Sturmliouville problems with eigenparameter dependent. Orthogonality sturmliouville problems eigenvalues and eigenfunctions sturmliouville equations a sturmliouville equation is a second order linear di. The eigenvalue problem of the sturm liouville systems depends on the boundary conditions. Characteristic functions for sturmliouville problems. These boundary conditions are called periodic boundary conditions. Just as a symmetric matrix has orthogonal eigenvectors, a selfadjoint sturmliouville operator has orthogonal eigenfunctions. Introduction to sturm liouville theory we have learned various techniques for solving certain odes, and in a rst course in di erential equations, such odes are generally accompanied by one or more initial conditions. Even for the regular case, a general discussion of separated and coupled selfadjoint boundary conditions is not easy to find in the existing literature. There are two important type boundary conditions, periodic type and separated type. A sturmliouville sl problem consists of a sturmliouville equation on an interval. Sturmliouville problem, nonlocal boundary conditions. R is a second order ode with boundary conditions at a and b.
Dirichlet conditions correspond to applying a zero temperature at the ends, neumann means insulating the ends, etc. Assuming that homogeneous boundary conditions are speci. The eigenvalues of a sturm liouville boundary value problem are nonnegative real numbers. L x what may seem a little surprising at rst is that the fact that the fouriersine functions are solutions to a. The eigenvalue problem for speriodic boundary value problem, that. This is a more complicated set of boundary conditions than our earlier examples, but we can approach it using the same ideas. Although the subject matter of slp is briefly discussed in many books, these discussions tend to be sketchy, particularly in the singular case. The nontrivial nonzero solutions, of the sturmliouville boundary value problem only exist at certain. Special cases are dirichlet bc d a d b 0 and neumann bc c a c b 0 periodic boundary conditions boundary conditions of the form ya yb y0a y0b 3 are called eriopdic boundary conditions. A second ordered di erential equation of the form d dx. Application of two parameter eigencurves to sturmliouville problems with eigenparameterdependent boundary conditions. The solution will always be locally squareintegrable, and the condition is a restriction on the largeasymptotic behaviour of. Sturmliouville problems and orthogonality eigenfunction expansions the hanging chain recall.
Pdf fractionalorder boundary value problem with sturm. For many physical applications of the sturmliouville problem a1, the spectrum of the associated differential operator with dirichlet boundary condition is either purely discrete e. Nonhomogeneous sturmliouville problems can arise when trying to solve nonhomogeneous pdes. Eigenfunctions of a regular sturmliouville problem satisfy an orthogonality property, just like the eigenfunctions in section 4. Oscillation theory for indefinite sturmliouville problems. Problem 5 isnt, although some authors use a definition of sturmliouville problem that does include it. For the literature in nbody problem, readers can refer to 5,8. Equation 7 together with the boundary conditions u0 0.
Proposition 3 let v 1 and v 2 be eigenfunctions of a regular sturmliouville operator 1 with boundary conditions 2 corresponding to distinct eigenvalues 1. Conversion of a linear second order differential equation to sturm liouville form. Sturmliouville problems with boundary conditions rationally. For the classical solution, a priori estimate is established and unique existence is proved. Introduction to sturmliouville theory we have learned various techniques for solving certain odes, and in a rst course in di erential equations, such odes are generally accompanied by one or more initial conditions. The differential equation is said to be in sturmliouville form or selfadjoint form. In addition, the associated eigenfunctions are orthogonal to each other with respect. Introduction to sturmliouville theory and the theory of generalized fourier series we start with some introductory examples. Regular sturmliouville problems slp with boundary conditions requiring a jump discontinuity at an interior point of the underlying interval are a very active current research area. Just as a symmetric matrix has orthogonal eigenvectors, a selfadjoint sturm liouville operator has orthogonal eigenfunctions. Oscillation theory for indefinite sturmliouville problems with eigenparameterdependent boundary conditions volume 127 issue 6 p. Many of these properties hold, as well, for solutions to. 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.
In this paper, we study the inverse spectral problems of a class of sturmliouville problems, l, defined on interval 0, 1 with separated boundary conditions and with transmission conditions at an interior point \ a\in 0,1\, which are rationally dependent on the eigenparameter. Request pdf sturmliouville problems with boundary conditions rationally dependent on the eigenparameter, ii necessary and sufficient conditions are given for two sequences. Given a selfadjoint sturmliouville eigenvalue problem, the eigenvalues form a countable set of real numbers n. Proposition 3 let v 1 and v 2 be eigenfunctions of a regular sturm liouville operator 1 with boundary conditions 2 corresponding to distinct eigenvalues 1. Characterization of the spectrum of the sturmliouville operator with irregular boundary conditions.
Math 5311 notes on sturmliouville equations many problems in physics, engineering, and chemistry fall in a general class of equations of the form 1 wx d dx wxpx du dx. We need an appropriate set to form a basis in the function space. The appropriate sturmliouville problem is solved, nding the eigenvalues nand the corresponding eigenfunctions u nx. Eigenvalue problems, sturmliouville boundary value problems. Sturmliouville eigenvalue problem 8, 910 is called regular if the coe. Oscillation theory for indefinite sturm liouville problems with eigenparameterdependent boundary conditions volume 127 issue 6 p. However, many odes that arise naturally in physics are accompanied with boundary conditions, in which the value. Its proof is very similar to the analogous theorem 4. The eigenvalue problem of the sturmliouville systems depends on the boundary conditions.
Sturm liouville operator with second kind of nonlocal boundary value conditions is considered. Since this is a sturmliouville problem, we know that the only possible values of with. Variational techniques for sturmliouville eigenvalue problems. 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. Regular sturm liouville problems slp with boundary conditions requiring a jump discontinuity at an interior point of the underlying interval are a very active current research area. Introduction to sturmliouville theory and the theory of. Properties of sturmliouville eigenfunctions and eigenvalues. As in the previous section we have boundary conditions of the form b ay. The nontrivial nonzero solutions, of the sturm liouville boundary value problem only exist at certain.
Such equations are common in both classical physics e. Most of our proofs are adapted from 1 and are given using variational methods. The appropriate sturm liouville problem is solved, nding the eigenvalues nand the corresponding eigenfunctions u nx. Nonlocal boundary value problem in terms of flow for sturm. Request pdf sturmliouville problems with boundary conditions rationally dependent on the eigenparameter, ii necessary and sufficient conditions are. Eigenvalue problem of sturmliouville systems with separated. Such an equation is said to be in sturmliouville form.
In this case, the sturm liouville eigenvalue problem consists of equation 7 together with the boundary conditions u0 0. The solution v is required to satisfy boundary conditions of the type. There is a corresponding sequence of eigenfunctions that form an orthonormal basis for l2. 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. 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. Forexample, consider aradiallysymmetric nonhomogeneousheat.
Given a selfadjoint sturmliouville system as above, there is an orthonormal basis for the space l2. We prove the uniqueness theorem for the inverse problem by three spectra. The beautiful feature of these sturmliouville operators is that they are selfadjoint with respect the inner product f,g b a fx. Sturmliouville operator with second kind of nonlocal boundary value conditions is considered. However, we are not guaranteed a nice set of eigenfunctions. Associated finitedifference scheme is proposed on uniform mesh and secondorder accuracy for approximation is proved. In this case, the sturmliouville eigenvalue problem consists of equation 7 together with the boundary conditions u0 0. If one imposes a separated condition b 1u 0, this restricts the initial conditions ua.
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