Homework 3 solutions Math 136 Gyu Eun Lee 2016 April 15. R = b a
|
|
- Godwin Dennis Bennett
- 6 years ago
- Views:
Transcription
1 Homework 3 solutions Math 136 Gyu Eun Lee 2016 April 15 A problem may have more than one valid method of solution. Here we present just one. Arbitrary functions are assumed to have whatever regularity properties are required, unless otherwise specified A rotation of the xy-plane is a linear transformation whose matrix is of the form ( ) a b R = b a where a 2 + b 2 = Each rotation defines a change of variable like so: ( ) ( ) ( ) ξ x ax by = R =. ν y bx + ay Given the equation a 11 u xx + 2a 12 u xy + a 22 u xx + b 1 u x + b 2 u y + cu = 0, (1) and a rotation R, replacing u(x,y) with u(ξ,η) = u(r(x,y)) gives us a new equation d 11 u ξ ξ + 2d 12 u ξ η + d 22 u ηη + e 1 u ξ + e 2 u η + f u = 0, (2) where the coefficients d i j, e i, and f depend on R. To say that the PDE is invariant under rotations means that the coefficients must actually be independent of R; 3 that is, d i j = a i j, b i = e i, and c = f for all choices of R (equivalently, for all choices of rotation angle θ), and our problem is to determine which conditions on a i j, b i, and c make this true. Since the second-order derivatives in Equation (1) correspond under the change of variable exactly to the second-order derivatives in Equation (2), and similarly for the first-order and zero-th order derivatives, it suffices to work with each order of derivatives separately. For the zero-th order term, clearly changing from xy-variables to ξ η-variables does not change the coefficient in front of u. Therefore we must have c = f for the zero-th order terms to be invariant. For the first-order term, a calculation with the chain rule shows that u x = a u ξ + b u η, c 2016, Gyu Eun Lee. Licensed under the Creative Commons Attribution-NonCommercial 4.0 International license (CC BY-NC 4.0), Last updated: April 20, Such a matrix is said to be a special orthogonal matrix, i.e. a matrix of determinant 1 with orthogonal rows and columns. The collection of all such matrices is called the special orthogonal group, denoted SO(2). 2 In particular, we can take a = cosθ, b = sinθ for some angle θ. However, we will see that expressing the matrix in this form only makes the notation cumbersome. 3 Equivalently, if u(x,y) is a solution to (1), then so is u(ξ,η) = u(r(x,y)) for all rotations R. 1
2 Therefore u u = b y ξ + a u η. Au x + Bu y = (aa bb)u ξ + (ba + ab)u η. If the equation is to be invariant under rotation, then we must have ( ) ( ) ( )( ) A aa bb a b A = = B ba + ab b a B for all choices of a,b with a 2 + b 2 = 1; that is, the 2-d vector (A,B) must be invariant under all rotations. But the only vector that is invariant under all rotations is the zero vector; therefore A = B = 0. For the second-order terms, we find the partial derivatives to be 2 u x 2 = 2 u a2 ξ 2 + 2ab 2 u ξ η + 2 u b2 η 2, Therefore 2 u x y = ab 2 u ξ 2 + (a2 b 2 ) 2 u ξ η + ab 2 u η 2, 2 u y 2 = 2 u b2 ξ 2 2ab 2 u η ξ + 2 u a2 η 2. Au xx + 2Bu xy +Cu yy = (a 2 A 2abB + b 2 C)u ξ ξ + (2abA + 2(a 2 b 2 )B 2abC)u ξ η + (b 2 C + 2abB + a 2 C)u ηη. If the second-order terms are to be invariant under all rotations, then A a 2 2ab b 2 A B = 2ab 2(a 2 b 2 ) 2ab B C b 2 2ab a 2 C for all choices of a,b with a 2 +b 2 = 1. Necessary conditions on A,B,C can now be obtained by taking particular choices of a and b. Choosing a = 1, b = 0 gives us A A B = B, C C and in particular B = 2B, which tells us B = 0. Choosing a = 0, b = 1 now gives us A A 0 = , C C 2
3 or equivalently A = C. Therefore we conclude that the only choice of coefficients making Equation (1) invariant under all rotations is a(u xx + u yy ) + cu = 0. Remark: Another way one could prove this statement is to rephrase the invariance of coefficients in terms of matrices. For example, the statement that the second-order coefficients are invariant is equivalent to: if A = (a i j ) is the matrix of second-order coefficients, then BAB t = A for all rotation matrices B. Equivalently, since all rotation matrices satisfy B t = B 1, BA = AB; i.e. A commutes with all rotations. Then one needs to find necessary and sufficient conditions on A for this to be true. Several people tried this method, but failed in one key step: it is not true that for a general matrix A, if BAB t = A for all rotation matrices B then A is a multiple of the identity. In 2 dimensions, a counterexample is when A is a rotation matrix, because rotations about the same axis always commute. The key is to use the fact that A is also a symmetric matrix, and a result called the spectral theorem. A proof of this sort that did not invoke the symmetry of A did not receive full credit By d Alembert s formula, u(x,t) = 1 2 [φ(x + ct) + φ(x ct)] + 1 2c x+ct x ct ψ(s) ds = 1 2c x+ct x ct ψ(s) ds. Since ψ is zero outside the interval [ a,a], this integral is equal to the length of the intersection of [ a, a] and [x ct, x + ct]. When t = ka/2c for k some positive integer, Then we have the following cases: [x ct,x + ct] = [x ka 2,x + ka 2 ]. 3
4 (a) If x > a + ka 2, then [x ka 2,x + ka 2 ] and [ a,a] are disjoint intervals. So for x in this region, u(x, ka 2c ) = 0. (b) If a x ka 2 a x + ka 2, then (c) If then (d) If u(x,t) = a x + ka 2. x ka 2 a x + ka 2 a, u(x,t) = x + ka 2 + a. a x ka 2 < x + ka 2 a, then u(x,t) = ka. (e) If x ka 2 a < a x + ka 2, then u(x,t) = 2a. Whether or not there exist x that fall into any of these cases depends on the value of k. We look at just the example of k = 1, i.e. t = a/2c. In this case, [x a 2,x + a 2 ] is an interval of length a. Since [ a,a] has length 2a, case (e) cannot be valid for any x, but all other cases are valid, and the so the string profile for u(x, 2c a ) is a trapezoid: 0 x < 3a 2, 1 2c (x + 3a 2 ) 3a 2 x < a 2, u(x,t) = a 2c a 2 x < 2 a, 1 2c ( 3a 2 x) a 2 x < 3a 2, 0 x 3a 2. The string profiles should be: (a) t = 2c a : a trapezoid of height 2c a (b) t = a c : a triangle of height a c (c) t = 3a 2c : a trapezoid of height a c (d) t = 2a c : a wider trapezoid of height a c (e) t = 5a c : an even wider trapezoid of height a c 4
5 2.1.8 (a) Suppose u solves the spherical wave equation u tt = c 2 (u rr + 2 r u r ), and set v = ru. Then v tt = ru tt and v rr = ru rr + 2u r. Then v tt = ru tt = c 2 (ru rr + 2u r ) = c 2 v rr. (b) The general solution is where f and g are arbitrary functions. v(r,t) = f (r + ct) + g(r ct) (c) The initial conditions u(r,0) = φ(r), u t (r,0) = ψ(r) give us v(r,0) = rφ(r), v t (r,0) = rψ(r). Therefore by d Alembert s formula Then v(r,t) = 1 2 [(r + ct)φ(r + ct) + (r ct)φ(r ct)] + 1 2c r+ct r ct sψ(s) ds. r+ct u(r,t) = v(r,t) = 1 1 [(r + ct)φ(r + ct) + (r ct)φ(r ct)] + sψ(s) ds. r 2r 2cr r ct Remark: The reason we assume φ and ψ are even is because u is a solution of the spherical wave equation in 3 dimensions. In principle u(r,t) is then defined only for r 0, since the distance r to the origin cannot be negative. Then when we change to v = ru, we cannot solve the IVP as we have been doing in this chapter, because the domain of the IVP is (r,t) [0, ) (, ) instead of (, ) (, ). However, we can extend the initial conditions u(r,0) = φ(r) and u t (r,0) = ψ(r) to the negative r-axis by reflecting φ and ψ across the y-axis; the resulting reflections are even, and now the domain of the IVP is (, ) (, ), allowing us to use the techniques of section 2.1. Other than this, the assumption that φ and ψ are even does not add much to this problem. This technique is known as even reflection, and we will see it again in chapter For the sake of ease of notation, we will assume c = ρ = T = 1. The general case should be obvious from this case; in fact, by scaling the wave equation and making some linear changes of variable, it is not hard to see that the general case can be reduced to this case. (Exercise.) When c = ρ = T = 1, the wave equation is and the energy is E(t) = 1 2 u tt = u xx (u 2 t + u 2 x) dx. 5
6 In principle this is a function of t; however, by conservation of energy, de/dt = 0, and therefore E is a constant function of t. By the initial condition u(x,0) = φ(x) 0, we have u x (x,0) 0. We also know that u t (x,0) = ψ(x) 0. Therefore E(0) = 1 2 (u t (x,0) 2 + u x (x,0) 2 ) dx = dx = 0. Since E is constant in t, this implies E(t) = 0 for all t. But since the integrand u t (x,t) 2 + u x (x,t) 2 is non-negative for all x and t, this implies that u t (x,t) 0 and u x (x,t) 0. The only continuously differentiable functions u that satisfy both these equations are the constant functions, so u must be constant. Since we know u(x,0) = 0, this constant must be 0; that is, u(x,t) Since u solves the wave equation, we know that for some functions f and g. Then u(x,t) = f (x +t) + g(x t) u(x + h,t + k) = f (x + h +t + k) + g(x + h t k), u(x h,t k) = f (x h +t k) + g(x h t + k), u(x + k,t + h) = f (x + k +t + h) + g(x + k t h), u(x k,t h) = f (x k +t h) + g(x k t + h). Comparing the sum of the first two lines, and the sum of the last two lines, we see that they are indeed equal. 6
First order wave equations. Transport equation is conservation law with J = cu, u t + cu x = 0, < x <.
First order wave equations Transport equation is conservation law with J = cu, u t + cu x = 0, < x
More informationMA 201: Partial Differential Equations D Alembert s Solution Lecture - 7 MA 201 (2016), PDE 1 / 20
MA 201: Partial Differential Equations D Alembert s Solution Lecture - 7 MA 201 (2016), PDE 1 / 20 MA 201 (2016), PDE 2 / 20 Vibrating string and the wave equation Consider a stretched string of length
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 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 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 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 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 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
More informationMATH 173: PRACTICE MIDTERM SOLUTIONS
MATH 73: PACTICE MIDTEM SOLUTIONS This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve all of them. Write your solutions to problems and in blue book #, and your
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 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 informationMathematical Methods - Lecture 9
Mathematical Methods - Lecture 9 Yuliya Tarabalka Inria Sophia-Antipolis Méditerranée, Titane team, http://www-sop.inria.fr/members/yuliya.tarabalka/ Tel.: +33 (0)4 92 38 77 09 email: yuliya.tarabalka@inria.fr
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 6
Strauss PDEs 2e: Section 3 - Exercise Page of 6 Exercise Carefully derive the equation of a string in a medium in which the resistance is proportional to the velocity Solution There are two ways (among
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 informationLINEAR SECOND-ORDER EQUATIONS
LINEAR SECON-ORER EQUATIONS Classification In two independent variables x and y, the general form is Au xx + 2Bu xy + Cu yy + u x + Eu y + Fu + G = 0. The coefficients are continuous functions of (x, y)
More informationMATH 31BH Homework 5 Solutions
MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [
More informationPDE and Boundary-Value Problems Winter Term 2014/2015
PDE and Boundary-Value Problems Winter Term 2014/2015 Lecture 13 Saarland University 5. Januar 2015 c Daria Apushkinskaya (UdS) PDE and BVP lecture 13 5. Januar 2015 1 / 35 Purpose of Lesson To interpretate
More informationSummer 2017 MATH Solution to Exercise 5
Summer 07 MATH00 Solution to Exercise 5. Find the partial derivatives of the following functions: (a (xy 5z/( + x, (b x/ x + y, (c arctan y/x, (d log((t + 3 + ts, (e sin(xy z 3, (f x α, x = (x,, x n. (a
More informationMath 320, spring 2011 before the first midterm
Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,
More informationPartial Differential Equations, Winter 2015
Partial Differential Equations, Winter 215 Homework #2 Due: Thursday, February 12th, 215 1. (Chapter 2.1) Solve u xx + u xt 2u tt =, u(x, ) = φ(x), u t (x, ) = ψ(x). Hint: Factor the operator as we did
More informationThe second-order 1D wave equation
C The second-order D wave equation C. Homogeneous wave equation with constant speed The simplest form of the second-order wave equation is given by: x 2 = Like the first-order wave equation, it responds
More informationHomework 1/Solutions. Graded Exercises
MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both
More informationChapter 3. Second Order Linear PDEs
Chapter 3. Second Order Linear PDEs 3.1 Introduction The general class of second order linear PDEs are of the form: ax, y)u xx + bx, y)u xy + cx, y)u yy + dx, y)u x + ex, y)u y + f x, y)u = gx, y). 3.1)
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
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 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 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 informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationCONVERGENCE OF EXTERIOR SOLUTIONS TO RADIAL CAUCHY SOLUTIONS FOR 2 t U c 2 U = 0
Electronic Journal of Differential Equations, Vol. 206 (206), No. 266, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CONVERGENCE OF EXTERIOR SOLUTIONS TO RADIAL CAUCHY
More informationWeek 4 Lectures, Math 6451, Tanveer
1 Diffusion in n ecall that for scalar x, Week 4 Lectures, Math 6451, Tanveer S(x,t) = 1 exp [ x2 4πκt is a special solution to 1-D heat equation with properties S(x,t)dx = 1 for t >, and yet lim t +S(x,t)
More informationMathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine
Lecture 2 The wave equation Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine V1.0 28/09/2018 1 Learning objectives of this lecture Understand the fundamental properties of the wave equation
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 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 informationUNIVERSITY OF MANITOBA
Question Points Score INSTRUCTIONS TO STUDENTS: This is a 6 hour examination. No extra time will be given. No texts, notes, or other aids are permitted. There are no calculators, cellphones or electronic
More informationA FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
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 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 informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationName: Math Homework Set # 5. March 12, 2010
Name: Math 4567. Homework Set # 5 March 12, 2010 Chapter 3 (page 79, problems 1,2), (page 82, problems 1,2), (page 86, problems 2,3), Chapter 4 (page 93, problems 2,3), (page 98, problems 1,2), (page 102,
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 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 informationAlgebra I. Book 2. Powered by...
Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........
More informationx 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3
Version: 4/1/06. Note: These notes are mostly from my 5B course, with the addition of the part on components and projections. Look them over to make sure that we are on the same page as regards inner-products,
More informationMAY THE FORCE BE WITH YOU, YOUNG JEDIS!!!
Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other
More informationAMATH 353 Lecture 9. Weston Barger. How to classify PDEs as linear/nonlinear, order, homogeneous or non-homogeneous.
AMATH 353 ecture 9 Weston Barger 1 Exam What you need to know: How to classify PDEs as linear/nonlinear, order, homogeneous or non-homogeneous. The definitions for traveling wave, standing wave, wave train
More informationProve proposition 68. It states: Let R be a ring. We have the following
Theorem HW7.1. properties: Prove proposition 68. It states: Let R be a ring. We have the following 1. The ring R only has one additive identity. That is, if 0 R with 0 +b = b+0 = b for every b R, then
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 informationHOMEWORK 5. Proof. This is the diffusion equation (1) with the function φ(x) = e x. By the solution formula (6), 1. e (x y)2.
HOMEWORK 5 SHUANGLIN SHAO. Section 3.. #. Proof. This is the diffusion equation with the function φx e x. By the solution formula 6, vx, t e x y e x+y φydy e x y e x+y e x y y dy e y dy e x+y y dy To compute
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 informationPart 1. For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form
Commutative Algebra Homework 3 David Nichols Part 1 Exercise 2.6 For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form m 0 + m
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 informationAutumn 2015 Practice Final. Time Limit: 1 hour, 50 minutes
Math 309 Autumn 2015 Practice Final December 2015 Time Limit: 1 hour, 50 minutes Name (Print): ID Number: This exam contains 9 pages (including this cover page) and 8 problems. Check to see if any pages
More informationMATH 532, 736I: MODERN GEOMETRY
MATH 532, 736I: MODERN GEOMETRY Test 2, Spring 2013 Show All Work Name Instructions: This test consists of 5 pages (one is an information page). Put your name at the top of this page and at the top of
More information18 Green s function for the Poisson equation
8 Green s function for the Poisson equation Now we have some experience working with Green s functions in dimension, therefore, we are ready to see how Green s functions can be obtained in dimensions 2
More informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
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 informationNumerical Methods for PDEs
Numerical Methods for PDEs Partial Differential Equations (Lecture 1, Week 1) Markus Schmuck Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt University, Edinburgh
More informationExtreme Values and Positive/ Negative Definite Matrix Conditions
Extreme Values and Positive/ Negative Definite Matrix Conditions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 016 Outline 1
More informationHomework 3/ Solutions
MTH 310-3 Abstract Algebra I and Number Theory S17 Homework 3/ Solutions Exercise 1. Prove the following Theorem: Theorem Let R and S be rings. Define an addition and multiplication on R S by for all r,
More informationStrauss PDEs 2e: Section Exercise 3 Page 1 of 13. u tt c 2 u xx = cos x. ( 2 t c 2 2 x)u = cos x. v = ( t c x )u
Strauss PDEs e: Setion 3.4 - Exerise 3 Page 1 of 13 Exerise 3 Solve u tt = u xx + os x, u(x, ) = sin x, u t (x, ) = 1 + x. Solution Solution by Operator Fatorization Bring u xx to the other side. Write
More informationAM 205: lecture 14. Last time: Boundary value problems Today: Numerical solution of PDEs
AM 205: lecture 14 Last time: Boundary value problems Today: Numerical solution of PDEs ODE BVPs A more general approach is to formulate a coupled system of equations for the BVP based on a finite difference
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationLECTURE NOTES FOR MATH 124A PARTIAL DIFFERENTIAL EQUATIONS
LECTURE NOTES FOR MATH 124A PARTIAL DIFFERENTIAL EQUATIONS S. SETO 1. Motivation for PDEs 1.1. What are PDEs? An algebraic equation is an equation which only involves algebraic operations, e.g. x 2 1 =.
More informationType II hidden symmetries of the Laplace equation
Type II hidden symmetries of the Laplace equation Andronikos Paliathanasis Larnaka, 2014 A. Paliathanasis (Univ. of Naples) Type II symmetries Larnaka, 2014 1 / 25 Plan of the Talk 1 The generic symmetry
More informationExtrema of Functions of Several Variables
Extrema of Functions of Several Variables MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background (1 of 3) In single-variable calculus there are three important results
More informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationPartial differential equations (ACM30220)
(ACM3. A pot on a stove has a handle of length that can be modelled as a rod with diffusion constant D. The equation for the temperature in the rod is u t Du xx < x
More informationPDE (Math 4163) Spring 2016
PDE (Math 4163) Spring 2016 Some historical notes. PDE arose in the context of the development of models in the physics of continuous media, e.g. vibrating strings, elasticity, the Newtonian gravitational
More informationMATH 124A Solution Key HW 05
3. DIFFUSION ON THE HALF-LINE Solutions prepared by Jon Tjun Seng Lo Kim Lin, TA Math 24A MATH 24A Solution Key HW 5 3. DIFFUSION ON THE HALF-LINE. Solve u t ku x x ; u(x, ) e x ; u(, t) on the half-line
More informationENGI 4430 PDEs - d Alembert Solutions Page 11.01
ENGI 4430 PDEs - d Alembert Solutions Page 11.01 11. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationMath 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt
Math 251 December 14, 2005 Final Exam Name Section There are 10 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationUNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics and Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4006 SEMESTER: Spring 2011 MODULE TITLE:
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More information(The) Three Linear Partial Differential Equations
(The) Three Linear Partial Differential Equations 1 Introduction A partial differential equation (PDE) is an equation of a function of 2 or more variables, involving 2 or more partial derivatives in different
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
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 informationHyperbolic PDEs. Chapter 6
Chapter 6 Hyperbolic PDEs In this chapter we will prove existence, uniqueness, and continuous dependence of solutions to hyperbolic PDEs in a variety of domains. To get a feel for what we might expect,
More informationA Motivation for Fourier Analysis in Physics
A Motivation for Fourier Analysis in Physics PHYS 500 - Southern Illinois University November 8, 2016 PHYS 500 - Southern Illinois University A Motivation for Fourier Analysis in Physics November 8, 2016
More informationx ct x + t , and the characteristics for the associated transport equation would be given by the solution of the ode dx dt = 1 4. ξ = x + t 4.
. The solution is ( 2 e x+ct + e x ct) + 2c x+ct x ct sin(s)dx ( e x+ct + e x ct) + ( cos(x + ct) + cos(x ct)) 2 2c 2. To solve the PDE u xx 3u xt 4u tt =, you can first fact the differential operat to
More informationMath 51, Homework-2. Section numbers are from the course textbook.
SSEA Summer 2017 Math 51, Homework-2 Section numbers are from the course textbook. 1. Write the parametric equation of the plane that contains the following point and line: 1 1 1 3 2, 4 2 + t 3 0 t R.
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationMath 51, Homework-2 Solutions
SSEA Summer 27 Math 5, Homework-2 Solutions Write the parametric equation of the plane that contains the following point and line: 3 2, 4 2 + t 3 t R 5 4 By substituting t = and t =, we get two points
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationFOURIER METHODS AND DISTRIBUTIONS: SOLUTIONS
Centre for Mathematical Sciences Mathematics, Faculty of Science FOURIER METHODS AND DISTRIBUTIONS: SOLUTIONS. We make the Ansatz u(x, y) = ϕ(x)ψ(y) and look for a solution which satisfies the boundary
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 informationMatrix Theory and Differential Equations Homework 6 Solutions, 10/5/6
Matrix Theory and Differential Equations Homework 6 Solutions, 0/5/6 Question Find the general solution of the matrix system: x 3y + 5z 8t 5 x + 4y z + t Express your answer in the form of a particulaolution
More informationMathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.
Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u
More informationPDE and Boundary-Value Problems Winter Term 2014/2015
PDE and Boundary-Value Problems Winter Term 2014/2015 Lecture 12 Saarland University 15. Dezember 2014 c Daria Apushkinskaya (UdS) PDE and BVP lecture 12 15. Dezember 2014 1 / 24 Purpose of Lesson To introduce
More information1.1 The classical partial differential equations
1 Introduction 1.1 The classical partial differential equations In this introductory chapter, we give a brief survey of three main types of partial differential equations that occur in classical physics.
More informationEnergy method for wave equations
Energy method for wave equations Willie Wong Based on commit 5dfb7e5 of 2017-11-06 13:29 Abstract We give an elementary discussion of the energy method (and particularly the vector field method) in the
More informationBefore you begin read these instructions carefully:
NATURAL SCIENCES TRIPOS Part IB & II (General Friday, 30 May, 2014 9:00 am to 12:00 pm MATHEMATICS (2 Before you begin read these instructions carefully: You may submit answers to no more than six questions.
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
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 informationKevin James. MTHSC 412 Section 3.1 Definition and Examples of Rings
MTHSC 412 Section 3.1 Definition and Examples of Rings A ring R is a nonempty set R together with two binary operations (usually written as addition and multiplication) that satisfy the following axioms.
More informationHomework 7 Math 309 Spring 2016
Homework 7 Math 309 Spring 2016 Due May 27th Name: Solution: KEY: Do not distribute! Directions: No late homework will be accepted. The homework can be turned in during class or in the math lounge in Pedelford
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More information