# (PDE) u t k(u xx + u yy ) = 0 (x, y) in Ω, t > 0, (BC) u(x, y, t) = 0 (x, y) on Γ, t > 0, (IC) u(x, y, 0) = f(x, y) (x, y) in Ω.

Size: px
Start display at page:

Download "(PDE) u t k(u xx + u yy ) = 0 (x, y) in Ω, t > 0, (BC) u(x, y, t) = 0 (x, y) on Γ, t > 0, (IC) u(x, y, 0) = f(x, y) (x, y) in Ω."

## Transcription

1 Seprtion of Vriles for Higher Dimensionl Het Eqution 1. Het Eqution nd Eigenfunctions of the Lplcin: An 2-D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed oundry temperture on Γ: (PDE u t k(u xx + u yy = 0 (x, y in Ω, t > 0, (BC u(x, y, t = 0 (x, y on Γ, t > 0, (IC u(x, y, 0 = f(x, y (x, y in Ω. (IDEA: Seprtion of Vriles We look for solutions of (PDE-(BC whose sptil structure of the temprture distriution remins invrint with time. In these solutions only the mplitudes of the temprture distriution my chnge with time. Mthemticlly, we try to clssify ll nontrivil solutions of the following form: u(x, y, t = T(tΦ(x, y 0. (THE EQUATIONS FOR PRODUCT SOLUTIONS Plugging u = T(tΦ(x, y in (PDE-(BC we see: There is constnt α such tht ( T (t kαt(t = 0 t > 0; ( Φ xx + Φ yy αφ = 0 (x, y in Ω, ( Φ = 0 (x, y on Γ. We look for solutions of (*-(**-(*** tht re 0. Here, (* nd (** re deduced from (PDE, nd (*** follows from (BC. Eqution ( is esy to solve. The min issue now is to solve the eigenvlue prolem ( -(. (EIGENVALUES AND EIGENFUNCTIONS Now focus on the oundry vlue prolem (**-(*** for Φ(x, y. The solution structure of this prolem depends on the prmeter vlue α. It cn e shown tht for most choices of α, (**-(*** only hs the trivil solution Φ(x, y 0. The specil vlues of α dmiting nontrivil solutions re clled eigenvlues of (**-(***, nd in tht cse, the corresponding nontrivil solutions Φ(x, y 0 re clled eigenfunctions. Unfortuntely, for generl domin Ω, it s impossile to write down the eigenvlues nd eigenfunctions of ( -( in explicit form. We do, however, hve mthemticl theorem out this eigenvlue prolem: THEOREM. (i The eigenvlues of ( -( form sequence of negtive numers {α n } such tht 0 > α 1 > α 2 α 3 α 4, lim n α n =. (ii The corresponding eigenfunctions Φ n (x, y re smooth functions in Ω. (iii Φ 1 > 0 does not chnge sign in Ω. (iv Ω Φ m(x, yφ n (x, ydxdy = 0 for m n. Let α = α n e one of the eigenvlues. We now solve eqution (*, which ecomes T (t kα n T(t = 0 t > 0. 1

2 The solutions re T(t = Constnt e kαnt. (CLASSIFICATION OF PRODUCT SOLUTIONS Thus, product solution of (PDE-(BC must e constnt multiple of u n (x, t = e kαnt Φ n (x, y, n = 1, 2, 3,. (FROM PRODUCT SOLUTIONS TO GENERAL SOLUTIONS The generl solutions of (PDE-(BC re liner comintions of the product solutions: u(x, t = n e kαnt Φ n (x, y, n re constnts. (SOLUTION FORMULA FOR THE INIT-BDRY VAL PROBLEM To otin the solution of the initil-oundry vlue prolem (PDE-(BC-(IC, we now need to use (IC to determine the vlues of n. Let t = 0: (FOURIER f(x, y = n Φ n (x, y. The coefficients n cn e expressed in terms of f(x, y. The derivtion is similr to the 1-D cse. Detils follow: Multiply oth sides of (FOURIER y Φ m (x, y nd integrte over Ω: Ω f(x, yφ m (x, ydxdy = n = m Ω Ω Φ n (x, yφ m (x, ydxdy Φ m (x, y 2 dxdy, y Property (iv of the ove theorem. This finlly gives the solution of the initil-oundry vlue prolem (PDE-(BC-(IC: u(x, t = n e kαnt Φ n (x, y, n = f(x, yφ Ω n(x, ydxdy Φ Ω n(x, y 2 dxdy n = 1, 2, 3,. 2

3 2. The Cse of Rectngulr Ω Ojective: Solve the initil-oundry vlue prolem (PDE-(BC-(IC, for the cse Ω is the rectngle {0 < x <, 0 < y < }: (PDE u t k(u xx + u yy = 0 0 < x <, 0 < y <, t > 0, { u(x, 0, t = 0, u(x,, t = 0 0 < x <, t > 0, (BC u(0, y, t = 0, u(, y, t = 0 0 < y <, t > 0, (IC u(x, y, 0 = f(x, y 0 < x <, 0 < y <. THE EIGENVALUE PROBLEM ( -( cn e solved y further seprtion of vriles: Thnks to the specil symmetry of the rectngulr domin, we cn otin explicit expressions for the eigenvlues nd eigenfunctions. As mtter of fct, the eigenfunctions of the product form: Φ(x, y = X(xY (y will produce ll eigenvlues. Although, in generl there might e eigenfunctions of non-product forms, those non-product eigenfunctions re liner comiintions of product eigenfunctions. In this sense, we only need to solve ( -( for product functions Φ(x, y = X(xY (y. This will seprte the 2-D eigenvlue prolem ( -( to two 1-D eigenvlue prolems which we know how to solve. An ALTERNATIVE APPROACH is to strt from the eginnning with product solutions of the form: u(x, y, t = T(tX(xY (y 0. Plugging u = T(tX(xY (y in (PDE-(BC we see: there re constnts λ nd µ such tht ( T (t + k(λ + µt(t = 0 t > 0; ( x X (x + λx(x = 0 (0 < x <, X(0 = X( = 0; ( y Y (y + µy (y = 0 (0 < y <, Y (0 = Y ( = 0. We know how to solve 1-D eigenvlue prolem ( x : If nd only if λ is one of the following vlues λ m = (mπ/ 2, m = 1, 2, prolem ( x hs nontrivil solutions which re constnt multiples of X m (x = sin(mπx/. Another 1-D eigenvlue prolem ( y is solved similrly: If nd only if µ is one of the following vlues µ n = (nπ/ 2, n = 1, 2, prolem ( y hs nontrivil solutions which re constnt multiples of Y n (y = sin(nπy/. For λ = λ m nd µ = µ n, we now solve eqution (*, which ecomes T (t + k(λ m + µ n T(t = 0 t > 0. 3

4 The solutions re T(t = Constnt e k(λm+µnt = Constnt e k[(mπ/2 +(nπ/ 2 ]t. (CLASSIFICATION OF PRODUCT SOLUTIONS Thus, (triple-fctor product solution of (PDE-(BC must e constnt multiple of u m,n (x, y, t = sin(mπx/ sin(nπy/e k[(mπ/2 +(nπ/ 2 ]t, m, n = 1, 2, 3,. (SOLUTION FORMULA FOR THE INIT-BDRY VAL PROBLEM The solution of the initil-oundry vlue prolem (PDE-(BC-(IC is given y: u(x, t = m,n = m=1 = 4 m,n sin(mπx/ sin(nπy/e k[(mπ/2 +(nπ/ 2 ]t, f(x, y sin(mπx/ sin(nπy/dxdy {sin(mπx/ sin(nπy/}2 dxdy f(x, y sin sin dxdy. 4

5 EXERCISES [1] Consider the 2-D het eqution in rectngle, with top nd ottom sides insulted, nd left nd right oundry temperture fixed t 0: (1 u t u xx u yy = 0 0 < x <, 0 < y <, t > 0, (2i u y (x, 0, t = 0, u y (x,, t = 0 0 x, 0 y, t > 0, (2ii u(0, y, t = 0, u(, y, t = 0 0 y, t > 0, (3 u(x, y, 0 = f(x, y 0 < x <, 0 < y <. ( Find ll nontrivil solutions u(x, y, t to (1-(2 of the form u(x, y, t = X(xY (yt(t. ( Find the Fourier series solution formul for (1-(2-(3 with the generl initil dt f(x, y. (c Find the Fourier series formul in the cse f(x, y = xy. [2] Consider the 3-D het eqution in 3-D ox with insulted oundry conditions: (4 u t u xx u yy u zz = 0 0 < x <, 0 < y <, 0 < z < c, t > 0, (5 u x (0, y, z, t = u x (, y, z, t = u y (x, 0, z, t = u y (x,, z, t = u z (x, y, 0, t = u z (x, y, c, t = 0 0 < x <, 0 < y <, 0 < z < c, t > 0, (6 u(x, y, z, 0 = f(x, y, z 0 < x <, 0 < y <, 0 < z < c. ( Find ll nontrivil solutions u(x, y, z, t to (4-(5 of the form u(x, y, z, t = X(xY (yz(zt(t. ( Find the Fourier series solution formul for (4-(5-(6 with the generl initil dt f(x, y, z. ANSWERS: [1] ( u(x, y, t = C sin e [(mπ/2 +(nπ/ 2 ]t, m is ny positive integer, n is ny nonnegtive integer, nd C is ny nonzero constnt. 5

6 ( u(x, y, t = m,0 = 2 m,n = 4 (c u(x, y, t = + m=1 m=1 n=0 m=1 m,n sin f(x, y sin f(x, y sin e [(mπ/2 +(nπ/ 2 ]t, dxdy (m 1 ( 1 m+1 sin e (mπ/2 t mπ 4( 1 m {1 ( 1 n } mn 2 π 3 sin dxdy (m 1, n 1 e [(mπ/2 +(nπ/ 2 ]t ( kπz [2] ( u(x, y, z, t = C c m, n, k re ny nonnegtive integers nd C is ny nonzero constnt. ( kπz ( u(x, y, z, t = m,n,k c 0,0,0 = 1 m,0,0 = 2 0,n,0 = 2 0,0,k = 2 m,n,0 = 4 0,n,k = 4 m,0,k = 4 m,n,k = 8 m=0 n=0 k=0 z=0 f(x, y, zdxdydz f(x, y, z f(x, y, z ( kπz f(x, y, z c f(x, y, z ( kπz f(x, y, z c ( kπz f(x, y, z c f(x, y, z (m 1, n 1, k 1 e [(mπ/2 +(nπ/ 2 +(kπ/c 2 ]t, dxdydz (m 1 dxdydz (n 1 dxdydz (k 1 e [(mπ/2 +(nπ/ 2 +(kπ/c 2 ]t, dxdydz (m 1, n 1 dxdydz (n 1, k 1 dxdydz (m 1, k 1 ( kπz c dxdydz 6

### u(x, y, t) = T(t)Φ(x, y) 0. (THE EQUATIONS FOR PRODUCT SOLUTIONS) Plugging u = T(t)Φ(x, y) in (PDE)-(BC) we see: There is a constant λ such that

Seprtion of Vriles for Higher Dimensionl Wve Eqution 1. Virting Memrne: 2-D Wve Eqution nd Eigenfunctions of the Lplcin Ojective: Let Ω e plnr region with oundry curve Γ. Consider the wve eqution in Ω

### The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

### Wave Equation on a Two Dimensional Rectangle

Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c

### Differential Equations 2 Homework 5 Solutions to the Assigned Exercises

Differentil Equtions Homework Solutions to the Assigned Exercises, # 3 Consider the dmped string prolem u tt + 3u t = u xx, < x , u, t = u, t =, t >, ux, = fx, u t x, = gx. In the exm you were supposed

### 10 Elliptic equations

1 Elliptic equtions Sections 7.1, 7.2, 7.3, 7.7.1, An Introduction to Prtil Differentil Equtions, Pinchover nd Ruinstein We consider the two-dimensionl Lplce eqution on the domin D, More generl eqution

### Module 9: The Method of Green s Functions

Module 9: The Method of Green s Functions The method of Green s functions is n importnt technique for solving oundry vlue nd, initil nd oundry vlue prolems for prtil differentil equtions. In this module,

### MA 201: Partial Differential Equations Lecture - 12

Two dimensionl Lplce Eqution MA 201: Prtil Differentil Equtions Lecture - 12 The Lplce Eqution (the cnonicl elliptic eqution) Two dimensionl Lplce Eqution Two dimensionl Lplce Eqution 2 u = u xx + u yy

### Summary: Method of Separation of Variables

Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section

### M344 - ADVANCED ENGINEERING MATHEMATICS

M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

### Section 3.2 Maximum Principle and Uniqueness

Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point

### Lecture 24: Laplace s Equation

Introductory lecture notes on Prtil Differentil Equtions - c Anthony Peirce. Not to e copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 24: Lplce s Eqution

### 1 2-D Second Order Equations: Separation of Variables

Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +

### Consequently, the temperature must be the same at each point in the cross section at x. Let:

HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the

### AMS 212A Applied Mathematical Methods I Lecture 06 Copyright by Hongyun Wang, UCSC. ( ), v (that is, 1 ( ) L i

AMS A Applied Mthemticl Methods I Lecture 6 Copyright y Hongyun Wng, UCSC Recp of Lecture 5 Clssifiction of oundry conditions Dirichlet eumnn Mixed Adjoint opertor, self-djoint opertor Sturm-Liouville

### 1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

### 1 1D heat and wave equations on a finite interval

1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion

### 1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

### 1 E3102: a study guide and review, Version 1.0

1 E3102: study guide nd review, Version 1.0 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in very

### AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h

AM Mthemticl Anlysis Oct. Feb. Dte: October Exercises Lecture Exercise.. If h, prove the following identities hold for ll x: sin(x + h) sin x h cos(x + h) cos x h = sin γ γ = sin γ γ cos(x + γ) (.) sin(x

### Mathematics. Area under Curve.

Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

### (9) P (x)u + Q(x)u + R(x)u =0

STURM-LIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

### k and v = v 1 j + u 3 i + v 2

ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended

### DEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.

398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts

### Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

### APM346H1 Differential Equations. = u x, u = u. y, and u x, y =?. = 2 u t and u xx= 2 u. x,t, where u t. x, y, z,t u zz. x, y, z,t u yy.

INTRODUCTION Types of Prtil Differentil Equtions Trnsport eqution: u x x, yu y x, y=, where u x = u x, u = u y, nd u x, y=?. y Shockwve eqution: u x x, yu x, yu y x, y=. The virting string eqution: u tt

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### dy ky, dt where proportionality constant k may be positive or negative

Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### Partial Differential Equations

Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

### Green function and Eigenfunctions

Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions

### Chapter 8.2: The Integral

Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

### u t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx

Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know

### 3 Mathematics of the Poisson Equation

3 Mthemtics of the Poisson Eqution 3. Green functions nd the Poisson eqution () The Dirichlet Green function stisfies the Poisson eqution with delt-function chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd

### Abstract inner product spaces

WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

### Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### REVIEW Chapter 1 The Real Number System

Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

### Torsion in Groups of Integral Triangles

Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### Math Fall 2006 Sample problems for the final exam: Solutions

Mth 42-5 Fll 26 Smple problems for the finl exm: Solutions Any problem my be ltered or replced by different one! Some possibly useful informtion Prsevl s equlity for the complex form of the Fourier series

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### Variational Techniques for Sturm-Liouville Eigenvalue Problems

Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment

### Suggested Solution to Assignment 5

MATH 4 (5-6) prti diferenti equtions Suggested Soution to Assignment 5 Exercise 5.. () (b) A m = A m = = ( )m+ mπ x sin mπx dx = x mπ cos mπx + + 4( )m 4 m π. 4x cos mπx dx mπ x cos mπxdx = x mπ sin mπx

### Bridging the gap: GCSE AS Level

Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

### Thomas Whitham Sixth Form

Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

### Math 5440 Problem Set 3 Solutions

Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 213 1: (Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping

### Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

### STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES

STURM-LIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 1. Regulr Sturm-Liouville Problem The method of seprtion of vribles to solve boundry vlue problems leds to ordinry differentil equtions on intervls

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### Heat flux and total heat

Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### Math 124A October 04, 2011

Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model

### Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

### Review of Gaussian Quadrature method

Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### 2. THE HEAT EQUATION (Joseph FOURIER ( ) in 1807; Théorie analytique de la chaleur, 1822).

mpc2w4.tex Week 4. 2.11.2011 2. THE HEAT EQUATION (Joseph FOURIER (1768-1830) in 1807; Théorie nlytique de l chleur, 1822). One dimension. Consider uniform br (of some mteril, sy metl, tht conducts het),

### Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

### Part II. Analysis of PDE

Prt II. Anlysis of PDE Lecture notes for MA342H P. Krgeorgis pete@mths.tcd.ie 1/38 Second-order liner equtions Consider the liner opertor L(u) which is defined by L(u) = 1 u xx + 2 u yy +b 1 u x +b 2 u

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### 13: Diffusion in 2 Energy Groups

3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Bases for Vector Spaces

Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

### Chapter 6 Techniques of Integration

MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

### Review SOLUTIONS: Exam 2

Review SOUTIONS: Exm. True or Flse? (And give short nswer ( If f(x is piecewise smooth on [, ], we cn find series representtion using either sine or cosine series. SOUTION: TRUE. If we use sine series,

### Waveguide Guide: A and V. Ross L. Spencer

Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

### The usual algebraic operations +,, (or ), on real numbers can then be extended to operations on complex numbers in a natural way: ( 2) i = 1

Mth50 Introduction to Differentil Equtions Brief Review of Complex Numbers Complex Numbers No rel number stisfies the eqution x =, since the squre of ny rel number hs to be non-negtive. By introducing

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Brief Notes For Math 3710

Brief Notes For Mth 371 Afshin Ghoreishi Fll 13 Contents Prefce iii -1 Hndouts 1 -.9 Hndouts.............................................. 1 Introduction 15 1 Fourier Series 18 1.1 Periodic Functions nd

### Math 5440 Problem Set 3 Solutions

Mth 544 Mth 544 Problem Set 3 Solutions Aron Fogelson Fll, 25 1: Logn, 1.5 # 2) Repet the derivtion for the eqution of motion of vibrting string when, in ddition, the verticl motion is retrded by dmping

### 5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship

5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is

### Recitation 3: More Applications of the Derivative

Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### Problem set 1: Solutions Math 207B, Winter 2016

Problem set 1: Solutions Mth 27B, Winter 216 1. Define f : R 2 R by f(,) = nd f(x,y) = xy3 x 2 +y 6 if (x,y) (,). ()Show tht thedirectionl derivtives of f t (,)exist inevery direction. Wht is its Gâteux

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### Sturm-Liouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1

Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

### Improper Integrals, and Differential Equations

Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

### ( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

### f = ae b e , i.e., ru + P = (r + P )(u + P ) = (s + P )(t + P ) = st + P. Then since ru st P and su P we conclude that r s t u = ru st

Mth 662 Spring 204 Homewor 2 Drew Armstrong Problem 0. (Drwing Pictures) The eqution y 2 = x 3 x defines curve in the complex plne C 2. Wht does it loo lie? Unfortuntely we cn only see rel things, so we

### Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

### Math 154B Elementary Algebra-2 nd Half Spring 2015

Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know

### Elliptic Equations. Laplace equation on bounded domains Circular Domains

Elliptic Equtions Lplce eqution on bounded domins Sections 7.7.2, 7.7.3, An Introduction to Prtil Differentil Equtions, Pinchover nd Rubinstein 1.2 Circulr Domins We study the two-dimensionl Lplce eqution

### Conservation Law. Chapter Goal. 5.2 Theory

Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

### 1 E3102: A study guide and review, Version 1.2

1 E3102: A study guide nd review, Version 1.2 Here is list of subjects tht I think we ve covered in clss (your milege my vry). If you understnd nd cn do the bsic problems in this guide you should be in

### 10 Vector Integral Calculus

Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve

### Calculus II: Integrations and Series

Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### 7.2 The Definite Integral

7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where