Differential Equations 2 Homework 5 Solutions to the Assigned Exercises
|
|
- Sabina Harvey
- 5 years ago
- Views:
Transcription
1 Differentil Equtions Homework Solutions to the Assigned Exercises, # 3 Consider the dmped string prolem u tt + 3u t = u xx, < x <, t >, u, t = u, t =, t >, ux, = fx, u t x, = gx. In the exm you were supposed to write out the solution in the form of terms u n x, t = T n tx n t. A solution to the exm question cn e found t the end of these pges. Use the solution to the prolem to complete the following tsks; n, n refer to the coefficients in my solution.. Find formul for the coefficients n, n in terms of the initil conditions f, g.. Write out the solution for the cse of fx = x + 3x, gx =. 3. For the concrete initil conditions of, sketch u, t s function of t in the intervl t. Solution. It gets pretty messy, which is why I did not wnt you to do it in the exm. The solution is copied from elow ux, t = e 3 + e 3+ x n 9 t n 9 t + n cos nx. n= n= The derivtive with respect to time of the solution is 3 u t x, t = e e 3+ x n 9 n 9 t n 9 t + n + n cos nx n= 3 n 9 t n 9 t n cos nx. Setting t = nd equting to fx, gx, respectively, we get fx = + x + n nx, gx = 3 n= x + Ug the formuls for the coefficients of Fourier e series, we get + = n = = 3 n 9 n + n = n 9 n 3 n nx. n= fx x dx, fx nx dx, n =, 3,,..., gx x dx, gx nx dx, n =, 3,,....
2 Solving we get = = n = n = fx x dx + fx x dx fx nx dx, n =, 3,,..., 6 n 9 Se non è vero, è en trovto fx nx dx + n 9 gx x dx, gx x dx,, #. Consider the het eqution for rectngulr region < x <, < y < u t = κ u x + u y, < x <, < y <, t >, gx nx dx, n =, 3,,.... suject to the initil condition ux, y, = fx, y. Find n expression for the solution if the oundry conditions re s given elow. In every cse, find lim t ux, y, t. { ux, y, t =, u x, y, t =, < y <, t >, Solution. u y x,, t =, u y x,, t =, < x <, t > I will e rief. The solution is given y ux, y, t = with the coefficients determined y m,n= mn e m fx, y cos mx + n κt mx cos ny cos, ny cos dy dx, m >, n >, mn = fx, y cos ny fx, y cos mx dy dx, m =, n >, dy dx, m >, n =, fx, y dy dx, m = n =. One hs lim ux, y, t = fx, y dy dx. t d { u, y, t =, ux, y, t =, < y <, t >, u y x,, t =, u y x,, t =, < x <, t > Solution. By seprtion of vriles, of course. Becuse t every prt of the oundry either u or the norml derivtive of u vnishes, n nlysis ug Ryleigh s quotient tells us there re no non-positive eigenvlues. Tht is looking for solutions u = XY T, plugging this into the eqution, the spce nd time vriles seprte s T κt = X X + Y Y = λ
3 3 nd the only vlues of λ for which we will hve non-zero solutions of the Sturm Liouville prolem will turn out to e positive, We now split the spce vriles s X X = Y Y λ = µ. The spce Sturm Liouville prolem unfolds into the two prolems X + µx =, X =, X =, Y + λ µy =, Y = Y =. The X prolem is seen to hve eigenvlues µ m = m, with corresponding eigenfunctions X m = m x, m =,, 3,.... We consider the Y prolem with µ = λ m. The vlues of λ for which there re non-zero solutions re then λ mn = µ m + n with eigenfunction Y n y = cos ny, n =,,,.... The solution is this given y ux, y, t = m= n= The coefficients re found y ug tht m x The coefficients work out to: cos nx c mn = c mn e k x = = m + κt n m x cos nx. cos lx m x dy dx, k m orl n,, k = m, l = n =,, k = m, l = n >. fx, y fx, y m x m x In this cse, lim t ux, y, t =., #. Consider the wve eqution for rectngulr memrne k x cos ny dx cos nx cos lx dy dx dy dx, m, n >, dy dx, m >, n =. u tt = c u xx + u yy, < x <, < y <, t >, suject to the oundry conditions { u, y, t =, u, y, t =, < y <, t >, ux,, t =, ux,, t =, < x <, t > Find n expression for the solution if the initil conditions re:. ux, y, = xy x y, u t x, y =.. ux, y, =, u t x, y, = x 7y.
4 3. Solution. ux, y, = xy x y, In ll cses, the solution of the PDE hs the form m ux, t = mn cos + n m ct + mn m,n= u t x, y = x 7y. + n ct mx ny. Turning to the three prticulr cses, we get:. In this cse nm = for ll n, m nd nm = xyx y mx ny dy dx = 6 m n 6 n 3 m 3. Notice tht if either n or m is even, nm =, so tht ux, t = 6 j 6 j 3 k 3 cos + j,k= k ct j x k y.. The solution is ux, t = c c x 7y. 3. The solution is the sum of the solutions of Prolem nd Prolem. It works out to: ux, t = 6 j 6 j 3 k 3 cos k j x + ct j,k= + c c x 7y., #. Solve suject to the oundry condition u tt = c u, r <, θ, t >, u, θ, t =, θ, t >, with the following initil conditions. Simplify the solution s much s possile.. ur, θ, =, u t r, θ, = r.. ur, θ, = r θ, u t r, θ =. Solution. k y From my notes on the circulr plte nd memrne, the solution to oth prolems hs the form ur, θ, t = J j kr A,k cos cj k t + C,k cj k t k= + J n j nkr A nk cos nθ + B nk nθ cos cj nk t + n,k= n,k= where j nk is the k-th positive zero of the Bessel function J n, J n j nkr C nk cos nθ + D nk nθ cj nk t, γ nk = rj n j nkr dr
5 SOLUTION TO EXERCISE OF EXAM for n =,,,... ; k =,, 3,... nd the coefficients A nk, B nk, C nk, D nk re given y A nk = B nk = C nk = D nk = γ nk γ nk γ nk cj n,k γ nk cj n,k fr, θrj n j nkr cos nθ dr dθ, n =,,,..., fr, θrj n j nkr nθ dr dθ, n =,,..., gr, θrj n j nkr cos nθ dr dθ, n =,,,..., gr, θrj n j nkr nθ dr dθ, n =,,.... Now to solve the prolems.. The fct tht f = gives A n,k = B n,k = for ll vlues of n, k. For the other coefficients it will e etter to differentite the expression for the solution with respect to t, set t = nd equte to r insted of ug the formuls. We get r = k= J j kr cj k t C,k + n,k= J n j nkr cj nk C nk cos nθ + D nk nθ. Since there is no dependence on θ, we get C nk = D nk = for n, ll k. So ll tht remins is C k. The solution in its simplest form is where ur, θ, t = k= C k = γ k cj,k C,k J j kr cj k t, r 3 J j kr dr.. Since g = ll coefficients C nk, D nk re. The only other non-zero coefficient is the one involving term with θ. The solution is thus ur, θ, t = B k J n j kr θ cos cj k t, where B k = γ k k= r 3 θj n j nkr dr dθ = γ k Solution to Exercise of Exm r 3 J n j nkr dr. If ux, t = XxT t solves the eqution nd the oundry conditions, then seprtion of vriles leds to the eqution/sl prolem T + 3T + λt =, X + λx =, X = X =. The eigenvlues/eigenfunctions of the SL prolem re λ n = n, X n = nx, respectively; n =,,.... For ech n, the T eqution is T + 3T + n T = ; to find the generl solution we need to solve the chrcteristic eqution r + 3r + n =. The solutions of this qudrtic eqution re r = 3 ± 9 n. If n =, then 9 n = > nd the generl solution is T t = e 3 + e 3+,
6 SOLUTION TO EXERCISE OF EXAM 6 for constnts, ; if n then 9 n < nd the generl solution is n 9 t n 9 t T n t = n cos, where n, n re constnts. It follows tht the solution of the PDE nd oundry conditions is given y ux, t = e 3 + e 3+ x n 9 t n 9 t + n cos nx. n=
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
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 2-D Exmple Ojective: Let Ω e plnr region with oundry curve Γ. Consider het conduction in Ω with fixed
More informationThe 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 informationMath 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 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 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 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 informationGreen 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 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 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 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 information1 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 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 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 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 informationAMS 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
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 two-dimensionl Lplce eqution on the domin D, More generl eqution
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 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 informationPHYSICS 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 informationSection 3.2 Maximum Principle and Uniqueness
Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point
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: 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 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 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 informationSTURM-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 informationAPM346H1 Differential Equations. = u x, u = u. y, and u x, y =?. = 2 u t and u xx= 2 u. x,t, where u t. x, y, z,t u zz. x, y, z,t u yy.
INTRODUCTION Types of Prtil Differentil Equtions Trnsport eqution: u x x, yu y x, y=, where u x = u x, u = u y, nd u x, y=?. y Shockwve eqution: u x x, yu x, yu y x, y=. The virting string eqution: u tt
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
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 two-dimensionl Lplce eqution
More information( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationSturm-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
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
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 informationVariational 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 informationChapter 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 informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
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 informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 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 informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
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 informationPartial Differential Equations
Partial Differential Equations Spring Exam 3 Review Solutions Exercise. We utilize the general solution to the Dirichlet problem in rectangle given in the textbook on page 68. In the notation used there
More informationPartial 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
More informationSTURM-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
More informationThomas 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(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 informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationHomework Assignment 6 Solution Set
Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know
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 informationPolynomials 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
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
More informationSuggested Solution to Assignment 5
MATH 4 (5-6) prti diferenti equtions Suggested Soution to Assignment 5 Exercise 5.. () (b) A m = A m = = ( )m+ mπ x sin mπx dx = x mπ cos mπx + + 4( )m 4 m π. 4x cos mπx dx mπ x cos mπxdx = x mπ sin mπx
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 informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationMA123, 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.
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationSTURM-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 informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More informationMA FINAL EXAM INSTRUCTIONS
MA 33 FINAL EXAM INSTRUCTIONS NAME INSTRUCTOR. Intructor nme: Chen, Dong, Howrd, or Lundberg 2. Coure number: MA33. 3. SECTION NUMBERS: 6 for MWF :3AM-:2AM REC 33 cl by Erik Lundberg 7 for MWF :3AM-:2AM
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationFunctions of Several Variables
Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x
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 informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
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 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 informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationCAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.
Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More information(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
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 informationM597K: Solution to Homework Assignment 7
M597K: Solution to Homework Assignment 7 The following problems re on the specified pges of the text book by Keener (2nd Edition, i.e., revised nd updted version) Problems 3 nd 4 of Section 2.1 on p.94;
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 informationSturm-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 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 informationNotes 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
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationMath 473: Practice Problems for the Material after Test 2, Fall 2011
Mth 473: Prctice Problems for the Mteril fter Test, Fll SOLUTION. Consider the following modified het eqution u t = u xx + u x. () Use the relevnt 3 point pproximtion formuls for u xx nd u x to derive
More informationChapter 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
More informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationMath 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 informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationDefinition 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)
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
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 informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
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 informationFinal Exam - Review MATH Spring 2017
Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationSECTION 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
More informationMath 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas
Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationPressure 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
More information7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!
7. Problem 7. We hve two semi-innite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being
More informationA Brief Note on Quasi Static Thermal Stresses In A Thin Rectangular Plate With Internal Heat Generation
Americn Journl of Engineering Reserch (AJER) 13 Americn Journl of Engineering Reserch (AJER) e-issn : 3-847 p-issn : 3-936 Volume-, Issue-1, pp-388-393 www.jer.org Reserch Pper Open Access A Brief Note
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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis - January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More information