Applied Prtil Differentil Equtions with Fourier Series nd Boundry Vlue Problems 5th Edition Richrd Hbermn
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If we directly pply (5.30), then Sturm Liouville Eigenvlue Problems (λ 2 λ 1 )u v =0. Thus, if λ 1 λ 2 (different eigenvlues), the corresponding eigenvectors re orthogonl in the sense tht u v =0. (5.31) We leve s n exercise the proof tht the eigenvlues of symmetric rel mtrix re rel. EXAMPLE 6 2 The eigenvlues of the rel symmetric mtrix re determined from (6 λ) 2 3 (3 λ) 4=λ 2 9λ +14=(λ 7)(λ 2) = 0. For λ = 2, the eigenvector stisfies nd hence x1 x 2 6x 1 +2x 2 =2x 1 nd 2x 1 +3x 2 =2x 2, 1 = x 1. For λ = 7, it follows tht 2 6x 1 +2x 2 =7x 1 nd 2x 1 +3x 2 =7x 2, x1 2 nd the eigenvector is = x x 2. As we hve just proved for ny rel symmetric 2 1 1 2 mtrix, the eigenvectors re orthogonl, =2 2=0. 2 1 Eigenvector expnsions. For rel symmetric mtrices it cn be shown tht if n eigenvlue repets R times, there will be R independent eigenvectors corresponding to tht eigenvlue. These eigenvectors re utomticlly orthogonl to ny eigenvectors corresponding to different eigenvlue. The Grm Schmidt procedure cn be pplied so tht ll R eigenvectors corresponding to the sme eigenvlue cn be constructed to be mutully orthogonl. In this mnner, for rel symmetric n n mtrices, n orthogonl eigenvectors cn lwys be obtined. Since these vectors re orthogonl, they spn the n-dimensionl vector spce nd my be chosen s bsis vectors. Any vector v my be represented in series of the eigenvectors: v = c i φ i, (5.32) where φ i is the ith eigenvector. For regulr Sturm Liouville eigenvlue problems, the eigenfunctions re complete, mening tht ny (piecewise smooth) function cn be represented in terms of n eigenfunction expnsion f(x) c i φ i (x). (5.33) 190
Sturm Liouville Eigenvlue Problems This is nlogous to (5.32). In (5.33) the Fourier coefficients c i re determined by the orthogonlity of the eigenfunctions. Similrly, the coordintes c i in (5.32) re determined by the orthogonlity of the eigenvectors. We dot Eqution (5.32) into φ m : v φ m = c i φ i φ m = c m φ m φ m, since φ i φ m =0,i m, determining c m. Liner systems. Sturm Liouville eigenvlue problems rise in seprting vribles for prtil differentil equtions. One wy in which the mtrix eigenvlue problem occurs is in seprting liner homogeneous system of ordinry differentil equtions with constnt coefficients. We will be very brief. A liner homogeneous first-order system of differentil equtions my be represented by dv dt = Av, (5.34) where A is n n n mtrix nd v is the desired n-dimensionl vector solution. v usully stisfies given initil conditions, v(0) = v 0. We seek specil solutions of the form of simple exponentils: v(t) =e λt φ, (5.35) where φ is constnt vector. This is nlogous to seeking product solutions by the method of seprtion of vribles. Since dv/dt = λe λt φ, it follows tht Aφ = λφ. (5.36) Thus, there exist solutions to (5.34) of the form (5.35) if λ is n eigenvlue of A nd φ is corresponding eigenvector. We now restrict our ttention to rel symmetric mtrices A. There will lwys be n mutully orthogonl eigenvectors φ i. We hve obtined n specil solutions to the liner homogeneous system (5.34). A principle of superposition exists, nd hence liner combintion of these solutions lso stisfies (5.34): v = c i e λit φ i. (5.37) We ttempt to determine c i so tht (5.37) stisfies the initil conditions, v(0) = v 0 : v 0 = c i φ i. 191
Sturm Liouville Eigenvlue Problems Here, the orthogonlity of the eigenvectors is helpful, nd thus, s before, c i = v 0 φ i φ i φ i. EXERCISES 5 APPENDIX 5A.1. Prove tht the eigenvlues of rel symmetric mtrices re rel. 5A.2. () Show tht the mtrix 1 0 A = 2 1 hs only one independent eigenvector. (b) Show tht the mtrix 1 0 A = 0 1 hs two independent eigenvectors. 5A.3. Consider the eigenvectors of the mtrix A = 6 4 1 3 () Show tht the eigenvectors re not orthogonl. (b) If the dot product of two vectors is defined s follows,. b = 1 4 1b 1 + 2 b 2, show tht the eigenvectors re orthogonl with this dot product. 5A.4. Solve dv/dt = Av using mtrix methods if 6 2 1 *() A =, v(0) = 2 3 2 1 2 2 (b) A =, v(0) = 2 4 3 5A.5. Show tht the eigenvlues re rel nd the eigenvectors orthogonl: 2 1 () A = 1 4 3 1 i *(b) A = (see Exercise 5A.6) 1+i 1 5A.6. FormtrixA whose entries re complex numbers, the complex conjugte of the trnspose is denoted by A H. For mtrices in which A H = A (clled Hermitin): () Prove tht the eigenvlues re rel. (b) Prove tht eigenvectors corresponding to different eigenvlues re orthogonl (in the sense tht φ i φ m =0,where denotes the complex conjugte). 192
Sturm Liouville Eigenvlue Problems 6 RAYLEIGH QUOTIENT The Ryleigh quotient cn be derived from the Sturm Liouville differentil eqution, d p(x) dφ + q(x)φ + λσ(x)φ =0, (6.1) by multiplying (6.1) by φ nd integrting: φ d ( p dφ ) + qφ 2 Since φ2 σ>0, we cn solve for λ: λ = φ d ( p dφ + λ φ2 σ ) + qφ 2 φ 2 σ=0.. (6.2) Integrtion by prts udv= uv vdu, where u = φ, dv = d/(pdφ/) nd hence du = dφ/, v = pdφ/ yields n expression involving the function φ evluted t the boundry: λ = pφ dφ b + b p ( ) 2 dφ qφ 2 φ2 σ, (6.3) known s the Ryleigh quotient. In Sections 3 nd 4 we hve indicted some pplictions of this result. Further discussion will be given in Section 7. Nonnegtive eigenvlues. Often in physicl problems, the sign of λ is quite importnt. As shown in Section 2.1, dh/dt + λh = 0 in certin het flow problems. Thus, positive λ corresponds to exponentil decy in time, while negtive λ corresponds to exponentil growth. On the other hnd, in certin vibrtion problems (see Section 7), d 2 h/dt 2 = λh. There, only positive λ corresponds to the usully expected oscilltions. Thus, in both types of problems we often expect λ 0: The Ryleigh quotient (6.3) directly proves tht λ 0if () pφ dφ 0, nd (b) q 0. b (6.4) We clim tht both () nd (b) re physiclly resonble conditions for nonnegtive λ. Consider the boundry constrint, pφ dφ/ b 0. The simplest types of homogeneous boundry conditions, φ = 0nddφ/ = 0, do not contribute to this boundry 193
Sturm Liouville Eigenvlue Problems term, stisfying (). The condition dφ/ = hφ (for the physicl cses of Newton s lw of cooling or the elstic boundry condition) hs h>0 t the left end, x =. Thus,it will hve positive contribution t x =. The sign switch t the right end, which occurs for this type of boundry condition, will lso cuse positive contribution. The periodic boundry condition e.g., φ() = φ(b) nd p() dφ/() = p(b) dφ/(b) s well s the singulrity condition φ() bounded, if p() = 0 lso do not contribute. Thus, in ll these cses pφ dφ/ b 0. The source constrint q 0 lso hs mening in physicl problems. For het flow problems, q 0 corresponds (q = α, Q = αu) to n energy-bsorbing (endothermic) rection, while for vibrtion problems, q 0 corresponds (q = α, Q = αu) to restoring force. Minimiztion principle. The Ryleigh quotient cnnot be used to determine explicitly the eigenvlue (since φ is unknown). Nonetheless, it cn be quite useful in estimting the eigenvlues. This is becuse of the following theorem: The minimum vlue of the Ryleigh quotient for ll continuous functions stisfying the boundry conditions (but not necessrily the differentil eqution) is the lowest eigenvlue: λ 1 = min pu du/ b p + (du/) 2 qu 2 u 2 σ, (6.5) where λ 1 represents the smllest eigenvlue. The minimiztion includes ll continuous functions tht stisfy the boundry conditions. The minimum is obtined only for u = φ 1 (x), the lowest eigenfunction. For exmple, the lowest eigenvlue is importnt in het flow problems (see Section 4). Tril functions. Before proving (6.5), we will indicte how (6.5) is pplied to obtin bounds on the lowest eigenvlue. Eqution (6.5) is difficult to pply directly since we do not know how to minimize over ll functions. However, let u T be ny continuous function stisfying the boundry conditions; u T is known s tril function. We compute the Ryleigh quotient of this tril function, RQu T : pu T du T / b + b p (du T /) 2 qu 2 T λ 1 RQu T = u2 T σ. (6.6) We hve noted tht λ 1 must be less thn or equl to the quotient since λ 1 is the minimum of the rtio for ll functions. Eqution (6.6) gives n upper bound for the lowest eigenvlue. 194