1 E3102: A study guide and review, Version 1.2
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1 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 very good shpe. This guide is probbly over-thorough. The test itself will hve bout 6 questions covering the whole course but emphsizing the second hlf (which implicitly includes ll of the first hlf). I ll try to void nything overly tricky. 1.1 Homogeneous Liner PDE s in 2 vribles Seprtion of vribles -go-go Be ble to solve the following seprble problems with homogeneous boundry conditions nd no forcing terms. When in doubt use full-blown seprtion of vribles. Alterntively, if you know the pproprite eigenfunctions you cn solve these by eigenfunction expnsion (the first three problems ll hve eigenfunctions tht re some combintions of sines nd cosines determined by the boundry conditions) 1-D time dependent het flow eqution 1-D vibrting string u 2 t = κ u x 2 2 u t 2 = c2 2 u x 2 Lplce s Eqution 2 u =0in crtesin coordintes (rectngles) nd polr coordintes (disks) 2 u = 2 u x u y 2 =0 2 u = 1 r r r u r + 1 u r 2 θ =0 Note: For the disk, the eigenfunctions re in θ. The seprted equtions for f(r) is n equidimensionl eqution with tril solutions of form f(r) =r p 1
2 Helmholtz Eqution 2 u + λu = 0 in crtesin nd polr coordintes. Note: Helmholtz eqution will give Eigenfunctions in both directions for λ>0. In crtesin coordintes you ll get combintions of sines nd cosines in x nd y. In polr coordintes you ll get sines nd cosines in θ nd the dreded Bessel functions in r. Ifλ<0 (modified helmholtz eqution) you ll only get eigenfunctions in one direction. 1.2 Sturm-Liouville Boundry Vlue problems Regulr Sturm-Liouville Boundry Vlue problems re of the form d df p(x) + q(x)f + λσ(x)f =0 dx dx with generl homogeneous Boundry conditions t x = nd x = b df β 1 f()+β 2 () =0 dx df β 3 f(b)+β 4 (b) =0 dx nd p>0, σ>0 for x b. (This eigenvlue problem cn lso be written s L(f) = λσf.) Know: the bsic properties of these problems (pge 157 in Hbermn). 1. They hve n infinite number of Rel eigenvlues λ 1 <λ 2 <... < λ n n 2. For ech eigenvlue λ n there is corresponding unique eigenfunction φ n (x) (note: uniqueness is only for 1-D problems without periodic BC s). 3. The eigenfunctions re orthogonl under weight σ,i.e. b φ n φ m σdx =0 form n 4. The eigenfunctions re complete in the sense tht ny piecewise smooth function g(x, t) cn be written in terms of n infinite series of the eigenfunctions. i.e. g(x, t) = n (t)φ n (x) 2 n=1
3 where the coefficients n (t) re defined by the integrls n (t) = b g(x, t)φ n(x)σ(x)dx b φ2 n(x)σ(x)dx Also know Green s formul for SL problems (nd where it cn be useful) b ( ulv vlu = p(x) u dv dx v du ) b dx for ny functions u(x) nd v(x). Also know for which boundry conditions, the right hnd side vnishes (nd therefore L is self-djoint). Ryleigh Quotient nd how to use it to estimte eigenvlues (or show if positive) b λ = φlφdx b φ2 n(x)σ(x)dx 1.3 Fourier Series/Generlized Fourier Series Understnd how Fourier series re specil cse of Sturm-Liouville theory Be ble to sketch full Fourier Series, Fourier Sine Series nd Fourier Cosine series Relize tht Fourier-Bessel series work the sme wy. I.e. for disk 0 < r<icn expnd ny function g(r) (bounded t r =0) in terms of bessel functions,e.g. g(r) = n J m (z mn r/) n=1 where 0 n = g(r)j m(z mn r/)rdr J 0 m(z 2 mn r/)rdr i.e. φ n (r) =J m (z m nr/) nd σ = r. Understnd when, nd when not, to differentite these infinite series termby-term. (i.e. it is oky for continuous functions g with the sme boundry conditions s the eigenfunctions). 3
4 1.4 Fourier Trnsforms The fourier trnsform f(k) =F [f(x)] = 1 2π nd inverse fourier trnsform f(x) =F 1 [ f(k) ] = f(x)e ikx dx f(k)e ikx dk re the nturl extension of the full fourier series for the infinite line <x<. Note, in generl f is complex number even if f is rel. Some other useful properties of the fourier trnsform tht we tlked bout in clss re tht they cn be extended to functions of multiple vribles nd their derivtives. For exmple if u(x, t) is well behved such tht u(x, t)dx < then ũ(k, t) =F [u(x, t)] = 1 u(x, t)e ikx dx 2π [ ] u F = ũ (1) t t [ ] u F = ikũ (2) x [ ] 2 u F = k 2 ũ (3) x 2 which cn be used to trnsform PDE s nd their boundry condions into ODES Homogeneous Liner PDE s in 3 or more vribles These problems re just more complicted versions of the first set of problems. In generl they cn be solved by either seprtion of vribles or eigenfunction expnsion. For eigenfunction expnsion, however, it is most useful to use the 2-D eigenfunctions of Helmholtz eqution. Bsic problems re Time dependent het-flow on rectngle u(t, x, y) or disk u(t, r, θ) u t = κ 2 u 4
5 Time dependent vibrtions of 2-d membrne 2 u t 2 = c2 2 u 3-D Lplce eqution 2 u =0on rectngulr solid or cylinder For the first two problems you cn lwys seprte out the time dependent prts using u(t, x) =h(t)w(x) where x =(x, y) for crtesin problems nd x =(r, θ) for polr problems. In both cses, w will stisfy Helmholtz Eqution 2 w + λw =0 Properties of Helmholtz eqution For λ > 0 nd w hving homogeneous boundry conditions on some domin R (e.g. rectngle or disk), then mny of the properties of the 1-D Sturm-Liouville theory re relevnt to the 2-D (or 3- D) Eigenfunction problem defined by Helmholtz Eqution (See Sections , pges ). Importnt exmples re There re n infinite number of rel eigenvlues λ 1 < λ 2 <... < λ n n For ech eigenvlue there my be multiple orthogonl eigenfunctions (this is different from the 1-D cse). Eigenfunctions with different eigenvlues re orthogonl with regrd to the re integrl over the domin R φ i φ j dxdy =0 fori j R this cn lso be mde generlly true for ny two eigenfunctions with the sme eigenvlue. (see below) The eigenfunctions re complete in the sense tht ny piecewise-smooth 2-D function cn be written s n infinite sum of ppropritely weighted eigenfunctions g(x, y) = i φ i (x, y) i where i = g(x, y)φ R i(x, y)da (x, y)da 5 R φ2 i
6 Some exmple solutions of Helmholtz Eq. rectngulr region 0 x L, 0 y H with w =0on the boundry λ mn = ( nπ L ) 2 ( mπ ) 2 + φ mn (x, y) =sin nπx H L sin mπy H Note 1: if L=H (squre region), then φ mn nd φ nm re orthogonl but hve the sme eigenvlue λ mn = λ nm Note 2: if the boundries in x re homogeneous but insted were w/ x(0) = w/ x(l) =0, the eigenvlues would be the sme (nd lso include λ =0), but the eigenfunctions would be φ mn =cos nπx mπy sin L H circulr disk 0 r, π θ π with w(, θ) =0on the boundry (nd w(0,θ) is bounded). ( zmn ) 2 λ mn = with two orthogonl eigenfunctions for ech λ mn. ( φ 1 r ) mn(r, θ) =J m z mn cos mθ φ 2mn(r, ( r ) θ) =J m z mn sin mθ where J m (r) is the Bessel function of the first kind of order m nd z mn is the nth zero of the mth Bessel function. Note for the disk da = rdrdθ. 1.6 Non-Homogeneous PDE s nd method of eigenfunction expnsion Here we extended the homogeneous problems to problems with both non-homogeneous source terms nd non-homogeneous boundry conditions. For the ltter problems however it ws lwys possible to set u(t, x) =v(t, x) +r(t, x) where r is ny function tht stisfies the non-homogeneous boundry conditions. Substituting this into the originl PDE, will produce new eqution for v where v hs homogeneous boundry conditions. Given these reduced problems, there is generl recipe for solving the non-homogeneous source terms using the method of eigenfunction expnsion which I will illustrte with the simplified time-dependent problem v t = 1 Lv + Q(t, x) σ(x) 6
7 with v(x, 0) = f(x) nd v hs homogeneous boundry conditions nd L is 2nd order differentil opertor tht only includes sptil derivtives. Some exmples in 1-D would be Lv = k 2 v/ x 2, σ =1,orLv = k r f, σ = r for polr coordintes, or in 2-D r r nd 3-D Lv = k 2 v nd σ =1. 1. Use seprtion of vribles on the ssocited homogeneous problem (ssume Q = 0) to find the eigenvlues nd eigenfunctions of the sptil boundry vlue problem Lφ n = λ n σφ n nd φ n hs the sme homogeneous boundry conditions s v. 2. Expnd both the solution nd Q in terms of these eigenfunctions, e.g. v(x,t)= n n (t)φ n (x) Q(x,t)= n q n (t)φ n (x) 3. Substitute these sums into the PDE for v (nd you cn tke ll the derivtives term by term becuse v nd φ n ll hve the sme boundry conditions) to get [ ] dn dt + λ n n (t) q n (t) φ n (x) =0 i where we hve used the reltionship Lv = i n (t)lφ n (x) = i n (t)λ n σφ n (x) = using the definition of the eigenfunctions of L 4. Use orthogonlity of the φ n s to get the set of 1st-order Non-homogeneous Ordinry differentil equtions d n dt + λ n n = q n (t) 5. solve this using vrition of prmeters (nd initil conditions) to find n (t) (nd therefore v). Here you will need to use the initil conditions n (0) = f(x)φ R nσda R φ2 nσda 6. reconstruct the full solution u(x, t) =v(x, t)+r(x, t)...the end 7
8 simple problems with equilibrium solutions In ddition to the full eigenfunction expnsion technique, sometimes it is esier to solve problems with stedy forcing terms Q(x) by looking for stedy stte solution u e (x) tht stisfies Lu e = Q(x) nd non-homogeneous boundry conditions, then look for trnsient solution v(x, t) of the remining homogeneous problem with homogeneous BC s. nd reconstruct the full solution u(x, t) =v(x, t) +u e (x). For exmple you cn use this to solve the het flow eqution with stedy forcing nd fixed temperture boundry conditions. 1.7 Green s Functions Given boundry vlue problem of form Lu = f(x) with homogeneous boundry conditions, find the Green s Functions with the sme boundry conditions defined by LG(x, x 0 )=δ(x x 0 ) where δ(x x 0 ) is Dirc delt function t point x 0.GivenG(x, x 0 ) the generl solution for u is u(x, t) = f(x 0 )G(x, x 0 )dx 0 Bsic problems 1. Understnd bsic behviour of the delt function i.e. therefore f(x 0 )= 1= R f(x)δ(x x 0 )dx δ(x x 0 )dx nd thus the Heviside step function is defined by H(x x 0 )= 8 x δ(x x 0 )dx
9 2. Know how to find the 1-D green s functions for Sturm Louiville operters (in prticulr Lu = d 2 u/dx 2 or Lu = d 2 u/dx 2 +u) nd pproprite boundry conditions. The bsic recipe is () find the generl solution of the homogeneous problem wy from the delt spike (b) Find specific solution for either side of the delt spike tht stisfies the homogeneous eqution nd one boundry condition. (c) Mke the functions continuous t the delt function (x = x 0 ) (d) integrte the ODE for smll region round x 0 nd find the constrints on the derivtives dg/dx on either side of x 0. And tht should do it. 3. Use the sme pproch to find the infinite spce Green s functions for 2 u = f(x) in 2 nd 3-D. 1.8 Wve Equtions nd the method of chrcteristics Understnd How to find solutions of simple 1-D wve eqution w t + c w x =0 with initil conditions w(x, 0) = f(x) on <x<, using the method of chrcteristics. Know how to extend it to more generl liner problems like w t + c(x, t) w x = w nd to non-liner shock problems like with w(x, 0) = f(x) w t + w w x =0 for ech of these problems know how to qulittively sketch wht is going on in spce nd time. A grphicl nswer will go long wy. 9
10 1.9 P.S. Tht s it for now...wtch this spce for nything new nd/or corrections. if you hve ny questions come nd see me in office hours or send me e-mil t mspieg@ldeo.columbi.edu to set up n ppointment. Good luck nd relx. 10
1 E3102: a study guide and review, Version 1.0
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