Green function and Eigenfunctions
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1 Green function nd Eigenfunctions Let L e regulr Sturm-Liouville opertor on n intervl (, ) together with regulr oundry conditions. We denote y, φ ( n, x ) the eigenvlues nd corresponding normlized eigenfunctions of L. Tht is, (note the minus sign in front of the nd derivtive) L( u) + x p( x u q( x ) u nd L ( φ ( n, ) φ ( n, x ) () We know tht { φ ( n, x ) } mkes complete orthonorml set in L (, ). Let f e function in L (, ) we write f s the generlized fourier series f( x ) () where f n is the fourier coefficients of f with respect to the se { φ ( n, x ) }. Consider the nonhomogeneous eqution (with the sme oundry condition). L( u) f (3) We know tht the solution cn e expressed s u( G (, ) x z f( dz (4) where G ( x, is the Green function of the Sturm-Liouville oundry vlue prolem Lu 0. Let's ssume tht the solution u of the nonhomogeneous eqution cn e expressed s u( u n φ ( n, x ) (5) We wnt to compre the coefficients u n in (5) with those f n in (). Formlly, we pply the opertor L to the infinite sum (5) to compute L( u ). Using () nd the eqution (3) we hve u n φ (, ) n x f n We then equlize the coefficients of φ ( n, x ) on oth sides nd find u n. Using this in (5) we hve (see the note tht l n follows) u( Importnt: If (6) 0 for some n, tht is if 0 is n eigenvlue for L then the ove mkes no sense (dividing y 0) unless f n 0 ccordingly. We cn see this from the reltion u n f n. Thus, we must hve f n f( x ) φ ( n, x ) dx 0. In other words, The right hnd side f must e orthogonl to ll eigenfunctions which correspond to the 0 eigenvlue, in order the eqution Lu f hs solution. Moreover, if this is the cse then the corresponding u n cn e ritrry constnt. This lso sys tht the prolem Lu Pge f does
2 not hve n unique solution. Let's sustitute the formul for f n otin u( φ ( n, x ) φ ( n, f( d λ z n λ n f( φ ( n, dz into the ove nd interchnge the integrl nd the summtion to Compring the lst integrl nd tht in (5), we conclude tht (the sum must exclude the terms with l n 0) G ( x, φ (, ) n x φ ( n, (7) Note: In Power ook, the differentil eqution hs een tken to e of the form Lu f in the discussion of the Green functions. Thus, in our present sitution, the Green function of Lu f should insted e given s follows (note the minus signs) G ( x, u( u( x ) W( u( x ) u( W( z nd z x x z nd z EXAMPLE : Consider the Euler opertor with Dirichlet nd Neumnn conditions [Sturm-Liouville type for p( x ), q( 0, w( ] over the intervl I { x 0 < x < }. The oundry conditions re type t the left nd type t the right end points. Euler differentil opertor Boundry conditions L( y ) x x y( y( 0 ) 0 nd y ( ) 0 x We will compute / the Green function for the Euler differentil eqution. / the expnsion of the Green function in terms of the eigenfunctions (see (7). c/ let nd f e the piecewisedefined function 0 x 0 nd x 0 x 0 nd x 0 Solve the prolem L( u) f using the Green function nd the eigenfunction series (6). Compre the results y grphing. Solution: / Green function: We need to find the fundmentl solutions u, u which stisfy the oundry conditions t x 0, x respectively. Generl solution of L( y) 0 is > restrt:y:x->c + C*x; Pge
3 y : x C + C x Using the oundry conditions to find the equtions determining the constnts > sus(x0,y()0; > sus(x,diff(y(,)0; C 0 C 0 Therefore, we cn choose u ( x nd u ( to e two fundmentl solutions which stisfies respectively the oundry conditions t x 0,. > u:x->x; u:x->; u : x u : The Wronskin > W:x->u(*diff(u(, - u(*diff(u(,; W : x u( x ) diff ( u( x ), x ) u( x ) diff ( u(, x ) The Green function is then > G : (x,z)-> piecewise(z< x,-u(z)*u(/w(z),x < z,-u(*u(z)/w(z)); > evl(g(x,z)); G : ( x, > plot3d(g(x,z),x0..,z0..); u( u( piecewise z x,, x z, W( z) x { z z x x x z u( x ) u( z) W( z) / Eigen_expnsion of Green function: As we discussed the eigenvlue prolem for L with Dirichlet nd Neumnn oundry conditions in SL.mws, the eigenvlues nd eigenfunctions re > lmd[n] : /4*(*n-)^*Pi^/(^); phi:(n,->sin(/*(*n-)*pi*x/)*sqrt()/(sqrt()); : 4 ( n ) π Pge 3
4 φ : ( n, ( n ) π x Thus, following (7), the eigen_expnsion of the Green function is > G_series:(x,z)->sum(phi(n,*phi(n,z)/lmd[n],n..infinity); G_series : ( x, z) φ (, ) n x φ ( n, z) The prtil (truncted) series of the first m terms is > G_Pseries:(m,x,z)->sum(phi(n,*phi(n,z)/lmd[n],n..m); G_Pseries : ( m, x, Let's tke m nd plot the grph of this prtil sum. > ::evl(g_pseries(,x,z)); m φ (, ) n x φ ( n, π x π z 3 8 π x 3 π z 8 + π 9 π > plot3d(g_pseries(,x,z),x0..,z0..); We cn see tht the sum pproximtes the Green function pretty well. c/ Solving Lu f. > f:x->piecewise(0<x nd x</,0,/<x nd x<,); evl(f(); f : x piecewise 0 x nd x,,, 0 x nd x 0 x 0 nd x 0 x 0 nd x 0 Using the Green function we get (Mple seems unle to hndle this(!), Cn you get it mnully?) Pge 4
5 > u:x->int(g(x,z)*f(z),z0..); u : x 0 G ( x, f( z) In fct, y splitting the integrl from 0 to / nd / to, using pproprite formul of G we cn esily find tht u is given y x x 0 nd x 0 u( x ) x + x x 0 nd x 0 8 > u:x->piecewise(x>0 nd x</,x/,x>/ nd x<,-x^/+x-/8); x piecewise 0 x nd x,,, x x nd x + x x Using the formul (6). First, we need to compute the "Fourier" coefficients F n of f. > F[n]:int(f(*phi(n,,x0..); F n : Then the "Fourier" coefficients for u is given y F n > U[n]:F[n]/lmd[n]; U n : 8 dz cos + π n π n sin( π n ) ( n ) π cos + π n π n sin( π n) ( n ) 3 π 3 Thus, the (prtil) "Fourier" series for u is > up_series:(m,->sum(u[n]*phi(n,,n..m); up_series : ( m, m U n φ ( n, Finlly, we plot the result given y the integrl nd the prtil sum ove in the sme grph for comprison. It cn e seen tht, even with one term ( m ), the prtil sum pproximtes well the solution. > plot([u(,up_series(,],x0..,color[red,lue],title"green_sol: RED, Prtil Sum: BLUE"); > 8 Pge 5
6 > plot(piecewise(x</,x/,x>/,-x^/+x-/8),x0..); Pge 6
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