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

Save this PDF as:
Size: px
Start display at page:

## Transcription

1 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 good shpe. If I get my ct together I ll try to point out representtive problems from the homework for ech section. This guide is probbly over-thorough. The test itself will hve bout 6 questions covering the whole course but emphsizing the bsic concepts. I ll try to void nything tricky or 5pt lnd mines. 1.1 Homogeneous Liner PDE s in 2 vribles Seprtion of vribles out the wzoo Be ble to solve the following seprble problems with homogeneous boundry conditions nd no forcing terms. When in doubt use full-blown seprtion of vribles. Alterntively, if you know the pproprite eigenfunctions you cn solve these by eigenfunction expnsion (the first three problems ll hve eigenfunctions tht re some combintions of sines nd cosines determined by the boundry conditions) 1-D time dependent het flow eqution 1-D vibrting string u 2 t = κ u x 2 2 u t 2 = c2 2 u x 2 Lplce s Eqution 2 u =0in crtesin coordintes (rectngles) nd polr coordintes (disks) 2 u = 2 u x u y 2 =0 2 u = 1 r r r u r + 1 u r 2 θ =0 Note: For the disk, the eigenfunctions re in θ. The seprted equtions for f(r) is n equidimensionl eqution with tril solutions of form f(r) =r p 1

2 Helmholtz Eqution 2 u + λu = 0 in crtesin nd polr coordintes. Note: Helmholtz eqution will give Eigenfunctions in both directions for λ>0. In crtesin coordintes you ll get combintions of sines nd cosines in x nd y. In polr coordintes you ll get sines nd cosines in θ nd the dreded Bessel functions in r. Ifλ<0 (modified helmholtz eqution) you ll only get eigenfunctions in one direction. 1.2 Sturm-Liouville Boundry Vlue problems Regulr Sturm-Liouville Boundry Vlue problems re of the form d df p(x) + q(x)f + λσ(x)f =0 dx dx with generl homogeneous Boundry conditions t x = nd x = b df β 1 f()+β 2 () =0 dx df β 3 f(b)+β 4 (b) =0 dx nd p>0, σ>0 for x b. (This eigenvlue problem cn lso be written s L(f) = λσf.) Know: the bsic properties of these problems (pge 157 in Hbermn). 1. They hve n infinite number of Rel eigenvlues λ 1 <λ 2 <... < λ n n 2. For ech eigenvlue λ n there is corresponding unique eigenfunction φ n (x) (note: uniqueness is only for 1-D problems without periodic BC s). 3. The eigenfunctions re orthogonl under weight σ,i.e. φ n φ m σdx =0 form n 4. The eigenfunctions re complete in the sense tht ny piecewise smooth function g(x, t) cn be written in terms of n infinite series of the eigenfunctions. i.e. g(x, t) = n (t)φ n (x) 2 n=1

3 where the coefficients n (t) re defined by the integrls n (t) = g(x, t)φ n(x)σ(x)dx φ2 n(x)σ(x)dx Also know Green s formul for SL problems (nd where it cn be useful) ( ulv vlu = p(x) u dv dx v du ) b dx for ny functions u(x) nd v(x). Ryleigh Quotient nd how to use it to estimte eigenvlues (or show if positive) λ = φlφdx b φ2 n(x)σ(x)dx 1.3 Fourier Series/Generlized Fourier Series Understnd how Fourier series re specil cse of Sturm-Liouville theory Be ble to sketch full Fourier Series, Fourier Sine Series nd Fourier Cosine series Relize tht fourier-bessel series work the sme wy. I.e. for disk 0 < r<icn expnd ny function g(r) (bounded t r =0) in terms of bessel functions,e.g. g(r) = n J m (z mn r/) n=1 where 0 n = g(r)j m(z mn r/)rdr J 0 m(z 2 mn r/)rdr i.e. φ n (r) =J m (z m nr/) nd σ = r. Understnd when, nd when not, to differentite these infinite series termby-term. (i.e. it is oky for continuous functions g with the sme boundry conditions s the eigenfunctions). 3

4 1.4 Homogeneous Liner PDE s in 3 or more vribles These problems re just more complicted versions of the first set of problems. In generl they cn be solved by either seprtion of vribles or eigenfunction expnsion. For eigenfunction expnsion, however, it is most useful to use the 2-D eigenfunctions of Helmholtz eqution. Bsic problems re Time dependent het-flow on rectngle u(t, x, y) or disk u(t, r, θ) u t = κ 2 u Time dependent vibrtions of 2-d membrne 2 u t = 2 c2 2 u 3-D Lplce eqution 2 u =0on rectngulr solid or cylinder For the first two problems you cn lwys seprte out the time dependent prts using u(t, x) =h(t)w(x) where x =(x, y) for crtesin problems nd x =(r, θ) for polr problems. In both cses, w will stisfy Helmholtz Eqution 2 w + λw =0 Properties of Helmholtz eqution For λ > 0 nd w hving homogeneous boundry conditions on some domin R (e.g. rectngle or disk), then mny of the properties of the 1-D Sturm-Liouville theory re relevnt to the 2-D (or 3- D) Eigenfunction problem defined by Helmholtz Eqution (See Sections , pges ). Importnt exmples re There re n infinite number of rel eigenvlues λ 1 < λ 2 <... < λ n n For ech eigenvlue there my be multiple orthogonl eigenfunctions (this is different from the 1-D cse). Eigenfunctions with different eigenvlues re orthogonl with regrd to the re integrl over the domin R φ i φ j dxdy =0 fori j R this cn lso be mde generlly true for ny two eigenfunctions with the sme eigenvlue. (see below) 4

5 The eigenfunctions re complete in the sense tht ny piecewise-smooth 2-D function cn be written s n infinite sum of ppropritely weighted eigenfunctions g(x, y) = i φ i (x, y) i where i = g(x, y)φ R i(x, y)da (x, y)da R φ2 i Some exmple solutions of Helmholtz Eq. rectngulr region 0 x L, 0 y H with w =0on the boundry λ mn = ( nπ L ) 2 ( mπ ) 2 + φ mn (x, y) =sin nπx H L sin mπy H Note 1: if L=H (squre region), then φ mn nd φ nm re orthogonl but hve the sme eigenvlue λ mn = λ nm Note 2: if the boundries in x re homogeneous but insted were w/ x(0) = w/ x(l) =0, the eigenvlues would be the sme but the eigenfunctions would be φ mn =cos nπx mπy sin L H circulr disk 0 r, π θ π with w(, θ) =0on the boundry (nd w(0,θ) is bounded). ( zmn ) 2 λ mn = with two orthogonl eigenfunctions for ech λ mn. ( φ 1 r ) mn(r, θ) =J m z mn cos mθ φ 2mn(r, ( r ) θ) =J m z mn sin mθ where J m (r) is the Bessel function of the first kind of order m nd z mn is the nth zero of the mth Bessel function. 1.5 Non-Homogeneous PDE s nd method of eigenfunction expnsion Here we extended the homogeneous problems to problems with both non-homogeneous source terms nd non-homogeneous boundry conditions. For the ltter problems 5

6 however it ws lwys possible to set u(t, x) =v(t, x) +r(t, x) where r is ny function tht stisfies the non-homogeneous boundry conditions. Substituting this into the originl PDE, will produce new eqution for v where v hs homogeneous boundry conditions. Given these reduced problems, there is generl recipe for solving the non-homogeneous source terms using the method of eigenfunction expnsion which I will illustrte with the simplified time-dependent problem v = Lv + Q(t, x) t with v(x, 0) = f(x) nd v hs homogeneous boundry conditions nd L is 2nd order differentil opertor tht only includes sptil derivtives (e.g. Lv = k 2 v/ x 2 in 1-D or Lv = k 2 v in 2-D or 3-D.) 1. Use seprtion of vribles on the ssocited homogeneous problem (ssume Q = 0) to find the eigenvlues nd eigenfunctions of the sptil boundry vlue problem Lφ n = λ n φ n nd φ n hs the sme homogeneous boundry conditions s v. 2. Expnd both the solution nd Q in terms of these eigenfunctions, e.g. v(x,t)= n n (t)φ n (x) Q(x,t)= n q n (t)φ n (x) 3. Substitute these sums into the PDE for v (nd you cn tke ll the derivtives term by term becuse v nd φ n ll hve the sme boundry conditions) to get [ ] dn dt + λ n n (t) q n (t) φ n (x) =0 where we hve used the reltionship i Lv = i n (t)lφ n (x) = i n (t)λ n φ n (x) = using the definition of the eigenfunctions of L 6

7 4. Use orthogonlity of the φ n s to get the set of 1st-order Non-homogeneous Ordinry differentil equtions d n dt + λ n n = q n (t) 5. solve this using vrition of prmeters (nd initil conditions) to find n (t) (nd therefore v). Here you will need to use the initil conditions R n (0) = f(x)φ nσda R φ2 nσda 6. reconstruct the full solution u(x, t) =v(x, t)+r(x, t)...the end simple problems with equilibrium solutions In ddition to the full eigenfunction expnsion technique, sometimes it is esier to solve problems with stedy forcing terms Q(x) by looking for stedy stte solution u e (x) tht stisfies Lu e = Q(x) nd non-homogeneous boundry conditions, then look for trnsient solution v(x, t) of the remining homogeneous problem with homogeneous BC s. nd reconstruct the full solution u(x, t) =v(x, t) +u e (x). For exmple you cn use this to solve the het flow eqution with stedy forcing nd fixed temperture boundry conditions. 1.6 Green s Functions Given boundry vlue problem of form Lu = f(x) with homogeneous boundry conditions, find the Green s Functions with the sme boundry conditions defined by LG(x, x 0 )=δ(x x 0 ) where δ(x x 0 ) is Dirc delt function t point x 0.GivenG(x, x 0 ) the generl solution for u is u(x, t) = f(x 0 )G(x, x 0 )dx 0 Bsic problems R 7

8 1. Importnt: know how to find the 1-D green s functions for Lu = d 2 u/dx 2 nd pproprite boundry conditions. 2. you might lso wnt to know how to find the infinite spce green s functions for 2 u = f(x) in 2 nd 3-D. 1.7 Wve Equtions nd the method of chrcteristics Understnd How to find solutions of simple 1-D wve eqution w t + c w x =0 with initil conditions w(x, 0) = f(x) using the method of chrcteristics. Know how to extend it to more generl liner problems like nd to non-liner shock problems like with w(x, 0) = f(x) w t + c(x) w x = w w t + w w x =0 for ech of these problems know how to qulittively sketch wht is going on in spce nd time. A grphicl nswer will go long wy. 1.8 P.S. Tht s it for now...wtch this spce for nything new nd/or corrections. if you hve ny questions come nd see me in office hours or send me e-mil t to set up n ppointment. Good luck nd relx. 8

### 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

### 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

### 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

### 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

### 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 +

### 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

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### 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,

### 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

### 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

### 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

### 4 Sturm-Liouville Boundary Value Problems

4 Sturm-Liouville Boundry Vlue Problems We hve seen tht trigonometric functions nd specil functions re the solutions of differentil equtions. These solutions give orthogonl sets of functions which cn be

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Linearity, linear operators, and self adjoint eigenvalue problems

Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry

### 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 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

### Applied Partial Differential Equations with Fourier Series and Boundary Value Problems 5th Edition Richard Haberman

Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn Person Eduction Limited Edinburgh Gte Hrlow Essex CM20 2JE Englnd nd Associted Compnies throughout

### PHYSICS 116C Homework 4 Solutions

PHYSICS 116C Homework 4 Solutions 1. ( Simple hrmonic oscilltor. Clerly the eqution is of the Sturm-Liouville (SL form with λ = n 2, A(x = 1, B(x =, w(x = 1. Legendre s eqution. Clerly the eqution is of

### Chapter 28. Fourier Series An Eigenvalue Problem.

Chpter 28 Fourier Series Every time I close my eyes The noise inside me mplifies I cn t escpe I relive every moment of the dy Every misstep I hve mde Finds wy it cn invde My every thought And this is why

### 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

### c n φ n (x), 0 < x < L, (1) n=1

SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry

### Math Theory of Partial Differential Equations Lecture 2-9: Sturm-Liouville eigenvalue problems (continued).

Mth 412-501 Theory of Prtil Differentil Equtions Lecture 2-9: Sturm-Liouville eigenvlue problems (continued). Regulr Sturm-Liouville eigenvlue problem: d ( p dφ ) + qφ + λσφ = 0 ( < x < b), dx dx β 1 φ()

### Pressure Wave Analysis of a Cylindrical Drum

Pressure Wve Anlysis of Cylindricl Drum Chris Clrk, Brin Anderson, Brin Thoms, nd Josh Symonds Deprtment of Mthemtics The University of Rochester, Rochester, NY 4627 (Dted: December, 24 In this pper, hypotheticl

### 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

### 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.....................................

### 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

### 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,

### 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

### STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

### 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

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### 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

### 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

### 18 Sturm-Liouville Eigenvalue Problems

18 Sturm-Liouville Eigenvlue Problems Up until now ll our eigenvlue problems hve been of the form d 2 φ + λφ = 0, 0 < x < l (1) dx2 plus mix of boundry conditions, generlly being Dirichlet or Neumnn type.

### New Expansion and Infinite Series

Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

### Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

### 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

### Separation of Variables in Linear PDE

Seprtion of Vribles in Liner PDE Now we pply the theory of Hilbert spces to liner differentil equtions with prtil derivtives (PDE). We strt with prticulr exmple, the one-dimensionl (1D) wve eqution 2 u

### 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

### 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

### P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

### 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

### Orthogonal Polynomials

Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils

### ENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01

ENGI 940 ecture Notes 7 - Fourier Series Pge 7.0 7. Fourier Series nd Fourier Trnsforms Fourier series hve multiple purposes, including the provision of series solutions to some liner prtil differentil

### Math 100 Review Sheet

Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

### 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

### (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.

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

### 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.

### 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

### Sturm Liouville Problems

Sturm Liouville Problems More generl eigenvlue problems So fr ll of our exmple PDEs hve led to seprted equtions of the form X + ω 2 X =, with stndrd Dirichlet or Neumnn boundry conditions Not surprisingly,

### 1. On the line, i.e., on R, i.e., 0 x L, in general, a x b. Here, Laplace s equation assumes the simple form. dx2 u(x) = C 1 x + C 2.

Lecture 16 Lplce s eqution - finl comments To summrize, we hve investigted Lplce s eqution, 2 = 0, for few simple cses, nmely, 1. On the line, i.e., on R, i.e., 0 x L, in generl, x b. Here, Lplce s eqution

### 12.8 Modeling: Membrane,

SEC. 1.8 Modeling: Membrne, Two-Dimensionl Wve Eqution 575 1.8 Modeling: Membrne, Two-Dimensionl Wve Eqution Since the modeling here will be similr to tht of Sec. 1., you my wnt to tke nother look t Sec.

### Waveguides Free Space. Modal Excitation. Daniel S. Weile. Department of Electrical and Computer Engineering University of Delaware

Modl Excittion Dniel S. Weile Deprtment of Electricl nd Computer Engineering University of Delwre ELEG 648 Modl Excittion in Crtesin Coordintes Outline 1 Aperture Excittion Current Excittion Outline 1

### . Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

### (4.1) D r v(t) ω(t, v(t))

1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

### 7 Green s Functions and Nonhomogeneous Problems

7 Green s Functions nd Nonhomogeneous Problems The young theoreticl physicists of genertion or two erlier subscribed to the belief tht: If you hven t done something importnt by ge 3, you never will. Obviously,

### 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

### 21.6 Green Functions for First Order Equations

21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to

### Massachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6

Msschusetts Institute of Technology Quntum Mechnics I (8.) Spring 5 Solutions to Problem Set 6 By Kit Mtn. Prctice with delt functions ( points) The Dirc delt function my be defined s such tht () (b) 3

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### MATH Greens functions, integral equations and applications

MATH 3432 Greens functions, integrl equtions nd pplictions Willim J. Prnell Spring 213 1 Contents Pge 1 Introduction nd motivtion 6 2 Green s functions in 1D 11 2.1 Ordinry Differentil Equtions: review....................

### 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

### Section 7.1 Integration by Substitution

Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### 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

### 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

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### 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

### Unit #9 : Definite Integral Properties; Fundamental Theorem of Calculus

Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

### MA 124 January 18, Derivatives are. Integrals are.

MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,

### 13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes

Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the

### ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus

ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems

### 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

### 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

### Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1

Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

### 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

### Notes on the Eigenfunction Method for solving differential equations

Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: Wereconsieringtheinfinite-imensionlHilbertspceL 2 ([, b] of ll squre-integrble functions over the intervl [, b] (ie, b f(x 2

### SECTION PROBLEMS IN POLAR, CYLINDRICAL AND SPHERICAL COORDINATES

CHAPTER 9 SECTION 9. 353 PROBLEMS IN POLAR, CYLINDRICAL AND SPHERICAL COORDINATES 9. Homogeneous Problems in Polr, Cylindricl, nd Sphericl Coordintes In Section 6.3, seprtion of vribles ws used to solve

### df dx There is an infinite number of different paths from

Integrl clculus line integrls Feb 7, 18 From clculus, in the cse of single vrible x1 F F x F x f x dx, where f x 1 x df dx Now, consider the cse tht two vribles re t ply. Suppose,, df M x y dx N x y dy

### 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

### AMATH 731: Applied Functional Analysis Fall Some basics of integral equations

AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### LECTURE 1. Introduction. 1. Rough Classiæcation of Partial Diæerential Equations

LECTURE 1 Introduction 1. Rough Clssiction of Prtil Dierentil Equtions A prtil dierentil eqution is eqution relting function of n vribles x 1 ;::: ;x n, its prtil derivtives, nd the coordintes x =èx 1

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section Outline nd Introduction ecturer: Dr. Willim J. Prnell (room 2.238; Willim.Prnell@mnchester.c.uk Exercise sheets, solutions,

### 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

### 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,

### STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

### g i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f

1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

### . Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

### Chapter 0. What is the Lebesgue integral about?

Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

### Quantum Mechanics Qualifying Exam - August 2016 Notes and Instructions

Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis

### Calculus of Variations: The Direct Approach

Clculus of Vritions: The Direct Approch Lecture by Andrejs Treibergs, Notes by Bryn Wilson June 7, 2010 The originl lecture slides re vilble online t: http://www.mth.uth.edu/~treiberg/directmethodslides.pdf

### Orthogonal functions

Orthogonl functions Given rel vrible over the intervl (, b nd set of rel or complex functions U n (ξ, n =, 2,..., which re squre integrble nd orthonorml b U n(ξu m (ξdξ = δ n,m ( if the set of of functions

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second