A proof for the full Fourier series on [ π, π] is given here.
|
|
- Claribel Strickland
- 6 years ago
- Views:
Transcription
1 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 the boundary points. If the smooth function satisfies the same boundary conditions as the eigenfunctions used for the expansion, uniform convergence can be obtained. Theorem: Let A n X n represents the full Fourier series, or Fourier sine series, or Fourier cosine series. The series A n X n converges to f(x) absolutely and uniformly on [a, b] provided that: (i) f(x) and f (x) are continuous on [a, b], and (ii) f(x) satisfies the same boundary conditions as those of X n. A proof for the full Fourier series on [ π, π] is given here. Proof: Let A n, B n denote the Fourier coefficients of f, and A n, B n denote the Fourier coefficients of f. se integration by parts to get π A n = f(x) cos nx dx π π = 1 nπ f(x) sin nx π π π π f (x) sin nx dx nπ = 1 n B n 1
2 Similarly, π B n = f(x) sin nx dx π π = 1 π nπ f(x) cos nx π π + f (x) cos nx dx π nπ = 1 n A n in which the periodicity of f(x) is used. Bessel s inequality for the f series gives 1 2 A ( A n 2 + B n 2 ) 1 π f 2 dx <. π π Then A n = ( ) 1/2 ( 1 n 2 1 n B n B n 2 ) 1/2 < continuity of f used in which the Schwarz s inequality has been applied. This means that the series A n converges absolutely. A similar statement can be made on B n. Therefore, the Fourier series converges absolutely. The sum of the Fourier series converges pointwise to f(x) everywhere as f is continuous and f( π) = f(π) (no jump anywhere). Then, 2
3 max f(x) S N (x) { } max ( A n cosnx + B n sin nx ) n=n+1 n=n+1 ( A n + B n ) 0 as N because it is the tail of a convergent series of numbers. Therefore, S N (x) f(x) uniformly. 3
4 Inhomogeneous problems in a finite interval Consider the inhomogeneous problem with time-dependent boundary values: u tt c 2 u xx = f(x, t) u(x, 0) = φ(x) u(0, t) = h(t) 0 < x < L u t (x, 0) = ψ(x) u(l, t) = k(t). The boundary conditions of this problem generally cannot be satisfied by the full, sine, or cosine Fourier series. Yet, the function u(x, t) can be represented pointwise in (0, L) by one of these series with time-dependent coefficients. Since the boundary conditions are of Dirichlet type, we use the Fourier sine series: u(x, t) = u n (t) sin nπx L. The equations for computing the coefficients of this series cannot be obtained by simply plugging the series into the pde. The spatial differential operator 2 cannot be moved inside x 2 the summation as the convergence is not uniform. It is therefore necessary to use a modified approach. Expansion method The technique is to expand u, u tt, and u tt into separate Fourier series with time-dependent coefficients. Multiplying both sides of the pde by sin nπx and integrating from 0 to L 4
5 L, one can get the following set of equations that connect the coefficients of the different Fourier series v n (t) c 2 w n (t) = f n (t). v n (t), w n (t), and f n (t) are coefficients of the Fourier sine series of u tt and u xx, and f(x, t), respectively. v n and w n can be related to the Fourier coefficient u n as following. v n (t) = 2 L w n (t) = 2 L L 0 L 0 2 u nπx sin t2 L dx = d2 u n dt, 2 2 u nπx sin x2 L dx = λ n u n (t) + 2nπL 2 [h(t) ( 1) n k(t)] where λ n = (nπ/l) 2. Therefore, u n satisfies the ode: integration by parts d 2 u n dt 2 + c2 λ n u n (t) = c 2 2nπL 2 [h(t) ( 1) n k(t)] + f n (t). The initial conditions for this ode are u n (0) = φ n and du n dt (0) = ψ n where φ n and ψ n are Fourier sine coefficients for φ(x) and ψ(x), respectively. Subtraction method The boundary conditions can be made homogeneous by subtracting any known function that satisfies them. We use the function 5
6 ( (x, t) = 1 x ) h(t) + x L L k(t) which satisfies the BCs. With the subtraction ũ(x, t) = u(x, t) (x, t), the problem can be converted to the following problem for ũ: ũ tt c 2 ũ xx = f(x, t) tt ũ(0, t) = 0 ũ(l, t) = 0 0 < x < L ũ(x, 0) = φ(x) (x, 0) ũ t (x, 0) = ψ(x) t (x, 0). Since the boundary conditions for ũ is now Dirichlet, we are assured of the uniform convergence of the Fourier sine series expansion of ũ(x, t) on [0, L] at any t. In particular, the Dirichlet boundary conditions are satisfied all the time. The Fourier sine series of the continuous function converges pointwise to on (0, L), but generally not at the boundary points. As u n (t) = ũ n (t) + n (t), the Fourier sine series of u(x, t) tends to h(t) and k(t) as x gets close to 0 and L, respectively. Note, however, the Gibb s phenomenon generally occurs. The equation for the Fourier sine series coefficient ũ n is d 2 ũ n dt + 2 c2 λ n ũ n (t) = f n (t) d2 n dt. 2 Through the substitution ũ n = u n n, this equation can be rewritten as d 2 u n dt 2 + c2 λ n [u n (t) n ] 6
7 = d2 u n dt 2 + c2 λ n u n (t) c 2 2nπL 2 [h(t) ( 1) n k(t)] = f n (t). The term c 2 λ n n re-introduces the boundary conditions h(t) and k(t) as two terms in the ode. This equation is the same as the one derived by the expansion method. The initial conditions are also the same. The wave equation in higher dimensions The solution of the wave equation in 3D can be expressed as a surface integral. The Kirchhoff-Poisson solution The Cauchy problem (CP) is: u tt = c 2 u x R 3, t 0 u( x, 0) = φ( x), u t ( x, 0) = ψ( x). where = 2 xx + 2 yy + 2 zz. This problem can be split into two parts according to the nature of the initial conditions: (1) WCP (2) VCP w tt = c 2 w w( x, 0) = φ( x), w t ( x, 0) = 0 7
8 v tt = c 2 v v( x, 0) = 0, v t ( x, 0) = ψ( x) Clearly, u = w + v is the solution of the original CP. If one can solve VCP, then one can also solve WCP. Lemma: If v is a solution of VCP with v t ( x, 0) = ψ( x), then w = v t is a solution of WCP with w( x, 0) = ψ( x). Theorem: (Kirchhoff s formula) Let ψ be continuous with continuous first and second partial derivatives for all x. Then for t > 0 and x R 3, the solution of VCP is v( x, t) = 1 ψ( y)dσ c 2 t S( x;t) in which S( x; t) = S t is the sphere with radius ct centered at x and dσ represents a surface element on S t. Proof: Let y = x + ct ξ, the integral formula can be written as v( x, t) = 1 c 2 t S t ψ( y)dσ = t where is the unit sphere in the ξ-space and dτ represents a surface element on. ψ( x + ct ξ)dτ (1) 8
9 Then v = t At the same time v t = 1 ψ( x + ct ξ)dτ = 1 c 2 t ψ( x + ct ξ)dτ + ct which can further be written as = 1 t v + 1 ψ ns dσ = 1 ct S t t v + 1 ct S t ψdσ. ψ n dτ (2) B t ψdv. n is the outward pointing unit normal on, n S is the outward pointing unit normal on S t, and B t is the solid ball with radius ct centered at x. Let I = ψdv, then v t = 1 B t t v + I ct. v tt = 1 t 2v + 1 t v t 1 ct 2I + I t ct = I t ct. Since I t = c ψdσ, S t v tt = 1 ψdσ = c 2 v. t S t The equation is satisfied. Furthermore, Eq.(1) gives 9
10 v( x, 0) = 0 and Eq.(2) gives v t ( x, 0) = 1 = ψ( x). ψ( x + 0 ζ)dσ = 0, ψ( x + c 0 ξ)dτ + c 0 The initial conditions are satisfied. ψ n dτ As a result, the solution of the CP is u( x, t) = t [ 1 c 2 t ] φ( y)dσ S t + 1 ψ( y)dσ. c 2 t S t This expression is known as Poisson s formula for the solution of the CP problem for the wave equation in 3D. Example: Consider the simple cases: (i) φ = 1, ψ = 0 and (ii) φ = 1, ψ = 1. For a given point x and time t, u( x, t) depends only on the initial data given on the sphere S t of radius ct about the point x. S t is the domain of influence of the initial conditions at time t. This was noted by Huygens in the seventeenth century (Huygens Principle). 10
11 Hadamard s method of descent The idea is to use Poisson s 3D formula to solve the 2D problem. Suppose that the 3D functions all depend only on the coordinates x and y. The initial conditions become u(x, y, z, 0) = φ(x, y), u t (x, y, z, 0) = ψ(x, y). The solution u(x, y, z, t) also depends only on x, y, and t. [ 1 c 2 t ] + 1 c 2 t u(x, y, t) = φ(ξ, η)dσ t S t (ξ, η, ζ) represents (x, y, z) in the integral. The surface integral can be projected onto the xy-plane as dξdη = n k dσ = ζ ct dσ = 1 (ct)2 (ξ x) ct 2 (η y) 2 dσ. Then, as S t projects two times on D t, ψ(ξ, η)dσ = S t ψ(ξ, η) 2ct D t (ct)2 (ξ x) 2 (η y) 2dξdη where D t is the disk with radius ct centered at (x, y). The formula for u(x, y, t) becomes S t ψ(ξ, η)dσ. 11
12 [ ] u(x, y, t) = 1 φ(ξ, η) 2πc t D t (ct)2 (ξ x) 2 (η y) 2dξdη + 1 ψ(ξ, η) 2πc D t (ct)2 (ξ x) 2 (η y) 2dξdη. The domain of influence is now the whole of D t. A disturbance at another point will eventually be felt at (x, y) and will then be felt at all later times. Huygens principle does not hold in the plane. plot example Problem Set 7 12
Diffusion 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 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 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 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 informationG: Uniform Convergence of Fourier Series
G: Uniform Convergence of Fourier Series From previous work on the prototypical problem (and other problems) u t = Du xx 0 < x < l, t > 0 u(0, t) = 0 = u(l, t) t > 0 u(x, 0) = f(x) 0 < x < l () we developed
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 information1 Wave Equation on Finite Interval
1 Wave Equation on Finite Interval 1.1 Wave Equation Dirichlet Boundary Conditions u tt (x, t) = c u xx (x, t), < x < l, t > (1.1) u(, t) =, u(l, t) = u(x, ) = f(x) u t (x, ) = g(x) First we present the
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
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 informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
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 informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationPartial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation:
Chapter 7 Heat Equation Partial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation: u t = ku x x, x, t > (7.1) Here k is a constant
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 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationHomework for Math , Fall 2016
Homework for Math 5440 1, Fall 2016 A. Treibergs, Instructor November 22, 2016 Our text is by Walter A. Strauss, Introduction to Partial Differential Equations 2nd ed., Wiley, 2007. Please read the relevant
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 informationThe Fourier series for a 2π-periodic function
The Fourier series for a 2π-periodic function Let f : ( π, π] R be a bounded piecewise continuous function which we continue to be a 2π-periodic function defined on R, i.e. f (x + 2π) = f (x), x R. The
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 informationReview For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.
Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =
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 informationswapneel/207
Partial differential equations Swapneel Mahajan www.math.iitb.ac.in/ swapneel/207 1 1 Power series For a real number x 0 and a sequence (a n ) of real numbers, consider the expression a n (x x 0 ) n =
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationMathematical Modeling using Partial Differential Equations (PDE s)
Mathematical Modeling using Partial Differential Equations (PDE s) 145. Physical Models: heat conduction, vibration. 146. Mathematical Models: why build them. The solution to the mathematical model will
More informationProblem set 3: Solutions Math 207B, Winter Suppose that u(x) is a non-zero solution of the eigenvalue problem. (u ) 2 dx, u 2 dx.
Problem set 3: Solutions Math 27B, Winter 216 1. Suppose that u(x) is a non-zero solution of the eigenvalue problem u = λu < x < 1, u() =, u(1) =. Show that λ = (u ) 2 dx u2 dx. Deduce that every eigenvalue
More informationPartial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
More informationMath 311, Partial Differential Equations, Winter 2015, Midterm
Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There
More informationMATH 819 FALL We considered solutions of this equation on the domain Ū, where
MATH 89 FALL. The D linear wave equation weak solutions We have considered the initial value problem for the wave equation in one space dimension: (a) (b) (c) u tt u xx = f(x, t) u(x, ) = g(x), u t (x,
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationMATH 124B: HOMEWORK 2
MATH 24B: HOMEWORK 2 Suggested due date: August 5th, 26 () Consider the geometric series ( ) n x 2n. (a) Does it converge pointwise in the interval < x
More informationStarting from Heat Equation
Department of Applied Mathematics National Chiao Tung University Hsin-Chu 30010, TAIWAN 20th August 2009 Analytical Theory of Heat The differential equations of the propagation of heat express the most
More informationSeparation of Variables in Linear PDE: One-Dimensional Problems
Separation of Variables in Linear PDE: One-Dimensional Problems Now we apply the theory of Hilbert spaces to linear differential equations with partial derivatives (PDE). We start with a particular example,
More informationPartial Differential Equations Separation of Variables. 1 Partial Differential Equations and Operators
PDE-SEP-HEAT-1 Partial Differential Equations Separation of Variables 1 Partial Differential Equations and Operators et C = C(R 2 ) be the collection of infinitely differentiable functions from the plane
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 informationSAMPLE FINAL EXAM SOLUTIONS
LAST (family) NAME: FIRST (given) NAME: ID # : MATHEMATICS 3FF3 McMaster University Final Examination Day Class Duration of Examination: 3 hours Dr. J.-P. Gabardo THIS EXAMINATION PAPER INCLUDES 22 PAGES
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
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 informationAnalysis III (BAUG) Assignment 3 Prof. Dr. Alessandro Sisto Due 13th October 2017
Analysis III (BAUG Assignment 3 Prof. Dr. Alessandro Sisto Due 13th October 2017 Question 1 et a 0,..., a n be constants. Consider the function. Show that a 0 = 1 0 φ(xdx. φ(x = a 0 + Since the integral
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationDifferential equations, comprehensive exam topics and sample questions
Differential equations, comprehensive exam topics and sample questions Topics covered ODE s: Chapters -5, 7, from Elementary Differential Equations by Edwards and Penney, 6th edition.. Exact solutions
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 informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
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 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 informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
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 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 informationAPPM GRADUATE PRELIMINARY EXAMINATION PARTIAL DIFFERENTIAL EQUATIONS SOLUTIONS
Thursday August 24, 217, 1AM 1PM There are five problems. Solve any four of the five problems. Each problem is worth 25 points. On the front of your bluebook please write: (1) your name and (2) a grading
More informationSalmon: Lectures on partial differential equations
6. The wave equation Of the 3 basic equations derived in the previous section, we have already discussed the heat equation, (1) θ t = κθ xx + Q( x,t). In this section we discuss the wave equation, () θ
More informationThere are five problems. Solve four of the five problems. Each problem is worth 25 points. A sheet of convenient formulae is provided.
Preliminary Examination (Solutions): Partial Differential Equations, 1 AM - 1 PM, Jan. 18, 16, oom Discovery Learning Center (DLC) Bechtel Collaboratory. Student ID: There are five problems. Solve four
More informationMath 124B January 31, 2012
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
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 informationMA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series Lecture - 10
MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series ecture - 10 Fourier Series: Orthogonal Sets We begin our treatment with some observations: For m,n = 1,2,3,... cos
More informationMATH 251 Final Examination August 14, 2015 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 14, 2015 FORM A Name: Student Number: Section: This exam has 11 questions for a total of 150 points. Show all your work! In order to obtain full credit for partial credit
More informationMath 115 ( ) Yum-Tong Siu 1. Derivation of the Poisson Kernel by Fourier Series and Convolution
Math 5 (006-007 Yum-Tong Siu. Derivation of the Poisson Kernel by Fourier Series and Convolution We are going to give a second derivation of the Poisson kernel by using Fourier series and convolution.
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 informationMath 489AB A Very Brief Intro to Fourier Series Fall 2008
Math 489AB A Very Brief Intro to Fourier Series Fall 8 Contents Fourier Series. The coefficients........................................ Convergence......................................... 4.3 Convergence
More informationwhere u 0 is given. In all our applications it is possible to subtract a known function v(x, t) from u(x, t) so that w(x, t) = u(x, t) v(x, t)
CHAPTER PDE Partial Differential Equations in Two Independent Variables D. An Overview Drawing on the Sturm-Liouville eigenvalue theory and the approximation of functions we are now ready to develop the
More informationThe One-Dimensional Heat Equation
The One-Dimensional Heat Equation R. C. Trinity University Partial Differential Equations February 24, 2015 Introduction The heat equation Goal: Model heat (thermal energy) flow in a one-dimensional object
More informationLecture 19: Heat conduction with distributed sources/sinks
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 ecture 19: Heat conduction
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
More informationMath 251 December 14, 2005 Answer Key to Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Name Section Math 51 December 14, 5 Answer Key to Final Exam There are 1 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning
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 informationMathematics of Physics and Engineering II: Homework problems
Mathematics of Physics and Engineering II: Homework problems Homework. Problem. Consider four points in R 3 : P (,, ), Q(,, 2), R(,, ), S( + a,, 2a), where a is a real number. () Compute the coordinates
More informationThe 1-D Heat Equation
The 1-D Heat Equation 18.303 Linear Partial Differential Equations Matthew J. Hancock Fall 004 1 The 1-D Heat Equation 1.1 Physical derivation Reference: Haberman 1.1-1.3 [Sept. 8, 004] In a metal rod
More informationMath 220A - Fall 2002 Homework 5 Solutions
Math 0A - Fall 00 Homework 5 Solutions. Consider the initial-value problem for the hyperbolic equation u tt + u xt 0u xx 0 < x 0 u t (x, 0) ψ(x). Use energy methods to show that the domain of dependence
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 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 informationMATH FALL 2014
MATH 126 - FALL 2014 JASON MURPHY Abstract. These notes are meant to supplement the lectures for Math 126 (Introduction to PDE) in the Fall of 2014 at the University of California, Berkeley. Contents 1.
More informationMath 5440 Problem Set 7 Solutions
Math 544 Math 544 Problem Set 7 Solutions Aaron Fogelson Fall, 13 1: (Logan, 3. # 1) Verify that the set of functions {1, cos(x), cos(x),...} form an orthogonal set on the interval [, π]. Next verify that
More informationMath 220a - Fall 2002 Homework 6 Solutions
Math a - Fall Homework 6 Solutions. Use the method of reflection to solve the initial-boundary value problem on the interval < x < l, u tt c u xx = < x < l u(x, = < x < l u t (x, = x < x < l u(, t = =
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
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 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 informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationSome Aspects of Solutions of Partial Differential Equations
Some Aspects of Solutions of Partial Differential Equations K. Sakthivel Department of Mathematics Indian Institute of Space Science & Technology(IIST) Trivandrum - 695 547, Kerala Sakthivel@iist.ac.in
More informationLecture notes: Introduction to Partial Differential Equations
Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential
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 informationMATH 220: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY
MATH 22: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY When discussing separation of variables, we noted that at the last step we need to express the inhomogeneous initial or boundary data as
More information(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y
() Consider A = { q Q : q 2 2} as a subset of the metric space (Q, d), where d(x, y) = x y. Then A is A) closed but not open in Q B) open but not closed in Q C) neither open nor closed in Q D) both open
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 informationChapter 10: Partial Differential Equations
1.1: Introduction Chapter 1: Partial Differential Equations Definition: A differential equations whose dependent variable varies with respect to more than one independent variable is called a partial differential
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 informationLaplace equation. In this chapter we consider Laplace equation in d-dimensions given by. + u x2 x u xd x d. u x1 x 1
Chapter 6 Laplace equation In this chapter we consider Laplace equation in d-dimensions given by u x1 x 1 + u x2 x 2 + + u xd x d =. (6.1) We study Laplace equation in d = 2 throughout this chapter (excepting
More informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
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 informationSeparation of Variables. A. Three Famous PDE s
Separation of Variables c 14, Philip D. Loewen A. Three Famous PDE s 1. Wave Equation. Displacement u depends on position and time: u = u(x, t. Concavity drives acceleration: u tt = c u xx.. Heat Equation.
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 informationMA Chapter 10 practice
MA 33 Chapter 1 practice NAME INSTRUCTOR 1. Instructor s names: Chen. Course number: MA33. 3. TEST/QUIZ NUMBER is: 1 if this sheet is yellow if this sheet is blue 3 if this sheet is white 4. Sign the scantron
More informationMore on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
More informationSECTION (See Exercise 1 for verification when both boundary conditions are Robin.) The formal solution of problem 6.53 is
6.6 Properties of Parabolic Partial Differential Equations SECTION 6.6 265 We now return to a difficulty posed in Chapter 4. In what sense are the series obtained in Chapters 4 and 6 solutions of their
More informationProblem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j (0 t 2π). Solution: We assume a > b > 0.
MATH 64: FINAL EXAM olutions Problem 1. Use a line integral to find the plane area enclosed by the curve C: r = a cos 3 t i + b sin 3 t j ( t π). olution: We assume a > b >. A = 1 π (xy yx )dt = 3ab π
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationSection 12.6: Non-homogeneous Problems
Section 12.6: Non-homogeneous Problems 1 Introduction Up to this point all the problems we have considered are 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 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationPartial Differential Equations for Engineering Math 312, Fall 2012
Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant
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 informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
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 information