Math 110, Spring 2004 Notes

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Bruce K. DriverMath 110, Spring 2004 NotesMay 25, 2004 File:110notes.texSpringerBerlin Heidelberg NewYorkHong Kong LondonMilan Paris Tokyo

Contents1Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The spectral theorem for symmetric matrices . . . . . . . . . . . . . . . 11.2 Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102PDE Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.1 d’Alembert’s solution to the 1-dimensional waveequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Heat Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Other Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151821233Linear ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 First order linear ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 Solving for etA using the Spectral Theorem . . . . . . . . . . . . . . . . .3.3 Second Order Linear ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4 ODE Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27273235384Linear Operators and Separation of Variables . . . . . . . . . . . . . . 414.1 Introduction to Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Application / Separation of variables . . . . . . . . . . . . . . . . . . . . . . . 475Orthogonal Function Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 Generalities about inner products on function spaces . . . . . . . . .5.2 Convergence of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4 Proof of Theorem 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.1 Gibb’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1111535357587074

4Contents5.5 Fourier Series on Other Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 766Boundary value problem examples . . . . . . . . . . . . . . . . . . . . . . . . . 817Boundary value generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.1 Linear Algebra of the Strurm-Liouville Eigenvalue Problem . . . 877.2 General Elliptic PDE Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918PDE Applications and Duhamel’s Principle . . . . . . . . . . . . . . . . 978.1 Interpretation of d’Alembert’s solution to the 1-d wave equation 978.2 Solving 1st - order equations using 2nd - order solutions . . . . . . 978.2.1 The Solution to the Heat Equation on R . . . . . . . . . . . . . 998.3 Duhamel’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.4 Application of Duhamel’s principle to 1 - d wave and heatequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069Problems In other Coordinates Systems . . . . . . . . . . . . . . . . . . . 1159.1 A Heat equation in spherical coordinates . . . . . . . . . . . . . . . . . . . 1159.2 Problems with cylindrical symmetries . . . . . . . . . . . . . . . . . . . . . . 1169.3 Strum Liouville problem in cylindrical coordinates . . . . . . . . . . . 1199.4 Bessel Equation and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128ASome Complex Variables Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133BAssigned Homework Problems: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Page: 4job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

1Preliminaries1.1 The spectral theorem for symmetric matricesLet A be a real N N matrix, a11 a12 . . . a1N a21 a22 . . . a2N A . ,. . . aN 1 aN 2 . . . aN Nandf1f2. f fN(1.1) RN be a given vector. As usual we will let ei denote the vector in RN with allentries being zero except for the ith which is taken to be one.We will write(u, v) : u · v NXui vi utr v andi 12 u (u, u) NXu2i utr u.i 1NRecall that {vi }i 1 RN is said to be an orthonormal basis if 1 if i j(vi , vj ) δij : .0 if i 6 j(1.2)The following proposition and its infinite dimensional analogue will be thebasis for much of this course.

21 PreliminariesNNProposition 1.1. If {vi }i 1 RN satisfies Eq. (1.2) then {vi }i 1 is a basisfor RN and if u RN we haveu NX(u, vi ) vi .(1.3)i 1Proof. Suppose that u (u, vj ) NXPNi 1ai vi for some ai R. Then!ai vi , vji 1 NXai (vi , vj ) i 1NXai δij aj .i 1In particular if u 0 we learn that aj (u, vj ) 0 and we have shown that N{vi }i 1 is a linearly independent set. Since dim RN N, it now followsNthat {vi }i 1 is a basis for RN and hence every u RN may be written in thePNform u i 1 ai vi . By what we have just proved, we must have ai (u, vi ) ,i.e. Eq. (1.3) is valid.Definition 1.2. A matrix A as in Eq. (1.1) is symmetric A Atr , i.e. ifaij aji for all i, j.The following characterization of a symmetric matrix will be more usefulfor our purposes.Lemma 1.3. If A is a real N N matrix then, for all u, v RN , (Au, v) u, Atr v .(1.4)Moreover A is symmetric iff(Au, v) (u, Av) for all u, v RN .(1.5)Proof. Eq. (1.4) is a consequence of the following matrix manipulations tr(Au, v) (Au) v utr Atr v u, Atr vtrwhich are based on the fact that (AB) B tr Atr . Hence if A is symmetric,then Eq. (1.5) holds. Conversely, if Eq. (1.5) holds, by taking u ei andv ej in Eq. (1.5) we learn that a1ia1j aji . , ej (Aei , ej ) (ei , Aej ) ei , . aij .aN,iaN,jCorollary 1.4. Suppose that A Atr and v, w RN are eigenvectors of Awith eigenvalues λ and µ respectively. If µ 6 λ then v and w are orthogonal,i.e. (v, w) 0.Page: 2job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

1.1 The spectral theorem for symmetric matrices3Proof. If Av λv and Aw µw with λ 6 µ thenλ (v, w) (λv, w) (Av, w) (v, Aw) (v, µw) µ (v, w)or equivalently, (λ µ) (v, w) 0. Sine λ 6 µ, we must conclude that (v, w) 0.The following important theorem from linear algebra gives us a methodfor guaranteeing that a matrix is diagonalizable. Again much of this course isbased on an infinite dimensional generalization of this theorem.Theorem 1.5 (Spectral Theorem). If A in Eq. (1.1) is a symmetric matrix, then A has an orthonormal basis of eigenvectors, {v1 , . . . , vN } and thecorresponding eigenvalues, {λ1 , λ2 , . . . , λN } are all real.Example 1.6. Suppose that A : 123 2 32 ,12(1.6)then 1 λ 23 32 12 λ 1 λ2 2 2p (λ) det (A λI) det94 which we set equal to zero to learn or equivalently, λ 121 λ2 2 94 32 and hence A has eigenvalues, λ1 1 and λ2 2.Sinceand 32 1 1 0 0 23 23A 2I 32 23 11 00 we learn thatPage: 332 32 A I : job: 110notes32 1v1 λ1 11 1v2 λ2 2. 1macro: svmono.clsdate/time: 25-May-2004/11:28

41 PreliminariesNotice that (v1 , v2 ) 0 as is guaranteed by Corollary 1.4. The normalized1eigenvectors are given by 2 1/2 v1 and 2 2 v2 . Consequently if f R2 , wehave f 2 1/2 v1 , f 2 1/2 v1 2 1/2 v2 , f 2 1/2 v2 11(v1 , f ) v1 (v2 , f ) v2 .22(1.7)Remark 1.7. As above, it often happens that naturally we find a orthogonalNbut not orthonormal basis {vi }i 1 for RN , i.e. (vi , vj ) 0 if i 6 j but(vi , vi ) 6 1. We can still easily expand in terms of these vectors. Indeed,noN 1is an orthonormal basis for RN and therefore if f RN we vi vii 1haveN N XX(f, vi ) 1 1f f, vi vi vi vi 2 vi .i 1i 1 vi Example 1.8. Working as above, one shows the symmetric matrix, 1 7 2A : 7 1 2 , 2 2 10(1.8)has characteristic polynomial given by p (λ) det (A λI) λ3 12λ2 36λ 432 (λ 6) (λ 12) (λ 6) .Thus the eigenvalues of A are given by λ1 6, λ2 6 and λ3 12 and thecorresponding eigenvectors are 11 1v1 : 1 6, v2 : 1 6, v3 : 1 12.012Again notice that {v1 , v2 , v3 } is an orthogonal set as is guaranteed by Corollary1.4. Relative to this basis we have the expansionf (f, v1 ) v12 v1 (f, v2 )v22 v2 (f, v3 )v32 v3 111(f, v1 ) v1 (f, v2 ) v2 (f, v2 ) v3 .236trFor example if f (1, 2, 3) , thenf Page: 4job: 110notes11v1 2v2 v3 .22macro: svmono.cls(1.9)date/time: 25-May-2004/11:28

1.2 Cylindrical and Spherical Coordinates53Exercise 1.1. Verify that the vectors {vi }i 1 are eigenvectors of A in Eq.(1.8) which have the stated eigenvalues. Hint: you are only asked to verifynot solve from scratch.33Exercise 1.2. Find eigenvectors {vi }i 1 and corresponding eigenvalues {λi }i 1for the symmetric matrix, 2 1 1A : 1 2 1 .1 1 2Make sure you choose them to be orthogonal. Also express the followingvectors,trf (1, 0, 2)trand g (0, 1, 2)trand h ( 1, 1, 0) ,3as linear combinations of the {vi }i 1 that you have found.NExercise 1.3. Suppose that A is a N N symmetric matrix and {vi }i 1 isNa basis of eigenvectors of A with corresponding eigenvalues {λi }i 1 . SupposeNf R has been decomposed asf NXai vi .i 1Show:PN1. An f i 1 ai λni vi .2. More generally, suppose thatp (λ) a0 a1 λ a2 λ2 · · · an λnis a polynomial in λ, thenp (A) f NXai p (λi ) vii 1and in particular p (A) v p (λ) v is Av λv.1.2 Cylindrical and Spherical CoordinatesOur goal in this section is to work out the Laplacian in cylindrical and spherical coordinates. We will need these results later in the course. Our methodis to make use of the following two observations:Page: 5job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

61 Preliminaries31. If {ui }i 1 is any orthonormal basis for R3 then f · g 3X( f, ui ) ( g, ui ) i 13X ui f ui gi 1and2. if g has compact support in a region Ω, then by integration by partsZZ f gdV f · gdV.(1.10)ΩΩThe following theorem is a far reaching generalization of Eq. (1.10).Theorem 1.9 (Divergence Theorem). Let Ω Rn be an open boundedregion with smooth boundary, n : Ω Rn be the unit outward pointingnormal to Ω. If Z C 1 (Ω̄, Rn ), thenZZZ(x) · n(x)dσ(x) · Z(x) dx.(1.11)Ω ΩCorollary 1.10 (Integration by parts). Let Ω Rn be an open boundedregion with smooth boundary, n : Ω Rn be the unit outward pointingnormal to Ω. If Z C 1 (Ω̄, Rn ) and f f C 1 (Ω̄, R), thenZZZf (x) ·Z(x) dx f (x)·Z(x) dx f (x) Z(x)·n(x)dσ(x). (1.12)ΩΩ ΩAlso if g C 2 (Ω̄, R), thenZZZf (x) g(x) dx f (x)· g(x) dx f (x) g(x)·n(x)dσ(x). (1.13)ΩΩ ΩProof. Eq. (1.12) follows by applying Theorem 1.9 with Z replaced by f Zmaking use of the fact that · (f Z) f · Z f · Z.Eq. (1.13) follows from Eq. (1.12) by taking Z g.1.2.1 Cylindrical coordinatesRecall that cylindrical coordinates, see Figures 1.1, are determined by(x, y, z) R(ρ, θ, z) (ρ cos θ, ρ sin θ, z).In these coordinates we havedV ρdρdθdz.Page: 6job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

1.2 Cylindrical and Spherical Coordinates7Fig. 1.1. Cylindrical and polar coordinates.Proposition 1.11 (Laplacian in Cylindrical Coordinates). The Laplacian in cylindrical coordinates is given by f 11 ρ (ρ ρ f ) 2 θ2 f z2 f.ρρ(1.14)Proof. We further observe thatRρ (ρ, θ, z) (cos θ, sin θ, 0)Rθ (ρ, θ, z) ( ρ sin θ, ρ cos θ, 0)Rz (ρ, θ, z) (0, 0, 1)so that Rρ (ρ, θ, z), ρ 1 Rθ (ρ, θ, z), Rz (ρ, θ, z)is an orthonormal basis for R3 . Therefore,( f, g) ( f, Rρ ) ( g, Rρ ) f, ρ 1 Rθ1 f g f g f g . ρ ρ ρ2 θ θ z z g, ρ 1 Rθ ( f, Rz ) ( g, Rz )If g has compact support in a region Ω, then by integration by parts,Page: 7job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

81 PreliminariesZZ f gdV f · gdV ZΩ 1 f g f g f g 2 · ρdρdθdz ρ θ θ z zΩ ρ ρ Z 1 ρ (ρ ρ f ) θ2 f ρ z2 f g · dρdθdzρΩ Z 11 22 ρ (ρ ρ f ) 2 θ f z f gρdρdθdzρΩ ρ Z 11 22 ρ (ρ ρ f ) 2 θ f z f gdV.ρΩ ρΩSince this formula holds for arbitrary g with small support, we conclude that f 11 ρ (ρ ρ f ) 2 θ2 f z2 f.ρρ1.2.2 Spherical coordinatesWe will now work out the Laplacian in spherical coordinates by a similarmethod. Recall that spherical coordinates, see Figures 1.2, are determined byFig. 1.2. Defining spherical coordinates of a point in R3 .(x, y, z) R(r, θ, ϕ) (r sin ϕ cos θ, r sin ϕ sin θ, r cos ϕ).In this coordinate systems we havedV r2 sin ϕdrdθdϕ.See Figure 1.3.Page: 8job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

1.2 Cylindrical and Spherical Coordinates9Fig. 1.3. A picture proof that dxdydz r2 sin φdrdθdφ, where r2 sin φdrdθdφ shouldbe viewed as (r sin φdθ)(rdφ)dr.Proposition 1.12 (Laplacian in spherical coordinates). The Laplacianin spherical coordinates is given by f 111 r (r2 r f ) 2 ϕ (sin ϕ ϕ f ) 2 2 θ2 f.r2r sin ϕr sin ϕ(1.15)A simple computation shows (as will be needed later) that11 r (r2 r f ) r2 (rf ).2rr(1.16)Proof. SinceRr (r, θ, ϕ) (sin ϕ cos θ, sin ϕ sin θ, cos ϕ)Rθ (r, θ, ϕ) ( r sin ϕ sin θ, r sin ϕ cos θ, 0)Rϕ (r, θ, ϕ) (r cos ϕ cos θ, r cos ϕ sin θ, r sin ϕ)it is easily verified that Rr (r, θ, ϕ), 11Rθ (ρ, θ, z), Rϕ (ρ, θ, z)r sin ϕris an orthonormal basis for R3 . Therefore( f, g) ( f, Rr ) ( g, Rr ) 1( f, Rθ ) ( g, Rθ )r2 sin2 ϕ1( f, Rϕ ) ( g, Rϕ )r2 f g1 f g1 f g 2 2 2. r rr sin ϕ θ θ r ϕ ϕ If g has compact support in a region Ω, thenPage: 9job: 110notesmacro: svmono.clsdate/time: 25-May-2004/11:28

101 PreliminariesZΩZ f gdV f · gdVΩ Z 11 f g f g f g 2 2 · r2 sin ϕdrdϕdθ r sin ϕ θ θ r2 ϕ ϕΩ r r Z 1 f g f g f g2 sin ϕ· drdϕdθr sin ϕ r rsin ϕ θ θ ϕ ϕΩ Z 1 2 r r2 r g sin ϕ f ϕ (sin ϕ · ϕ f ) g · drdϕdθsinϕ θΩ Z 11122 r r g 2 2 θ f 2 ϕ (sin ϕ ϕ f ) g · r2 sin ϕdrdϕdθ2 rr sin ϕr sin ϕΩ r Z 11122 r r g 2 2 θ f 2 ϕ (sin ϕ ϕ f ) g · dV.2 rr sin ϕr sin ϕΩ rSince this formula holds for arbitrary g we conclude that f 111 r (r2 r f ) 2 ϕ (sin ϕ ϕ f ) 2 2 θ2 f.2rr sin ϕr sin ϕ1.2.3 ExercisesIn the following two exercises, I am using the conventions in the Lecture notesand not the book.Exercise 1.4. Compute f where f is given in cylindrical coordinates as:f ρ3 cos θ zρExercise 1.5. Compute f where f is given in spherical coordinates as:f r 1 cos θ sin ϕ.Page: 10 job: 110notes macro: svmono.cls date/time: 25-May-2004/11:28

2PDE Examples2.1 The Wave EquationExample 2.1 (Wave Equation for a String). Suppose that we have a stretchedstring supported at x 0 and x L and y 0. Suppose that the string onlyundergoes vertical motion (pretty bad assumption). Let u(t, x) and T (t, x)denote the height and tension respectively of the string at (t, x), δ(x) denotethe density in equilibrium and T0 be the equilibrium string tension. Let J Fig. 2.1. A piece of displace string[x, x x] [0, L], thenZPJ (t) : ut (t, x)δ(x)dxJis the momentum of the piece of string above J. (Notice that δ(x)dx is theweight of the string above x.) Newton’s equations stateZdPJ (t) utt (t, x)δ(x)dx Force on String.dtJSince the string is to only undergo vertical motion we require

122 PDE ExamplesT (t, x x) cos(αx x ) T (t, x) cos(αx ) 0for all x and therefore that T (t, x) cos(αx ) : H for some constant H, i.e.the horizontal component of the tension is constant. Looking at Figure 2.2,the tension on the piece of string above J [a, b] at the right endpoint b mustFig. 2.2. Computing the net vertical force due to tension on the part of the stringabove [a, b].be given by H (1, ux (t, b)) while the tension at the left endpoint, a, must begiven by H ( 1, ux (t, a)) . So the net tension force on the string above J isZ bH [ux (t, b) ux (t, a)] Huxx (t, x) dx.aFinally there may be a component due to gravity and air resistance, sayZ bgravity gδ(x)dx andZair resistance abk(x)ut (t, x)dx.aSo Newton’s equations becomeZ bZ butt (t, x)δ(x)dx [Huxx (t, x) gδ (x) k(x)ut (t, x)] dx.aaDifferentiating this equation in b at b x then showsutt (t, x)δ(x) Huxx (t, x) gδ(x) k(x)ut (t, x)or equivalently thatutt (t, x) k(x)Huxx (t, x) g ut (t, x).δ(x)δ(x)(2.1)Page: 12 job: 110notes macro: svmono.cls date/time: 25-May-2004/11:28

2.1 The Wave Equation13Example 2.2 (Wave equation. for a drum head). Suppose that u (t, x) represents the height at time t of a drum head over a point x Ω — Ω being thebase of the drum head, see Figure 2.3. As for the string we will make the sim-Fig. 2.3. A deformed membrane attached to a “wire” base. We are also computethe tension density on a region of the membrane above a region V in the plane.plifying assumption that the membrane only moves vertically or equivalentlythat the horizontal component of tension/unit-length is a constant value, H.Let V Ω be a test region and consider the membrane which lie above Vas in Figure 2.3. ThenZPV (t) : ut (t, x)δ(x)dxVis the momentum of the piece of string above V where δ(x)dx is the weightof the membrane above x. Newton’s equations stateZdPV (t) utt (t, x)δ(x)dx Force on membrane.dtVTo find the vertical force on the membrane above V, let x V, then(n (x) , u (t, x) · n (x)) d 0 (x sn (x) , u(t, x sn (x))dsis a vector orthogonal to the boundary of the region above V and by assumption the tension/unit-length at x is H (n (x) , u (t, x) · n (x)) . Thus thevertical component of the force on the membrane above V is given byZZZH u (t, x) · n (x) d (x) H · u (t, x) dx H u (t, x) dx. VVVPage: 13 job: 110notes macro: svmono.cls date/time: 25-May-2004/11:28

142 PDE ExamplesFinally there may be a component due to gravity and air resistance, sayZgravity gδ(x)dx andZVair resistance k(x)ut (t, x)dx.VSo Newton’s equations becomeZZutt (t, x)δ(x)dx [H u (t, x) gδ(x) k(x)ut (t, x)] dx.VVSince V is arbitrary, this impliesδ(x)utt (t, x) H u (t, x) gδ(x) k(x)ut (t, x)or equivalently thatutt (t, x) k(x)H u (t, x) g ut (t, x).δ (x)δ(x)(2.2)Example 2.3 (Wave equation for a metal bar). Suppose that have a metal wirewhich we is going to be deformed and then released. We would like to find theequation that the displacement u (t, x) of the section of the bar originally atlocation x must solve, see Figure 2.4 below.Fig. 2.4. The picture represents an elastic bar in its un-deformed state and then in adeformed state. The quantity y (t, x) represents the displacement of the section thatwas originally at location x in the un-deformed bar. In the above figure y (t, x) 0which y (t, x x) 0.Page: 14 job: 110notes macro: svmono.cls date/time: 25-May-2004/11:28

2.1 The Wave Equation15To do this will write down Newton’s equation of motion. First off, thelongitudinal force on the left face of the section which was originally betweenx and x x is approximately, AEu (t, x x) u(t, x), xwhere E is Young’s modulus of elasticity and A is the area of the bar. (Theminus represents the fact that we must pull to the left to get the currentconfiguration in the figure.) Letting x 0, we find the force of the sectionthat was originally at x is given by AEux (t, x) . Now suppose that x isnot necessarily small. Then we have the momentum of the region of the baroriginally between x and x x is given byZx xut (t, x) δ (x) dxxwhere δ (x) i

8.1 Interpretation of d’Alembert’s solution to the 1-d wave equation 97 8.2 Solving 1st - order equations using 2nd - order solutions . 97 8.2.1 The Solution to the Heat Equation on R . 99 8.3 Duhamel’s Principle .100 8.4 Application of D

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