Transcription of Sturm-Liouville Problems
1 47 Sturm-Liouville Problems Sturm-Liouville Problems are boundary-value Problems that naturally arise when solving cer-tain partial differential equation Problems using a separation of variables method that will bediscussed in a later chapter. It is the theory behind Sturm-Liouville Problems that, ultimately,justifies the separation of variables method for these partial differential equation simplest applications lead to the various Fourier series, and less simple applications lead togeneralizations of Fourier series involving Bessel functions, Hermite polynomials, , there are several difficulties with our studying Sturm-Liouville .
2 Had we the time, we would first discuss partial differential equation problemsand develop the separation of variables method for solving certain important types ofproblems involving partial differential equations. We would then see how these Sturm-Liouville Problems arise and why they are so important. Butwe don t have time. Instead,I ll briefly remind you of some results from linear algebra that are analogous to the resultswe will, eventually, difficulty is that the simplest examples (which are very important since they leadto the Fourier series) are too simple to really illustrate certain elements of the theory,while the other standard examples tend to get complicated and require additional trickswhich distract from illustrating the theory.
3 We ll deal with this as well as we material gets theoretical. Sorry, there is no way aroundthis. The end results, however,are very useful in computations, especially now that we havecomputers to do the tediouscomputations. I hope we get to that , I must warn you that, in most texts, the presentation of the sturm -Liouvilletheory stinks. In most introductory ordinary differential equation texts, this material isusually near the end, which usually means that the author just wants to finish the damnbook, and get it published. In texts on partial differentialequations, the results are usuallyquoted with some indication that the proofs can be found in a good text on & Page: 47 Linear Algebraic AntecedentsLet us briefly review some elements from linear algebra whichwe will be recreating usingfunctions instead of finite-dimensional vectors.
4 But first,let me remind you of a little complexvariable algebra . This is because our column vectors and matrices may have complex From (Real) Linear AlgebraFor the rest of this section, letNbe some positive integer. Following fairly standard convention,we ll letRNdenote the vector space of allN 1 matrices with real components. That is, sayingthatvandware vectors inRNwill simply meanv= andw= where thevk s andwk s are real that the norm (or length ) of the abovev,kvk, is computed bykvk=q|v1|2+|v2|2+ |vN|2and that, computationally, the classic dot product of the abovevandwisv w=v1w1+v2w2+ +vNwN= [v1, v2.]
5 , vN] =vTwwherevTwis the matrix product withvTbeing the transpose of matrixv( , the matrixconstructed fromvby switching rows with columns).Observe that, sincev2=|v|2whenvis real,v v=v1v1+v2v2+ +vNvN=|v1|2+|v2|2+ +|vN|2= recall that a set of vectors{v1,v2,v2,..}is said to be orthogonal if and only ifvj vk=0wheneverj6= finally, recall that a matrixAissymmetricif and only ifAT=A, and that, forsymmetric matrices, we have the following theorem from linear algebra (and briefly mentionedin chapter 39):Theorem a symmetricN Nmatrix (with real-valued components). Then both of the followinghold:Linear Algebraic AntecedentsChapter & Page: 47 the eigenvalues ofAare is an orthogonal basis forRNconsisting of eigenvectors , we will obtain an analog to this theorem involving a linear differential operatorinstead of a Algebra with Complex ComponentsComplex Conjugates and MagnitudesWe ve certainly used complex numbers before in this text, but I should remind you that a complexnumberzis something that can be written asz=x+iywherexandyare real numbers therealandimaginary parts, respectively ofz.
6 Thecorresponding complex conjugatez and magnitude|z|ofzare then given byz =x iyand|z|=px2+ ifxis a real number positive, negative or zero thenx2=|x|2. However, you caneasily verify thatz26=|z|2. Instead, we have|z|2=x2+y2=x2 (iy)2=(x iy)(x+iy)=z and Matrices with Complex-Valued ComponentsEverything mentioned above can be generalized toCN, the vector space of allN 1 matriceswith complex components. In this case, saying thatvandware vectors inCNsimply meansv= andw= where thevk s andwk s are complex numbers. Since the components are complex, wedefinethe complex conjugate ofvin the obvious way,v = v 1v N.
7 The natural norm (or length ) ofvis still given bykvk=q|v1|2+|v2|2+ |vN|2 Chapter & Page: 47 4 sturm -LiouvilleHowever, when we try to relate this to the classic dot productv v, we getkvk2=|v1|2+|v2|2+ +|vN|2=v1 v1+v2 v2+ +vN vN=(v ) v6=v suggests that, instead of using the classic dot product, we use the(standard vector) innerproductofvwithw, which is denoted byhv|wiand defined by byhv|wi=v w=v1 w1+v2 w2+ +vN wN= [v 1, v 2, .. , v N] =(v ) |vi=v v= also adjust our notion of orthogonality by saying that any set{v1,v2, ..}of vectors inCNisorthogonalif and only if vm vn =0wheneverm6= inner product for vectors with complex components is themathematically natural ex-tension of the standard dot product for vectors with real components.
8 Some easily verified (anduseful) properties of this inner product are given in the next theorem. Verifying it will be left asan exercise (see exercise ).Theorem (properties of the inner product)Suppose and are two (possibly complex) constants, andv,w, anduare vectors |wi=hw|vi , | v+ wi= hu|vi+ hu|wi, v+ w|ui= hv|ui+ hw|ui, |vi= , we ll define other inner products for functions. These inner products will have verysimilar properties to those in given in the last any matrixA denotedA is the transpose of the complex conjugate ofA,A =(A )T(equivalently, AT ) .Linear Algebraic AntecedentsChapter & Page: 47 5 That is,A is the matrix obtained from matrixAby replacing each entry inAwith its complexconjugate, and then switching the rows and columns (or first switch the rows and columns andthen replace the entries with their complex conjugates youget the same result either way).
9 ! Example :IfA="1+2i3 4i5i 6i7 8i#,thenA = "1+2i3 4i5i 6i7 8i# !T="1 2i3+4i 5i6i78i#T= 1 2i6i3+4i7 5i8i .This adjoint turns out to be more useful than the transposewhen we allow vectors to havecomplex matrixAisself adjoint1if and only ifA =A. Note self-adjoint matrix is automatically a square matrix with just real components, thenA =AT, and Ais self adjoint means the same as Ais symmetric .If you take the proof of theorem and modify it to take intoaccount the possibility ofcomplex-valued components, you getTheorem a self-adjointN Nmatrix. Then both of the following the eigenvalues ofAare is an orthogonal basis forCNconsisting of eigenvectors theorem is noteworthy because it will help explain the source of some of the terminologythat we will later be is more noteworthy is what the above theorem says about computingAvwhenAisself adjoint.
10 To see this, let b1,b2,b3, .. ,bN be any orthogonal basis forCNconsisting of eigenvectors forA(remember, the theorem saysthere is such a basis), and let{ 1, 2, 3, .. , N}be the corresponding set of eigenvectors (soAbk= kbkfork=1,2,..,N). Since the setofbk s is a basis, we can expressvas a linear combination of these basis +v2b2+v3b3+ +vNbN=NXk=1vkbk,( )1the termHermitianis also usedChapter & Page: 47 6 sturm -Liouvilleand can reduce the computation ofAvtoAv=A v1b1+v2b2+v3b3+ +vNbN =v1Ab1+v2Ab2+v3Ab3+ +vNAbN=v1 1b1+v2 2b2+v3 3b3+ +vN large, this could be a lot faster than doing the basic component-by-component matrixmultiplication.
