Chapter Five  Eigenvalues, Eigenfunctions, and All That


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1 Chpter Five  Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl tux,t Fx,t for x D n t, in which u is specifie on the ounry of D s re initil conitions t t. We ssume tht L 1 x is liner ifferentil opertor in x n L t is liner ifferentil opertor in t. In the previous chpter, L 1 xx x, n we were fortunte tht this opertor h n infinite orthogonl collection of eigenfunctions which pproximte functions in nice wy. We shll see wht hppens in more generl setting. First it of liner lger review. Definition. Suppose V is liner spce together with n inner prouct. A liner opertor L : V V such thtlf,g f,lg for ll f,g V is si to e selfjoint. Let V e liner spce of nice functions efine on n intervl x with the inner prouct f,g fxgx. Define the liner opertor L y L px qxx. We ssume tht p n q re rel n continuous n tht p is continuously ifferentile n positive. Such n opertor is clle SturmLiouville opertor. Definition. Suppose r is rel continuous n positive function on x.a sclr such tht L r for some nonzero V is clle n eigenvlue of L, n the function is n eigenfunction. For resons tht will soon e cler, we woul very much like to hve our liner opertor L e selfjoint. Thus, we wntlf,g f,lg for ll f n g in the spce V. Let s compute: Lf,gf,Lg LfxgxfxLgx px f px g gxqxfxgx fxqxgxfx 1
2 px f px gx f gx fx g px g fx pgf fg p gf fg. In other wors, L will e selfjoint if we choose V to e liner spce of functions such tht p gf fg p gf fg. Exmples. The opertor L is SturmLiouville opertor on the intervl x, with px 1 n qx.the selfjoint ounry conitions then ecome gf fg gf fg Thus, if we choose V to e the spce of functions such tht, the opertor L is selfjoint. So why shoul we cre out selfjoint opertors? Let s see. Proposition 1. Every eigenvlue of selfjoint opertor is rel. Proof: Suppose L r for some nonzero. Then L,,L ecuse L is selfjoint, n this ecomesr,,r, orr,,r. In other wors, rx x rx x. The integrl is not zero, n so we hve. Proposition. Eigenfunctions corresponing to ifferent eigenvlues of selfjoint opertor L re orthogonl with respect to the inner prouct f,g r rf,g f,rg rxfxgx. Proof: Suppose L r n L r for. Agin, from the fct tht L is selfjoint, we know tht L,,L. Thus, r,,r.from the previous proposition we know
3 tht n re rel. Thus, r,,r, or, r. But n so it must e true tht, r. Nothing we hve one gurntees the existence of eigenvlues of L. There re, in fct, n infinite numer of them, giving n infinite orthogonl set of eigenfunctions. A proof is too much for these notes. The celerte SturmLiouville Theorem sys even more. It sys tht every nice function f cn e expne in series f n n n1 of the eigenfunctions n, n this series converges in the men to f. Exmple. Consier the prolem u xx u t, for x n t u,t n u t,t ux, fx In solving this prolem, I hope it is cler why we egin with the eigenvlue prolem,. Note this is n exmple of the SturmLiouville prolem we hve just iscusse, so we know wht to expect. First, we know tht we must in orer to hve nonzero solutions to the prolem. To remin us of tht, let.then x Acosx Bsinx, n the ounry conitions ecome n A 3
4 Bcos. If or B re zero, then, so it must e true tht cos. Thus must e n o multiple of / : n n1 1 n 1, for n 1,,3, Our eigenvlue re thus n n n 1,n he corresponing eigenfunctions re n x sin n 1 x. Continuing, let ux,t n tsin n x.from here on, things look fmilir. n1 u xx u t n n t n t sin n x. n1 As usul, n n t n t tells us tht n t n e nt n e n t, n so ux,t n e n t sin n x. n1 The constnts n re etermine from the initil conition ux, fx : Thus, ux, fx n sin n x. n1 f,sin n n x sin n x,sin n x fxsin n 4
5 Exercises 1. Fin ll eigenvlues n eigenfunctions for,.. Fin ll eigenvlues n eigenfunctions:, Let n e nonnegtive integer, n consier the SturmLiouville prolem x n x x xx, x c. Note tht this opertor oes not quite fit our efinitions. The coefficient qx n x of is not nice t the left en point of the intervl uner consiertion n px x is zero t the left en point of the intervl. This is wht is known s singulr SturmLiouville prolem. If, however, we consier functions tht, together with their erivtives, re resonly wellehve t the left en point, then our selfjoint ounry conitions ecome p gf fg p gf fg c gf fg, n re stisfie y functions in the spce of those nice functions tht vnish t x. The eqution x n x x xx, or x x n x is the worlfmous Bessel s Eqution. We shll seek solutions y mens of the Froenius metho. Tht is, ssume solutionof the form 5
6 Then x c k x k, c. k k k k c k x k, kc k x k1, kk1c k x k, n we hve x x n x k k Writing c k x k c k x k gives us k k kk1kn c k x k c k x k x x n x k n c k x k k k c k x k n c x 1 n c 1 x 1 k n c k c kx k. k In orer for the ifferentil eqution to e stisfie, ech of the coefficients of the powers of x must e zero: n c 1 n c 1 k n c k c k, for k,3,4,. In the first eqution, c n so it must e true tht n, giving us n, or n. Let s strt with n. The reminer of the equtions ecome 6
7 n1c 1 knkc k c k, for k,3,4,. Thus, c 1 c 3 c 5, n c k knk c k, for k,3,4,. Letting k m results in Then c m mnm c m1 mnm c m1. c m mnm c m1 1 m m n! m m!nm! c where c is ny constnt. For the ske of netness, choose c 1 n! n. Then c m 1 m m n! m m!nm! c n 1 m m!nm! m, n our solutions, t lst, x n m x x n m 1 m m!nm! 1 m m!nm! m x mn,or x m. The function J n x x n m 1 m m!nm! x m is the Bessel function of the first kin of orer n. Our solution is thus x J n x. 7
8 A secon linerly inepenent solution is foun reucing the orer of the originl eqution. It turns out not to e nice t x, n so we hve solutionsx AJ n x.the ounry conition is thus J n. It is known tht J n hs n infinite numer of nonnegtive zeros. These n ll sorts of other informtion out Bessel functions cn e foun in most hnooks or y mens of most computer lger systems; e.g., Mple or Mthemtic. Here re some pictures. First, J : x .4 Now here re J 1,J,n J 3 : x It is cler then tht our eigenvlues re nm, with nm z nm /, where z nm is the m th zero of J n. Here is short tle of some of these vlues: m n n 1 n n The corresponing eigenfunctions re, of course, J n nm x. Note there is nonconstnt weight function here, so our orthogonlity ecomes 8
9 xj n nk xj n nl x, for k l. Exmple. Consier the prolem x xu x u tt, x L, t ul,t ux, fx, u tt x, gx. The solution u gives the isplcement of hnging chin of length L : It shoul e cler y now why we re intereste in the eigenvlue prolem x L, n nice. This is tntlizingly close to Bessel s eqution of orer zero. We coul use the metho of Froenius to fin solutions, etc. Inste, let s mke chnge of vrile. Let z x, or x z /4.Define z z /4. Then, Our ifferentil eqution thus ecomes z z, or z z. z x z z z z. z z 4 z z, or 9
10 Now we hve Bessel s eqution with n. The solution is J z J x.the ounry conitionl is J L, n so we hve the eigenvlues m z m L, where z m is the m th zero of J. We set ux,t m tj m m1 x. Then x xu x u tt m m t m tj m x. m1 This gives us m t m cos m t m sin m t. Thus, ux,t m cos m t m sin m tj m m1 x Use the initil conitions to fin the constnts m, m : ux, m J m m1 x fx. Hence, m L fxj m L J m x. x Also, n so, u t x, m m J m m1 x, 1
11 m L gxj m L m J m x x Exercise 3. Fin the first four terms of the series for u in cse L 1, fx x1x, n gx. Drw some pictures of your pproximtion for ifferent vlues of t. 11
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