Sturm-Liouville Systems Research Articles

The Stieltjes Summability Method and Summing Sturm–Liouville Expansions

The Stieltjes summability method for divergent integrals is defined. The name is derived from its close relationship with the Stieltjes transform. After proving some basic properties, the Stieltjes summability method is compared with Cesàro and Abel summability methods. The main result proves the Stieltjes summability of eigenfunction expansions associated with a certain class of singular Sturm–Liouville systems, and develops, using the Stieltjes means, a stable method of summing such expansions under perturbations of the coefficients.

Branching from Large Eigenvalues of Nonlinear Sturm–Liouville Systems

The behavior of solutions branching from large eigenvalues of certain nonlinear Sturm–Liouville systems is studied. A perturbation method is presented which permits the solution branches to be extended beyond the region of validity of standard perturbation and iteration techniques. Numerical calculations support the asymptotic results.

Direct Solutions for Sturm-Liouville Systems with Discontinuous Coefficients

Direct Solutions for Sturm-Liouville Systems with Discontinuous Coefficients

Distribution of Eigenvalues of A Two-Parameter System of Differential Equations

In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval $0 \leqslant {x_1} \leqslant 1$, the second for the interval $0 \leqslant {x_{2 }} \leqslant 1$, and each containing the parameters $\lambda$ and $\mu$. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $({\lambda _{j,k}},{\mu _{j,k}})$ and ${\psi _{j,k}}({x_{1,}}{x_2})$, respectively, $j, k = 0, 1, \ldots$, asymptotic methods are employed to derive asymptotic formulae for these expressions, as $j + k \to \infty$ when $(j, k)$ is restricted to lie in a certain sector of the $(x, y)$ -plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the ${\psi _{j,k}} ({x_1}, {x_2})$.

Distribution of eigenvalues of a two-parameter system of differential equations

In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval 0 ⩽ x 1 ⩽ 1 0\, \leqslant \,{x_1}\, \leqslant \,1 , the second for the interval 0 ⩽ x 2 ⩽ 1 0\, \leqslant \,{x_{2\,}}\, \leqslant \,1 , and each containing the parameters λ \lambda and μ \mu . Denoting the eigenvalues and eigenfunctions of the simultaneous systems by ( λ j , k , μ j , k ) ({\lambda _{j,k}},{\mu _{j,k}}) and ψ j , k ( x 1 , x 2 ) {\psi _{j,k}}({x_{1,}}{x_2}) , respectively, j , k = 0 , 1 , … j,\,k\, = \,0,\,1,\, \ldots \, , asymptotic methods are employed to derive asymptotic formulae for these expressions, as j + k → ∞ j + k \to \infty when ( j , k ) (j,\,k) is restricted to lie in a certain sector of the ( x , y ) (x,\,y) -plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the ψ j , k ( x 1 , x 2 ) {\psi _{j,k}}\,({x_1},\,{x_2}) .

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract The oscillation theorem for two simultaneous Sturm-Liouville systems in two parameters is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theorem under another important definiteness condition.

Spectral properties and oscillation theorems for mixed boundary-value problems of Sturm-Liouville type

This paper presents analogs, for certain mixed boundary-value problems, of the spectral and oscillatory properties exhibited by classical Sturm-Liouville systems. These mixed boundary-value problems have Green's functions which are sign consistent for all even and/or odd orders.

On the Distribution of the Eigenvalues of a Two-Parameter System of Ordinary Differential Equations of the Second Order

In this paper two simultaneous Sturm–Liouville systems are considered, the first defined for the interval $0 \leqq x_1 \leqq 1$, the second for the interval $0 \leqq x_2 \leqq 1$, and each containing the parameters $\lambda $ and $\mu $. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $(\lambda _{j,k} ,\mu _{j,k} )$ and $\psi _{j,k} (x_1 ,x_2 )$, respectively, $j,k = 0,1, \cdots $, the principle of the argument and asymptotic methods are employed to derive asymptotic formulas for these expressions, as $j^2 + k^2 \to \infty $, when $(j,k)$ is restricted to lie in each of several sectors of the $(x,y)$-plane. These results partially resolve a problem posed by Atkinson concerning the behavior of the eigenvalues and eigenfunctions of multiparameter Sturm–Liouville systems and constitute a further stage in the development of the theory related to these questions.

Multiple solutions of coupled Sturm-Liouville systems

This paper deals with the coupled Sturm-Liouville system − ( pu′)′ + Pu + rv = λ 1 u + λ 1 N 11( u, v) + λ 2 N 21( u, v), − ( qv′)′ + Qv + ru = λ 2 v + λ 1 N 12( u, v) + λ 2 N 22( u, v), α 11 u(0) + α 12 u′(0) = 0 = α 21 v(0) + α 22 v′(0), β 11 u(1) + β 12 u′(1) = 0 = β 21 v(1) + β 22 v′(1). The functions p, P, q, Q, r are smooth; λ 1 and λ 2 are eigenparameters; N ij ( u, v) is analytic and of higher order. The linearized problem, all N ij &z.tbnd; 0 , is shown to have eigenvalues ( λ 1, λ 2) which are continuously distributed along a sequence of monotonically decreasing curves in the λ 1 λ 2-plane. A generalized Lyapunov-Schmidt method establishes that if ( λ 1, λ 2) is near a simple eigenvalue of the linearized problem, then the number of small solutions of the nonlinear problem corresponds to the number of real roots of a certain polynomial.

Numerical approximation of eigenvalues of Sturm-Liouville systems

A new technique based on the Prüfer transformations is proposed. This technique is illustrated using several examples of Sturm-Liouville systems offering varying degrees of computational difficulty. For some of the examples, other techniques have been used to calculate eigenvalues, and in these cases, results obtained are compared. The technique appears capable of generalization to multiparameter systems of differential equations.

Upper and lower bounds on eigenvalues of second-order Sturm-Liouville systems

Separation and comparison theorems have attracted the attention of mathematicians for a long time. The first results of this kind go back to Sturm [l]. Additional results were obtained by Picone [2] and Bother [3,4]; an account of this theory is to be found in the book of Ince [S]. More recently, a vast literature on this subject has developed of which Swanson [6] and Barret [7] both contain an extensive list of references. The separation and comparison theorems have been principal tools used to obtain oscillation results and upper and lower bounds on eigenvalues for Sturm-Liouville systems. Usually the bounds on the eigenvalues are determined in terms of the eigenvalues of a related system, although explicit, analytic values of the eigenvalues or explicit equations for the eigenvalues of the comparison systems are not necessarily known; see for example St. Mary [8] and Howard [9]. Some explicit bounds for eigenvalues have been obtained by Breuer and Gottlieb [lo]. In their paper, bounds are given for the first eigenvalue for a variety of homogeneous conditions. However, explicit analytic bounds for more than the first eigenvalue are given only under the condition that the eigenfunction vanish at the endpoints of the interval. In this paper, we consider the eigenvalue problem

A Prüfer transformation for the equation of the vibrating beam

In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows. The eigenfunction and its first three derivatives are considered as a four-dimensional vector, ( u , u ′ , p u , ( p u ) ′ ) (u,u’,pu,(pu)’) . This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Prüfer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives, u ′ , p u , ( p u ) ′ u’,pu,(pu)’ and the relation of these zeros to the n − 1 n - 1 zeros of the n n th eigenfunction. The method also can be used to establish comparison theorems of a given type.

Estimating the Eigenvalues of Sturm–Liouville Problems by Approximating the Differential Equation

This paper is concerned with computing accurate approximations to the eigenvalues and eigenfunctions of regular Sturm–Liouville differential equations. The method consists of replacing the coefficient functions of the given problem by piecewise polynomial functions and then solving the resulting simplified problem. Error estimates in terms of the approximate solutions are established and numerical results are displayed. Since the asymptotic properties for Sturm–Liouville systems are preserved by the approximation, the relative error in the higher eigenvalues is much more uniform than is the case for finite difference or Rayleigh–Ritz methods.

Asymptotic Behavior of Sturm-Liouville Systems

It is shown that the bound-state solutions of regular and singular Sturm-Liouville systems have the same asymptotic behavior, in the limit of infinitely deep potential wells. This conclusion is illustrated explicity by showing the profound similarity that exists between two seemingly unconnected problems: Mathieu's equation in a finite domain and Schrödinger's equation with Morse's potential.

Upper and lower bounds on eigenvalues of Sturm-Liouville systems

It is shown that within the scope of ordinary differential equations, the unknown eigenvalues of second order Sturm-Liouville systems, under various homogeneous boundary conditions, can be bounded from above and below in terms of known eigenvalues of judiciously constructed, associated Sturm-Liouville systems and in terms of the coefficients of the underlying system.

A new type of boundary value coupling for second order Sturm-Liouville systems

A new type of boundary value coupling for second order Sturm-Liouville systems

Equivalence of Sturm-Liouville-Type Distributed-Parameter Systems

This paper analyzes equivalence properties of linear distributed-parameter systems characterized by Sturm-Liouville-type differential equations. The approach taken here may be regarded as an application of the method of independent-variable transformation to those fundamental equations. In accord with the meaning of equivalence used for lumped-parameter systems, distributed-parameter systems are considered here as equivalent when they possess the same terminal behavior. The principal result is an equivalence theorem which establishes a general method of equivalent transformation of Sturm-Liouville systems. The fact that we can generate any number of equivalent parameter distributions from a given one is not only of theoretical interest but also of practical importance, because it offers a flexibility in the design of distributed-parameter structures. The derivation of the transformation function which plays a central role in the equivalent transformation is briefly described. Discussion is also given to the quantities which are invariant under the equivalent transformation. In the later part of the paper applications of the general theory to several problems of practical interest are considered, which include guiding systems for electromagnetic and space-charge waves such as nonuniform transmission lines, lens-like media, and nonuniform drift regions. The physical meanings of the transformation function, the parameters introduced, and various transformation invariants are discussed in detail. A simple example of equivalent systems is also illustrated in the case of nonuniform transmission lines.

Maximal theorems for some orthogonal series II

If { u n } is the orthonormal sequence of eigenfunctions arising from a nonsingular (or sometimes a singular) Sturm-Liouville system with separated boundary conditions defined on (0, π), it has long been known that the eigenfunction expansion with respect to { u n } of a function φ on (0, π) has properties similar to those of the Fourier-cosine expansion of φ. For instance, there is the classical equiconvergence theorem of Haar ([5]; see [3] pp. 1616–1622 for a general survey). In this paper, by restricting attention to two specific classes of singular Sturm-Liouville systems, we shall establish a much more precise relationship between the corresponding eigenfunction expansions and the Fourier-cosine expansions. Many results for Fourier series can then be carried over to these eigenfunction expansions. The method of proof includes as special cases Fourier-Bessel functions, ultraspherical polynomials, Jacobi polynomials and any nonsingular Sturm-Liouville system.

Free and forced torsional motion of joined concentric circular shells

This paper describes the free and forced motion of two, thin, concentric, circular cylindrical shells joined at their ends by rigid diaphragms with inertial properties. Mathematically the problem is described by two special Sturm-Liouville systems with common eigenvalues coupled only through the boundary conditions. The systems possess a modified orthogonality principle and give rise to a modified expansion principle for the associated non-homogeneous systems. These principles are derived in detail and several free motion problems are presented to illustrate the theory.

Recursively generated Sturm-Liouville polynomial systems

Recursively generated Sturm-Liouville polynomial systems