1 2D Second Order Equations: Separation of Variables


 Daisy Sharp
 1 years ago
 Views:
Transcription
1 Chpter 12 PDEs in Rectngles 1 2D 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 + F u = G. (1) y 2. If G = 0 we sy the problem is homogeneous otherwise it is nonhomogeneous. 3. A Solution is function u(x, y) tht hs the required differentibility nd stisfies the eqution. Unlike the cse of ODEs the ide of generl solution is not very cler. We will look for prticulr solutions. 4. A very useful method for looking for solutions is the Method of Seprtion of Vribles in which we look for solution in the form u(x, y) = ψ(y). 5. (Principle of Superposition) Since the problem is liner, if we find severl solutions, sy {u j (x, y)} n j=1 then for ll constnts {c j } n j=1 is lso solution. u(x, y) = n c j u j (x, y) j=1 6. Equtions in the form (1) cn be clssified s one three types of equtions by () Hyperbolic if B 2 4AC > 0 (b) Prbolic if B 2 4AC = 0 (c) Elliptic if B 2 4AC < 0 7. Min Clssicl Exmples: (replcing y by t in the hyperbolic nd prbolic cses) () Hyperbolic Wve eqution 2 u t (x, t) = 2 u 2 2 (x, t) x2 (b) Prbolic Het eqution u t (x, t) = k 2 u (x, t) x2 (c) Elliptic Lplce s eqution 2 u x (x, y) + 2 u (x, y) = 0 2 y2 8. Exmples of seprtion of vribles for min exmples: () For 2 u t 2 = 2 u 2 x seek u(x, t) = ψ(t) ψ (t) 2 2 ψ(t) = ϕ (x) (b) For u t = k 2 u x seek u(x, t) = ψ(t) ψ (t) 2 kψ(t) = ϕ (x) (c) For 2 u x + 2 u 2 y = 0 seek u(x, y) = ψ(y) ϕ (x) 2 + ψ (y) ψ(y) = λ where λ is constnt. = λ where λ is constnt. = λ where λ is constnt. 1
2 9. Boundry Conditions re given on the physicl boundry of the sptil domin, i.e., t the ends of heted rod or vibrting string. The most common BCs re () Dirichlet Homogeneous BC t x = is u(, t) = 0, Nonhomogeneous BC is u(, t) = γ(t) (b) Neumnn Homogeneous BC t x = is u x (, t) = 0, Nonhomogeneous BC is u x (, t) = γ(t) (c) Robin Homogeneous BC t x = is u x (, t) ± ku(, t) = 0, Nonomogeneous BC is u x (, t) ± ku(, t) = γ(t) where k > 0 is constnt nd (±) is determined by which end of the rod or string. 2 Regulr SturmLiouville problem One of the most importnt ides in functionl nlysis is contined in the following discussion which is very much relted to the ide of eigenvlues nd n orthonorml bsis of eigenvectors. Consider the boundry vlue problem. ϕ (x) q(x) = λ, α 1 ϕ () + α 2 ϕ() = 0, (2) β 1 ϕ (b) + β 2 ϕ(b) = 0 This is n eigenvlue problem which is referred to s Regulr SturmLiouville problem. Theorem 2.1. The problem (2) hs infinitely mny eigenpirs {λ n, ϕ n (x)} which stisfy the following properties: 1. The eigenvlues re ll simple, i.e. they re eigenvlues of multiplicity one which mens tht λ j λ k for j k. 2. The eigenvlues re ll rel nd ll but finite number re negtive. If α 1 + α 2 0, β 1 + β 2 0 then ll (but possibly one) of the eigenvlues re less thn or equl to zero. 3. If we order the eigenvlues in decresing order by λ n < λ n 1 < < λ 2 < λ 1, nd λ n s n. 4. The eigenfunctions re ll rel nd ϕ n (x) hs exctly (n 1) zeros in the intervl (, b). 5. The eigenfunctions form Complete Orthonorml Set in the following sense { b 0, n m ϕ n (x)ϕ m (x) dx = δ n,m = 1, n = m nd if f(x) is piecewise smooth (P C (1) (, b) in my nottion in clss), then (f(x+) + f(x )) = c n ϕ n (x), < x < b, where c n = f(x)ϕ n (x) dx. 2 ( 1/2 N For ny f such tht f = f 2 (x) dx) < we hve f(x) N c n ϕ n (x) 0, i.e., f(x) = c n ϕ n (x) in the sense of L 2 (, b). 2
3 3 1D Het Eqution: Eigenvlues nd Eigenvectors Our first PDE is the het eqution on finite rod x b. u t (x, t) = ku xx (x, t), < x < b, t > 0 u(x, 0) = f(x) There re three min types of boundry conditions imposed t the ends of the rod. The two min conditions re u(, t) = 0, u(b, t) = 0 Dirichlet Conditions u x (, t) = 0, u x (b, t) = 0 Neumnn Conditions α 1 u x (, t) + α 2 u(, t) = 0, β 1 u x (b, t) + β 1 u(b, t) = 0 Robin Conditions We cn lso hve ny combintion of these conditions, i.e., we could hve Dirichlet condition t x = nd Neumnn condition t x = b. Notice tht Dirichlet nd Neumnn BCs re specil cses of the Robin BCs. There is lso more generl problem involving two extr terms tht correspond to het conduction nd convection. u t (x, t) = k ( u xx (x, t) 2u(x, t) x + bu(x, t) ), 0 < x < l, t > 0 u(x, 0) = f(x). Let us consider the het eqution on x b. u t (x, t) = ku xx (x, t), < x < b, t > 0 α 1 u x (, t) + α 2 u(, t) = 0, β 1 u x (b, t) + β 2 u(b, t) = 0 u(x, 0) = Applying seprtion of vribles we seek simple solutions in the form This gives u(x, t) = ψ(t). ψ (t) kψ(t) = ϕ (x) nd since the left side is independent of x nd the right side is independent of t, it follows tht the expression must be constnt: ψ (t) kψ(t) = ϕ (x) = λ We seek to find ll possible constnts λ nd the corresponding nonzero functions ϕ nd ψ. The eqution ψ kλψ = 0 hs generl solution ψ(t) = Ce kλt (3) where C is n rbitrry constnt. 3
4 We lso obtin ϕ λϕ = 0 Furthermore, the boundry conditions give (α 1 ϕ () α 2 ϕ())ψ(t) = 0, (β 1 ϕ (b) + β 2 ϕ(b))ψ(t) = 0 for ll t. Since ψ(t) is not identiclly zero we obtin the desired eigenvlue problem ϕ (x) λ = 0, α 1 ϕ () + α 2 ϕ() = 0, β 1 ϕ (b) + β 2 ϕ (b) = 0. By the SturmLiouville Theorem there re infinitely mny eigenpirs {λ n, ϕ n (x)} with the {ϕ n } re orthonorml, i.e., { b 0, n m ϕ n, ϕ m = ϕ n (x)ϕ m (x) dx = δ n,m = 1, n = m. We seek solution to het problem in the form u(x, t) = c n e λnt ϕ n (x). (4) We need two things: 1. When t = 0 we find vlues for {c n } so tht given by f(x) = u(x, 0) = c n = c n ϕ n (x) f(x)ϕ n (x) dx. 2. Next we need for the infinite sum in (7) to represent solution to the eqution. This is formlly true since u t (x, t) = de λnt c n ϕ n (x) dt = c n λ n e λnt ϕ n (x) = c n e λnt d2 ϕ n dx (x) = 2 u (x, t) 2 x2 A rigorous proof in the cse f is smooth or if we use convergence in L 2 (, b) nd study wek solutions ( topic beyond the scope of this clss). 4
5 4 1D Wve Eqution: Eigenvlues nd Eigenvectors The one dimensionl wve eqution modeling the displcement of vibrting string of length l = (b ) covering the intervl < x < b is u tt (x, t) = c 2 u xx (x, t), < x < b, t > 0 (5) u(x, 0) = f(x) u t (x, 0) = g(x) Once gin we hve the sme three min types of boundry conditions imposed t the ends of the string: Dirichlet Conditions, Neumnn Conditions, nd Robin Conditions. These re ll specil cses of the generl BCs α 1 ϕ () + α 2 ϕ() = 0, β 1 ϕ (b) + β 2 ϕ(b) = 0. Let us consider the het eqution on x b. u tt (x, t) = c 2 u xx (x, t), < x < b, t > 0 α 1 u x (, t) + α 2 u(, t) = 0, β 1 u x (b, t) + β 2 u(b, t) = 0 u(x, 0) = f(x), u t (x, 0) = g(x) Applying seprtion of vribles we seek simple solutions in the form u(x, t) = ψ(t). This gives ψ (t) c 2 ψ(t) = ϕ (x) nd since the left side is independent of x nd the right side is independent of t, it follows tht the expression must be constnt: ψ (t) c 2 ψ(t) = ϕ (x) = λ We seek to find ll possible constnts λ nd the corresponding nonzero functions ϕ nd ψ. In the x vrible we hve ϕ λϕ = 0 Furthermore, the boundry conditions give (α 1 ϕ() + α 2 ϕ())ψ(t) = 0, (β 1 ϕ(b) + β 2 ϕ(b))ψ(t) = 0 for ll t. Since ψ(t) is not identiclly zero we obtin the desired eigenvlue problem ϕ (x) λ = 0, α 1 ϕ () + α 2 ϕ() = 0, β 1 ϕ (b) + β 2 ϕ(b) = 0. 5
6 By the SturmLiouville Theorem there re infinitely mny eigenpirs {λ n, ϕ n (x)} with the eigenfunctions forming complete orthonorml set. In most prcticl problems we hve λ n = µ 2 n so the problem for ψ becomes ψ + c 2 µ 2 nψ = 0 hs generl solution ψ n (t) = n cos(cµ n t) + b n sin(cµ n t) (6) where n nd b n re rbitrry constnts. At lest for continuous initil conditions ϕ we obtin solution to wve eqution in the form where u(x, t) = ( n cos(cµ n t) + b n sin(cµ n t)) ϕ n (x). (7) 1. f(x) = u(x, 0) = 2. g(x) = u t (x, 0) = n ϕ n (x) with n = f(x)ϕ n (x) dx. (cµ n )b n ϕ n (x) with b n = (cµ n ) 1 g(x)ϕ n (x) dx. 3. Next we need for the infinite sum in (7) to represent solution to the eqution. This is formlly true since 2 u t (x, t) = d 2 c 2 n dt ( 2 n cos(cµ n t) + b n sin(cµ n t)) ϕ n (x) = = c n ( cµ n ) 2 ( n cos(cµ n t) + b n sin(cµ n t)) ϕ n (x) c n (c 2 λ n ) ( n cos(cµ n t) + b n sin(cµ n t)) ϕ n (x) = c 2 = c 2 2 u (x, t) x2 c n ( n cos(cµ n t) + b n sin(cµ n t)) d2 ϕ n dx 2 (x) 6
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 informationSTURMLIOUVILLE BOUNDARY VALUE PROBLEMS
STURMLIOUVILLE 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 informationMA 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 informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville 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 informationMath Fall 2006 Sample problems for the final exam: Solutions
Mth 425 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 informationGreen function and Eigenfunctions
Green function nd Eigenfunctions Let L e regulr SturmLiouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions
More informationMath 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 informationThe 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
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L13. 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 informationPhysics 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 informationWave 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
More informationSturmLiouville 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
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant 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 informationApplied 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 informationMath Theory of Partial Differential Equations Lecture 29: SturmLiouville eigenvalue problems (continued).
Mth 412501 Theory of Prtil Differentil Equtions Lecture 29: SturmLiouville eigenvlue problems (continued). Regulr SturmLiouville eigenvlue problem: d ( p dφ ) + qφ + λσφ = 0 ( < x < b), dx dx β 1 φ()
More informationMath 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
More informationElliptic 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 twodimensionl Lplce eqution
More informationu 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 timeindependent problems Introductory emples Onedimensionl het eqution Consider the onedimensionl het eqution with boundry conditions nd initil condition We lredy know
More information1 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 informationMATH34032: 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 informationg 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 informationModule 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 informationSTURMLIOUVILLE THEORY, VARIATIONAL APPROACH
STURMLIOUVILLE THEORY, VARIATIONAL APPROACH XIAOBIAO LIN. Qudrtic functionl nd the EulerJcobi Eqution The purpose of this note is to study the SturmLiouville problem. We use the vritionl problem s
More informationPDE 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.....................................
More information1.1. 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
More information1 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 informationNotes 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(9) P (x)u + Q(x)u + R(x)u =0
STURMLIOUVILLE 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 informationLECTURE 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
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationThe area under the graph of f and above the xaxis 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 xxis etween nd is denoted y f(x) dx nd clled the
More informationu(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: 2D Wve Eqution nd Eigenfunctions of the Lplcin Ojective: Let Ω e plnr region with oundry curve Γ. Consider the wve eqution in Ω
More informationMatrix 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 informationThe 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
More informationConservation 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
More informationM344  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(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 2D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationDifferential 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
More informationMath 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, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More information4 SturmLiouville Boundary Value Problems
4 SturmLiouville 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
More informationSummary: 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 informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationBest 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
More information4402 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
4402 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More information18 SturmLiouville Eigenvalue Problems
18 SturmLiouville 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.
More information1.3 The Lemma of DuBoisReymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBoisReymond We needed extr regulrity to integrte by prts nd obtin the Euler Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics ENotes 8(8)  c IN 675 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationPartial 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 righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationOrthogonal 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
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More information1. 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
More informationLinearity, 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 informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More informationProblem 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
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationSTURMLIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES
STURMLIOUVILLE PROBLEMS: GENERALIZED FOURIER SERIES 1. Regulr SturmLiouville Problem The method of seprtion of vribles to solve boundry vlue problems leds to ordinry differentil equtions on intervls
More information20 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
More informationSeparation 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 onedimensionl (1D) wve eqution 2 u
More information1 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
More informationc 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 informationGeneralizations of the Basic Functional
3 Generliztions of the Bsic Functionl 3 1 Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 3.1 Functionls with Higher Order Derivtives.......... 3 3 3.2 Severl Dependent Vribles...............
More informationLecture 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 informationReview 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,
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationODE: 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() =
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information63. 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 informationTheoretical 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
More information3 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 deltfunction chrge 2 G D (r, r o ) = δ 3 (r r o ) (3.) nd
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationSTURMLIOUVILLE PROBLEMS
STURMLIOUVILLE PROBLEMS Mrch 8, 24 We hve seen tht in the process of solving certin liner evolution equtions such s the het or wve equtions we re led in very nturl wy to n eigenvlue problem for second
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information10 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 twodimensionl Lplce eqution on the domin D, More generl eqution
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 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
More informationAbstract 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
More information21.6 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 informationHeat 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
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationExample Sheet 6. Infinite and Improper Integrals
Sivkumr Exmple Sheet 6 Infinite nd Improper Integrls MATH 5H Mteril presented here is extrcted from Stewrt s text s well s from R. G. Brtle s The elements of rel nlysis. Infinite Integrls: These integrls
More informationChapter 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
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationMath 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
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More information1 Wave Equation on Finite Interval
1 Wave Equation on Finite Interval 1.1 Wave Equation Dirichlet Boundary Conditions u tt (x, t) = c u xx (x, t), < x < l, t > (1.1) u(, t) =, u(l, t) = u(x, ) = f(x) u t (x, ) = g(x) First we present the
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationProperties 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
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pn.pl/ tzielins/ Tble of Contents 1 Introduction 1 1.1 Motivtion nd generl concepts.............
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationSolutions of Klein  Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSRJM) eissn: 22785728, pissn: 2319765X. Volume 13, Issue 6 Ver. IV (Nov.  Dec. 2017), PP 1924 www.iosrjournls.org Solutions of Klein  Gordn equtions, using Finite Fourier
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationMA 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 informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationIn 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 TWOPOINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationDefinite 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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More information