Properties of Fourier Cosine and Sine Transforms
|
|
|
- Marjory Sparks
- 5 years ago
- Views:
Transcription
1 International Journal of Sciences: Basic and Applied Research (IJSBAR) ISSN (Print & Online) Properties of Fourier Cosine and Sine Transforms Shiferaw Geremew Kebede* Department of Mathematics, Madda Walabu University, Bale Robe, Ethiopia, PO Box: Abstract The time and frequency domains are alternative ways of representing signals. The Fourier transform is the mathematical relationship between these two representations. These transformations are of interest mainly as tools for solving ODEs, PDEs and integral equations, and they often also help in handling and applying special functions [8,9]. In this article, I have outlined the main features of properties of Fourier cosine and sine Transforms. These properties demand the implementation of representation of a function in integral form, known as Fourier cosine and sine transforms. The purpose of this paper is to provide a brief representation any function in integral form, Fourier cosine and sine transforms, after multiplying the given function by power functions; xx, xx 2, xx 3, xx nn and provide the relation between Fourier Cosine transforms and Fourier Sine transforms. Keywords: Fourier transforms; Fourier cosine and sine transforms. 1. Introduction On 21 December 187, in one of the most memorable sessions of the French Academy, Jean Baptiste Joseph Fourier, a 21-year old Mathematician and engineer announced a thesis which began a new chapter in the history of Mathematics. Fourier claimed that an arbitrary function, defined in a finite interval by an arbitrary and capricious graph, can always be resolved into a sum of pure sine and cosine. Fourier series in the work of Euler and D. Bernoulli on vibrating strings, but the theory of Fourier series truly began with the profound work of Fourier on heat conduction at the beginning of the 19th century * Corresponding author. 184
2 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp He wanted to use this form to come up with solution to certain linear partial differential equations (specially the heat equation) because sines and cosines behave nicely under differentiation. For instance, he said the derivative of the above functions should be this sum. Fourier deals with the problem of describing the evolution of the temperature TT (xx, tt) of a thin wire of length, stretched between xx = and xx =, with a constant zero temperature at the ends: TT (, tt) = and TT (, tt) =. He proposed that the initial temperature TT (xx, ) = ff(xx) could be expanded in a series of sine functions. 1828, Dirichlet formulated conditions for a function f(x) to have the Fourier transform f(x) must be single valued have a finite number of discontinuities in any given interval have a finite number of extrema in any given interval be square-integrable [1,2,7] Definition The Fourier cosine transform of the function ff (xx ) is given by Where, c suggests cosine. [3], [4], [5]. [6]: 1.2. Definition The Fourier sine transform of the function ff(xx) is given by: Where, s suggests sine [3,4,5,6]: 1.3. Definition The Fourier transform of the function ff(xx) is given by: F ff(xx) = 1 ff(xx)ee iiiiii dddd 2 References [3,4,5,6]. 2. Properties of Fourier Cosine and Sine Transforms Theorem: (Properties of Cosine and Sine Transforms) A) Suppose ff(xx) has a Fourier cosine transforms: 185
3 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp Then, if ff(xx) is multiplied by xx then, the Fourier cosine transform of a new function xxxx(xx) is F cc ff(xx) = dd dddd F ss ff(xx) Where, F cc is the Fourier cosine transform of xxxx(xx). If ff(xx) is multiplied by xx 2 then, the Fourier cosine transform of a new function xx 2 ff(xx) is F cc ff(xx) = dd2 ddωω 2 F cc ff(xx) Where, F cc is the Fourier cosine transform of xx 2 ff(xx). In general, For nn even and if ff(xx) is multiplied by xx nn, then the Fourier cosine transform of xx nn ff(xx) is: F cc xx nn ff(xx) = ( 1) mm ddωω nn [F ccff(xx)] For nn odd and if ff(xx) is multiplied by xx nn, then the Fourier cosine transform of xx nn ff(xx) is: F cc xx nn ff(xx) = ( 1) mm+1 ddωω nn [F ssff(xx)] Proof Suppose ff(xx) has a Fourier cosine and sine transforms; such that And 186
4 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp Let, multiply ff(xx) by xx, then by definition; the Fourier cosine transforms of xxxx(xx) is: F cc (xxxx(xx)) = 2 xxxx(xx) cos xxxxxxxx Let, F cc = F cc (xxxx(xx)), then from the Fourier sine transform we have: by taking differentiation to both sides with respect to ωω, we have dd dddd [F ss(ff(xx))] = dd dddd [ 2 ff(xx) sin xxxxxxxx] [ff(xx) sin xxxx]dddd dddd ff(xx) [sin xxxx]dddd dddd = 2 xxxx(xx) cos xxxxxxxx = F cc (xxxx(xx)) = F cc Therefore, F cc ff(xx) = dd dddd F ss ff(xx) Let, multiply ff(xx) by xx 2, then by definition: The Fourier cosine transforms of xx 2 ff(xx)) is 187
5 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp F cc (xx 2 ff(xx))) = 2 xx2 ff(xx)) cos xxxxxxxx Then, from the Fourier cosine transform we have: by taking differentiation to both sides with respect to ωω twice, we have: dd dddd F cc ff(xx) = dd dddd [ 2 ff(xx) cos xxxxxxxx] [ff(xx) cos xxxx]dddd dddd ff(xx) [cos xxxx]dddd dddd = 2 xxxx(xx) sin xxxxxxxx = 2 xxxx(xx) sin xxxxxxxx And dd 2 ddωω 2 F cc ff(xx) = dd2 ddωω 2 [ 2 ff(xx) cos xxxxxxxx] = dd dddd [ dd dddd ( 2 ff(xx) cos xxxxxxxx)] = dd dddd [ 2 xxxx(xx) sin xxxxxxxx] 188
6 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp = 2 dd [xxxx(xx) sin xxxx]dddd dddd = 2 xx2 ff(xx) cos xxxxxxxx = F cc (xx 2 ff(xx)) = F cc Therefore, F cc = dd2 ddωω 2 F cc ff(xx) Continuing the process; For nn even and if ff(xx) is multiplied by xx nn, then the Fourier cosine transform of xx nn ff(xx) is: F cc xx nn ff(xx) = ( 1) mm ddωω nn [F ccff(xx)] For nn odd and if ff(xx) is multiplied by xx nn, then the Fourier cosine transform of xx nn ff(xx) is: F cc xx nn ff(xx) = ( 1) mm ddωω nn [F ssff(xx)] B) Suppose ff(xx) has Fourier sine transforms: Then, if ff(xx) is multiplied by xx then, the Fourier cosine transform of a new function xxxx(xx) is F ss ff(xx) = dd dddd F cc ff(xx) Where, F ss is the Fourier sine transform of xxxx(xx). If ff(xx) is multiplied by xx 2 then, the Fourier sine transform of a new function xx 2 ff(xx) is 189
7 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp F ss ff(xx) = dd2 ddωω 2 F ss ff(xx) Where, F ss is the Fourier cosine transform of xx 2 ff(xx). In general, For nn even and if ff(xx) is multiplied by xx nn, then the Fourier sine transform of xx nn ff(xx) is: F ss xx nn ff(xx) = ( 1) mm+1 ddωω nn [F ssff(xx)] For nn odd and if ff(xx) is multiplied by xx nn, then the Fourier sine transform of xx nn ff(xx) is: F ss xx nn ff(xx) = ( 1) mm+1 ddωω nn [F ccff(xx)] Proof Suppose ff(xx) has a Fourier cosine and sine transforms: And Let, multiply ff(xx) by xx, then by definition; the Fourier sine transforms xxxx(xx) is, F ss (xxxx(xx)) = 2 xxxx(xx) sin xxxxxxxx Let, F ss = F ss (xxxx(xx)), then from the Fourier cosine transform we have: 19
8 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp by taking differentiation to both sides with respect to ωω dd dddd [F cc(ff(xx))] = dd dddd [ 2 ff(xx) cos xxxxxxxx] [ff(xx) cos xxxx]dddd dddd ff(xx) [cos xxxx]dddd dddd = 2 xxxx(xx) sin xxxxxxxx = 2 xxxx(xx) sin xxxxxxxx = F ss (xxxx(xx)) = F ss Therefore, F ss ff(xx) = dd dddd F cc ff(xx) Let, multiply ff(xx) by xx 2, then by definition; the Fourier cosine transforms of xx 2 ff(xx)) is, F ss (xx 2 ff(xx))) = 2 xx2 ff(xx)) sin xxxxxxxx Then, from the Fourier sine transform we have: by taking differentiation to both sides with respect to ωω twice, we have: 191
9 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp dd dddd F ss ff(xx) = dd dddd [ 2 ff(xx) sin xxxxxxxx] [ff(xx) sin xxxx]dddd dddd ff(xx) [sin xxxx]dddd dddd = 2 xxxx(xx) cos xxxxxxxx And dd 2 ddωω 2 F ss ff(xx) = dd2 ddωω 2 [ 2 ff(xx) sin xxxxxxxx] = dd dddd [ dd dddd ( 2 ff(xx) sin xxxxxxxx)] = dd dddd [ 2 xxxx(xx) cos xxxxxxxx] [xxxx(xx) cos xxxx]dddd dddd = 2 xx2 ff(xx) sin xxxxxxxx = 2 xx2 ff(xx) sin xxxxxxxx = F ss (xx 2 ff(xx)) = F ss Therefore, F ss = dd2 ddωω 2 F ss ff(xx) 192
10 International Journal of Sciences: Basic and Applied Research (IJSBAR) (217) Volume 35, No 3, pp Continuing the process; For nn even and if ff(xx) is multiplied by xx nn, then the Fourier sine transform of xx nn ff(xx) is: F ss xx nn ff(xx) = ( 1) mm+1 ddωω nn [F ssff(xx)] For nn odd and if ff(xx) is multiplied by xx nn, then the Fourier sine transform of xx nn ff(xx) is: F ss xx nn ff(xx) = ( 1) mm+1 ddωω nn [F ccff(xx)] 3. Conclusions The purpose of this paper is to provide a brief representation any function in integral form, Fourier cosine and sine transforms, after multiplying the given function by power functions; xx, xx 2, xx 3, xx nn and provide the relation between Fourier Cosine transforms and Fourier Sine transforms. It is hoped that this implementation of the Fourier cosine and sine transforms will help representing the solutions of ODEs, PDEs, and integral equations that involving power functions terms in the integral form of simpler functions Cosine and Sine. References [1] Remarks on history of abstract harmonic analysis, Radomir S. Stankovic, Staakko T. Astola 1, Mark G. Karpovsky 2 [2] James S. Walker, University of Wisconsin-Eau clarie [3] Linear Partial Differential Equations for Scientists and Engineers, Tyn Myint-U Lokenath Debnath, 27 [4] Differential Equation and Integral Equations, Peter J. Collins, 26. [5] Differential Equations, James R. Brannan, William E. Boyce, 2 nd edition. [6] Advanced Engineering Mathematics 7th Edition, PETER V. ONEIL. [7] Historically, how and why was the Laplace Transform invented? Written 18 Oct 215 From Wikipedia: [8] The frequency domain Introduction: [9] JPNM Physics Fourier Transform: 193
FOURIER 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
An improper integral and an infinite series
An improper integral and an infinite series A. Baltimore one of the old cities in the United States Yue Kwok Choy In summer of 2010, I had great time visiting my daughter, Wendy, who is living in a small
MATH 1080: Calculus of One Variable II Fall 2018 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of One Variable II Fall 2018 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart Unit 2 Skill Set Important: Students should expect test questions that require
Signals and Systems Lecture (S2) Orthogonal Functions and Fourier Series March 17, 2008
Signals and Systems Lecture (S) Orthogonal Functions and Fourier Series March 17, 008 Today s Topics 1. Analogy between functions of time and vectors. Fourier series Take Away Periodic complex exponentials
3 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
Tyn 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
Analysis II: Fourier Series
.... Analysis II: Fourier Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American May 3, 011 K.Maruno (UT-Pan American) Analysis II May 3, 011 1 / 16 Fourier series were
PHYS 502 Lecture 3: Fourier Series
PHYS 52 Lecture 3: Fourier Series Fourier Series Introduction In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating
FOURIER 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
The wave equation in one dimension
The wave equation in one dimension Prof. Joyner 1 The theory of the vibrating string touches on musical theory and the theory of oscillating waves, so has likely been a concern of scholars since ancient
Linear 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
CHAPTER 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
Fast Fourier Transform
Fast Fourier Transform December 8, 2016 FFT JPEG RGB Y C B C R (luma (brightness), chroma 2 (color)) chroma resolution is reduced image is split in blocks 8 8 pixels JPEG RGB Y C B C R (luma (brightness),
Application of Series in Heat Transfer - transient heat conduction
Application of Series in Heat Transfer - transient heat conduction By Alain Kassab Mechanical, Materials and Aerospace Engineering UCF EXCEL Applications of Calculus Part I background and review of series
Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition
Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.
Math 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
SATHYABAMA 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
1 A complete Fourier series solution
Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider
Chapter 10: Partial Differential Equations
1.1: Introduction Chapter 1: Partial Differential Equations Definition: A differential equations whose dependent variable varies with respect to more than one independent variable is called a partial differential
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
f (t) K(t, u) d t. f (t) K 1 (t, u) d u. Integral Transform Inverse Fourier Transform
Integral Transforms Massoud Malek An integral transform maps an equation from its original domain into another domain, where it might be manipulated and solved much more easily than in the original domain.
Strauss 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
Topic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
Lecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
Lecture 5 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 1 -. 8 -. 6 -. 4 -. 2-1 -. 8 -. 6 -. 4 -. 2 -. 2. 4. 6. 8 1
Lecture # 06. Image Processing in Frequency Domain
Digital Image Processing CP-7008 Lecture # 06 Image Processing in Frequency Domain Fall 2011 Outline Fourier Transform Relationship with Image Processing CP-7008: Digital Image Processing Lecture # 6 2
Computer Problems for Fourier Series and Transforms
Computer Problems for Fourier Series and Transforms 1. Square waves are frequently used in electronics and signal processing. An example is shown below. 1 π < x < 0 1 0 < x < π y(x) = 1 π < x < 2π... and
[Engineering Mathematics]
![[Engineering Mathematics]](/thumbs/82/85860352.jpg "[Engineering Mathematics]")
[MATHS IV] [Engineering Mathematics] [Partial Differential Equations] [Partial Differentiation and formation of Partial Differential Equations has already been covered in Maths II syllabus. Present chapter
Continuous-time Fourier Methods
ELEC 321-001 SIGNALS and SYSTEMS Continuous-time Fourier Methods Chapter 6 1 Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity
Brenda Rubenstein (Physics PhD)
PHYSICIST PROFILE Brenda Rubenstein (Physics PhD) Postdoctoral Researcher Lawrence Livermore Nat l Lab Livermore, CA In college, Brenda looked for a career path that would allow her to make a positive
Math 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
Chapter 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
Rotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition
Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr
The domain and range of lines is always R Graphed Examples:
Graphs/relations in R 2 that should be familiar at the beginning of your University career in order to do well (The goal here is to be ridiculously complete, hence I have started with lines). 1. Lines
CODE: 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
MA6351-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS. Question Bank. Department of Mathematics FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY
MA6351-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS Question Bank Department of Mathematics FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY MADURAI 625 020, Tamilnadu, India 1. Define PDE and Order
CHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L
CHAPTER 4 FOURIER SERIES 1 S A B A R I N A I S M A I L Outline Introduction of the Fourier series. The properties of the Fourier series. Symmetry consideration Application of the Fourier series to circuit
Overview of Fourier Series (Sect. 6.2). Origins of the Fourier Series.
Overview of Fourier Series (Sect. 6.2. Origins of the Fourier Series. Periodic functions. Orthogonality of Sines and Cosines. Main result on Fourier Series. Origins of the Fourier Series. Summary: Daniel
Midterm 2: Sample solutions Math 118A, Fall 2013
Midterm 2: Sample solutions Math 118A, Fall 213 1. Find all separated solutions u(r,t = F(rG(t of the radially symmetric heat equation u t = k ( r u. r r r Solve for G(t explicitly. Write down an ODE for
Fourier Analysis Partial Differential Equations (PDEs)
PART C Fourier Analysis. Partial Differential Equations (PDEs) CHAPTER CHAPTER Fourier Analysis Partial Differential Equations (PDEs) Chapter and Chapter are directly related to each other in that Fourier
12.7 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
23.4. Convergence. Introduction. Prerequisites. Learning Outcomes
Convergence 3.4 Introduction In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily
Bernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
Weighted SS(E) = w 2 1( y 1 y1) y n. w n. Wy W y 2 = Wy WAx 2. WAx = Wy. (WA) T WAx = (WA) T b
6.8 - Applications of Inner Product Spaces Weighted Least-Squares Sometimes you want to get a least-squares solution to a problem where some of the data points are less reliable than others. In this case,
10 Motivation for Fourier series
10 Motivation for Fourier series Donkeys and scholars in the middle! French soldiers, forming defensive squares in Egypt, during Napoleon s campaign The central topic of our course is the so-called Fourier
MATH 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
Math 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
DEPARTMENT OF MATHEMATICS FACULTY OF ENGINERING AND TECHNOLOGY SRM UNIVERSITY MA0211- MATHEMATICS III SEMESTER III ACADEMIC YEAR:
DEPARTMENT OF MATHEMATICS FACULTY OF ENGINERING AND TECHNOLOGY SRM UNIVERSITY MA0211- MATHEMATICS III SEMESTER III ACADEMIC YEAR: 2011 2012 LECTURE SCHEME / PLAN The objective is to equip the students
1.1 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.
Chapter Thirteen: Traveling Waves and the Wave Equation in Air, Water and the Ether
Chapter Thirteen: Traveling Waves and the Wave Equation in Air, Water and the Ether We have been discussing the waves that are found in the string of a guitar or violin and how they are the physical reality
Second Order Linear Equations
Second Order Linear Equations Linear Equations The most general linear ordinary differential equation of order two has the form, a t y t b t y t c t y t f t. 1 We call this a linear equation because the
Examples 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:
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) =
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
Differential Equations
Electricity and Magnetism I (P331) M. R. Shepherd October 14, 2008 Differential Equations The purpose of this note is to provide some supplementary background on differential equations. The problems discussed
MA 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,
THE 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
FOURIER 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
FOURIER ANALYSIS. (a) Fourier Series
(a) Fourier Series FOURIER ANAYSIS (b) Fourier Transforms Useful books: 1. Advanced Mathematics for Engineers and Scientists, Schaum s Outline Series, M. R. Spiegel - The course text. We follow their notation
MATH 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
VALLIAMMAI 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
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
Time-Frequency Analysis
Time-Frequency Analysis Basics of Fourier Series Philippe B. aval KSU Fall 015 Philippe B. aval (KSU) Fourier Series Fall 015 1 / 0 Introduction We first review how to derive the Fourier series of a function.
Fourier Transform and its Applications
Fourier Transform and its Applications Prof. (Dr.) K.R. Chowdhary, Director COE Email: kr.chowdhary@iitj.ac.in webpage: http://www.krchowdhary.com JIET College of Engineering August 18, 2017 kr chowdhary
Mathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.
Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...
Math 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
Chapter 2 Joseph Fourier
Chapter 2 Joseph Fourier Joseph Fourier (1768 1830) 2.1 Introduction Fourier series are trigonometric series used to represent a function, and they are widely used throughout pure and applied mathematics.
PDE 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
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7
ENGIN 211, Engineering Math. Fourier Series and Transform
ENGIN 11, Engineering Math Fourier Series and ransform 1 Periodic Functions and Harmonics f(t) Period: a a+ t Frequency: f = 1 Angular velocity (or angular frequency): ω = ππ = π Such a periodic function
Functions of Several Variables
Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend
ELEC273 Lecture Notes Set 11 AC Circuit Theorems
ELEC273 Lecture Notes Set C Circuit Theorems The course web site is: http://users.encs.concordia.ca/~trueman/web_page_273.htm Final Exam (confirmed): Friday December 5, 207 from 9:00 to 2:00 (confirmed)
3.2 A2 - Just Like Derivatives but Backwards
3. A - Just Like Derivatives but Backwards The Definite Integral In the previous lesson, you saw that as the number of rectangles got larger and larger, the values of Ln, Mn, and Rn all grew closer and
More on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
Fourier Transforms and its Applications in Engineering Field
Fourier Transforms and its Applications in Engineering Field U S Hegde 1, Uma S 2, Aravind P N 3, Malashri S 4 Associate Professor & HOD, Department of Mathematics, SIR MVIT, Bangalore, India 1 Associate
MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series Lecture - 10
MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series ecture - 10 Fourier Series: Orthogonal Sets We begin our treatment with some observations: For m,n = 1,2,3,... cos
A 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
B.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
EE 341 A Few Review Problems Fall 2018
EE 341 A Few Review Problems Fall 2018 1. A linear, time-invariant system with input, xx(tt), and output, yy(tt), has the unit-step response, cc(tt), shown below. a. Find, and accurately sketch, the system
Lecture6. Partial Differential Equations
EP219 ecture notes - prepared by- Assoc. Prof. Dr. Eser OĞAR 2012-Spring ecture6. Partial Differential Equations 6.1 Review of Differential Equation We have studied the theoretical aspects of the solution
A proof for the full Fourier series on [ π, π] is given here.
![A proof for the full Fourier series on [ π, π] is given here.](/thumbs/75/72296360.jpg "A 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
u 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
Lecture 13: Series Solutions near Singular Points
Lecture 13: Series Solutions near Singular Points March 28, 2007 Here we consider solutions to second-order ODE s using series when the coefficients are not necessarily analytic. A first-order analogy
Math 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
Chapter 5: Spectral Domain From: The Handbook of Spatial Statistics. Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell
Chapter 5: Spectral Domain From: The Handbook of Spatial Statistics Dr. Montserrat Fuentes and Dr. Brian Reich Prepared by: Amanda Bell Background Benefits of Spectral Analysis Type of data Basic Idea
SAMPLE COURSE OUTLINE MATHEMATICS METHODS ATAR YEAR 11
SAMPLE COURSE OUTLINE MATHEMATICS METHODS ATAR YEAR 11 Copyright School Curriculum and Standards Authority, 2017 This document apart from any third party copyright material contained in it may be freely
Radial Basis Function (RBF) Networks
CSE 5526: Introduction to Neural Networks Radial Basis Function (RBF) Networks 1 Function approximation We have been using MLPs as pattern classifiers But in general, they are function approximators Depending
MATH 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
Partial Differential Equations
Partial Differential Equations Spring Exam 3 Review Solutions Exercise. We utilize the general solution to the Dirichlet problem in rectangle given in the textbook on page 68. In the notation used there
Today s lecture. The Fourier transform. Sampling, aliasing, interpolation The Fast Fourier Transform (FFT) algorithm
Today s lecture The Fourier transform What is it? What is it useful for? What are its properties? Sampling, aliasing, interpolation The Fast Fourier Transform (FFT) algorithm Jean Baptiste Joseph Fourier
Partial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
Math 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
Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018
Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent
CHAPTER 10 NOTES DAVID SEAL
CHAPTER 1 NOTES DAVID SEA 1. Two Point Boundary Value Problems All of the problems listed in 14 2 ask you to find eigenfunctions for the problem (1 y + λy = with some prescribed data on the boundary. To
MATH 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
Wave 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
MATH 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
MATH 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
Lecture Notes for Ch 10 Fourier Series and Partial Differential Equations
ecture Notes for Ch 10 Fourier Series and Partial Differential Equations Part III. Outline Pages 2-8. The Vibrating String. Page 9. An Animation. Page 10. Extra Credit. 1 Classic Example I: Vibrating String
X 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
17 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
KREYSZIG E Advanced Engineering Mathematics (10th ed., Wiley 2011) Chapter 11 - Fourier analysis
KREYSZIG E Advanced Engineering Mathematics (th ed., Wiley ) Chapter - Fourier analysis . CHAPTER Fourier Analysis 474 This chapter on Fourier analysis covers three broad areas: Fourier series in Secs...4,