The resulting operator is referred to as a sturmliouville operator. Green function and eigenfunctions let l be a regular sturmliouville operator on an interval a, b together with regular boundary conditions. Aim is to nd all aluesv for which a nontrivial solution y. Then the subspace spanned by the eigenfunctions corresponding to admits an orthonormal basis of realvalued functions. Introduction 179 any solution of the homogeneous sturmliouville equation linearly independent of. In sturmliouville theory, we say that the multiplicity of an eigenvalue of a sturmliouville problem l. Then we have a solution formula for ux for any fx we want to utilize. Integral equations and greens functions ronald b guenther and john w lee, partial di. The eigenfunctions of a sturmliouville problem can be chosen to be real. Darboux transformation of the greens function of a regular sturmliouville problem article pdf available in russian physics journal 4810. Now, if i have a scalarvalued function vx, we can also form the operator rvru. Apart from their use in solving inhomogeneous equations, green functions play an important role in many areas. Such an equation is said to be in sturmliouville form. The result is trivially true if is a simple eigenvalue.
Hassana aa 2017 greens function solution of nonhomogenous. In recent years sturmliouville problems with boundary conditions requiring dis continuous eigenfunctions or discontinuous derivatives of. We sought to determine a greens function for a general sturmliouville problem with homogeneous bound. Greens functions greens function of the sturmliouville equation consider the problem of. Such problems are called sturmliouville problems and their solutions have a rich structure as we shall see. The tool we use is the green function, which is an integral kernel representing the inverse operator l1. Greens functions for sturmliouville problems on directed. 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 is a vector. Values of for which 1,2 has a nontrivial solution are called eigenvalues and a nontrivial solution u corresponding to is called an eigenfunction. Just as the 183637 papers of sturm and liouville started the study of regular slp, the 1910 paper of weyl initiated the investigation of singular slp. Sturmliouville examples math 322 spring 2014 we will go through two examples of solving sturmliouville problems. Proposition 4 let be an eigenvalue of a regular or periodic sturmliouville problem. The sturmliouville expansion for the kummer green function. In this video, i prove the sturmliouville theorem and explain the ideas of eigenvalues and eigenfunctions.
Lagranges method for sturm liuoville problems see for example 2. Pdf greens functions for sturmliouville problems on. 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. Sturmliouville twopoint boundary value problems 3 we bring 28. The eigenvalues of a sturmliouville problem are all. Well see later how this compares to the eigenfunction expansion for inverting sturmliouville operators that we gave in 2. In 1910 hermann weyl published one of the most widely quoted papers in analysis 607. Greens functions for sturmliouville problems on directed tree graphs. The sturmliouville operator on the axis is considered.
The green function for the kummer differential equation is cal culated by means of the sturmliouvi le method. Chapter 5 green functions in this chapter we will study strategies for solving the inhomogeneous linear di erential equation ly f. Greens function for twointerval sturmliouville problems aiping wang, anton zettl dedicated to john w. The resulting operator is referred to as a sturm liouville operator. We will highlight some of the properties of these operators and see how they are used in applications. Greens functions for sturmliouville problems on directed tree. Lagranges method for sturmliuoville problems see for example 2. We sought to determine a green s function for a general sturm liouville problem with homogeneous bound. Consider the problem of finding a function u ux, x. Greens function for regular sturmliouville problems we are interested in in solving problems like ly. Finally, section 3 is devoted to the construction and the formula for the the green s function. Greens function for regular sturm liouville problems we are interested in in solving problems like ly.
We construct the greens function and the characteristic function for twointerval regular sturmliouville problems with separated and coupled, selfadjoint and nonselfadjoint, boundary conditions. As byproduct we obtain laguerre, hermite and whittaker green functions. Greens functions, oscillation, sturmliouville eigenvalues. Greens function solution of nonhomogenous regular sturm. However, we are not guaranteed a nice set of eigenfunctions. Dozens of papers are published on sturmliouville problems slp every year. Variational techniques for sturmliouville eigenvalue problems. 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. We consider the dirichlet bvp for the general secondorder operator.
Green function for finite stretched string with periodic forcing. R is a second order ode with boundary conditions at a and b. Orthogonality sturmliouville problems eigenvalues and eigenfunctions sturmliouville equations a sturmliouville equation is a second order linear di. Earlier we noted the symmetry of the green function for the sturmliouville.
We are interested in the sturmliouville eigenvalue problem slep with dirichlet boundary conditions. Given a selfadjoint sturmliouville system as above, there is an orthonormal basis for the space l2. That is, note the minus sign in front of the 2nd derivative lu. A survey on stationary problems, greens functions and. The differential equation is said to be in sturmliouville form or selfadjoint form. On greens function for boundary value problem with. We assume no smoothness condition on the potential. Among the triumphs of nineteenthcentury mathematics was the realization that these sequences of eigenfunctions can be used to represent arbitrary functions1 via in nite series. Eigenvalue problems, integral equations, and greens functions. Greens functions green s function of the sturm liouville equation consider the problem of.
Pdf darboux transformation of the greens function of a. Finally, section 3 is devoted to the construction and the formula for the the greens function. In this paper we obtain greens function for a regular sturmliouville problem having the eigenparameter in all boundary conditions in which the left one is in quadratic form. Consider the following nonhomogeneous sturmliouville problem. The sturmliouville theory provides an in nite sequence of eigenvalueeigenfunction pairs.
1146 1435 1264 1426 686 271 1317 66 55 1423 363 329 231 1333 220 1000 1030 1079 831 44 271 857 1041 1487 937 1328 631 1019 664 693 307 1070 448 190 1223 789 1415 601 653 1262 351 529