3150 Review Problems for Final Exam. (1) Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos(x) 0 x π f(x) =
|
|
- Teresa Barnett
- 5 years ago
- Views:
Transcription
1 350 Review Problems for Final Eam () Find the Fourier series of the 2π-periodic function whose values are given on [0, 2π) by cos() 0 π f() = 0 π < < 2π (2) Let F and G be arbitrary differentiable functions of one variable. Then u(, t) = F ( + ct) + G( ct) is a solution to the wave equation 2 u t = 2 u 2 c2, provided that F and G are sufficiently smooth. Use this result to solve the wave equation 2 with initial data as follows. (a) u(, 0) = e 2, u(, 0) = 0, < <. t (b) u(, 0) = 0, u(, 0) =, < <. t ( 2 + ) 4 (c) u(, 0) = e 2, u(, 0) =, < <. t ( 2 + ) 4 (Use the superposition principle along with the results of the previous two parts.) (3) Find the sine series epansion of f() = on [0, 2π). (4) Find the requested Fourier series of the given function: (a) The Fourier series of the 6π-periodic function f() = 2 + cos(3) + sin(9) (b) The Fourier sine series of the 3-periodic function f() = 4 sin( 2π ) + 7 sin(8π) 3 (c) The Fourier cosine series of the 2π-periodic function f() = 5 7 cos() + 3 cos(3) 7 cos(4) (d) The comple Fourier series of the 2π-periodic function f() = 5 7e 3i + 9e 5i (e) The Fourier double sine series of the (2, 5)-periodic function f(, y) = 8 sin() sin( 4π y) + 9 sin(2) sin( 6π y) 5 5 (5) Find a period of the given function. (a) cos(6) (b) sin(5π) (c) sin( 4 ) 5 (d) cos( 5π ) p (6) Let f be the 5-periodic function defined on [0, 5) by 2 0 < 2 f() = 2 2 < < 5 To what does the Fourier series of f converge when = 2? (7) (a) Solve the equation 7u + u y = 0 by an appropriate change of variables. (b) Find the solution u that is equal to /(5 + 2 ) along the -ais. (8) Consider the equation 6u t + 0u = 0 for u = u(, t). (a) Verify that u(, t) = f(3 5t) is a solution, where f is any differentiable function of a single variable. (b) Find a solution of the equation which equals sin() on the -ais. (9) Find the Fourier series of the 2π-periodic function f() = sin( ) cos( 3 ). Hint: use trig identities. 2 2 (0) Use the method of characteristic curves to find solutions u = u(, y) to the equation 3y 2 u 2u y = 0. Check your answer by plugging back into the equation. () (a) Solve the equation 3u + 5u t = 9 by an appropriate change of variables. (b) Find the solution that equals e t2 along the t-ais (2) Let g() = 3 on the interval 0 2π. (a) Sketch the graph of the odd 4π-periodic etension of g. (b) Find the Fourier sine series of g(). (c) By evaluating at = π, deduce a series identity. (3) Let g() = 3 on the interval 0 2π. (a) Sketch the graph of the even 4π-periodic etension of g. (b) Find the Fourier cosine series of g(). (4) Verify that u = e 2t cos(3) is a solution of the heat equation for a suitable value of c. (5) (a) Find the Fourier series of the 2-periodic function whose values on [, ] are given by f() = 2. (b) Derive the value of series
2 2 (6) Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here, all functions are defined on [, ] and have f(0) = 0. (a) f() = cos(/ 2 ) (b) f() = cos(/) (c) f() = sin()/ (d) f() = 2 cos(/) (7) Solve the equation 5u t + 4u = 3u by an appropriate change of variables. (8) Show that if f() has period T, then f(3) has period T/3. (9) Let u and u 2 be two solutions of 3u + 2 u y = 0. Show that u = au + bu 2 is also a solution for any constants a and b. (20) Solve the boundary value wave problem where L = π, f() = sin() cos(), g() = 0, c = 5. (2) Find and plot the nodal lines for the function u(, y, t) = sin( 4π ) sin( π y) sin( 3t), 0 < < 7, 0 < y < (22) (a) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L = π, c =, f() = 6 sin() 4 sin(4). (b) Solve the same problem, but with non-homogeneous boundary conditions and temperature distribution as follows: u(0, t) = 0, u(π, t) = 00, t > 0 f() = 20 sin() + 30 sin(4)) + 00 π (c) Solve the same problem, but with insulated endpoints and temperature distribution as follows: u (0, t) = 0, u (π, t) = 0, t > 0 f() = 20 cos() + 2 cos(4) (23) Verify that the function u(, y, t) = sin(2) sin(3y)e 65t satisfies the two dimensional heat equation u t = 5 2 u. (24) Use d Alembert s solution of the vibrating string problem to solve where L = 2π, f() = 0, g() = sin() 9 sin(3), c =. Completely describe f and G (an antiderivative of g ). Also, determine u(π/4, π/8). (25) Find the comple Fourier series of the 2π-periodic function given by f() = e 3 on the interval π < π. (26) Solve the boundary value problem u tt = c 2 (u + u yy) u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), u t(, y, 0) = g(, y), (, y) R if a =, b = 4, c = 2, f(, y) = ( 2 ) sin( 5π y), and g(, y) = 5 sin(3π) sin(πy). 4 2 (27) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L = π, c =, f() = 20 sin() + 30 sin(4).
3 350 Review Problems for Final Eam 3 (28) Use Fourier series methods to solve the boundary value problem where L = 5, f() = 0, g() = 3 sin 2π 2 sin 6π, c = 2. Also, determine the deflection at the mid point of the string 5 5 at time t = 7.37 seconds. (29) Determine whether the given partial differential equation and boundary or initial conditions are linear or nonlinear, and, if linear, whether they are homogeneous or nonhomogeneous. (a) u + y 2 u y = 2, u(0, y) = 0, u (0, y) = 0. (b) u + u 2 t = 5, u(0, t) = u(, t). (c) e u + e y u y = e y u y, u(, 0) =. + 2 (d) u 5u tt = 4u cos(t), u(, 0) = u(, 2). (e) 2y(u + 3u y) = 4u t, u(, y, 0) = + y 2. (f) (u + 3u y)u t = 0, u (0, y, t) = 0, u (, y, t) = 0. (g) u ttt = 6(u + u yyy), u(0, y, t) = 0, u(2, y, t) = y, u(, 0, t) = 0, u(, 5, t) = + 2. (30) Use d Alembert s solution of the vibrating string problem to solve where L =, f() = sin(2π), g() = 0, c = 2. Completely describe f and G (an antiderivative of g ). Also, determine u(/4, 5). (3) (a) Solve the boundary value heat problem u(, 0) = f(), 0 < < L where L =, c =, f() = 0. π (b) Solve the same problem, but with non-homogeneous boundary conditions and temperature distribution as follows: u(0, t) = 0, u(π, t) = 5, t > 0 f() = 0 (c) Solve the same problem, but with insulated endpoints and temperature distribution as follows: u (0, t) = 0, u (π, t) = 0, t > 0 f() = 0 (32) Solve the boundary value problem u tt = u + u u(0, t) = 0, u(π, t) = 0, t > 0 u(, 0) = sin(3), u t(, 0) = 0, 0 < < π (Use the method of separation of variables.) (33) Verify that the given function satisfies the two dimensional Laplace equation. (a) u = 3 3y 2 (b) u = e 3y sin(3) (34) Find and plot the nodal lines for the function u(, y, t) = sin(3π) sin(5πy) cos( 7π t), 0 < <, 0 < y <. 2 (35) Find the comple Fourier series of the 2π-periodic function given by f() = cos(5) on the interval π < π. (36) Solve the boundary value wave problem u tt = c 2 (u + u yy) u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), u t(, y, 0) = g(, y), (, y) R if a =, b =, c = 2, f(, y) = 8 sin(π) sin(2πy), and g(, y) = 0 sin(3π) sin(πy).
4 4 (37) Solve the boundary value heat problem u(, 0, t) = 0, u(, b, t) = 0, t > 0 u(0, y, t) = 0, u(a, y, t) = 0, t > 0 u(, y, 0) = f(, y), (, y) R if a =, b =, c = 2, f(, y) = 8 sin(π) sin(2πy). (38) Let b() be a function with graph y (a) Describe in qualitative terms what would happen if an elastic string of length with c = had initial displacement f() = b() and initial velocity g() = 0. (b) For what initial displacement and velocity would the solution model a single blip propagating back and forth across the string. (c) If two blips on an elastic string moving in opposite directions collided, what would happen? < 2 (39) Let h() = 0 otherwise. (a) Find the Fourier integral representation of h. (b) Epress h() as a comple Fourier integral. (c) Suppose an infinite wire with thermal diffusivity c 2 = has initial temperature h(). Obtain an epression for the temperature u(, t) in integral form. (d) Find the solution of the wave equation on an infinite string, < <, t > 0 u(, 0) = f() u t(, 0) = g() where c = 5, f() = 0, g() = h(). (40) Solve the vibrating membrane problem u tt = c 2 ( u rr + r ur ) where a = 3, c =, f(r) =.5, and g(r) = 0. (4) Find solutions u(, y) to 2yu u y = 0 by separating variables. 2 < < 2 (42) Let f() = 0 otherwise (a) Sketch the graph of f. (b) Find the fourier integral representation of f. ( 2 sin(2w) (c) Deduce the identity π = 4 cos(2w) ) sin(2w) dw. 0 w 2 w ( 2 sin(2w) (d) What is the value of 4 cos(2w) ) sin(37w) dw? 0 w 2 w 2e 2 0 (43) Let f() = 0 < 0. (a) Find the Fourier transform of f(). (b) Epress f() as a (comple) Fourier integral. (c) By evaluating the Fourier integral of f() at = 0 and at = 37, derive two integral identities.
5 350 Review Problems for Final Eam 5 e 2 0 (d) Let g() = and h() = 2π 2 e 2 0. Show that f g() = h(). 0 < 0 0 < 0 (e) Use the above result to find the Fourier transform of h(), given that F (g()) (ω) = 2π (2 + iω). (44) Consider a vibrating string with length L = 6, ends fied, and c = 7, corresponding to the wave equation u tt = 49u. (a) Find u(, t) if the initial velocity is zero and the initial displacement is f() = 3 3 = 2 π 2 [sin( π6 ) 9 sin( 3π6 ) + 25 sin( 5π6 ) ]. (b) Find u(, t) if the initial displacement is zero and the initial velocity is g() = 3 3 = 2 π 2 [sin( π6 ) 9 sin( 3π6 ) + 25 sin( 5π6 ) ]. (45) Using Fourier methods, find the solution of the wave equation on an infinite string, < <, t > 0 u(, 0) = f() u t(, 0) = g() where c = 2, f() = e 2, g() = 0. (46) Find the temperature u(, t) in a wire (length 8 cm, thermal diffusivity c 2 = 4 cm2 /sec) whose ends are perfectly insulated, assuming that u(, 0) = f() = 5 cos( 2π 3π ) 7 cos( 8 8 ). (47) Solve the vibrating membrane problem u tt = c 2 ( u rr + r ur ) where a = 2, c = 0, f(r) = 4 r 2, and g(r) = 0. (48) (a) Solve the equation 5u + 4u y = 0 by an appropriate change of variables. (b) Find the solution u that is equal to e along the -ais. (49) Using Fourier methods, find the solution of the heat equation on an infinite rod where c = 5, f() = 2 +. (50) Solve the vibrating membrane problem, < <, t > 0 u(, 0) = f() u tt = c 2 ( u rr + r ur ) where a =, c =, f(r) =, and g(r) = 5. (5) Two circular drumheads have the same tension and density, but the second has a radius one third as large as the first. How much higher is the frequency of the fundamental mode of the second drumhead? Eplain carefully. (52) Let g() = on the interval 0 2π. (a) Find the Fourier sine series of g(). (b) Find the double Fourier sine series of f(, y) = 9y on the rectangle 0 2π, 0 y 2π. (c) Find the displacement u(, y, t) of a square vibrating membrane of sides a = b = 2π and c = 2 if the initial displacement is 9y and the initial velocity is 0. (53) Compute the Laplacian in an appropriate coordinate system and decide if the function is harmonic, i.e. satisfies Laplace s equation 2 u = 0: (54) (a) Let f() = e 2 0 u(, y) = 2 + y 2.. Find the Fourier transform of f. 0 < 0
6 6 (b) Show that the convolution h() = f f() is h() = 2π e < 0. (c) Find the Fourier transform of h in three was: a) directly, b) using the behavior of Fourier transform under convolution, and c) using the rule for F (f()). (55) Find the deflection u(r, t) of the circular membrane of radius a = 2 if c =, the initial velocity is zero, and the initial deflection is a function f(r) with Fourier-Bessel series f(r) = J 0( α 2 r) + α2 J0( 4 2 r) + α3 J0( 9 2 r) +. (56) Compute the Laplacian in an appropriate coordinate system and decide if the function satisfies Laplace s equation 2 u = 0: u(, y, z) = y + z 2 + y. 2 (57) Verify that u = e 2t cos(3) is a solution of the heat equation for a suitable value of c. (58) Solve the vibrating membrane problem ( u tt = c 2 u rr + ) r ur where a =, c = 2, f(r) = ( r 2 ) 2, and g(r) = 0. (59) Use the method of separation of variables to find solutions u = u(, t) to the equation 3t 2 u 4u t = 0 on 0 2π satisfying u(0, t) = u(π, t) = 0 and u(, 0) = 3 sin(2) + sin(5). (60) Find the Fourier integral representation of the function e > 2 f() = 0 otherwise
Boundary value problems for partial differential equations
Boundary value problems for partial differential equations Henrik Schlichtkrull March 11, 213 1 Boundary value problem 2 1 Introduction This note contains a brief introduction to linear partial differential
More information1. Partial differential equations. Chapter 12: Partial Differential Equations. Examples. 2. The one-dimensional wave equation
1. Partial differential equations Definitions Examples A partial differential equation PDE is an equation giving a relation between a function of two or more variables u and its partial derivatives. The
More 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 informationVALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur DEPARTMENT OF MATHEMATICS QUESTION BANK
SUBJECT VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 63 3. DEPARTMENT OF MATHEMATICS QUESTION BANK : MA6351- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS SEM / YEAR : III Sem / II year (COMMON
More informationChapter 12 Partial Differential Equations
Chapter 12 Partial Differential Equations Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 12.1 Basic Concepts of PDEs Partial Differential Equation A
More informationVibrating Strings and Heat Flow
Vibrating Strings and Heat Flow Consider an infinite vibrating string Assume that the -ais is the equilibrium position of the string and that the tension in the string at rest in equilibrium is τ Let u(,
More informationHomework 7 Solutions
Homework 7 Solutions # (Section.4: The following functions are defined on an interval of length. Sketch the even and odd etensions of each function over the interval [, ]. (a f( =, f ( Even etension of
More informationName of the Student: Fourier Series in the interval (0,2l)
Engineering Mathematics 15 SUBJECT NAME : Transforms and Partial Diff. Eqn. SUBJECT CODE : MA11 MATERIAL NAME : University Questions REGULATION : R8 WEBSITE : www.hariganesh.com UPDATED ON : May-June 15
More informationFOURIER SERIES, TRANSFORMS, AND BOUNDARY VALUE PROBLEMS
fc FOURIER SERIES, TRANSFORMS, AND BOUNDARY VALUE PROBLEMS Second Edition J. RAY HANNA Professor Emeritus University of Wyoming Laramie, Wyoming JOHN H. ROWLAND Department of Mathematics and Department
More informationTHE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t)))
THE WAVE EQUATION The aim is to derive a mathematical model that describes small vibrations of a tightly stretched flexible string for the one-dimensional case, or of a tightly stretched membrane for the
More informationPartial Differential Equations (PDEs)
C H A P T E R Partial Differential Equations (PDEs) 5 A PDE is an equation that contains one or more partial derivatives of an unknown function that depends on at least two variables. Usually one of these
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 informationSATHYABAMA UNIVERSITY
DEPARTMENT OF MATHEMATICS ENGINEERING MATHEMATICS IV (SMTX ) SATHYABAMA UNIVERSITY DEPARTMENT OF MATHEMATICS Engineering Mathematics-IV (SMTX) Question Bank UNIT I. What is the value of b n when the function
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 informationFOURIER TRANSFORMS. At, is sometimes taken as 0.5 or it may not have any specific value. Shifting at
Chapter 2 FOURIER TRANSFORMS 2.1 Introduction The Fourier series expresses any periodic function into a sum of sinusoids. The Fourier transform is the extension of this idea to non-periodic functions by
More informationMATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:
MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
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 informationAPPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems
APPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fourth Edition Richard Haberman Department of Mathematics Southern Methodist University PEARSON Prentice Hall PEARSON
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
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 informationB.E./B.TECH. DEGREE EXAMINATION, CHENNAI-APRIL/MAY TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS(COMMON TO ALL BRANCHES)
B.E./B.TECH. DEGREE EXAMINATION, CHENNAI-APRIL/MAY 2010. TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS(COMMON TO ALL BRANCHES) 1. Write the conditions for a function to satisfy for the existence of a Fourier
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 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 3150 Problems Chapter 3
Name Math 15 Problems Chapter Due date: See the internet due date. Problems are collected once a week. Records are locked when the stack is returned. Records are only corrected, never appended. Submitted
More information25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes
Applications of PDEs 25.2 Introduction In this Section we discuss briefly some of the most important PDEs that arise in various branches of science and engineering. We shall see that some equations can
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 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 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 informationCircular Membranes. Farlow, Lesson 30. November 21, Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431
The Problem Polar coordinates Solving the Problem by Separation of Variables Circular Membranes Farlow, Lesson 30 Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL 33431 November
More informationPartial Differential Equations
++++++++++ Partial Differential Equations Previous year Questions from 017 to 199 Ramanasri Institute 017 W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 8 7 5 0 7 0 6 6 /
More informationCODE: GR17A1003 GR 17 SET - 1
SET - 1 I B. Tech II Semester Regular Examinations, May 18 Transform Calculus and Fourier Series (Common to all branches) Time: 3 hours Max Marks: 7 PART A Answer ALL questions. All questions carry equal
More informationMATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:
MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 points. In order to obtain full credit for partial credit problems, all work must
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 informationFourier Analysis Fourier Series C H A P T E R 1 1
C H A P T E R Fourier Analysis 474 This chapter on Fourier analysis covers three broad areas: Fourier series in Secs...4, more general orthonormal series called Sturm iouville epansions in Secs..5 and.6
More informationSolutions to Math 41 Final Exam December 9, 2013
Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain
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 informationDepartment of Mathematics
INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY Department of Mathematics MA 04 - Complex Analysis & PDE s Solutions to Tutorial No.13 Q. 1 (T) Assuming that term-wise differentiation is permissible, show that
More informationPARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS NAKHLE H. ASMAR University of Missouri PRENTICE HALL, Upper Saddle River, New Jersey 07458 Contents Preface vii A Preview of Applications and
More informationu tt = a 2 u xx u tt = a 2 (u xx + u yy )
10.7 The wave equation 10.7 The wave equation O. Costin: 10.7 1 This equation describes the propagation of waves through a medium: in one dimension, such as a vibrating string u tt = a 2 u xx 1 This equation
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationThe Fundamental Theorem of Calculus Part 3
The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative
More informationBoundary. DIFFERENTIAL EQUATIONS with Fourier Series and. Value Problems APPLIED PARTIAL. Fifth Edition. Richard Haberman PEARSON
APPLIED PARTIAL DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fifth Edition Richard Haberman Southern Methodist University PEARSON Boston Columbus Indianapolis New York San Francisco
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 informationTyn Myint-U Lokenath Debnath. Linear Partial Differential Equations for Scientists and Engineers. Fourth Edition. Birkhauser Boston Basel Berlin
Tyn Myint-U Lokenath Debnath Linear Partial Differential Equations for Scientists and Engineers Fourth Edition Birkhauser Boston Basel Berlin Preface to the Fourth Edition Preface to the Third Edition
More informationSolution to Problems for the 1-D Wave Equation
Solution to Problems for the -D Wave Equation 8. Linear Partial Differential Equations Matthew J. Hancock Fall 5 Problem (i) Suppose that an infinite string has an initial displacement +, u (, ) = f ()
More informationFinal Examination Linear Partial Differential Equations. Matthew J. Hancock. Feb. 3, 2006
Final Examination 8.303 Linear Partial ifferential Equations Matthew J. Hancock Feb. 3, 006 Total points: 00 Rules [requires student signature!]. I will use only pencils, pens, erasers, and straight edges
More informationCHAPTER 1 Introduction to Differential Equations 1 CHAPTER 2 First-Order Equations 29
Contents PREFACE xiii CHAPTER 1 Introduction to Differential Equations 1 1.1 Introduction to Differential Equations: Vocabulary... 2 Exercises 1.1 10 1.2 A Graphical Approach to Solutions: Slope Fields
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 informationAND NONLINEAR SCIENCE SERIES. Partial Differential. Equations with MATLAB. Matthew P. Coleman. CRC Press J Taylor & Francis Croup
CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES An Introduction to Partial Differential Equations with MATLAB Second Edition Matthew P Coleman Fairfield University Connecticut, USA»C)
More informationMa 530. Partial Differential Equations - Separation of Variables in Multi-Dimensions
Ma 530 Partial Differential Equations - Separation of ariables in Multi-Dimensions Temperature in an Infinite Cylinder Consider an infinitely long, solid, circular cylinder of radius c with its axis coinciding
More 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 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 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 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 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 informationX b n sin nπx L. n=1 Fourier Sine Series Expansion. a n cos nπx L 2 + X. n=1 Fourier Cosine Series Expansion ³ L. n=1 Fourier Series Expansion
3 Fourier Series 3.1 Introduction Although it was not apparent in the early historical development of the method of separation of variables what we are about to do is the analog for function spaces of
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationAB Calculus 2013 Summer Assignment. Theme 1: Linear Functions
01 Summer Assignment Theme 1: Linear Functions 1. Write the equation for the line through the point P(, -1) that is perpendicular to the line 5y = 7. (A) + 5y = -1 (B) 5 y = 8 (C) 5 y = 1 (D) 5 + y = 7
More informationBranch: Name of the Student: Unit I (Fourier Series) Fourier Series in the interval (0,2 l) Engineering Mathematics Material SUBJECT NAME
13 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE UPDATED ON : Transforms and Partial Differential Equation : MA11 : University Questions :SKMA13 : May June 13 Name of the Student: Branch: Unit
More information4 The Harmonics of Vibrating Strings
4 The Harmonics of Vibrating Strings 4. Harmonics and Vibrations What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task
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 informationSpotlight on Laplace s Equation
16 Spotlight on Laplace s Equation Reference: Sections 1.1,1.2, and 1.5. Laplace s equation is the undriven, linear, second-order PDE 2 u = (1) We defined diffusivity on page 587. where 2 is the Laplacian
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More informationFirst Year Physics: Prelims CP1 Classical Mechanics: DR. Ghassan Yassin
First Year Physics: Prelims CP1 Classical Mechanics: DR. Ghassan Yassin MT 2007 Problems I The problems are divided into two sections: (A) Standard and (B) Harder. The topics are covered in lectures 1
More informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
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 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:
MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 points. In order to obtain full credit for partial credit problems, all work must
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 informationMath 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
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 informationSubject: Mathematics III Subject Code: Branch: B. Tech. all branches Semester: (3rd SEM) i) Dr. G. Pradhan (Coordinator) ii) Ms.
Subject: Mathematics III Subject Code: BSCM1205 Branch: B. Tech. all branches Semester: (3 rd SEM) Lecture notes prepared by: i) Dr. G. Pradhan (Coordinator) Asst. Prof. in Mathematics College of Engineering
More informationPart 1: Integration problems from exams
. Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating
More information24 Solving planar heat and wave equations in polar coordinates
24 Solving planar heat and wave equations in polar coordinates Now that all the preparations are done, I can return to solving the planar heat and wave equations in domains with rotational symmetry. 24.1
More informationFOURIER SERIES. Chapter Introduction
Chapter 1 FOURIER SERIES 1.1 Introduction Fourier series introduced by a French physicist Joseph Fourier (1768-1830), is a mathematical tool that converts some specific periodic signals into everlasting
More informationA Guided Tour of the Wave Equation
A Guided Tour of the Wave Equation Background: In order to solve this problem we need to review some facts about ordinary differential equations: Some Common ODEs and their solutions: f (x) = 0 f(x) =
More informationVibrations of string. Henna Tahvanainen. November 8, ELEC-E5610 Acoustics and the Physics of Sound, Lecture 4
Vibrations of string EEC-E5610 Acoustics and the Physics of Sound, ecture 4 Henna Tahvanainen Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering November 8,
More informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
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 information13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs)
13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs) A prototypical problem we will discuss in detail is the 1D diffusion equation u t = Du xx < x < l, t > finite-length rod u(x,
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 informationMATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:
MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work
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 informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationQuestions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.
Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,
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 informationIntroduction to the Wave Equation
Introduction to the Ryan C. Trinity University Partial Differential Equations ecture 4 Modeling the Motion of an Ideal Elastic String Idealizing Assumptions: The only force acting on the string is (constant
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 informationDifferential Equations with Mathematica
Differential Equations with Mathematica THIRD EDITION Martha L. Abell James P. Braselton ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore
More information1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)
MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given
More informationBessel Functions. A Touch of Magic. Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2. SMILE REU Summer Louisiana State University
Bessel s Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2 1 Department of Mathematics Louisiana State University 2 Department of Mathematics University of Arkansas SMILE REU Summer
More informationLinear Partial Differential Equations for Scientists and Engineers
Tyn Myint-U Lokenath Debnath Linear Partial Differential Equations for Scientists and Engineers Fourth Edition Birkhäuser Boston Basel Berlin Tyn Myint-U 5 Sue Terrace Westport, CT 06880 USA Lokenath Debnath
More informationPreCalculus First Semester Exam Review
PreCalculus First Semester Eam Review Name You may turn in this eam review for % bonus on your eam if all work is shown (correctly) and answers are correct. Please show work NEATLY and bo in or circle
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
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 informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationTHE UNIVERSITY OF WESTERN ONTARIO. Applied Mathematics 375a Instructor: Matt Davison. Final Examination December 14, :00 12:00 a.m.
THE UNIVERSITY OF WESTERN ONTARIO London Ontario Applied Mathematics 375a Instructor: Matt Davison Final Examination December 4, 22 9: 2: a.m. 3 HOURS Name: Stu. #: Notes: ) There are 8 question worth
More informationCHAPTER 4. Introduction to the. Heat Conduction Model
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS
More information