One way to solve such problems is using the eigenfunction expansion for the correspondinghomogeneoussturmliouvilleproblem. In addition, the unknown function y is required to satisfy boundary conditions. Pdf regular and singular sturmliouville problems with coupled. Dependence of eigenvalues on the boundary conditions of.
The fourier cosine transform of a function fx will be denoted by. Regular approximations of singular sturm liouville problems p. The history of boundary value problems for differential equations starts with the wellknown studies of d. Sturm liouville theory is actually a generalization for infinite dimensional case the famous eigenvalueeigenvector problems for finite square matrices that we discussed in part i of this tutorial. For classical regular twopoint selfadjoint sturm liouville problems slp the dependence of the eigenvalues on the boundary conditions is well understood because of some surprisingly recent results. For bi 0, we have what are called dirichlet boundary. Namely, we want to solve the eigenvalue problem ly xy. For different values, one has special types of boundary conditions. Application of two parameter eigencurves to sturmliouville problems with eigenparameterdependent boundary conditions. The aim is to analyze the numerical solution of various regular and singular sturmliouville problems by means of.
The eigenvalues of a sturm liouville problem are all of multiplicity one. Computation of eigenvalues of singular sturmliouville systems by d. Secondorder nonlinear singular sturmliouville problems. Dependence of eigenvalues on the boundary conditions of sturm.
Sturmliouville problem, or eigenvalue problem, in mathematics, a certain class of partial differential equations pdes subject to extra constraints, known as boundary values, on the solutions. A singular sturmliouville problem with limit circle. In this case, the sturmliouville eigenvalue problem consists of equation 7 together with the boundary conditions u0 0. Basedontheseresults,someresearchersstudiedtheregularsturmliouvilleproblemswitheigen. The user has complete freedom in specifying the boundary conditions as long as these are formulated appropriately in a manner to be described below. A singular sturmliouville problem with limit circle endpoints and eigenparameter dependent boundary conditions. Secondorder nonlinear singular sturmliouville problems with integral boundary conditions. Such problems are called singular sturmliouville problems, and are the. It happens at any time by using the approach of separation of variables to solve suitable partial differential.
Solving singular boundary value problems for ordinary differential. We need an appropriate set to form a basis in the function space. Wupositive solutions of a singular boundary value problem for systems of secondorder differential equations. Boundary value problems and sturmliouville theory at the endpoints.
Positive solutions for singular sturmliouville boundary. In this article, we consider singular sturm liouville problems with transmission conditions. Here, we research a singular sturm liouville problem with two limitcircle endpoints and the parameter is not only in the equation but also in the boundary conditions. Introduction to sturmliouville theory trinity university. The sturmliouville boundary value problem on the half line. It is well known that the sturmliouville problem with eigenparameter dependent boundary conditions is of great importance for its wide applications in mechanics, mathematical physics, and engineering see 14 and the corresponding references cited therein. A sturm liouville sl boundary value problem consists of a second order linear. The singular characterisation is an exact parallel of the regular one and reduces to it. The eigenvalues of a sturmliouville problem are all of multiplicity one. In this case, the sturm liouville eigenvalue problem consists of equation 7 together with the boundary conditions u0 0. The methods and notions that originated during studies of the sturmliouville problem played an important role in the development of many directions in mathematics and physics.
Problem 5 isnt, although some authors use a definition of sturm liouville problem that does include it. 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. Although a sturm liouville problem can be formulated in operator form as l y. In this paper, we study the secondorder nonlinear singular sturmliouville boundary. Pdf eigenvalues of both regular and singular sturmliouville sl problems with general coupled selfadjoint boundary conditions are.
Here we extend the known classical results to such problems. In recent years, various problems of the spectral theory of regular sturmliouville problem whose boundary conditions depend on spectral parameter have been examined in 1822. The appropriate sturm liouville problem is solved, nding the eigenvalues nand the corresponding eigenfunctions u nx. Regular approximations of singular sturmliouville problems p. A sturmliouville equation is a second order linear differential equation that can. Recall that when put in sturmliouville form we had px rx x and qx. A general sturm liouville problem can be written as pxy0 0. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturm liouville problem l. A matrix method for the solution of sturmliouville problems. Nonselfadjoint singular sturmliouville problems with. The problem studied here differs from the standard sturmliouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter l appears not only in the differential equation, but also in the boundary conditions.
For example heat propagation in a rod of length l whose end points are kept at temperature 0 leads to the ode problem. Eigenvalue problems, sturm liouville boundary value problems. In the present paper, we shall show that the isolated eigenvalues of a selfadjoint sturmliouville problem with one singular endpoint are continuously dependent on the boundary conditions, and thus form the continuous eigenvalue branches on a subset of the space of boundary conditions in both the limit circle case and the limit point case. Computation of eigenvalues of singular sturmliouville systems. This is an example of a singular sturmliouville problem.
Given a selfadjoint sturmliouville system as above, there is an orthonormal basis for the space l2. Regular approximation of singular sturmliouville problems with eigenparameter dependent boundary conditions article pdf available in boundary value problems 20201 december 2020 with 33 reads. Sturmliouville problems 55 this has nontrivial solution for the pair a, b if and only if. A classical sturmliouville equation, is a real secondorder linear differential. A study of the eigenfunctions of the singular sturmliouville. Typically, sturmliouville problems arise in the context of the separation of variables method for partial di. Eigenvalues of both regular and singular sturmliouville sl problems with general coupled selfadjoint boundary conditions are characterised. Sturmliouville problem encyclopedia of mathematics. Pdf sturmliouville problems and discontinuous eigenvalues. The greatest success in spectral theory of ordinary differential operators has been achieved for sturmliouville problems. Such equations are common in both classical physics e. Orthogonality sturm liouville problems eigenvalues and eigenfunctions example if m.
Introduction to sturmliouville theory and the theory of. We were able to find the eigenvalues of problems 14 explicitly because in each problem the coefficients in the boundary conditions satisfy \\alpha\beta0\ and \\rho\delta. These problems may be regular or singular at each endpoint of the underlying interval. A study of the eigenfunctions of the singular sturm. The solution will always be locally squareintegrable, and the condition is a restriction on the largeasymptotic behaviour of. Regular and singular sturmliouville problems with coupled boundary conditions article pdf available in proceedings of the royal society of edinburgh section a mathematics 1263 january 1996. A gentle introduction to sturmliouville problems ryan walker march 10, 2010 ryan walker a gentle introduction to sturmliouville problems introduction the nonsingular problem the singular problem references a sturmliouville problem a sturmliouville problem slbvp on a. Sturmliouville problems with eigenparameter dependent. We seek the eigenfunctions of the operator found in example 6. Eigenvalue problems, sturmliouville boundary value problems. In this article, we consider singular sturmliouville problems with transmission conditions.
Regular sturmliouville problem the method of separation of variables to solve boundary value problems leads to ordinary di. The eigenvalues of the sturmliouville operator may be characterized as those for which the differential equation has a nontrivial solution satisfying both the boundary condition and the condition. This is a more complicated set of boundary conditions than our earlier examples, but we can approach it using the same ideas. Zettl abstract given any selfadjoint realization sof a singular sturm liouville sl problem, it is possible to construct a sequence fs rgof regular sl problems with the properties. We saw that the eigenvalues of this problem are n n2. These eigenvalueeigenfunction problems lie at the heart of many formulations. Zettl abstract given any selfadjoint realization sof a singular sturmliouville sl problem, it is possible to construct a sequence fs rgof regular sl problems with the properties. Positive solutions for singular sturmliouville boundary value problems with integral boundary conditions. Such conditions are known by various names including transmission conditions, interface conditions, point interactions in the physics literature, etc. Note that, the boundary conditions used in 117 are independent of spectral parameter. Introduction to sturmliouville theory and the theory of generalized. Most of our proofs are adapted from 1 and are given using variational methods.
See 24, an online reference, and successor to the classical handbook 1. Variational techniques for sturmliouville eigenvalue problems. The spectrum of singular sturmliouville problems with eigenparameter dependent boundary conditions and its approximation. Since this is a sturmliouville problem, we know that the only possible values of with. Eigenvalues of sturm liouville problems with discontinuous boundary conditions aiping wang, anton zettl communicated by jerome goldstein abstract. The spectrum of the boundary value problem in the singular case is simple but may. This characterisation, although elementary, appears to be new even in the regular case. R is a second order ode with boundary conditions at a and b. Boundary conditions for a solution y of a differential equation on interval a, b are classified as follows. Introduction regular sturmliouville problems slp with boundary conditions requiring a. Xiping liu, yu xiao, jianming chen college of science, university of shanghai for science and technology, shanghai, 200093, china abstract. Here, we research a singular sturmliouville problem with two limitcircle endpoints and the parameter is not only in the equation but also in the boundary conditions. Solving singular boundary value problems for ordinary di.
Regular approximations of singular sturmliouville problems. Regular and singular sturmliouville problems with coupled. A discontinuous sturmliouville problem with boundary. Regular approximation of singular sturmliouville problems. Equation 3 with these boundary conditions is a sturmliouville eigenvalue problem. Although the subject matter of slp is briefly discussed in many books, these discussions tend to be sketchy, particularly in the singular case. Introduction to sturmliouville theory and the theory of generalized fourier series we start with some introductory examples.
Regular and singular sturmliouville problems with coupled boundary conditions article pdf available in proceedings of the royal society of edinburgh section a mathematics 1263. Pdf regular approximation of singular sturmliouville. In this context, the idea is to transform the continuous problem 1 in a discrete one approximating each derivative by means of an. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. A singular sturm liouville problem with limit circle endpoints and eigenparameter dependent boundary conditions. Eigenfunctions of a regular sturmliouville problem satisfy an orthogonality property, just like the eigenfunctions in section 4. Such problems are called sturmliouville problems and their solutions have a rich structure as we shall see. Regular and singular sturmliouville problems with coupled boundary conditions.
Based on the modified inner product, we define a new selfadjoint operator such that the eigenvalues of such a problem are coincided with those of. Eigenvalues of both regular and singular sturm liouville sl problems with general coupled selfadjoint boundary conditions are characterised. Its proof is very similar to the analogous theorem 4. The boundary conditions for the ordinary boundary value problem are then lim.
Let hbe the subspace of functions that satisfy the boundary conditions of sl problem. Numerical solution of singular eigenvalue problems for odes. Pdf regular and singular sturmliouville problems with. The appropriate sturmliouville problem is solved, nding the eigenvalues nand the corresponding eigenfunctions u nx.
844 1536 450 29 922 796 114 1488 707 617 741 1305 709 1157 413 589 88 420 695 828 1173 906 119 868 963 1633 567 600 105 66 200 712 1317 1297 612 970 493 809 1290 484 1163 434 960 1241 453 1431 1000 380