Partial Differential Equations

Size: px
Start display at page:

Download "Partial Differential Equations"

Transcription

1 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 vlue problem in much the sme wy tht n inverse mtri provides generl solution for systems of liner equtions. In this section the reen s function is introduced in the contet of simple one-dimensionl problem. Some of the proofs use the identity b (uv vu ) d b uv vu. This cn be obtined by integrting (uv vu ) uv vu. A singulrity function K(, ξ) of the opertor L defined by Lu() u () c()u() is chrcterised by three properties:. K is continuous; 2. K is continuous in < ξ nd in > ξ, nd K ( +, ) K (, ) ; 3. K is continuous nd LK for ξ. Note tht the three properties do not define singulrity function uniquely: if K is singulrity function then so is K + H, where H(, ξ) is ny function with continuous H nd H nd with LH. The reen s function (, ξ) for the opertor L nd the domin (, b) with Dirichlet boundry conditions is the singulrity function tht stisfies the homogeneous Dirichlet conditions (, ξ) nd (b, ξ). The reen s function provides the solution to the boundry vlue problem with Dirichlet boundry conditions: Theorem If u stisfies the differentil eqution u + cu f on (, b) nd the boundry conditions u(), u(b), then for ll ξ (, b). u(ξ) b (, ξ)f() d () Proof. Letting v() (, ξ), we hve b vf d ξ b uv vu d + uv vu d ξ + ξ uv vu + b ξ + uv vu b uv vu u(ξ) ξ+ ξ v + ξ+ ξ vu, which completes the proof.

2 The lod (or source) function f in the differentil eqution u cu f cn be thought of s superposition of point lods f() δ( ξ)f(ξ)dξ, where δ( ξ) is concentrted t ξ nd hs unit mgnitude (i.e. δ d ). Then formul () represents the solution s weighted sum of reen s functions, where ech (ξ, ) is the solution to u () δ( ξ). The reen s function cn thus be thought of s the response to unit point lod. The reen s function lso provides the solution of the boundry vlue problem with nonhomogeneous boundry conditions: Theorem 2 The solution of u +cu with boundry conditions u() h nd u(b) h stisfies u(ξ) (, ξ)h (b, ξ)h (2) for ll ξ (, b). Proof. Denoting v() (, ξ), we hve ξ which completes the proof. b (uv vu ) d + (uv vu ) d ξ + ξ (uv vu ) + b ξ + (uv vu ) b (uv vu ) ξ+ ξ (uv vu ) b uv + u(ξ), The following result tells us tht the response t to point lod pplied t ξ is equl to the response t ξ to point lod pplied t. Theorem 3 (Reciprocity Principle) (, ξ) (ξ, ) for ll, ξ (, b). Proof. Let y nd η be distinct points in (, b) with y < η, let u() (, η) nd v() (, y). Then y y b uv vu η b uv vu d + uv vu d + uv vu d y + η + uv vu + η y + uv vu + b η + uv vu y+ y uv vu η+ η uv vu, u(y) v(η) leving u(y) v(η), tht is, (y, η) (η, y). The proof for y > η is similr. As consequence of Theorem 2, we cn rewrite formul () s nd formul (2) s u() b (, ξ)f(ξ) dξ. u() ξ (, )h ξ (, b)h. A singulrity function of d2 is given by d 2 K(, ξ) ξ, 2 s cn redily be verified. The reen s function for the intervl (, ) cn be 2

3 found by solving H with boundry conditions H(, ξ) K(, ξ) nd H(, ξ) K(, ξ), then setting K + H. This yields.25 (, ξ) ξ + ( + ξ) ξ 2 2 { ( )ξ for ξ < ( ξ) for < ξ. The solution of u f stisfying u() h nd u() h is given by ξ u() ( ) (, ξ)f(ξ) dξ + ξ (, )h ξ (, )h ξf(ξ) dξ + ( ξ)f(ξ) dξ + ( )h + h..2 reen s Function for Two-Dimensionl Poisson Eqution Now we go through the sme discussion in two dimensions. Some of the proofs use reen s second identity u v v u da (u v v u) n dl, Ω This cn be derived by writing reen s first identity twice, with u nd v interchnged the second time, nd subtrcting. A singulrity function K(, ) of the opertor is chrcterised by the three properties. For ny fied, lim disk centred t ; ɛ 2. For ny fied, lim ɛ outwrd unit norml to B ɛ; 3. K is hrmonic s function of for. K(, ) dl, where B ɛ denotes the rdius-ɛ K(, ) n dl, where n denotes the Note tht these three properties do not define singulrity function uniquely: if K is singulrity function then so is K + H, where H(, ) is ny function tht is hrmonic s function of. A singulrity function of is given by K(, ) 2π ln. (3) This ssertion cn be verified s follows. Without loss of generlity we cn tke. In polr coordintes, we hve K ln r with r, which is 2π hrmonic in R 2 \ {} (see section 9.4). Also, K dl 2π ln ɛ ɛ dθ ɛ ln ɛ 2π nd (becuse K n K/ r) K n dl 2π 3 2π ɛ dθ. ɛ

4 The reen s function for nd domin Ω with Dirichlet boundry conditions is singulrity function tht stisfies (, ) for. The reen s function provides the solution to the Poisson eqution with homogeneous Dirichlet boundry conditions: Theorem 4 If u f in Ω nd u on then u( ) (, )f() da ( Ω). (4) Ω Proof. Letting v() (, ), we hve vf da u v v u da Ω\B ɛ Ω\B ɛ (u v v u) n dl (u v v u) n dl u( ) v n dl + vu n dl, } {{ } nd the second term goes to zero becuse min u n v dl which completes the proof. vu n dl m u n v dl, The formul (4) represents the solution s the superposition of reen s functions, which cn be thought of s point source responses. The following result tells us tht the response t to point source locted t is equl to the response t to point source locted t. Theorem 5 (Reciprocity Principle) (, ) (, ) for. Proof. Let y nd y be distinct points in Ω, let B ɛ nd B ɛ denote ɛ-rdius disks centred t y nd y, nd let u() (, y ) nd v() (, y). Then u v v u da Ω\B ɛ\b ɛ (u v v u) n dl B ɛ u(y) v n dl + vu n dl B } ɛ B {{}} ɛ {{} uv n dl + v(y ) u n dl, leving us with u(y) v(y ), tht is, (y, y ) (y, y). 4

5 As consequence of Theorem 4, formul (4) cn be written s u() (, )f( ) da. Ω The singulrity function (3) is clled the free-spce reen s function for Poisson s eqution. It is reen s function for Poisson s eqution with the boundry condition u() s. The free-spce reen s function doesn t stisfy this boundry condition, but the boundry condition does ensure tht the term (u v v u) n dl in the proof of Theorem 3 goes to zero when Ω is tken to be disk of rdius R centred t nd R. The method of imges cn be used to find reen s functions for other domins. For emple, using the ide tht unit point source locted t (, y ) with y > nd point source of strength locted t (, y ) will cncel ech other on the -is, we find the reen s function (, ) 2π ln ( ( ) 2 + (y y ) 2) 2 + 2π ln ( ( ) 2 + (y + y ) 2) 2 for the Poisson eqution in the hlf-plne y >. Using similr ides, one cn derive the formul for reen s function for the origin-centred disk of rdius s (, ) 2π ln π ln. Representing nd in polr coordintes s (r, θ) nd (ρ, θ ), nd using the cosine lw cos γ with γ θ θ, the disk reen s function cn be written (, ) 2π ln 4 + r 2 ρ rρ cos γ 2 (r 2 + ρ 2 2rρ cos γ). (5) The reen s function ((, y), (,.5)) for the hlf-plne nd the unit-rdius disk re shown below y.5 y The reen s function lso provides the solution of Lplce s eqution with nonhomogeneous Dirichlet boundry conditions: 5

6 Theorem 6 If u in Ω nd u h on then u( ) h() (, ) n dl. Proof. Similr to the proof of Theorem 4. For the hlf-plne y > the unit norml points in the negtive y direction nd we hve n y /π y y ( ) 2 + (y ). 2 The solution of Lplce s eqution u in the hlf-plne with u(, ) h() is therefore u(, y ) y h() π ( ) 2 + (y ) d. 2 For the origin-centred disk, the unit outwrd norml points in the rdil direction nd we hve n r r 2π 2 ρ ρ 2 2ρ cos γ. The solution of Lplce s eqution on the disk with u h on the boundry r is therefore u(ρ, θ ) 2π ( 2 ρ 2 )h(θ) 2π 2 + ρ 2 2ρ cos(θ θ ) dθ, which is Poisson s integrl formul (section.)..3 reen s functions from eigenfunctions The Poisson problem cn lso be solved by the method of eigenfunctions. To introduce the technique, we strt with one-dimensionl problem. Consider the differentil eqution u f with boundry conditions u(), u(). The ssocited eigenvlue problem is φ + λφ with the sme boundry conditions. The eigenvlues nd eigenfunctions re λ m m2 π 2 2, φ m () sin mπ. Substituting tril solution of the form m A m φ m () into the differentil eqution, multiplying through by φ m (), nd integrting gives A m λ m sin 2 mπ d /2 m f() sin mπ Thus the A m re λ m the Fourier sine coefficients of f, nd the solution t point ξ (, ) is [ ] 2 mπ u(ξ) f() sin m 2 π2 sin mπξ d. d, 6

7 Compring this with formul (), we deduce the reen s function to be the term in brckets, tht is, (, ξ) m 2 mπ mπξ sin sin m 2 π2. When, this is the Fourier sine epnsion of the reen s function presented in section. Net, consider the two-dimensionl Poisson eqution u f on the domin (, ) (, b), with u on the boundry. The ssocited eigenvlue problem is u + λu, with the sme boundry conditions. Assuming solution of the form u(, y) X()Y (y) yields, with α 2 s seprtion constnt, the two eigenvlue problems X + α 2 X, Y + (λ α 2 )Y with homogeneous boundry conditions X() X() Y () Y (b). The eigenfunctions of the one-dimensionl problems re nd the eigenvlues re X m () sin mπ, Y n(y) sin nπy b. λ mn π 2 ( m n2 b 2 Substituting tril solution of the form m the Poisson eqution, multiplying through by X m Y n, nd integrting gives A mn λ mn sin 2 mπ d sin 2 nπy dy b /2 b b/2 ). n A m,n X m ()Y n (y) into b f(, y) sin mπ nπy sin d dy b Thus the A mn re λ mn the two-dimensionl Fourier sine coefficients of f, nd the solution t point is with (, y,, y ) 4b π 2 u(, y ) m n b f(, y)(, y,, y ) d dy mπ sin m 2 b 2 + n 2 2 sin nπy sin mπ b b sin nπy. b The reen s function (, y, /2, /4) for squre domin is plotted below y/.25.5 / 7

Summary: Method of Separation of Variables

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

More information

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

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

More information

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

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

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

21.6 Green Functions for First Order Equations

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

More information

Elliptic Equations. Laplace equation on bounded domains Circular Domains

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

More information

Math Fall 2006 Sample problems for the final exam: Solutions

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

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

PHYSICS 116C Homework 4 Solutions

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

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

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

More information

Module 9: The Method of Green s Functions

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,

More information

df dt f () b f () a dt

df dt f () b f () a dt Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem

More information

MA 201: Partial Differential Equations Lecture - 12

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

More information

3 Mathematics of the Poisson Equation

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

More information

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

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

More information

Green function and Eigenfunctions

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

More information

1 2-D Second Order Equations: Separation of Variables

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 +

More information

Plates on elastic foundation

Plates on elastic foundation Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

1 1D heat and wave equations on a finite interval

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

More information

1 E3102: a study guide and review, Version 1.0

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

More information

Variational Techniques for Sturm-Liouville Eigenvalue Problems

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

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Msschusetts Institute of Technology Deprtment of Physics Physics 8.07 Fll 2005 Problem Set 3 Solutions Problem 1: Cylindricl Cpcitor Griffiths Problems 2.39: Let the totl chrge per unit length on the inner

More information

10 Elliptic equations

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

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

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

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

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl

More information

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

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

More information

r = cos θ + 1. dt ) dt. (1)

r = cos θ + 1. dt ) dt. (1) MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr

More information

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

(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

More information

Notes on length and conformal metrics

Notes on length and conformal metrics Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

More information

The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

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

More information

Candidates must show on each answer book the type of calculator used.

Candidates must show on each answer book the type of calculator used. UNIVERSITY OF EAST ANGLIA School of Mthemtics My/June UG Exmintion 2007 2008 ELECTRICITY AND MAGNETISM Time llowed: 3 hours Attempt FIVE questions. Cndidtes must show on ech nswer book the type of clcultor

More information

Math 100 Review Sheet

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

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

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.

1.2. 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

More information

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. 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

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

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

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

More information

In Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)

In Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4) 678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWO-POINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution

More information

Kai Sun. University of Michigan, Ann Arbor

Kai Sun. University of Michigan, Ann Arbor Ki Sun University of Michign, Ann Arbor How to see toms in solid? For conductors, we cn utilize scnning tunneling microscope (STM) to see toms (Nobel Prize in Physics in 1986) Limittions: (1) conductors

More information

(6.5) Length and area in polar coordinates

(6.5) Length and area in polar coordinates 86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +

More information

Improper Integrals, and Differential Equations

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

More information

Mathematics. Area under Curve.

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

More information

Waveguide Guide: A and V. Ross L. Spencer

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

More information

, the action per unit length. We use g = 1 and will use the function. gψd 2 x = A 36. Ψ 2 d 2 x = A2 45

, the action per unit length. We use g = 1 and will use the function. gψd 2 x = A 36. Ψ 2 d 2 x = A2 45 Gbriel Brello - Clssicl Electrodynmics.. For this problem, we compute A L z, the ction per unit length. We use g = nd will use the function Ψx, y = Ax x y y s the form of our pproximte solution. First

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

Chapter 8: Methods of Integration

Chapter 8: Methods of Integration Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

More information

Chapter 28. Fourier Series An Eigenvalue Problem.

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

More information

c n φ n (x), 0 < x < L, (1) n=1

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

More information

Linearity, linear operators, and self adjoint eigenvalue problems

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

More information

Problems for HW X. C. Gwinn. November 30, 2009

Problems for HW X. C. Gwinn. November 30, 2009 Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

Lecture 24: Laplace s Equation

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

More information

Chapter 6 Techniques of Integration

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

More information

Thomas Whitham Sixth Form

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

More information

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

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

More information

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4

Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4 WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix Eigenvalues and Eigenvectors September 13, 2017 Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

More information

The Algebra (al-jabr) of Matrices

The Algebra (al-jabr) of Matrices Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

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

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),

More information

Kirchhoff and Mindlin Plates

Kirchhoff and Mindlin Plates Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory,

More information

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

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

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent? 1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the

More information

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence

(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll

More information

Sturm-Liouville Theory

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

More information

This final is a three hour open book, open notes exam. Do all four problems.

This final is a three hour open book, open notes exam. Do all four problems. Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

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

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

More information

APPM 1360 Exam 2 Spring 2016

APPM 1360 Exam 2 Spring 2016 APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

Phys 6321 Final Exam - Solutions May 3, 2013

Phys 6321 Final Exam - Solutions May 3, 2013 Phys 6321 Finl Exm - Solutions My 3, 2013 You my NOT use ny book or notes other thn tht supplied with this test. You will hve 3 hours to finish. DO YOUR OWN WORK. Express your nswers clerly nd concisely

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

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

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()

More information

Note 16. Stokes theorem Differential Geometry, 2005

Note 16. Stokes theorem Differential Geometry, 2005 Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion

More information

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

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,

More information

) 4n+2 sin[(4n + 2)φ] n=0. a n ρ n sin(nφ + α n ) + b n ρ n sin(nφ + β n ) n=1. n=1. [A k ρ k cos(kφ) + B k ρ k sin(kφ)] (1) 2 + k=1

) 4n+2 sin[(4n + 2)φ] n=0. a n ρ n sin(nφ + α n ) + b n ρ n sin(nφ + β n ) n=1. n=1. [A k ρ k cos(kφ) + B k ρ k sin(kφ)] (1) 2 + k=1 Physics 505 Fll 2007 Homework Assignment #3 Solutions Textbook problems: Ch. 2: 2.4, 2.5, 2.22, 2.23 2.4 A vrint of the preceeding two-dimensionl problem is long hollow conducting cylinder of rdius b tht

More information

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

Definite integral. Mathematics FRDIS MENDELU

Definite integral. Mathematics FRDIS MENDELU Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the

More information

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

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

More information

ENGI 9420 Lecture Notes 7 - Fourier Series Page 7.01

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

More information

Lecture 1: Electrostatic Fields

Lecture 1: Electrostatic Fields Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: nyyeri@iust.c.ir Clss web site: http://webpges.iust.c. ir/nyyeri/courses/bee 1.1. Coulomb s Lw Something known from the ncient time (here

More information

HW3, Math 307. CSUF. Spring 2007.

HW3, Math 307. CSUF. Spring 2007. HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

Jim Lambers MAT 280 Spring Semester Lecture 26 and 27 Notes

Jim Lambers MAT 280 Spring Semester Lecture 26 and 27 Notes Jim Lmbers MAT 280 pring emester 2009-10 Lecture 26 nd 27 Notes These notes correspond to ection 8.6 in Mrsden nd Tromb. ifferentil Forms To dte, we hve lerned the following theorems concerning the evlution

More information

Unit 5. Integration techniques

Unit 5. Integration techniques 18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

Math 5440 Problem Set 3 Solutions

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

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

More information

Math 32B Discussion Session Session 7 Notes August 28, 2018

Math 32B Discussion Session Session 7 Notes August 28, 2018 Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls

More information

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s

More information