Partial Differential Equations - part of EM Waves module (PHY2065)
|
|
- Julia Scott
- 5 years ago
- Views:
Transcription
1 Partial Differential Equations - part of EM Waves module (PHY2065) Richard Sear February 7, 2013 Recommended textbooks 1. Mathematical methods in the physical sciences, Mary Boas. 2. Essential mathematical methods for the physical sciences, Riley and Hobson. 3. Advanced Engineering Mathematics, Erwin Kreyszig All 3 are in the library. Riley and Hobson is also available as an ebook. Overview 1. Introduction to PDEs: more than one variable; the wave equation; Laplace s equation etc; boundary conditions (BCs). 2. The wave equation (in one dimension). 3. The diffusion equation (two dimensions, steady state). 4. Spherical and circular polar coordinates. 5. Schrödinger equation for the hydrogen atom (s wavefunctions spherically symmetric solutions). 1
2 1 Introduction to Partial Differential Equations Partial differential equations (PDEs) differ from ordinary differential equations (ODEs) in that there is more than one variable. Thus the derivatives are partial derivatives, and so they are called partial differential equations. In physics when we have more than one variable this is almost always either because the functions depend on more than one spatial dimension, e.g., not on x only, but on say both x and y, or because they depend on space and time, e.g., x and t. As they contain more than one variable, PDEs can be trickier than ODEs, and in both cases unless they are linear they are often impossible to solve analytically nonlinear equations are typically solved using a computer. Here we will only consider linear PDEs. In this part of the course, we will introduce the most common simple PDEs in physics, and show how to solve them. As it happens these are all second-order, i.e., contain second derivatives, and so we will only study second-order PDEs. 1.1 Time-independent PDEs Firstly, there are three common named PDEs that depend on the position coordinates x, y and z, but not on time. For a scalar function u(x,y,z) these are: Laplace s equation 2 u = 0 Poisson s equation 2 u = f(x,y,z) with f some specified function of x and y and z, e.g., f = x 2 + 2y 2 + z 2, and Helmholtz s equation 2 u + au(x,y,z) = 0 with a a constant. They all include 2 which is in three dimensions 2 = 2 x y z 2 Thus in three dimensions Laplace s equation is ( ) 2 2 u = x y + 2 u = 2 u 2 z 2 x + 2 u 2 y + 2 u 2 z = 0 2 In, for example, one dimension, this simplifies to d 2 u(x) dx 2 = 0 where the derivatives are no longer partial derivatives as u(x) is now a function only of x. Note that Poisson s equation is the most general, as if f = 0 it reduces to Laplace s equation and if f = au it reduces to Helmholtz s equation. Poisson s equation appears in electromagnetism. There u = φ(x, y, z) the electrostatic potential, and f = ρ(x,y,z)/ǫ 0, with ρ the charge density as a function of position and ǫ 0 the permittivity. Then Poisson s equation is 2 φ = ρ/ǫ 0 2
3 which is one of Maxwell s four equations that govern electromagnetism, in the electrostatic case. Thus, Poisson s equation occurs very frequently in electromagnetism. Also, in a region of space that is free of charges, i.e., where ρ = 0, then Poisson s equation for the electrostatic potential φ simplifies to 2 φ = 0 which is Laplace s equation. Thus Laplace s equation also appears frequently in electromagnetism. Systems like semiconductors, salty water etc have free charges, electrons and/or ions. At thermal equilibrium the charge density of these charges ρ is often is proportional to the electrostatic potential. Then we have ρ φ and Poisson s equation becomes Helmholtz s equation. Thus Helmholtz s equation is common whenever we are dealing with electrons or ions. Another important time-independent PDE that you have already come across is Schrödinger s time-independent equation for the quantum mechanical wavefunction ψ 1.2 Time-dependent PDEs 2 2m 2 ψ(x,y,z) + V (x,y,z)ψ(x,y,z) = Eψ(x,y,z) So much for the time independent PDEs. Time dependent PDEs need to be solved for the function of position and time v(x,y,z,t). Most of the common simple time dependent PDEs in physics are either of the form of the diffusion equation or the wave equation. The diffusion equation is diffusion equation D 2 v = v t where D is a constant, the diffusion constant. D has dimensions of a length squared over time. The wave equation is wave equation 2 v = 1 2 v c 2 t 2 c is the speed of the wave and has dimensions of length over time. Note that the difference between the two PDEs is that the diffusion equation has the first derivative with respect to time, while the wave equation has the second derivative with respect to time. As we will see, this apparently small difference will result in their solutions being very different. As their names suggest the diffusion equation describes diffusion, e.g., of molecules in a gas or liquid, whereas the wave equation describes waves: light, sound etc. Also note that Schrödinger s time-dependent equation looks very like a diffusion equation. For the time-dependent wave function ψ(x,y,z,t) of an electron in a potential V (x,y,z) it is 2 2m 2 ψ(x,y,z,y) + V (x,y,z)ψ(x,y,z,t) = i ψ(x,y,z,t) t If the potential V = 0 then the equation is the diffusion equation i 2m 2 ψ(x,y,z,t) = ψ(x,y,z,t) t where we divided by i. This is almost the diffusion equation, but with D = i /2m, i.e., with an imaginary diffusion constant. This looks a bit strange but the maths works out fine. 3
4 Also, when we study diffusion, sometimes, we are studying it at steady-state, which means (by definition) that v is not a function of time. This is typically diffusion down a constant concentration gradient. Then as v is not a function of time, the time derivative is of course zero and the diffusion equation reduces to Laplace s equation. Thus, Laplace s equation also occurs whenever diffusion occurs down a time-independent concentration gradient. 1.3 Boundary conditions for PDEs Time-independent PDEs To place the BCs for second-order PDEs in context, let us start by recalling how the BCs work for second-order ODEs. For a second order ODE we require two boundary conditions, typically either the value of the function at two points or the value of the function and its first derivative at a point. For example, if we have the general solution of an ODE, call it f(x), and want to know f(x) for x between 0 and 1, then a possible set of boundary conditions consists of the values of f(x) at both x = 0 and x = 1 i.e., the values of f at the boundaries of the range of x values we are interested in. The boundary conditions for a PDE are analogous except that if we have a function of, say, two variables x and y, f(x,y), then we will be interested not in a function along a line but a function over an area of the xy plane. This is illustrated in Fig. 1, where we show a possible area A over which we want the solution f(x,y) to a PDE. See the caption for possible BCs for that area. A particularly simple area to work with is that of a rectangle. A rectangular area is bounded by four straight lines. Once we have the general solution of a PDE, f(x,y), then possible boundary conditions are the values of the function f(x,y) along all four sides of the rectangle. Other possible BCs are the values of the derivative of f(x,y) along these four sides. This derivative of f(x,y) must be the derivative normal to the direction of the boundary 1, at all points along this boundary. It is also possible to have boundary conditions that combine values of f and its derivative along the boundary of the area. Here either f or its derivative must be specified at every point on the boundary. An example set of BCs for the rectangle is that f(x,y) = 1 along all four sides of the rectangle. In three dimensions, we have three variables, x, y and z, and then we will require the function f(x,y,z) over some volume and the boundary conditions are be the values of the function or its derivative over the surface that encloses this volume. For example, if we want the f(x,y,z) inside a cubic volume then one possible BC is the function s value over all six faces of this cube. One final small point, if along the boundary of an area (or a volume in three dimensions) we only specify the derivative (i.e., we don t specify the function itself anywhere), then the function is only specified up to an unknown constant, i.e., we end up with f(x,y) + C with f precisely defined and C an unknown constant. Often not knowing C does not matter Time-dependent PDEs For time-dependent PDEs, the boundary conditions often include initial conditions, i.e., the function at time t = 0. For example, for the wave equation, the boundary condition might be the function f(x,t) and its time derivative, at t = 0. If the function is defined all along the x axis that may be enough, but sometimes boundary conditions at the ends of some length of the x axis are imposed as well. For example, if the wave is on a taught string, the boundary conditions may be fixed values of 1 Recall from your vector calculus lectures that the derivative of a scalar f in two or three dimensions, f, is a vector. 4
5 Figure 1: Schematic of the xy plane, to illustrate boundary conditions for a time-independent PDE. In two dimensions we will be interested in the solution, f(x,y), within some area A. A possible area A is shown as the yellow (grey) shaded area bounded by the black curve. The boundary conditions for a solution of f(x,y) in A can then be: 1) the values of function f(x,y) along the complete boundary (shown as black curve); 2) the values of the derivative of f(x,y) normal to the boundary, along the complete boundary; or 3) some combination of f and its derivative, along the complete boundary. y area A f(x,y) x f at the two ends of the string, applied at at all times, plus the position and velocity of the string at time t = 0. In general, the BCs here can be quite varied, but in this course only a few simple examples will be discussed. I do not expect you to know the general definition of the BCs of a time-dependent PDE Superposition for homogeneous PDEs When you studied ODEs, you came across homogeneous ODEs. These were ODEs in which all terms were linear in either the function itself or a derivative of the function, i.e., if the function is F(x), each term is proportional to F(x) or df(x)/dx, etc, and there are no F 2, F 3, exp(f), (df(x)/dx) 2, etc terms, and no terms that are just functions of x. Homogeneous ODEs have the very useful property that the sum of any two solutions of the ODE, is also a solution to the ODE, i.e., if F 1 (x) and F 2 (x) are solutions are an ODE, then a 1 F 1 (x)+a 2 F 2 (x) is also a solution to the same ODE. Here a 1 and a 2 are any two constants. Homogeneous PDEs are exactly analogous to homogeneous ODEs. They are PDEs in which every term is proportional to F(x,y,z,t) or df(x)/dx, d 2 F(x)/dxdy, etc, and there are no F 2, F 3, exp(f), (df(x)/dx) 2, etc terms, and no terms that are just functions of x, y, z and/or t. Homogeneous PDEs have the same useful property as homogeneous ODEs. The sum of any two solutions of a homogeneous PDE is also a solution to the same PDE, i.e., if F 1 (x,y,z,t) and F 2 (x,y,z,t) are both solutions to a PDE, then F(x,y,z,t) = a 1 F 1 (x,y,z,t) + a 2 F 2 (x,y,z,t) F, F 1 and F 2 solutions of a homogeneous PDE is also a solution to the same PDE. This is true for any values of the constants a 1 and a 2. This property is extremely useful. In particular, often sines and cosines are solutions of a homogeneous PDE, and then we can construct a solution of the PDE that satisfies the imposed BCs, by summing lots of sine and cosine wave solutions to the PDE (which works because the PDE is homoge- 5
6 neous). This is what is called a Fourier series solution of a PDE, and these solutions are very widely used and very useful in the study of physics problems with PDEs. 6
Introduction 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 informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Ryan C. Trinity University Partial Differential Equations Lecture 1 Ordinary differential equations (ODEs) These are equations of the form where: F(x,y,y,y,y,...)
More information1D Wave PDE. Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) Richard Sear.
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) November 12, 2018 Wave equation in one dimension This lecture Wave PDE in 1D Method of Separation of
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Spring 2017 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u =
More informationPDEs in Spherical and Circular Coordinates
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) This lecture Laplacian in spherical & circular polar coordinates Laplace s PDE in electrostatics Schrödinger
More informationLecture 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
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More 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 informationLecture6. 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
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 informationMathematics of Chemistry: Techniques & Applications (CHEM-UA 140)
Mathematics of Chemistry: Techniques & Applications (CHEM-UA 140) Professor Mark E. Tuckerman Office: 1166E Waverly Phone: 8-8471 Email: mark.tuckerman@nyu.edu Class Time & Location: Tuesday, Thursday:
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
More informationPH2130. Week 1. Week 2. Week 3. Questions for contemplation. Why differential equations? Why usually linear diff eq n s? Why usually 2 nd order?
PH130 week 3, page 1 PH130 Questions for contemplation Week 1 Why differential equations? Week Why usually linear diff eq n s? Week 3 Why usually nd order? PH130 week 3, page Aims of Wk 3 Lect 1 Recognise
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 informationThe integrating factor method (Sect. 1.1)
The integrating factor method (Sect. 1.1) Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Overview
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 information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationLecture Introduction
Lecture 1 1.1 Introduction The theory of Partial Differential Equations (PDEs) is central to mathematics, both pure and applied. The main difference between the theory of PDEs and the theory of Ordinary
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 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 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 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 informationFall 2016 Spring 2017
BE 603: Partial differential equations (Spring 2017) Instructor: TA: Andy Fan (fana@bu.edu) TBD Class: Time is TBD (Spring 2016 only) Recitation: Should be on Fri afternoons (Spring 2016 only) Office Hours:
More informationDifferential 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
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More informationStaple or bind all pages together. DO NOT dog ear pages as a method to bind.
Math 3337 Homework Instructions: Staple or bind all pages together. DO NOT dog ear pages as a method to bind. Hand-drawn sketches should be neat, clear, of reasonable size, with axis and tick marks appropriately
More informationMATH3203 Lecture 1 Mathematical Modelling and ODEs
MATH3203 Lecture 1 Mathematical Modelling and ODEs Dion Weatherley Earth Systems Science Computational Centre, University of Queensland February 27, 2006 Abstract Contents 1 Mathematical Modelling 2 1.1
More informationIntroduction and some preliminaries
1 Partial differential equations Introduction and some preliminaries A partial differential equation (PDE) is a relationship among partial derivatives of a function (or functions) of more than one variable.
More informationParticle in a 3 Dimensional Box just extending our model from 1D to 3D
CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-1 Particle in a 3 Dimensional Box just extending our model from 1D to 3D A 3D model is a step closer to reality than a 1D model. Let s increase the
More informationMATH 308 COURSE SUMMARY
MATH 308 COURSE SUMMARY Approximately a third of the exam cover the material from the first two midterms, that is, chapter 6 and the first six sections of chapter 7. The rest of the exam will cover the
More informationPartial Differential Equations Summary
Partial Differential Equations Summary 1. The heat equation Many physical processes are governed by partial differential equations. temperature of a rod. In this chapter, we will examine exactly that.
More informationSolutions to Laplace s Equations
Solutions to Laplace s Equations Lecture 14: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We have seen that in the region of space where there are no sources of charge,
More informationWe will begin by first solving this equation on a rectangle in 2 dimensions with prescribed boundary data at each edge.
Page 1 Sunday, May 31, 2015 9:24 PM From our study of the 2-d and 3-d heat equation in thermal equlibrium another PDE which we will learn to solve. Namely Laplace's Equation we arrive at In 3-d In 2-d
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 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 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 informationSecond 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
More informationPartial Differential Equations and Complex Variables
EE-22, Spring 29 p. /25 EE 22 Partial Differential Equations and Complex Variables Ray-Kuang Lee Institute of Photonics Technologies, Department of Electrical Engineering and Department of Physics, National
More informationMcGill University April 20, Advanced Calculus for Engineers
McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) 1D Heat Equation: Derivation Current Semester 1 / 19 Introduction The derivation of the heat
More informationPartial Differential Equations with MATLAB
CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES An Introduction to Partial Differential Equations with MATLAB Second Edition Matthew P. Coleman CHAPMAN & HALL/CRC APPLIED MATHEMATICS
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 informationModule 7: The Laplace Equation
Module 7: The Laplace Equation In this module, we shall study one of the most important partial differential equations in physics known as the Laplace equation 2 u = 0 in Ω R n, (1) where 2 u := n i=1
More informationNumerical Solutions of Partial Differential Equations
Numerical Solutions of Partial Differential Equations Dr. Xiaozhou Li xiaozhouli@uestc.edu.cn School of Mathematical Sciences University of Electronic Science and Technology of China Introduction Overview
More informationMath 5587 Lecture 2. Jeff Calder. August 31, Initial/boundary conditions and well-posedness
Math 5587 Lecture 2 Jeff Calder August 31, 2016 1 Initial/boundary conditions and well-posedness 1.1 ODE vs PDE Recall that the general solutions of ODEs involve a number of arbitrary constants. Example
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 informationDifferential Equations
Differential Equations Problem Sheet 1 3 rd November 2011 First-Order Ordinary Differential Equations 1. Find the general solutions of the following separable differential equations. Which equations are
More informationPDE: The Method of Characteristics Page 1
PDE: The Method of Characteristics Page y u x (x, y) + x u y (x, y) =, () u(, y) = cos y 2. Solution The partial differential equation given can be rewritten as follows: u(x, y) y, x =, (2) where = / x,
More informationIf you must be wrong, how little wrong can you be?
MATH 2411 - Harrell If you must be wrong, how little wrong can you be? Lecture 13 Copyright 2013 by Evans M. Harrell II. About the test Median was 35, range 25 to 40. As it is written: About the test Percentiles:
More informationNumerical Methods for Partial Differential Equations: an Overview.
Numerical Methods for Partial Differential Equations: an Overview math652_spring2009@colorstate PDEs are mathematical models of physical phenomena Heat conduction Wave motion PDEs are mathematical models
More informationc2 2 x2. (1) t = c2 2 u, (2) 2 = 2 x x 2, (3)
ecture 13 The wave equation - final comments Sections 4.2-4.6 of text by Haberman u(x,t), In the previous lecture, we studied the so-called wave equation in one-dimension, i.e., for a function It was derived
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 informationModule 2: First-Order Partial Differential Equations
Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order
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 informationPhysics 342 Lecture 23. Radial Separation. Lecture 23. Physics 342 Quantum Mechanics I
Physics 342 Lecture 23 Radial Separation Lecture 23 Physics 342 Quantum Mechanics I Friday, March 26th, 2010 We begin our spherical solutions with the simplest possible case zero potential. Aside from
More informationPartial Differential Equations. Examples of PDEs
Partial Differential Equations Almost all the elementary and numerous advanced parts of theoretical physics are formulated in terms of differential equations (DE). Newton s Laws Maxwell equations Schrodinger
More informationConnection to Laplacian in spherical coordinates (Chapter 13)
Connection to Laplacian in spherical coordinates (Chapter 13) We might often encounter the Laplace equation and spherical coordinates might be the most convenient 2 u(r, θ, φ) = 0 We already saw in Chapter
More informationSeparation of Variables
Separation of Variables A typical starting point to study differential equations is to guess solutions of a certain form. Since we will deal with linear PDEs, the superposition principle will allow us
More informationLecture 16: Relaxation methods
Lecture 16: Relaxation methods Clever technique which begins with a first guess of the trajectory across the entire interval Break the interval into M small steps: x 1 =0, x 2,..x M =L Form a grid of points,
More informationMATH20411 PDEs and Vector Calculus B
MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411
More informationWeek 01 : Introduction. A usually formal statement of the equality or equivalence of mathematical or logical expressions
1. What are partial differential equations. An equation: Week 01 : Introduction Marriam-Webster Online: A usually formal statement of the equality or equivalence of mathematical or logical expressions
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define
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 informationLecture 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
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
More informationfree space (vacuum) permittivity [ F/m]
Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charge distributions. The mathematical definition of the electrostatic field is derived
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationThe Heat Equation John K. Hunter February 15, The heat equation on a circle
The Heat Equation John K. Hunter February 15, 007 The heat equation on a circle We consider the diffusion of heat in an insulated circular ring. We let t [0, ) denote time and x T a spatial coordinate
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 informationThe Schrödinger Equation
Chapter 13 The Schrödinger Equation 13.1 Where we are so far We have focused primarily on electron spin so far because it s a simple quantum system (there are only two basis states!), and yet it still
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 informationUpon successful completion of MATH 220, the student will be able to:
MATH 220 Matrices Upon successful completion of MATH 220, the student will be able to: 1. Identify a system of linear equations (or linear system) and describe its solution set 2. Write down the coefficient
More informationDifferential Operators and the Divergence Theorem
1 of 6 1/15/2007 6:31 PM Differential Operators and the Divergence Theorem One of the most important and useful mathematical constructs is the "del operator", usually denoted by the symbol Ñ (which is
More informationLecture 1, August 21, 2017
Engineering Mathematics 1 Fall 2017 Lecture 1, August 21, 2017 What is a differential equation? A differential equation is an equation relating a function (known sometimes as the unknown) to some of its
More informationMATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES
MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes
More information1 Basics of Quantum Mechanics
1 Basics of Quantum Mechanics 1.1 Admin The course is based on the book Quantum Mechanics (2nd edition or new international edition NOT 1st edition) by Griffiths as its just genius for this level. There
More informationIn what follows, we examine the two-dimensional wave equation, since it leads to some interesting and quite visualizable solutions.
ecture 22 igher-dimensional PDEs Relevant section of text: Chapter 7 We now examine some PDEs in higher dimensions, i.e., R 2 and R 3. In general, the heat and wave equations in higher dimensions are given
More informationClassification of partial differential equations and their solution characteristics
9 TH INDO GERMAN WINTER ACADEMY 2010 Classification of partial differential equations and their solution characteristics By Ankita Bhutani IIT Roorkee Tutors: Prof. V. Buwa Prof. S. V. R. Rao Prof. U.
More information4 Power Series Solutions: Frobenius Method
4 Power Series Solutions: Frobenius Method Now the ODE adventure takes us to series solutions for ODEs, a technique A & W, that is often viable, valuable and informative. These can be readily applied Sec.
More informationChapter 2. Vector Calculus. 2.1 Directional Derivatives and Gradients. [Bourne, pp ] & [Anton, pp ]
Chapter 2 Vector Calculus 2.1 Directional Derivatives and Gradients [Bourne, pp. 97 104] & [Anton, pp. 974 991] Definition 2.1. Let f : Ω R be a continuously differentiable scalar field on a region Ω R
More informationLecture 7. Heat Conduction. BENG 221: Mathematical Methods in Bioengineering. References
BENG 221: Mathematical Methods in Bioengineering Lecture 7 Heat Conduction References Haberman APDE, Ch. 1. http://en.wikipedia.org/wiki/heat_equation Lecture 5 BENG 221 M. Intaglietta Heat conduction
More informationMath Partial Differential Equations
Math 531 - Partial Differential Equations to Partial Differential Equations Joseph M. Mahaffy, jmahaffy@sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences
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 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 informationMathematical Notes for E&M Gradient, Divergence, and Curl
Mathematical Notes for E&M Gradient, Divergence, and Curl In these notes I explain the differential operators gradient, divergence, and curl (also known as rotor), the relations between them, the integral
More informationElectromagnetic waves in free space
Waveguide notes 018 Electromagnetic waves in free space We start with Maxwell s equations for an LIH medum in the case that the source terms are both zero. = =0 =0 = = Take the curl of Faraday s law, then
More informationElements of Vector Calculus : Line and Surface Integrals
Elements of Vector Calculus : Line and Surface Integrals Lecture 2: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay In this lecture we will talk about special functions
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationToday in Physics 217: electric potential
Today in Physics 17: electric potential Finish Friday s discussion of the field from a uniformly-charged sphere, and the gravitational analogue of Gauss Law. Electric potential Example: a field and its
More informationPhysics 6303 Lecture 2 August 22, 2018
Physics 6303 Lecture 2 August 22, 2018 LAST TIME: Coordinate system construction, covariant and contravariant vector components, basics vector review, gradient, divergence, curl, and Laplacian operators
More informationINTRODUCTION TO ELECTRODYNAMICS
INTRODUCTION TO ELECTRODYNAMICS Second Edition DAVID J. GRIFFITHS Department of Physics Reed College PRENTICE HALL, Englewood Cliffs, New Jersey 07632 CONTENTS Preface xi Advertisement 1 1 Vector Analysis
More information1. Differential Equations (ODE and PDE)
1. Differential Equations (ODE and PDE) 1.1. Ordinary Differential Equations (ODE) So far we have dealt with Ordinary Differential Equations (ODE): involve derivatives with respect to only one variable
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 6
Strauss PDEs 2e: Section 3 - Exercise Page of 6 Exercise Carefully derive the equation of a string in a medium in which the resistance is proportional to the velocity Solution There are two ways (among
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More informationELECTROSTATIC THEOREMS
ELECTROSTATIC THEOREMS In these notes I prove several important theorems concerning the electrostatic potential V(x,y,z), namely the Earnshaw theorem, the mean-value theorem, and two uniqueness theorems.
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 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 informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More information