Computing eigenvalues of regular sturmliouville problems. Computing eigenvalues of singular sturmliouville problems. Pdf computational procedure for sturmliouville problems. Sturmliouville problem an overview sciencedirect topics. Next, a concrete implementation to the inverse sturmliouville problem algorithm proposed by barcilon 1974 is provided. Our approach to the sturmliouville eigenvalue problem shall be via a very general abstract method, coming from functional analysis. Zettl abstract an algorithm is presented for computing eigenvalues of regular selfadjoint sturmliouville problems with matrix coe cients and arbitrary coupled boundary conditions. An eigenvalue of the the sl problem 1 is a value of for which a nontrivial solution exist. Continuous dependence of eigenvalues on the problem. On the complexity of the multivariate sturmliouville. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. The eigenvalues of a sturmliouville problem are all of multiplicity one.
These problems will appear when we solve pdes on the nite interval using separation of variables. Computing eigenvalues of regular sturmliouville problems h. Orthogonality sturmliouville problems eigenvalues and eigenfunctions eigenvalues and eigenfunctions a nonzero function y that solves the sturmliouville problem pxy. An efficient technique for finding the eigenvalues of. Multiparameter eigenvalue problems sturm liouville theory pdf. Pdf the solution of the 1d sturmliouville problem using the control volume method is discussed. An efficient technique for finding the eigenvalues of fourth. The simplest applications lead to the various fourier series, and less simple applications lead to generalizations of fourier series involving bessel. Pdf the solution of the 1d sturm liouville problem using the control volume method is discussed. Introduction sledge 8, sleign 3,2 see also 6 and the nag library code 9 are. For this, we conveniently reformulate the condition 1. Although the subject of sturmliouville problems is over 170 years old, a sur. Royden, real analysis, third edition, macmillan, new yorklondon, 1988.
A nonzero function y that solves the sturmliouville problem pxy. Properties of sturmliouville eigenfunctions and eigenvalues. However, the discreteness and simplicity of the eigenvalues. Sturmliouville problems arise directly as eigenvalue problems in one space dimension. Note that sl differential equation is essentially an eigenvalue problem since. Sturmliouville problems by regular problems, and to the theoretical aspects of the sleign2 computer program. If m and n are two distinct eigenvalues of a sl system, with orrcesponding eigenfunctions y m and y n, then y m and y n. Furthermore, computational feasibility and applicability of this algorithm to solve inverse sturmliouville problems of higher order for n 2,4 are veri. In the previous lecture i gave four examples of different boundary value problems for a second. Sturmliouville problems arise directly as eigenvalue problems in one space. Greens identity and selfadjointness of the sturmliouville operator 11 2. Eigenvalue problems, sturmliouville boundary value. In this paper the solution of an eigenvalue problem of the sl type with ordi.
Goh fourier series and sturmliouville eigenvalue problems. We are interested in the sturmliouville eigenvalue problem slep with dirichlet boundary conditions. While solving these equations we used the method separation of variables which reduces the problem to one of the following types of sturmliouville problems sturmliouville eigenvalue problem. Show that for any smooth functions ux and vx the following properties hold. The sturmliouville problem appears directly as the eigenvalue problem in a onedimensional space. Introduction this paper follows on from the results in the previous paper in this subject 5. For a more detailed study of the integer order sturmliouville theory, we refer the reader to 15. In all examples we have analyzed so far the boundary value problem that. Sturmliouville eigenvalue problems selfadjoint operators like sl have nice properties. We can multiply this equation by mx a2x 1 x2 e r dx x 1 x, to put the equation in sturmliouville form. Eigenvalue problems, sturmliouville boundary value problems.
Sturmliouville problems sturmliouville problems are boundaryvalue problems that naturally arise when solving certain. Conversion of a linear second order differential equation to sturm liouville form. The sturm liouville theory deals with the secondorder differential equation. For classical regular twopoint selfadjoint sturmliouville problems slp the dependence of the eigenvalues on the boundary conditions is well understood because of some surprisingly recent results.
Pdf uniform estimation of the eigenvalues of sturm. Comparison methods for solving nonlinear sturmliouville. Variational techniques for sturmliouville eigenvalue problems. Introductionthe solution of various problems in the field of mathematical physics and engineering is closely related to the solution of the corresponding onedimensional linear sturmliouville boundary value eigenproblem. Matrix methods for computing eigenvalues of sturmliouville. Proposition 6 the set of eigenvalues of a regular sturmliouville problem is countably in nite, and is a monotonically increasing sequence 0 problem, except that the sequence is monotonically nondecreasing. Such an eigenvalue problem is called a regular sturmliouville problem. As particular examples, one could imagine the onedimensional schrodinger equation with spacedependent potential function and numerous problems of physicochemical kinetics.
The following theorem characterizes the eigensolutions we obtain. The main result of sturmliouville theory states that, for the regular sturmliouville problem 1, 2, 3. In this lecture, we consider the problem of developing a solution. We say that the number is an eigenvalue if there is a nonzero solution u2c2a. We will merely list some of the important facts and focus on a few of the properties. The code sleign sturmliouville eigenvalue was introduced in 1978. That earlier paper on sturmliouville boundary value problems studies the continuous. Atkinson and others published multiparameter eigenvalue problems. The eigenvalues of a sturmliouville problem are the values of. It is the theory behind sturmliouville problems that, ultimately, justi. Fourier series and sturmliouville eigenvalue problems. The rst will have exact closed form solutions because the boundary condtions are very simple, the second will not have closed form solutions, and will. Eigenfunctions of a regular sturmliouville problem satisfy an orthogonality property, just like the eigenfunctions in section 4.
Values of for which 1,2 has a nontrivial solution are called eigenvalues and a nontrivial solution u corresponding to is called an eigenfunction. Sturm liouville problems are boundaryvalue problems that naturally arise when solving certain partial differential equation problems using a separation of. Sturmliouville theory find, read and cite all the research you need on researchgate. For any regular sturmliouville problem, the following theorems are valid. Abbasbandy and shirzadi 14 applied the homotopy analysis method ham to numerically approximate the eigenvalues of the second and fourth order sturmliouville problems. Such a value is called an eigenvalue and the corresponding nontrivial solutions y are called eigenfunctions. Greens function, boundary value problems, eigenvalue and eigenfunction. Sturmliouville examples math 322 spring 2014 we will go through two examples of solving sturmliouville problems. They also commonly arise from linear pdes in several space dimensions when the equations are separable in some coordinate system, such as cylindrical or spherical coordinates. There is an orthogonal basis for rn consisting of eigenvectors for a.
Shi and cao 15 presented a computational method for solving eigenvalue. We have found the method of separation of variables to be quite successful in solving some homogeneous partial differential equations with homogeneous boundary con ditions. The differential equation is said to be in sturmliouville form or selfadjoint form. Suppose that we have a regular sturmliouville problem.
The set of eigenvalues of the sl problem 1 forms an increasing sequence 1. Sturmliouville eigenvalue problems with mathematica journal of. In 18 they compute eigenvalues and eigenfunctions of singular twointerval sturmliouville problems. Eigenvalues of sturmliouville problems with discontinuous boundary conditions aiping wang, anton zettl communicated by jerome goldstein abstract.
Zettl abstract the eigenvalues of sturmliouville sl problems depend not only continuously but smoothly on the problem. While in 19, they solve fourth order linear differential equations. We also have the fredholm alternative theorem we talked about before for all regular sturmliouville problems. Its proof is very similar to the analogous theorem 4. Sep 01, 20 fliess series approach to the computation of the eigenvalues of fourthorder sturmliouville problems appl. It also arises when linear partial differential equations are separable in a certain coordinate system. In all examples we have analyzed so far the boundary value problem. Book multiparameter eigenvalue problems sturm liouville theory uploaded by karl may, ebook pdf multiparameter eigenvalue problems sturm liouville theory contains important information and a detailed explanation about ebook pdf multiparameter eigenvalue problems sturm liouville theory, its. This is also true for much more general sturmliouville operators, as we will see shortly.
If m and n are two distinct eigenvalues of a sl system, with orrcesponding eigenfunctions y m and y n, then y m and y n are orthogonal. Fractional sturmliouville problem based on the operational matrix method was presented in 20. The proof of the conclusions 1 and 2 of the theorem is quite simple and is a generalization of the proof of the corresponding theorem for eigenvalues and eigenvectors of a symmetric matrix which is proved in many physics courses and. The sturmliouville eigenvalue problem is then written l. All secondorder linear ordinary differential equations can be recast in the form on the lefthand side of by multiplying both sides of the equation by an appropriate integrating factor although the same is not true of secondorder partial differential equations, or if y. An expression for the derivative of the nth eigenvalue with respect to a given parameter.
Formulation of the homogeneous sturmliouville problem 10 2. The general form of the sturmliouville equation is an ode for ux of the form. Sturmliouville theory american mathematical society. The eigenvalues of sturmliouville sl problems depend not only continuously but smoothly on the problem.
Selfadjoint operators like sl have nice properties. Sturmliouville eigenvalue problem 8, 910 is called regular if the coe. The rst will have exact closed form solutions because the boundary condtions are very simple, the second will not have closed form solutions, and will need to be solved graphically andor numerically. For most values of the parameter, problem 1 has only the trivial solution.
1085 1151 1342 896 990 1584 383 1467 1629 1531 1257 1141 731 1342 1147 1468 415 854 1498 1262 1596 1617 285 1341 537 400 1133 1598 1559 681 1380 168 1088 750 594 872