Legendre

On an Eigenvalue Problem Involving Legendre Functions by Nicholas J. Rose North Carolina State University [email protected] 1. INTRODUCTION. The classical eigenvalue problem for the Legendre Polynomials is DE:¡ .1¡x2/y0¢0C‚yD0;¡1•x•1:. 1/ ThisisasingularSturm-Liouvilleproblem. Because xD§1aresingularpoints,noexplicitboundaryconditions aregivenatthesepoints; insteaditisrequiredthat y.x/andy0.x/remainboundedas x!§1. Itiswellknown ([1. p. 186], or [4, p. 325 ]) that this boundedness condition implies that the eigenvalues must be ‚Dn.nC1/;nD0;1;::: andthecorrespondingeigenfunctionsarethenon-zeromultiplesoftheLegendrepolynomials Pn.x/whichform an orthogonal set on the interval ¡1•x•1. Another proof of this fact will be given at the end of this paper. We are mainly interested in the eigenvalue problem DE:¡ .1¡x2/y0¢0C‚yD0;a•x•1 where¡1<a<1 BC:y.a/D0;y.x/andy0.x/bounded as x!1:.2/ Many books discuss the case when aD0 where it can be shown [1, p. 192] that the eigenvalues are given by ‚D.2kC1/.2kC2/;kD0;1;:::and the corresponding eigenfunctions are the odd Legendre polynomials P2kC1.x/whichformaorthogonalsetontheinterval0 •x•1. Inthispaperweconsiderthecasewhen a6D0. This problem is actually solved in the Hobson treatise [4, p. 444] but seems not to be considered in currenttextbooksforfirstcoursesinboundaryvalueproblemsor”AdvancedMathematicsforEngineers”. Thereasonisclear;thesolutionrequiresLegendrefunctionsofarbitrarydegreewhicharenotusuallyconsideredinsuchtexts. Aswillbeseenbelow,thisproblemprovidesaniceexampleoftheuseofFrobeniusseries,theLegendrefunc- tionsofarbitrarydegreecomeupnaturallyinthesolution.. Inaddition,usingamoderncomprehensivecomputingenvironmentsuchasMAPLE,wecanreadilyfindapproximationstotheeigenvalues,plottheeigenfunctionsandtest the eigenfunction expansion for specific functions. The classical problem (1) arises in finding the steady state temperature in a solid homogeneous sphere with given temperature on the surface, assuming the temperature is symmetrical about a diameter [1, p. 193-196].The problem (2) arises in finding the steady state temperatures in the portion of a sphere cut off by a concentricright-circular cone, where the temperature on the spherical surface is assumed to be symmetrical about the axisofsymmetryoftheconeandsphere;thetemperatureonthesurfaceofthesphereisgivenandthetemperatureonthe surface of the cone is 0 [4, p. 444]. 2. SOLUTION OF THE EIGENVALUE PROBLEM. It is shown in [2, p. 294] that the eigenvalues, ‚,o f problem (2) are nonnegative. It is easy to check that ‚D0 is not an eigenvalue so that the eigenvalues are all positive. Following the usual custom we write ‚D”.”C1/, where”>0 is a real number. DE:¡ .1¡x 2/y0¢0C”.”C1/yD0;a•x•1;.¡1<a<1/; BC:y.a/D0;y.x/andy0.x/bounded as x!1:.3/ SincexD1 is a regular singular point we look for a solution as a Frobenius series about this point. To facilitate computations let x¡1Dtand express the differential equation (3) in terms of t. The equation becomes ¡ .t2C2t/y0¢0¡”.”C1/yD0 .4/ 1 whereyis considered as a function of t. We look for a Frobenius solution of the form yD1X 0aktkCfiDa0tfiCa1tfiC1C¢¢¢;a06D0: Substitutinginequation(4)andsettingthecoefficientof a0tozeroyieldstheindicialequation fi2D0. Sincethe indicial equation has a double root of zero, we know that there is one power series solution yD1X 0aktk:. 5/ and a second linearly independent solution which has a logarithmic singularity at tD0. We are interested in solutions that are bounded as tDx¡1!0, and therefore need only consider the power series solution. Substituting the (5) into the DE (4) we find the recurrence relation akC1D.”¡k/.”CkC1/ 2.kC1/2ak;k‚0:. 6/: From this one finds that akD.”Ck/.”Ck¡1/¢¢¢.”C1/”.”¡1/¢¢¢.”¡.k¡1// 2k.k!/2a0; and the power series solution about xD1i si s yDa01X 0.”Ck/.”Ck¡1/¢¢¢.”C1/”.”¡1/¢¢¢.”¡.k¡1// 2k.k!/2.x¡1/k: If we seta0D1, which means y.1/D1 we obtain the Legendre function of the first kind of order ”: P”.x/D1X 0.”Ck/.”Ck¡1/¢¢¢.”C1/”.”¡1/¢¢¢.”¡.k¡1// 2k.k!/2.x¡1/k:. 7/ This is the result obtained in ([3], p. 312). From the recurrence relation (6) it is easy to see that that the series (7) converges forjx¡1j<2:If we assume that ”isnota positive integer we may multiply numerator and denominator of the coefficients by 0.”¡.k¡1//to obtain P”.x/D1X 00.”CkC1/ 2k.k!/20.”¡.k¡1//.x¡1/k: Nowwemaycompletethesolutionoftheboundaryvalueproblem. Weknowthat P”.x/satisfiestheDEfor arbitrary”>0 and is bounded at xD1. To satisfy the left hand boundary condition we need only find those values of”for which P”.a/D0: We assume that this equation has an infinite number of roots and denote the eigenvalues by ”kDkth positive root ofP”.a/D0;kD1;2;3;:::and the corresponding eigenfunctions by P”k.x/. The functions P”.x/are builtintoMAPLEandaretherecalledLegendreP .”;x/. Togetspecificnumericalresultslet aD1=2. Figure1. shows the result of using MAPLE to plot P”.1=2/from”D0t o”D40. 2 Figure 1. UsingFigure1. forinitialestimatesoftheeigenvalues,theequationsolvingpropertyofMAPLEcanbeused to get more refined estimates. The first four eigenvalues are ”1D1:777288270;”2D4:762779438;”3D7:758258853;”4D10:75608784: The corresponding eigenfunctions are shown in Figure 2, the number of zeros increases as the suffix increases. Figure 2. 3 The eigenfunction expansion of an arbitrary piecewise smooth function is f.xC0/Cf.x¡0/ 2D1X 1AkP”k.x/;1=2<x<1;. 8/ where AkDR1 1=2f.x/P”k.x/dx R1 1=2P2”k.x/dx: To check out the eigenfunction expansion consider the function f.x/D‰x¡1=2;1=2<x>3=4; x¡1;3=4<x<1:.9/ The result of the first 40 terms of (8) compared with f.x/is shown in Figure 3. Figure 3. 3. Concluding Remarks. Eigenvalues and eigenfunctions for any value of ain (2) may be found in a similar manner. Also, the boundary condition y.a/D0 may be changed to y0.a/D0o ry0.a/Dhy.a/;h>0 and similar results may be obtained. In Figure 1. the difference between successive eigenvalues is approximately 3. This is consistent with the asymptotic expansion [4, p. 303]: P”.cosµ/D2p 2sinµ.2”/! 22n.”!/2cos‡… 4¡¡ ”C1 2¢ µ· COµ1 ”¶ :. 10/ Consideredasafunctionof ”,thefirsttermin(10)showsthatthedistancebetweensuccessivezerosofthecosine function is approximately …=µ. In the case of Figure 1, we have µD…=3. so that…=µD3: WhenvDn,anon-negativeinteger,equation(6)showsthat akD0;k>nandthesolutionisapolynomial In this case (7) reduces to a formula for the Legendre Polynomials: Pn.x/DnX 0.nCk/! 2k.k!/2.n¡k/!.x¡1/k:. 11/ 4 Thisformulaisinterestinginthatitisnotnecessarytohavedifferentformulaefor nevenandnodd. Ontheother handitisnotobviousfrom(11)that Pnisanevenpolynomialif nisevenandoddif nisodd. Equation(11)may also be written as Pn.x/DnX 01 2kµn k¶µnCk n¶ .x¡1/k:. 12/ In conclusion we give a proof of the fact stated in the introduction: The only values of ‚for which (1) has a solution which is continuous for ¡1•x•1are‚Dn.nC1/;nD 0;1;2;¢¢¢. Letting‚D”.”C1/, where”‚0 is an arbitrary real number we obtain the differential equation DE: ¡ .1¡x2/y0¢0C”.”C1/yD0: Intheprevioussectionitwasshownthattheonlysolutionsthatareboundedat xD1aremultiplesoftheLegendre functionPv.x/. From the recurrence formula (6.) see that P”.x/has a polynomial solution if and only if ”is a non-negative integer. Therefore it is sufficient to prove that P”.¡1/diverges unless vD0;1;. Assuming v=2f0;1;2;¢¢¢g, the Legendre functions are given by a non-terminating infinite series P”.x/D1X 0ak.x¡1/k; wherea0D1 and the aksatisify the recurrence relation (6). Therefore P”.¡1/D1X 0ak.¡2/k:. 13/ LetbkDak.¡2/k:. The ratio of successive terms is bkC1 bkDakC1.¡2/kC1 ak.¡2/kD.”¡k/.”CkC1/ 2.kC1/2.¡2/D.k¡”/.”CkC1/ .kC1/2: Ifk>”, the terms bkhave the same sign; assume they are positive. For k>”C1w eh a v e bkC1 bk‚.”CkC1/ .kC1/2‚k kC1: It follows that for jD2;3;¢¢¢we have b”Cj‚”C1 ”Cjb”C1;jD1;2;3¢¢¢: This implies the seriesP1 jD1bjdiverges and therefore P”.¡1/diverges also. References 1. Churchill, Ruel V. Fourier Series and Boundary Value Problems . McGraw Hill Book Company, New York, 1941. 2. Courant,R.andHilbert,D. MethodsofMathematicalPhysics VolumeI,IntersciencePublishers,NewYork, 1953. 3. Whittaker,E.T.andWatson,G.N. ACourseofModernAnalysis ,FourthEdition,CambridgePress,Cambridge, England, 1927. 4. Hobson,E.W. TheTheoryofSphericalandEllipsoidalHarmonics ,CambridgePress,Cambridge,England, 1931 5