AMS 212A Applied Mathematical Methods I Lecture 06 Copyright by Hongyun Wang, UCSC. ( ), v (that is, 1 ( ) L i

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1 AMS A Applied Mthemticl Methods I Lecture 6 Copyright y Hongyun Wng, UCSC Recp of Lecture 5 Clssifiction of oundry conditions Dirichlet eumnn Mixed Adjoint opertor, self-djoint opertor Sturm-Liouville prolem in self-djoint form: Lu ( ) r( x)u, x (, ) u( )+ u( ) u ( )+ u( ) with respect to the inner product u, v u( x)v( x)r( x)dx where the liner differentil opertor is Lu ( ) p( x)u x ( )u ( ) x + q x Results of Sturm-Liouville theory: u, r( x) L( v) r( x) Lu All eigenvlues re rel. ( ), v (tht is, r x ( ) L i For ech eigenvlue, we cn find rel eigenfunction. () is self-djoint). Eigenfunctions for different eigenvlues re orthogonl to ech other. Results of Sturm-Liouville theory (continued) Since eigenvlues nd eigenfunctions re oth rel, we only need to work with rel functions. All eigenvlues re simple. (i.e., one corresponding eigenfunction for ech eigenvlue.) - -

2 AMS A Applied Mthemticl Methods I Suppose Clim: Proof: u(x) nd v(x) re eigenfunctions corresponding to eigenvlue. u(x) nd v(x) re proportionl to ech other. Lu ( ) r( x)u L( v) r( x)v > ul( v) vl( u) u( r( x)v) v( r( x)u) On the other hnd, using the differentil form of L( ), we write ul( v) vl( u) u( ( pv x ) x + qv) v( ( pu x ) x + qu) u( pv x ) x vpu ( x ) x (Using Lemm of Lecture 5) ( ) x puv ( x vu x ) Comining these two results, we hve ( puv ( x vu x )) x > puv ( x vu x ) const Since oth u(x) nd v(x) stisfy the oundry conditions, we hve ( uv vu) x (Lemm of Lecture 5) [ ] > pu ( v vu), x, > uv vu, > x [, ] v u, x, v > u const End of proof [ ] Eigenvlue sequence Eigenvlues form n unounded strictly incresing infinite sequence < < < < n < - -

3 AMS A Applied Mthemticl Methods I lim n + n+ We will prove this fter we descrie the completeness of eigenfunctions. Completeness of eigenfunctions The set of eigenfunctions v ( x), v ( x), v ( x),, ( x), piecewise continuous functions on [, ]. Tht is, ny piecewise continuous function ƒ(x) cn e expnded s f( x) x ( ) where coefficient is given y f,, { } is complete sis for ll f( x) ( x)r( x)dx ( ( x) ) r x ( )dx Below we do 3 things: prove the eigenvlue sequence, introduce the Ryleigh quotient, nd then prove the completeness Proof of eigenvlue sequence Eigenvlue nd eigenfunction u(x) stisfy the Sturm-Liouville eqution ( p( x)u x ) x + q( x)u r( x)u, x (, ) nd oundry conditions u( )+ u( ) u ( )+ u( ) The key step of the proof is Prufer sustitution: ( ) p( x)u x ( x) ( x)cos ( x) u( x) ( x)sin( ( x) ) Before the Prufer sustitution, we hve second order liner ODE for u(x). After the Prufer sustitution, (s we will see) we hve - 3 -

4 AMS A Applied Mthemticl Methods I first order non-liner ODE system for (x) nd (x). Let us look t the dvntges of Prufer sustitution. Advntge #: [ ] ( x), x, Suppose ( x ) t some point x in [, ]. > p(x ) p( x )u x ( x ) u( x ) > u x ( x ) u( x ) Tht is, u(x) stisfies the zero initil conditions t x. The Sturm-Liouville eqution is second order ODE of u(x) > u( x) (due to the zero initil conditions t x ) In the Sturm-Liouville prolem, we re only interested in non-trivil solutions. Advntge #: Boundry conditions ffect only Boundry condition t x : u( )+ u( ) ( ) > ( )sin( ( ) )+ p ( ) > p ( )sin( ( ) )+ cos( ( ) ) > sin( ( ) ) cos ( ( )) where stisfies cos( ) ( ) ( ) p, sin( ) ( p) + The oundry condition ecomes ( ) n ( p ( )) + > ( ) + n We cn shift the whole function (x) to write the oundry condition t x s - 4 -

5 AMS A Applied Mthemticl Methods I BC t x : ( ), < After the shifting, we write the oundry condition t x s BC t x : ( ) μ + n, < μ ote: we use shifting nd pick suitle vlue of n to mke < nd < μ, which is importnt in the proof of Lemm nd proof of eigenvlue sequence. Advntge #3: The evolution eqution of (x) is independent of (x). (x) stisfies the first order non-liner ODE x ( r( x)+ q( x) )sin + p x ( ) cos Key steps in derivtion: equtions for (x) nd (x): Eqution : ( p( x)u x ) x + ( r( x)+ q( x) )u (the Sturm-Liouville eqution) > ( cos) x + ( r( x)+ q( x) )sin Eqution : cos p( x)u x p( x) ( sin ) x (See Exercise #6 for derivtion). Lemm : Let (x, ) e the solution of x ( r( x)+ q( x) )sin + p x ( ), < ( ) cos where p(x) > nd r(x) > for x in [, ] (from the Sturm-Liouville prolem). Then, (x, ) hs the 3 properties elow. For x >, ) (x, ) is strictly incresing function of. ) lim x, + c) lim x, (For proof of Lemm, see Appendices of Lecture 6 in seprte PDF file)

6 AMS A Applied Mthemticl Methods I ow we use the results of Lemm to prove the eigenvlue sequence. (, ) < μ (since < μ ) lim > μ < (, )< μ for negtive nd lrge. As increses to infinity, (, ) lso increses to infinity. At some vlue,, we will hve (, ) μ. > is the smllest vlue of for which (, ) μ + n is stisfies. > is the smllest eigenvlue. As increses further, t some vlue,, we will hve (, ) μ + ; > is the smllest eigenvlue fter. At some vlue,, we will hve (, ) μ + ; > is the smllest eigenvlue fter. At some vlue, 3, we will hve (, 3 ) μ + 3; > 3 is the smllest eigenvlue fter. Thus, we hve strictly incresing sequence of eigenvlues < < < < n < ext we prove lim n +. n+ Since { n } increses monotoniclly, we hve either lim n + or lim n finite. n+ n+ Suppose lim n ( c) finite. We hve n+ ( c) > n for ny n > (, ( c) )> (, n ) μ + n for ny n > (, ( c) )+, which contrdicts with ( c) finite. End of proof of eigenvlue sequence Ryleigh quotient Definition: - 6 -

7 AMS A Applied Mthemticl Methods I Ryleigh quotient is defined s Ru ( ) u, r( x) Lu ( ) u, u ul( u)dx u r( x)dx Ryleigh quotient on eigenfunctions: Suppose < < < < n < is the eigenvlue sequence nd { v ( x), v ( x), v ( x),, ( x), } is the sequence of corresponding eigenfunction. It is strightforwrd to verify tht Ryleigh quotient stisfies Rv ( n ) n, n,,, R min n { } R,, (See Exercise #6 for derivtion). This lst result motivtes the Ryleigh principle elow. Ryleigh principle: Let C BC u( x)u C nd stisfies u ( )+ u ( ) u ( )+ u( ) Consider the minimiztion of R(u) over C BC. Let u rg min Ru ( ) uc BC, u Clim : u is n eigenfunction corresponding to eigenvlue nd min Ru ( ) uc BC, u Proof of Clim : Consider function of sclr vrile s - 7 -

8 AMS A Applied Mthemticl Methods I gs ( ) Ru ( + su) where u C BC By definition of u, we hve g( ) for ny u C BC The derivtives of numertor nd denomintor of R(u + s u) re d ds u + su, u + su s u, u d ds u + su, r x ( ) Lu ( + su) s (See Exercise #6 for derivtion). u, r( x) Lu ( ) The derivtive of R(u + s u) hs the expression d ds Ru ( + su) s u, u, u r( x) Lu ( ) + Ru ( ) u, u u, u u, (See Exercise #6 for derivtion). r( x) Lu ( )+ Ru ( )u The derivtive of R(u + s u) is zero for ny u C BC. d ds Ru ( + su) for ny u C BC s > u, r( x) Lu ( )+ Ru ( )u for ny u C BC > r( x) ( )+ Ru ( )u > Lu ( ) R( u )r( x)u > R(u ) is n eigenvlue nd u is corresponding eigenfunction. ext we show Ru ( ). By definition of u, we hve Ru ( ) min Ru ( ) uc BC > Ru ( ) Rv ( ) (v is n eigenfunction corresponding to ). On the other hnd, R(u ) is n eigenvlue nd is the smllest eigenvlue. Thus, we hve - 8 -

9 AMS A Applied Mthemticl Methods I Ru ( ) Comining these two results, we conclude Ru ( ) min Ru ( ) End of proof of Clim uc BC ext we look t the minimum of Ryleigh quotient over suset of C BC. Let W spn{ v,, v } { nd u W } W uuc BC Consider the minimiztion of R(u) over W. Let u + rg min Ru ( ) uw, u Clim : u + is n eigenfunction corresponding to eigenvlue + nd + min Ru ( ) uw, u (For proof of Clim, see Appendices of Lecture 6 in seprte PDF file) Ryleigh principle provides wy of pproximting the lowest few eigenvlues, especilly the smllest eigenvlue. Exmple: u u u( ), u() This is Sturm Liouville prolem with p( x), r( x), q( x), Lu ( ) u We solved it previously. The eigenvlues re n ( n + ), n,,, ow we use Ryleigh principle to pproximte the smllest eigenvlue. (Skip the detils in Lecture) We try w( x) x( x) (it hs to stisfy the oundry conditions) - 9 -

10 AMS A Applied Mthemticl Methods I The numertor nd denomintor of Ryleigh quotient re w, r( x) L( w) w( x) w ( x)dx x( x)dx 3 ( ( )) dx w, w w x x x 3 ( ) dx > Rw ( ) w, r x w, w ( ) L( w) > min Ru ( ) Rw ( ) uc BC The true vlue is Proof of the completeness of eigenfunctions For piecewise continuous function ƒ(x), we show tht where lim f n f,, u, v u( x)v( x)r( x)dx u u, u The proof consists of results -4 elow. Result : f v j for j This follows directly from the definition of. Result : f min { c n } f c n - -

11 AMS A Applied Mthemticl Methods I This follows directly from Result ove f + v n n, f + n f, f v n n + n, n > f c n f Result 3: We only need to show tht for piecewise continuous function ƒ(x), we hve Result 4: for ny > there exists nd {c n } such tht f c n < (***) Since C BC is dense in the set of piecewise continuous functions, we only need to show tht for f( x)c BC, we hve for ny > there exists nd {c n } such tht f c n < (***) For proof of (***), see Appendices of Lecture 6 in seprte PDF file. - -

AMS 212A Applied Mathematical Methods I Appendices of Lecture 06 Copyright by Hongyun Wang, UCSC. ( ) cos2

AMS 212A Applied Mathematical Methods I Appendices of Lecture 06 Copyright by Hongyun Wang, UCSC. ( ) cos2 AMS 22A Applied Mathematical Methods I Appendices of Lecture 06 Copyright by Hongyun Wang UCSC Appendix A: Proof of Lemma Lemma : Let (x ) be the solution of x ( r( x)+ q( x) )sin 2 + ( a) 0 < cos2 where