1 E3102: a study guide and review, Version 1.0


 Evelyn Woods
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1 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 good shpe. If I get my ct together I ll try to point out representtive problems from the homework for ech section. This guide is probbly overthorough. The test itself will hve bout 6 questions covering the whole course but emphsizing the bsic concepts. I ll try to void nything tricky or 5pt lnd mines. 1.1 Homogeneous Liner PDE s in 2 vribles Seprtion of vribles out the wzoo Be ble to solve the following seprble problems with homogeneous boundry conditions nd no forcing terms. When in doubt use fullblown 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) 1D time dependent het flow eqution 1D 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 SturmLiouville Boundry Vlue problems Regulr SturmLiouville 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 1D problems without periodic BC s). 3. The eigenfunctions re orthogonl under weight σ,i.e. 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) = g(x, t)φ n(x)σ(x)dx φ2 n(x)σ(x)dx Also know Green s formul for SL problems (nd where it cn be useful) ( ulv vlu = p(x) u dv dx v du ) b dx for ny functions u(x) nd v(x). Ryleigh Quotient nd how to use it to estimte eigenvlues (or show if positive) λ = φlφdx b φ2 n(x)σ(x)dx 1.3 Fourier Series/Generlized Fourier Series Understnd how Fourier series re specil cse of SturmLiouville theory Be ble to sketch full Fourier Series, Fourier Sine Series nd Fourier Cosine series Relize tht fourierbessel 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 termbyterm. (i.e. it is oky for continuous functions g with the sme boundry conditions s the eigenfunctions). 3
4 1.4 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 2D eigenfunctions of Helmholtz eqution. Bsic problems re Time dependent hetflow on rectngle u(t, x, y) or disk u(t, r, θ) u t = κ 2 u Time dependent vibrtions of 2d membrne 2 u t = 2 c2 2 u 3D 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 1D SturmLiouville theory re relevnt to the 2D (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 1D 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) 4
5 The eigenfunctions re complete in the sense tht ny piecewisesmooth 2D 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 R φ2 i 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 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. 1.5 NonHomogeneous PDE s nd method of eigenfunction expnsion Here we extended the homogeneous problems to problems with both nonhomogeneous source terms nd nonhomogeneous boundry conditions. For the ltter problems 5
6 however it ws lwys possible to set u(t, x) =v(t, x) +r(t, x) where r is ny function tht stisfies the nonhomogeneous 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 nonhomogeneous source terms using the method of eigenfunction expnsion which I will illustrte with the simplified timedependent problem v = Lv + Q(t, x) t with v(x, 0) = f(x) nd v hs homogeneous boundry conditions nd L is 2nd order differentil opertor tht only includes sptil derivtives (e.g. Lv = k 2 v/ x 2 in 1D or Lv = k 2 v in 2D or 3D.) 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 φ 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 where we hve used the reltionship i Lv = i n (t)lφ n (x) = i n (t)λ n φ n (x) = using the definition of the eigenfunctions of L 6
7 4. Use orthogonlity of the φ n s to get the set of 1storder Nonhomogeneous 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 R n (0) = f(x)φ nσda R φ2 nσda 6. reconstruct the full solution u(x, t) =v(x, t)+r(x, t)...the end 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 nonhomogeneous 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. 1.6 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 R 7
8 1. Importnt: know how to find the 1D green s functions for Lu = d 2 u/dx 2 nd pproprite boundry conditions. 2. you might lso wnt to know how to find the infinite spce green s functions for 2 u = f(x) in 2 nd 3D. 1.7 Wve Equtions nd the method of chrcteristics Understnd How to find solutions of simple 1D wve eqution w t + c w x =0 with initil conditions w(x, 0) = f(x) using the method of chrcteristics. Know how to extend it to more generl liner problems like nd to nonliner shock problems like with w(x, 0) = f(x) w t + c(x) w x = w 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. 1.8 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 emil t to set up n ppointment. Good luck nd relx. 8
1 E3102: A study guide and review, Version 1.2
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