(9) P (x)u + Q(x)u + R(x)u =0

Transcription

1 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 Becuse the eqution is liner, ny liner comintion of solutions is gin solution: if u 1,u 2 re solutions of (9) nd c 1,c 2 re constnts then c 1 u 1 (x)+c 2 u 2 (x) is lso solution of (9). Assumtions. 1) It is ssumed ssume tht the coefficients P (x),q(x),r(x) re continuous on n intervl [, ]. (However, jum discontinuities do er in lictions, nd cn lso e ccommodted; we will discuss this lter.) 2) It is ssumed tht P (x) does not vnish in [, ]. Though, we will sometimes let P vnish t or. (Points where P (x) is zero re singulr, solutions re usully very secil t such oints, nd cre is needed.) 3) It cn e ssumed without loss of generlity tht P (x) > 0 on (, ). (Since P (x) is never zero on (, ), nd it is continuous, then P (x) iseither ositive on (, ) or negtive on (, ). If P < 0 we multily the eqution y 1.) Existence nd uniqueness of solution to the initil vlue rolem: given x 0 [, ], so tht P (x 0 ) = 0 nd given the numers u 0,u 0 then there exists unique solution u(x) of (9) so tht u(x 0 )=u 0 nd u (x 0 )=u 0. This solution u(x) is twice differentile (moreover, u is continuous, s it is seen from (9)), nd it deends continuously on the initil conditions. Generl solution. There exist two linerly indeendent solutions of (9): u 1 (x),u 2 (x) solutions for x (, ) so tht the vectors (u 1 (x),u 1 (x)) nd (u 2(x),u 2 (x)) re linerly indeendent t ll x (, ). In fct, u 1 (x),u 2 (x) re linerly indeendent t ll x (, ) is equivlent to u 1 (x 0 ),u 2 (x 0 ) re linerly indeendent t some x 0 (, ). For exmle, the solutions with the initil conditions u 1 (x 0 )=1,u 1 (x 0)= 0 nd u 2 (x 0 )=0,u 2 (x 0) = 1 re linerly indeendent. Any solution of (9) is liner comintion of two indeendent solutions: (10) u(x) =C 1 u 1 (x)+c 2 u 2 (x) for some constnts C 1,2. In fct, C 1 u 1 (x) +C 2 u 2 (x) withc 1,2 ritrry rmeters is clled the generl solution of (9). So the set of ll the solutions of (9) form liner sce of dimension two (the dimension equls the order of the eqution). Note: The generl solution deends on two rmeters, so it mkes sense tht two conditions re required to determine these rmeters. However, there is no riori gurntee tht solutions stisfying different tyes of rolems, like oundry conditions, do exist.

2 8 RODICA D. COSTIN An equivlent condition for two solutions to e linerly indeendent is tht their Wronskin W [u 1,u 2 ]=u 1u 2 u 1 u 2 stisfies W (x) = 0 for ll x (, ) (equivlently, t some x 0 (, )). Recll tht the Wronskin stisfies the differentil eqution (11) W (x) = Q(x) P (x) W (x) nd therefore W (x) =C ex Q(x) 2.2. The selfdjoint form of liner second order eqution. Consider eigenvlue rolems for equtions (9): (12) P (x)u + Q(x)u + R(x)u + λu =0 We will now show tht ny eqution (12) cn e written in self-djoint form: 1 (13) d w(x) dx (x)du dx + q(x) u = λu or, exnded, (14) u +( q + λw)u =0 where (x),q(x),w(x) re functions which we will determine now. Exnding the left hnd-side of (13) we otin which must e (12), therefore w u + w u + q w + λ u =0 w = P, w = Q, q w = R The first two equtions imly tht / = Q/P therefore Q(x) (15) (x) =ex Then since w = /P nd q = wr we otin (16) w(x) = 1 Q(x) P (x) ex, q(x) = R(x) P (x) ex Q(x)

3 STURM-LIOUVILLE THEORY Homogeneous oundry conditions. These conditions re usully inherited from the PDEs which roduced the ODE (12) y sertion of vriles. If the vlues on the oundry re not zero, sustitutions cn often e mde to ensure zero vlues on the oundry: these re clled homogeneous oundry conditions. These could hve the form: Dirichlet conditions: u() = 0, u() = 0, or Newmn conditions: u () =0, u () = 0, or Mixed Dirichlet-Newmn conditions: (17) B [u] αu()+α u () =0 B [u] βu()+β u () =0 where α, α,β,β re constnts. The mixed conditions re the most generl, s they hve the Dirichlet nd the Neumn conditions s rticulr cses (if α =0=β we otin Dirichlet conditions, nd if α = 0 = β we otin Newmn conditions). Therefore we work with the generl mixed Dirichlet-Newmn conditions. Note: B,B re liner functionls of u. It must e ssumed tht the oundry conditions re nontrivil: the liner functionls B,B re not identiclly zero; this mens tht t lest one of the numers α, α is not zero (note tht this condition cn e written s α + α = 0), nd similrly, t lest one of the numers β,β is not zero (i.e. β + β = 0) Another wy of writing B [u], B [u]. Clerly if we multily α nd α y the sme constnt, we otin the sme oundry condition B [u], nd similrly for β nd β in B [u]. It is sometimes convenient (nd lwys ossile!) to choose these in the form (18) B [u] cos(θ )u() sin(θ )()u () =0 B [u] cos(θ )u() sin(θ )()u () =0 which re very suitle for Prüfer coordintes. The trnsformtion which rings (17) in the form (18) is the following: dividing α nd α y the quntity ± α 2 +(α /()) 2 with the sign chosen to e oosite to the sign of α, we otin B [u] = 0 in the form α 1 u() α 2 ()u () = 0 where α α2 2 = 1 nd α 2 0 therefore there exists θ [0,π) so tht α 1 = cos(θ ) nd α 2 = sin(θ ) (we choose θ <π/2if α 1 > 0 nd θ >π/2ifα 1 < 0). A similr trnsformtion cn e erformed on B. Note tht we cn choose θ in [n, (n + 1)π) for ny integer n (if n is even, we roceed s for the condition t x =, whileifn is odd we choose the oosite sign in front of ± β 2 +(β /()) 2, nmely the sign of β ) Singulr oundry conditions. Other tye of conditions which er in lictions re:

4 10 RODICA D. COSTIN Periodic conditions: if () = () then it cn e required tht the solutions e eriodic: More generlly: u() =u(), nd u () =u () α 1 u()+α 1 u ()+β 1 u()+β 1 u () =0 α 2 u()+α 2 u ()+β 2 u()+β 2 u () =0 If vnishes t n endoint, sy () = 0: then the oundry condition t x = is droed Formultion of the homogeneous Sturm-Liouville rolem. We will consider rel-vlued rolems: the functions P, Q, R nd the numers α, α,β,β re rel. In cse comlex vlued functions re needed, then equtions cn e written nd sertely solved for the rel nd imginry rts of these functions. Note tht with this relity ssumtion we hve (x) > 0 nd w(x) > 0 (see (15), (16)). Given the functions, q, w continuous on [, ] nd, w > 0 on [, ], nd B [u], B [u] (nontrivil) find the numers λ so tht the following rolem hs nontrivil (i.e. nonzero) solution u(x) on [, ]: [(x)u ] +[ q(x)+λw(x)]u =0 (19) Boundry conditions t x = nd x = (20) The oundry conditions re one of the following: regulr conditions: B [u] αu()+α u () =0 ( α + α = 0) B [u] βu()+β u () =0 ( β + β = 0) singulr conditions: if () = 0: (21) B [u] αu()+α u () =0 ( α + α = 0) or, if () = 0: (22) B [u] βu()+β u () =0 ( β + β = 0) (23) or, if oth () = 0, () = 0, then no oundry conditions re ssumed eriodic conditions: (lso singulr) if () = () C[u] u() u() =0 C [u] u () u () =0 The numers λ re clled eigenvlues, nd the corresonding solutions - eigenfunctions.

5 STURM-LIOUVILLE THEORY Green s Identity nd self-djointness of the Sturm-Liouville oertor. We show here tht the rolem (19) is indeed selfdjoint. Let us first show generl formul: Lemm 1. Green s identity: (24) (u ) v u(v ) = (u v uv ) Reltion (24) follows esily using integrtion y rts: (u ) v u(v ) = (u ) v u(v ) = u v (u )v uv + u (v )=(u v uv )