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Sturm-Liouville Boundary Value Prob- lems

4 Sturm-Liouville Boundary Value Prob- lemsWe have seen that trigonometric functionsand special functionsare the solutions of differential equations. These solutions give orthogonalsets of functions which can be used to represent functions in generalizedFourier series expansions. At the same time we would like to generalizethe techniques we had first used to solve the heat equation in order to solvemore general initial- Boundary Value problems. Namely, we use separationof variables to separate the given partial differential equation into a set ofordinary differential equations. A subset of those equations provide us witha set of Boundary Value problems whose eigenfunctions are useful in repre-senting solutions of the partial differential equation. Hopefully, those solu-tions will form a useful basis in some function class of problems to which our previous examples belong are theSturm- liouville eigenvalue problems.

lems involving second order ordinary differential equations. For example, we will explore the wave equation and the heat equation in three dimen-sions. Separating out the time dependence leads to a three dimensional boundary value problem in both cases. Further separation of variables leads

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Transcription of Sturm-Liouville Boundary Value Prob- lems

1 4 Sturm-Liouville Boundary Value Prob- lemsWe have seen that trigonometric functionsand special functionsare the solutions of differential equations. These solutions give orthogonalsets of functions which can be used to represent functions in generalizedFourier series expansions. At the same time we would like to generalizethe techniques we had first used to solve the heat equation in order to solvemore general initial- Boundary Value problems. Namely, we use separationof variables to separate the given partial differential equation into a set ofordinary differential equations. A subset of those equations provide us witha set of Boundary Value problems whose eigenfunctions are useful in repre-senting solutions of the partial differential equation. Hopefully, those solu-tions will form a useful basis in some function class of problems to which our previous examples belong are theSturm- liouville eigenvalue problems.

2 These problems involve self-adjoint(differential) operators which play an important role in the spectral theoryof linear operators and the existence of the eigenfunctions needed to solvethe interesting physics problems described by the above initial-boundaryvalue problems. In this section we will introduce the Sturm-Liouville eigen- Value problem as a general class of Boundary Value problems containing theLegendre and Bessel equations and supplying the theory needed to solve avariety of OperatorsIn physics many problems arise in the formof Boundary Value Prob- lems involving second order ordinary differential equations. For example,we will explore the wave equation and the heat equation in three dimen-sions. Separating out the time dependence leads to a three dimensionalboundary Value problem in both cases. Further separation of variables leadsto a set of Boundary Value problems involving second order ordinary dif-ferential partial differential equationsIn general, we might obtain equations of the forma2(x)y +a1(x)y +a0(x)y=f(x)( )subject to Boundary conditions.

3 We can write such an equation in operatorform by defining the differential operatorL=a2(x)D2+a1(x)D+a0(x),whereD=d/ dx. Then, Equation ( ) takes the formLy= that we had solved such nonhomogeneous differential equations inChapter2. In this section we will show that these equations can be solvedusing eigenfunction expansions. Namely, we seek solutions to the eigen- Value problemL = with homogeneous Boundary conditions on and then seek a solution ofthe nonhomogeneous problem,Ly=f, as an expansion over these eigen-functions. Formally, we lety(x) = n=1cn n(x).However, we are not guaranteed a nice set of eigenfunctions. We need anappropriate set to form a basis in the function space. Also, it would benice to have orthogonality so that we can easily solve for the turns out that any linear second order differential operator can beturned into an operator that possesses just the right properties (self-adjointedness)to carry out this procedure.

4 The resulting operator is referred to as a Sturm-Liouville operator. We will highlight some of the properties of these opera-tors and see how they are used in define the Sturm-Liouville operator asThe Sturm-Liouville (x)ddx+q(x).( )The Sturm-Liouville eigenvalue problem is given by the differential equa-tionThe Sturm-Liouville eigenvalue (x)y,orddx(p(x)dydx)+q(x)y+ (x)y=0,( )forx (a,b),y=y(x), plus Boundary conditions. The functionsp(x),p (x),q(x)and (x)are assumed to be continuous on(a,b)andp(x)>0, (x)>0on[a,b]. If the interval is finite and these assumptions on the coefficientsare true on[a,b], then the problem is said to be a regular sturm -Liouvilleproblem. Otherwise, it is called a singular Sturm-Liouville Boundary Value problems 109We also need to impose the set of homogeneous Boundary conditionsTypes of Boundary conditions. 1y(a) + 1y (a) =0, 2y(b) + 2y (b) =0.( )The s and s are constants.

5 For different values, one has special typesof Boundary conditions. For i=0, we have what are called Dirichletboundary conditions. Namely,y(a) =0 andy(b) =0. For i=0, weDirichlet Boundary conditions - the so-lution takes fixed values on the bound-ary. These are named after Gustav Leje-une Dirichlet (1805-1859).have Neumann Boundary conditions. In this case,y (a) =0 andy (b) = terms of the heat equation example, Dirichlet conditions correspondNeumann Boundary conditions - thederivative of the solution takes fixed val-ues on the Boundary . These are namedafter Carl Neumann (1832-1925).to maintaining a fixed temperature at the ends of the rod. The Neumannboundary conditions would correspond to no heat flow across the ends, orinsulating conditions, as there would be no temperature gradient at thosepoints. The more general Boundary conditions allow for partially type of Boundary condition that is often encountered is the pe-riodic Boundary condition.

6 Consider the heated rod that has been bent toform a circle. Then the two end points are physically the same. So, wewould expect that the temperature and the temperature gradient shouldagree at those points. For this case we writey(a) =y(b)andy (a) =y (b). Boundary Value problems using these conditions have to be handled differ-ently than the above homogeneous conditions. These conditions leads todifferent types of eigenfunctions and equations of previously mentioned, equations of the form ( ) occur often. Wenow show that any second order linear operator can be put into the formof the Sturm-Liouville operator. In particular, equation ( ) can be put intothe formddx(p(x)dydx)+q(x)y=F(x).( )Another way to phrase this is provided in the theorem:The proof of this is straight forward as we soon show. Let s first considerthe equation ( ) for the case thata1(x) =a 2(x). Then, we can write theequation in a form in which the first two terms combine,f(x) =a2(x)y +a1(x)y +a0(x)y= (a2(x)y ) +a0(x)y.

7 ( )The resulting equation is now in Sturm-Liouville form. We just identifyp(x) =a2(x)andq(x) =a0(x).Not all second order differential equations are as simple to convert. Con-sider the differential equationx2y +xy +2y= this casea2(x) =x2anda 2(x) =2x6=a1(x). So, this does not fall intothis case. However, we can change the operator in this equation,x2D+xD, to a Sturm-Liouville operator,D p(x)Dfor ap(x)that depends on partial differential equationsIn the sturm liouville operator the derivative terms are gathered togetherinto one perfect derivative,D p(x)D. This is similar to what we saw in theChapter2when we solved linear first order equations. In that case wesought an integrating factor. We can do the same thing here. We seek amultiplicative function (x)that we can multiply through ( ) so that itcan be written in Sturm-Liouville first divide out thea2(x), givingy +a1(x)a2(x)y +a0(x)a2(x)y=f(x)a2(x).

8 Next, we multiply this differential equation by , (x)y + (x)a1(x)a2(x)y + (x)a0(x)a2(x)y= (x)f(x)a2(x).The first two terms can now be combined into an exact derivative( y ) if the second coefficient is (x). Therefore, (x)satisfies a first order , sepa-rable differential equation:d dx= (x)a1(x)a2(x).This is formally solved to give the sought integrating factor (x) =e a1(x)a2(x) , the original equation can be multiplied by factor (x)a2(x)=1a2(x)e a1(x)a2(x)dxto turn it into Sturm-Liouville summary,Equation ( ),a2(x)y +a1(x)y +a0(x)y=f(x),( )can be put into the Sturm-Liouville formddx(p(x)dydx)+q(x)y=F(x),( )wherep(x) =e a1(x)a2(x)dx,q(x) =p(x)a0(x)a2(x),F(x) =p(x)f(x)a2(x).( ) Sturm-Liouville Boundary Value problems x2y +xy +2y=0into Sturm-Liouville can multiply this equation by (x)a2(x)=1x2e dxx=1x,to put the equation in Sturm-Liouville form:Conversion of a linear second orderdifferential equation to sturm +y +2xy= (xy ) +2xy.

9 ( ) of Sturm-Liouville Eigenvalue ProblemsThere are several properties that can be provenfor the (regular) Sturm-Liouville eigenvalue problem in ( ). However, we will not provethem all here. We will merely list some of the important facts and focus ona few of the , countable The eigenvalues are real, countable, ordered and there is a smallesteigenvalue. Thus, we can write them as 1< 2<.. However,there is no largest eigenvalue andn , n .Oscillatory For each eigenvalue nthere exists an eigenfunction nwithn 1zeros on(a,b).3. Eigenfunctions corresponding to different eigenvalues are orthogonalwith respect to the weight function, (x). Defining the inner productoff(x)andg(x)as f,g = baf(x)g(x) (x)dx,( )then the orthogonality of the eigenfunctions can be written in theOrthogonality of n, m = n, n nm,n,m=1, 2, ..( )4. The set of eigenfunctions is complete; , any piecewise smooth func-tion can be represented by a generalized Fourier series expansion ofthe eigenfunctions,f(x) n=1cn n(x),wherecn= f, n n, n.

10 Actually, one needsf(x) L2 (a,b), the set of square integrable func-tions over[a,b]with weight function (x). By square integrable, wemean that f,f < . One can show that such a space is isomorphicto a Hilbert space, a complete inner product space. Hilbert spacesplay a special role in quantum basis of partial differential equations5. The eigenvalues satisfy the Rayleigh quotient n= p nd ndx ba+ ba[p(d ndx)2 q 2n]dx n, n .The Rayleigh quotient is named afterLord Rayleigh, John William Strutt,3rdBaron Raleigh (1842-1919).This is verified by multiplying the eigenvalue problemL n= n (x) nby nand integrating. Solving this result for n, we obtain the Rayleighquotient. The Rayleigh quotient is useful for getting estimates ofeigenvalues and proving some of the other some of these properties for the eigenvalue problemy = y,y(0) =y( ) = is a problem we had seen many times. The eigenfunctions for this eigenvalueproblem are n(x) =sinnx,with eigenvalues n=n2for n=1, 2.


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