HOMEWORK 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.
|
|
- Neal Miles
- 6 years ago
- Views:
Transcription
1 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 the first integral, e x y y dy e kt x e kt x π ekt x e x +y xy+y dy e x +y +kt xy+kt x kt x dy e 4k t x kt x kt x e y+kt x dy e y+kt x e y dy e y dy π ekt x π π ekt x kt x π Erfkt x Erf kt x. dy e y dy
2 Similarly for the second integral, So e x+y y dy ex+kt Erf kt + x. vx, t ekt x Erf kt x ekt+x Erf kt + x.. Section 3.. # Proof. Let vx, t ux, t. Then the system of equation is By 6, Hence vx, t v t kv xx, vx,, v, t. π + π. e x y e x y e x+y φydy e x+y e x y dy + ux, t. dy e x+y dy 3. Section 3.. #3. Proof. We make an even extension of φ: { φx, for x, φ even x φ x, for x. We solve the diffusion equation with the Neumann condition u t ku xx, ux, φ even x, u x, t. By the solution formula for the diffusion equation, ux, t Sx y, tφ even ydy.
3 Since φ even is an even function in y, so u x, t ux, t. Let wx, t ux, t for x. If we differentiate both sides and set x, we obtain u x x,. So w x x,. Next we replace φ even by φ in the formula above to derive a solution formula for the original problem. For x, wx, t Sx y, tφydy + Sx y, tφydy + Sx y, t + Sx + y, t φydy. Sx y, tφ ydy, Sx + y, tφ ydy, 4. Section 3.. #. Proof. We make an even extension of φ and ψ: { φx, for x, φ even x φ x, for x. and ψ even x { ψx, for x, ψ x, for x. Thus by the solution for the wave equation on the real line vx, t [φ evenx + ct + φ even x ct] + c Since φ even and ψ even are even in y, so This implies that v x, t vx, t. v x, t. Let wx, t vx, t for x.therefore w x, t, ψ even ydy. which satisfies the boundary condition. For x, we divide the discussion into 3 cases because x, c and < x <. 3
4 Case. x + ct and x ct. vx, t [φ evenx + ct + φ even x ct] + c [φx + ct + φx ct] + c Case. x + ct and x ct <. vx, t [φ evenx + ct + φ even x ct] + c [φx + ct + φct x] + c [φx + ct + φct x] + c ψydy. ψ even ydy ψ even ydy ψ ydy + c ct x ψydy + c ψydy ψydy Case 3. x ct x + ct <. vx, t [φ evenx + ct + φ even x ct] + c [φ x ct + φct x] + c [φ x ct + φct x] + c ct x ψ ydy ψydy. ψ even ydy Combining these three cases, we obtain the solution formula for v. 5. Section 3.. # 5. Proof. This is the Dirichlet problem for the wave equation over the half line with φx, ψx and v, t. By the solution formula on page 6, for x, vx, t φ odd x + ct + φ odd x ct + c φ odd x + ct + φ odd x ct. When x + ct and x ct, Thus φ odd x + ct, φ odd x ct. vx, t. 4 ψ odd ydy
5 When x + ct and x ct <, vx, t +. When x + ct x ct, vx, t Section 3.3. #. Proof. We make an odd extension of φ: φx, for x, φ odd x φ x, for x,, for x. and we make an odd extension of f: fx, for x, f odd x f x, for x,, for x. Now we solve the system of equations: u t ku xx fx, t, ux, φ odd x. By the solution formula, we obtain ux, t Sx y, tφ odd ydy + t Sx y, t sf odd y, sdyds. The first integral is an odd function of x since φ odd S is an even function. Indeed, S x y, tφ odd ydy S x + y, tφ odd ydy This implies that when x, Sx y, tφ odd ydy. S y, tφ odd ydy. 5 is an odd function, and
6 When x in the second integral, because f odd proves that t t. S y, t sf odd y, sdyds 4πks e y 4kt s f odd y, sdyds is an odd function and Sx, t is an even function in x. This u, t. The second integral is independent of φ; For the first integral, Thus ux, t Sx y, tφ odd ydy Sx y, tφydy + Sx y, tφydy Sx y, t φ ydy Sx + y, tφydy Sx y, t Sx + y, t φydy. Sx y, t Sx + y, t φydy+ t Sx y, t sfy, sdyds. 7. Section 3.3. #. Proof. Let ux, t vx, t ht. Then vx, t ux, t + ht. Then from the equations satisfied by v, we have u t ku xx fx, t h t, u, t, ux, φx h. Then by Exercise in this section, ux, t + Sx y, t Sx + y, t φy h dy t Sx y, t s fy, s h t dyds. 6
7 Then we have vx, t ux, t + ht + t +ht. 8. Section 3.4, #. Proof. By using the formula 3, since φ ψ, we have ux, t c c c x t x+ct s t t t s t x ct s x+ct s x ct s ysdyds ydy sxct sdyds st sds x t sds t 3 x t3 3 xt3 6. t s ds 7
8 9. Section 3.4. #. Proof. By using the formula 3, since φ ψ, we have ux, t c a a t x+ct s t t eax+act a eax+act a eax a c x ct s e ay x+ct s x ct s ds e ay dyds e ax+ct s e ax ct s ds t t e acs ds eax act e acs ds a ac e acs t eax act a ac eacs t e act eax e act a c eax a c + eax+act a c + eax act a. c. Section 3.4. # 3. Proof. We know that ux, sin x and u t x, + x. solution formula, we have Then by the ux, t [φx + ct + φx ct]+ c For the first term, ψydy+ c t x+ct s x ct s [φx + ct + φx ct] [sinx + ct + sinx ct]. For the second term, c For the third term, c + ydy 4c t x+ct s x ct s + x + ct + x ct t + x. cos ydyds cosx + ct cosx ct. c 8 cos ydyds.
9 Hence ux, t [sinx + ct + sinx ct] + t + x + cosx + ct cosx ct. c Department of Mathematics, KU, Lawrence, KS address: slshao@math.ku.edu 9
HOMEWORK 4 1. P45. # 1.
HOMEWORK 4 SHUANGLIN SHAO P45 # Proof By the maximum principle, u(x, t x kt attains the maximum at the bottom or on the two sides When t, x kt x attains the maximum at x, ie, x When x, x kt kt attains
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 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 informationPDEs, Homework #3 Solutions. 1. Use Hölder s inequality to show that the solution of the heat equation
PDEs, Homework #3 Solutions. Use Hölder s inequality to show that the solution of the heat equation u t = ku xx, u(x, = φ(x (HE goes to zero as t, if φ is continuous and bounded with φ L p for some p.
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 informationDiffusion on the half-line. The Dirichlet problem
Diffusion on the half-line The Dirichlet problem Consider the initial boundary value problem (IBVP) on the half line (, ): v t kv xx = v(x, ) = φ(x) v(, t) =. The solution will be obtained by the reflection
More 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 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 informationMATH 425, HOMEWORK 3 SOLUTIONS
MATH 425, HOMEWORK 3 SOLUTIONS Exercise. (The differentiation property of the heat equation In this exercise, we will use the fact that the derivative of a solution to the heat equation again solves the
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 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 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 informationDiffusion equation in one spatial variable Cauchy problem. u(x, 0) = φ(x)
Diffusion equation in one spatial variable Cauchy problem. u t (x, t) k u xx (x, t) = f(x, t), x R, t > u(x, ) = φ(x) 1 Some more mathematics { if x < Θ(x) = 1 if x > is the Heaviside step function. It
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 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 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 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 information9 More on the 1D Heat Equation
9 More on the D Heat Equation 9. Heat equation on the line with sources: Duhamel s principle Theorem: Consider the Cauchy problem = D 2 u + F (x, t), on x t x 2 u(x, ) = f(x) for x < () where f
More informationFirst 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 informationMath 126 Final Exam Solutions
Math 126 Final Exam Solutions 1. (a) Give an example of a linear homogeneous PE, a linear inhomogeneous PE, and a nonlinear PE. [3 points] Solution. Poisson s equation u = f is linear homogeneous when
More informationApplications of the Maximum Principle
Jim Lambers MAT 606 Spring Semester 2015-16 Lecture 26 Notes These notes correspond to Sections 7.4-7.6 in the text. Applications of the Maximum Principle The maximum principle for Laplace s equation is
More 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 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 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 informationMath 5440 Problem Set 5 Solutions
Math 5 Math 5 Problem Set 5 Solutions Aaron Fogelson Fall, 3 : (Logan,. # 3) Solve the outgoing signal problem and where s(t) is a known signal. u tt c u >, < t
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 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 informationAnalysis III Solutions - Serie 12
.. Necessary condition Let us consider the following problem for < x, y < π, u =, for < x, y < π, u y (x, π) = x a, for < x < π, u y (x, ) = a x, for < x < π, u x (, y) = u x (π, y) =, for < y < π. Find
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 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 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 informationIntroduction to Differential Equations
Chapter 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
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 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 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 informationHomework 3 solutions Math 136 Gyu Eun Lee 2016 April 15. R = b a
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
More informationHomework #3 Solutions
Homework #3 Solutions Math 82, Spring 204 Instructor: Dr. Doreen De eon Exercises 2.2: 2, 3 2. Write down the heat equation (homogeneous) which corresponds to the given data. (Throughout, heat is measured
More informationCHAPTER 3. Analytic Functions. Dr. Pulak Sahoo
CHAPTER 3 Analytic Functions BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-4: Harmonic Functions 1 Introduction
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 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 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 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 informationHeat Equation on Unbounded Intervals
Heat Equation on Unbounded Intervals MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 28 Objectives In this lesson we will learn about: the fundamental solution
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 informationMA 523: Homework 1. Yingwei Wang. Department of Mathematics, Purdue University, West Lafayette, IN, USA
MA 523: Homework Yingwei Wang Department of Mathematics, Purdue University, West Lafayette, IN, USA One dimensional transport equation Question: Solve the transport equation using the method of characteristics.
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 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 informationSOLUTIONS TO SELECTED EXERCISES. Applied Partial Differential Equations, 3rd Ed. Springer-Verlag, NY (2015)
SOLUTIONS TO SELECTED EXECISES Applied Partial Differential Equations, 3rd Ed. Springer-Verlag, NY (25) J. David Logan University of Nebraska Lincoln CHAPTE 2 Partial Differential Equations on Unbounded
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
More informationProblem Set 1. This week. Please read all of Chapter 1 in the Strauss text.
Math 425, Spring 2015 Jerry L. Kazdan Problem Set 1 Due: Thurs. Jan. 22 in class. [Late papers will be accepted until 1:00 PM Friday.] This is rust remover. It is essentially Homework Set 0 with a few
More informationPartial Differential Equations (TATA27)
Partial Differential Equations (TATA7) David Rule Spring 9 Contents Preliminaries. Notation.............................................. Differential equations...................................... 3
More informationMATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 6
MATH 412 Fourier Series and PDE- Spring 21 SOLUTIONS to HOMEWORK 6 Problem 1. Solve the following problem u t u xx =1+xcos t
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 informationLecture notes. May 2, 2017
Lecture notes May, 7 Preface These are ecture notes for MATH 47: Partia differentia equations with the soe purpose of providing reading materia for topics covered in the ectures of the cass. Severa exampes
More informationChapter 2: First Order DE 2.6 Exact DE and Integrating Fa
Chapter 2: First Order DE 2.6 Exact DE and Integrating Factor First Order DE Recall the general form of the First Order DEs (FODE): dy dx = f(x, y) (1) (In this section x is the independent variable; not
More informationMATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:
MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 points. In order to obtain full credit for partial credit problems, all work must be
More informationLecture 17: Section 4.2
Lecture 17: Section 4.2 Shuanglin Shao November 4, 2013 Subspaces We will discuss subspaces of vector spaces. Subspaces Definition. A subset W is a vector space V is called a subspace of V if W is itself
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 informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
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 informationA proof for the full Fourier series on [ π, π] is given here.
niform convergence of Fourier series A smooth function on an interval [a, b] may be represented by a full, sine, or cosine Fourier series, and pointwise convergence can be achieved, except possibly at
More informationLecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225
Lecture Notes on Partial Dierential Equations (PDE)/ MaSc 221+MaSc 225 Dr. Asmaa Al Themairi Assistant Professor a a Department of Mathematical sciences, University of Princess Nourah bint Abdulrahman,
More informationInstructor s Solutions Manual PARTIAL DIFFERENTIAL EQUATIONS. with FOURIER SERIES and BOUNDARY VALUE PROBLEMS. NAKHLÉ H. ASMAR University of Missouri
Instructor s Solutions Manual PARTIA DIFFERENTIA EQUATIONS with FOURIER SERIES and BOUNDARY VAUE PROBEMS Second Edition NAKHÉ H. ASMAR University of Missouri Contents Preface Errata v vi A Preview of Applications
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationIntroduction of Partial Differential Equations and Boundary Value Problems
Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions
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 informationIn this lecture we shall learn how to solve the inhomogeneous heat equation. u t α 2 u xx = h(x, t)
MODULE 5: HEAT EQUATION 2 Lecture 5 Time-Dependent BC In this lecture we shall learn how to solve the inhomogeneous heat equation u t α 2 u xx = h(x, t) with time-dependent BC. To begin with, let us consider
More informationSOLUTIONS TO PROBLEMS FROM ASSIGNMENT 5. Problems 3.1:6bd
SOLUTIONS TO PROBLEMS FROM ASSIGNMENT 5 Statement. Solve the problem Problems 3.1:6bd u t = u xx, (t ), with the initial condition u(x, ) = f(x), where the functions u(x, t) and f(x) are assumed to be
More informationMath 5440 Problem Set 6 Solutions
Math 544 Math 544 Problem Set 6 Solutions Aaron Fogelson Fall, 5 : (Logan,.6 # ) Solve the following problem using Laplace transforms. u tt c u xx g, x >, t >, u(, t), t >, u(x, ), u t (x, ), x >. The
More informationFunctional Analysis HW 2
Brandon Behring Functional Analysis HW 2 Exercise 2.6 The space C[a, b] equipped with the L norm defined by f = b a f(x) dx is incomplete. If f n f with respect to the sup-norm then f n f with respect
More informationPractice Problems for Final Exam
Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5,
More information(d). Why does this imply that there is no bounded extension operator E : W 1,1 (U) W 1,1 (R n )? Proof. 2 k 1. a k 1 a k
Exercise For k 0,,... let k be the rectangle in the plane (2 k, 0 + ((0, (0, and for k, 2,... let [3, 2 k ] (0, ε k. Thus is a passage connecting the room k to the room k. Let ( k0 k (. We assume ε k
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 32A: MIDTERM 2 REVIEW. sin 2 u du z(t) = sin 2 t + cos 2 2
MATH 3A: MIDTERM REVIEW JOE HUGHES 1. Curvature 1. Consider the curve r(t) = x(t), y(t), z(t), where x(t) = t Find the curvature κ(t). 0 cos(u) sin(u) du y(t) = Solution: The formula for curvature is t
More informationMATH 31B: MIDTERM 2 REVIEW. sin 2 x = 1 cos(2x) dx = x 2 sin(2x) 4. + C = x 2. dx = x sin(2x) + C = x sin x cos x
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate sin x and cos x. Solution: Recall the identities cos x = + cos(x) Using these formulas gives cos(x) sin x =. Trigonometric Integrals = x sin(x) sin x = cos(x)
More informationMath 212-Lecture 8. The chain rule with one independent variable
Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
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 informationChapter 4: Partial differentiation
Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of
More informationMath 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).
Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y).
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 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 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 informationMAGIC058 & MATH64062: Partial Differential Equations 1
MAGIC58 & MATH6462: Partial Differential Equations Section 6 Fourier transforms 6. The Fourier integral formula We have seen from section 4 that, if a function f(x) satisfies the Dirichlet conditions,
More information12.7 Heat Equation: Modeling Very Long Bars.
568 CHAP. Partial Differential Equations (PDEs).7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms Our discussion of the heat equation () u t c u x in the last section
More informationMath512 PDE Homework 2
Math51 PDE Homework October 11, 009 Exercise 1.3. Solve u = xu x +yu y +(u x+y y/ = 0 with initial conditon u(x, 0 = 1 x. Proof. In this case, we have F = xp + yq + (p + q / z = 0 and Γ parameterized as
More informationMath Practice Exam 3 - solutions
Math 181 - Practice Exam 3 - solutions Problem 1 Consider the function h(x) = (9x 2 33x 25)e 3x+1. a) Find h (x). b) Find all values of x where h (x) is zero ( critical values ). c) Using the sign pattern
More information1MA6 Partial Differentiation and Multiple Integrals: I
1MA6/1 1MA6 Partial Differentiation and Multiple Integrals: I Dr D W Murray Michaelmas Term 1994 1. Total differential. (a) State the conditions for the expression P (x, y)dx+q(x, y)dy to be the perfect
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 informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationFirst-Order ODE: Separable Equations, Exact Equations and Integrating Factor
First-Order ODE: Separable Equations, Exact Equations and Integrating Factor Department of Mathematics IIT Guwahati REMARK: In the last theorem of the previous lecture, you can change the open interval
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More informationThis practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II.
MTH 142 Practice Exam Chapters 9-11 Calculus II With Analytic Geometry Fall 2011 - University of Rhode Island This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus
More informationPartial Differential Equations
Partial Differential Equations Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical Engineering University of Connecticut xchen@engr.uconn.edu Contents 1
More informationFINAL REVIEW FOR MATH The limit. a n. This definition is useful is when evaluating the limits; for instance, to show
FINAL REVIEW FOR MATH 500 SHUANGLIN SHAO. The it Define a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. This definition is useful is when evaluating the its; for instance, to
More informationFind the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.
Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no
More informationMATH-UA 263 Partial Differential Equations Recitation Summary
MATH-UA 263 Partial Differential Equations Recitation Summary Yuanxun (Bill) Bao Office Hour: Wednesday 2-4pm, WWH 1003 Email: yxb201@nyu.edu 1 February 2, 2018 Topics: verifying solution to a PDE, dispersion
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More information