More informationSturm-Liouville Theory
More on Ryan C. Trinity University Partial Differential Equations April 19, 2012 Recall: A Sturm-Liouville (S-L) problem consists of A Sturm-Liouville equation on an interval: (p(x)y ) + (q(x) + λr(x))y
More informationTwo special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.
More informationLAPLACE EQUATION. = 2 is call the Laplacian or del-square operator. In two dimensions, the expression of in rectangular and polar coordinates are
LAPLACE EQUATION If a diffusion or wave problem is stationary (time independent), the pde reduces to the Laplace equation u = u =, an archetype of second order elliptic pde. = 2 is call the Laplacian or
More informationBessel Functions. A Touch of Magic. Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2. SMILE REU Summer Louisiana State University
Bessel s Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2 1 Department of Mathematics Louisiana State University 2 Department of Mathematics University of Arkansas SMILE REU Summer
More informationc2 2 x2. (1) t = c2 2 u, (2) 2 = 2 x x 2, (3)
ecture 13 The wave equation - final comments Sections 4.2-4.6 of text by Haberman u(x,t), In the previous lecture, we studied the so-called wave equation in one-dimension, i.e., for a function It was derived
More information1 Boas, p. 643, problem (b)
Physics 6C Solutions to Homework Set #6 Fall Boas, p. 643, problem 3.5-3b Fin the steay-state temperature istribution in a soli cyliner of height H an raius a if the top an curve surfaces are hel at an
More informationMATH 311 Topics in Applied Mathematics Lecture 25: Bessel functions (continued).
MATH 311 Topics in Applied Mathematics Lecture 25: Bessel functions (continued). Bessel s differential equation of order m 0: z 2 d2 f dz 2 + z df dz + (z2 m 2 )f = 0 The equation is considered on the
More information3.3 Unsteady State Heat Conduction
3.3 Unsteady State Heat Conduction For many applications, it is necessary to consider the variation of temperature with time. In this case, the energy equation for classical heat conduction, eq. (3.8),
More informationConnection to Laplacian in spherical coordinates (Chapter 13)
Connection to Laplacian in spherical coordinates (Chapter 13) We might often encounter the Laplace equation and spherical coordinates might be the most convenient 2 u(r, θ, φ) = 0 We already saw in Chapter
More informationExpansion of 1/r potential in Legendre polynomials
Expansion of 1/r potential in Legendre polynomials In electrostatics and gravitation, we see scalar potentials of the form V = K d Take d = R r = R 2 2Rr cos θ + r 2 = R 1 2 r R cos θ + r R )2 Use h =
More informationIntroduction to Sturm-Liouville Theory
Introduction to R. C. Trinity University Partial Differential Equations April 10, 2014 Sturm-Liouville problems Definition: A (second order) Sturm-Liouville (S-L) problem consists of A Sturm-Liouville
More information1 A complete Fourier series solution
Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider
More informationPhysics 116C Solutions to the Practice Final Exam Fall The method of Frobenius fails because it works only for Fuchsian equations, of the type
Physics 6C Solutions to the Practice Final Exam Fall 2 Consider the differential equation x y +x(x+)y y = () (a) Explain why the method of Frobenius method fails for this problem. The method of Frobenius
More informationNotes on Special Functions
Spring 25 1 Notes on Special Functions Francis J. Narcowich Department of Mathematics Texas A&M University College Station, TX 77843-3368 Introduction These notes are for our classes on special functions.
More informationChapter 5.8: Bessel s equation
Chapter 5.8: Bessel s equation Bessel s equation of order ν is: x 2 y + xy + (x 2 ν 2 )y = 0. It has a regular singular point at x = 0. When ν = 0,, 2,..., this equation comes up when separating variables
More informationPartial Differential Equations
Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives with respect to those variables. Most (but
More informationQualification Exam: Mathematical Methods
Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM-1-2 Solve the differential equation dy + y = sin
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationMath 300 Winter 2011 Advanced Boundary Value Problems I. Bessel s Equation and Bessel Functions
Math 3 Winter 2 Avance Bounary Value Problems I Bessel s Equation an Bessel Functions Department of Mathematical an Statistical Sciences University of Alberta Bessel s Equation an Bessel Functions We use
More informationMATH FALL 2014 HOMEWORK 10 SOLUTIONS
Problem 1. MATH 241-2 FA 214 HOMEWORK 1 SOUTIONS Note that u E (x) 1 ( x) is an equilibrium distribution for the homogeneous pde that satisfies the given boundary conditions. We therefore want to find
More informationDepartment of Mathematics
INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY Department of Mathematics MA 04 - Complex Analysis & PDE s Solutions to Tutorial No.13 Q. 1 (T) Assuming that term-wise differentiation is permissible, show that
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 Lecture Notes 8 - PDEs Page 8.01 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationAPMTH 105 Final Project: Musical Sounds
APMTH 105 Final Project: Musical Sounds Robert Lin, Katherine Playfair, Katherine Scott Instructor: Dr. Margo Levine Harvard University April 30, 2016 Abstract Strings, drums, and instruments alike are
More informationAmir Javidinejad Zodiac Aerospace, 7330 Lincoln Way, Garden Grove, CA USA,
Journal of Theoretical and Applied Mechanics, Sofia, 13, vol. 43, No. 1, pp. 19 6 VIBRATION MODAL SOLUTIONS DEVELOPING OF THE ELASTIC CIRCULAR MEMBRANE IN POLAR COORDINATES BASED ON THE FOURIER-BESSEL
More informationThe heat and wave equations in 2D and 3D
The heat and wave equations in and 3 18.303 Linear Partial ifferential Equations Matthew J. Hancock Fall 005 1 and 3 Heat Equation [Nov, 005] Ref: Haberman 1.5, 7.1 Consider an arbitrary 3 subregion V
More informationLaplace s equation in polar coordinates. Boundary value problem for disk: u = u rr + u r r. r 2
Laplace s equation in polar coordinates Boundary value problem for disk: u = u rr + u r r + u θθ = 0, u(a, θ) = h(θ). r 2 Laplace s equation in polar coordinates Boundary value problem for disk: u = u
More informationFINAL EXAM, MATH 353 SUMMER I 2015
FINAL EXAM, MATH 353 SUMMER I 25 9:am-2:pm, Thursday, June 25 I have neither given nor received any unauthorized help on this exam and I have conducted myself within the guidelines of the Duke Community
More informationVibrating-string problem
EE-2020, Spring 2009 p. 1/30 Vibrating-string problem Newton s equation of motion, m u tt = applied forces to the segment (x, x, + x), Net force due to the tension of the string, T Sinθ 2 T Sinθ 1 T[u
More informationSeparation of Variables
Separation of Variables A typical starting point to study differential equations is to guess solutions of a certain form. Since we will deal with linear PDEs, the superposition principle will allow us
More informationLecture Notes for MAE 3100: Introduction to Applied Mathematics
ecture Notes for MAE 31: Introduction to Applied Mathematics Richard H. Rand Cornell University Ithaca NY 14853 rhr2@cornell.edu http://audiophile.tam.cornell.edu/randdocs/ version 17 Copyright 215 by
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 informationVIBRATIONS OF A CIRCULAR MEMBRANE
VIBRATIONS OF A CIRCULAR MEMBRANE RAM EKSTROM. Solving the wave equation on the isk The ynamics of vibrations of a two-imensional isk D are given by the wave equation..) c 2 u = u tt, together with the
More informationFig.1: Non-stationary temperature distribution in a circular plate.
LECTURE Problem 1 13. 5 : Fi.1: Non-stationary temperature distribution in a circular plate. The non-stationary, radial symmetric temperature distribution ur, t in a circular plate of radius c is determined
More informationENGI 9420 Lecture Notes 8 - PDEs Page 8.01
ENGI 940 ecture Notes 8 - PDEs Page 8.0 8. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
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 information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
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 information3150 Review Problems for Final Exam. (1) Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos(x) 0 x π f(x) =
350 Review Problems for Final Eam () Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos() 0 π f() = 0 π < < 2π (2) Let F and G be arbitrary differentiable functions
More informationQUASI-SEPARATION OF THE BIHARMONIC PARTIAL DIFFERENTIAL EQUATION
QUASI-SEPARATION OF THE BIHARMONIC PARTIAL DIFFERENTIAL EQUATION W.N. EVERITT, B.T. JOHANSSON, L.L. LITTLEJOHN, AND C. MARKETT This paper is dedicated to Lawrence Markus Emeritus Regents Professor of Mathematics
More informationCircular Membranes. Farlow, Lesson 30. November 21, Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431
The Problem Polar coordinates Solving the Problem by Separation of Variables Circular Membranes Farlow, Lesson 30 Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431 November
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationTHE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t)))
THE WAVE EQUATION The aim is to derive a mathematical model that describes small vibrations of a tightly stretched flexible string for the one-dimensional case, or of a tightly stretched membrane for the
More informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationAn Introduction to Bessel Functions
An Introduction to R. C. Trinity University Partial Differential Equations March 25, 2014 Bessel s equation Given p 0, the ordinary differential equation x 2 y +xy +(x 2 p 2 )y = 0, x > 0 (1) is known
More informationSolutions to Exercises 8.1
Section 8. Partial Differential Equations in Physics and Engineering 67 Solutions to Exercises 8.. u xx +u xy u is a second order, linear, and homogeneous partial differential equation. u x (,y) is linear
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 informationAnalysis II: Fourier Series
.... Analysis II: Fourier Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American May 3, 011 K.Maruno (UT-Pan American) Analysis II May 3, 011 1 / 16 Fourier series were
More informationTHE METHOD OF SEPARATION OF VARIABLES
THE METHOD OF SEPARATION OF VARIABES To solve the BVPs that we have encountered so far, we will use separation of variables on the homogeneous part of the BVP. This separation of variables leads to problems
More informationPlot of temperature u versus x and t for the heat conduction problem of. ln(80/π) = 820 sec. τ = 2500 π 2. + xu t. = 0 3. u xx. + u xt 4.
10.5 Separation of Variables; Heat Conduction in a Rod 579 u 20 15 10 5 10 50 20 100 30 150 40 200 50 300 x t FIGURE 10.5.5 Example 1. Plot of temperature u versus x and t for the heat conduction problem
More informationt 2 + 2t dt = (t + 1) dt + 1 = arctan t x + 6 x(x 3)(x + 2) = A x +
MATH 06 0 Practice Exam #. (0 points) Evaluate the following integrals: (a) (0 points). t +t+7 This is an irreducible quadratic; its denominator can thus be rephrased via completion of the square as a
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationProject 10.5C Circular Membrane Vibrations
Projet 1.5C Cirular Membrane Vibrations In problems involving regions that enjoy irular symmetry about the origin in the plane (or the vertial z-axis in spae), the use of polar (or ylindrial) oordinates
More informationElectrodynamics I Midterm - Part A - Closed Book KSU 2005/10/17 Electro Dynamic
Electrodynamics I Midterm - Part A - Closed Book KSU 5//7 Name Electro Dynamic. () Write Gauss Law in differential form. E( r) =ρ( r)/ɛ, or D = ρ, E= electricfield,ρ=volume charge density, ɛ =permittivity
More informationThe General Dirichlet Problem on a Rectangle
The General Dirichlet Problem on a Rectangle Ryan C. Trinity University Partial Differential Equations March 7, 0 Goal: Solve the general (inhomogeneous) Dirichlet problem u = 0, 0 < x < a, 0 < y < b,
More informationSolving the Heat Equation (Sect. 10.5).
Solving the Heat Equation Sect. 1.5. Review: The Stationary Heat Equation. The Heat Equation. The Initial-Boundary Value Problem. The separation of variables method. An example of separation of variables.
More informationu tt = a 2 u xx u tt = a 2 (u xx + u yy )
10.7 The wave equation 10.7 The wave equation O. Costin: 10.7 1 This equation describes the propagation of waves through a medium: in one dimension, such as a vibrating string u tt = a 2 u xx 1 This equation
More informationKarlstad University FYGB08. Analytical Mechanics. Modelling of Drums. January 21, 2015
Karlstad University FYGB08 Analytical Mechanics Modelling of Drums Author: Wilhelm S. Wermelin Supervisors: Jürgen Fuchs Igor Buchberger January 21, 2015 Abstract In this project the motion of a membrane
More informationBoundary-value Problems in Rectangular Coordinates
Boundary-value Problems in Rectangular Coordinates 2009 Outline Separation of Variables: Heat Equation on a Slab Separation of Variables: Vibrating String Separation of Variables: Laplace Equation Review
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More information7-7 Multiplying Polynomials
Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) Example 1C: Multiplying Monomials Group factors with
More information3 Green s functions in 2 and 3D
William J. Parnell: MT34032. Section 3: Green s functions in 2 and 3 57 3 Green s functions in 2 and 3 Unlike the one dimensional case where Green s functions can be found explicitly for a number of different
More information( ) ( ) Math 17 Exam II Solutions
Math 7 Exam II Solutions. Sketch the vector field F(x,y) -yi + xj by drawing a few vectors. Draw the vectors associated with at least one point in each quadrant and draw the vectors associated with at
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 2 Approximation Theory. Antony Jameson
Advanced Computational Fluid Dynamics AA5A Lecture Approximation Theory Antony Jameson Winter Quarter, 6, Stanford, CA Last revised on January 7, 6 Contents Approximation Theory. Least Squares Approximation
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More informationFinal Examination Linear Partial Differential Equations. Matthew J. Hancock. Feb. 3, 2006
Final Examination 8.303 Linear Partial ifferential Equations Matthew J. Hancock Feb. 3, 006 Total points: 00 Rules [requires student signature!]. I will use only pencils, pens, erasers, and straight edges
More informationHigher Order Beam Equations
Higher Order Beam Equations Master s Thesis in Solid and Fluid Mechanics HOSSEIN ABADIKHAH Department of Applied Mechanics Division of Dynamics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2011 Master
More informationMath 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
More informationThermal Bending of Circular Plates for Non-axisymmetrical Problems
Copyrigt 2 Tec Science Press SL, vol.4, no.2, pp.5-2, 2 Termal Bending of Circular Plates for Non-axisymmetrical Problems Dong Zengzu Peng Weiong and Li Suncai Abstract: Due to te complexity of termal
More informationPHYS 404 Lecture 1: Legendre Functions
PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS 404 - LECTURE 1 DR. V. LEMPESIS 1 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions
More informationPhysics 6303 Lecture 11 September 24, LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation
Physics 6303 Lecture September 24, 208 LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation, l l l l l l. Consider problems that are no axisymmetric; i.e., the potential depends
More informationMathematics for Engineers II. lectures. Power series, Fourier series
Power series, Fourier series Power series, Taylor series It is a well-known fact, that 1 + x + x 2 + x 3 + = n=0 x n = 1 1 x if 1 < x < 1. On the left hand side of the equation there is sum containing
More informationCHAPTER 10 NOTES DAVID SEAL
CHAPTER 1 NOTES DAVID SEA 1. Two Point Boundary Value Problems All of the problems listed in 14 2 ask you to find eigenfunctions for the problem (1 y + λy = with some prescribed data on the boundary. To
More informationPHYSICS 116C Homework 5 Solutions. y 2 = 0. x T
PHYSICS 116C Homework 5 Solutions 1. a The temperature T, y satisfies aplace s equation T + T y. The boundary conditions are T along,, and y, with T T at y. We look for a separable solution, T,y XYy. Following
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More information10.2-3: Fourier Series.
10.2-3: Fourier Series. 10.2-3: Fourier Series. O. Costin: Fourier Series, 10.2-3 1 Fourier series are very useful in representing periodic functions. Examples of periodic functions. A function is periodic
More informationAn Introduction to Partial Differential Equations in the Undergraduate Curriculum
An Introduction to Partial Differential Equations in the Undergraduate Curriculum John C. Polking LECTURE 12 Heat Transfer in the Ball 12.1. Outline of Lecture The problem The problem for radial temperatures
More informationAnouncements. Assignment 3 has been posted!
Anouncements Assignment 3 has been posted! FFTs in Graphics and Vision Correlation of Spherical Functions Outline Math Review Spherical Correlation Review Dimensionality: Given a complex n-dimensional
More informationLaplace s Equation in Cylindrical Coordinates and Bessel s Equation (I)
Laplace s Equation in Cylindrical Coordinates and Bessel s Equation I) 1 Solution by separation of variables Laplace s equation is a key equation in Mathematical Physics. Several phenomena involving scalar
More informationOverview of Fourier Series (Sect. 6.2). Origins of the Fourier Series.
Overview of Fourier Series (Sect. 6.2. Origins of the Fourier Series. Periodic functions. Orthogonality of Sines and Cosines. Main result on Fourier Series. Origins of the Fourier Series. Summary: Daniel
More informationBESSEL FUNCTIONS APPENDIX D
APPENDIX D BESSEL FUNCTIONS D.1 INTRODUCTION Bessel functions are not classified as one of the elementary functions in mathematics; however, Bessel functions appear in the solution of many physical problems
More information