Regular Sturm-Liouville Problems Research Articles

Application of Lagrange Multiplier for Solving Non – Homogenous Differential Equations

The Lagrange Multiplier method has been applied in solving the regular Sturm Liouville (RSL) equation under a boundary condition of the first kind (Dirichlet boundary condition). This method is a very powerful tool for solving the RSL equation and it involves the solution of the RSL equation with the expansion of eigenfunctions into trigonometric series. The efficiency of this approach is emphasized by solving two examples of regular Sturm Liouville problem under homogenous Dirichlet boundary conditions. The methodology is effectively demonstrated, and the results show a high degree of accuracy of the solution in comparison with the exact solution and reasonably fast convergence.

EIGENVALUES OF STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER DEPENDENT BOUNDARY AND INTERFACE CONDITIONS

In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.

Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus

The purpose of this paper is to investigate some properties of multiplicative regular and periodic Sturm-Liouville problems given in general form. We first introduce regular and periodic Sturm-Liouville (S-L) problems in multiplicative analysis by using some algebraic structures. Then, we discuss the main properties such as orthogonality of different eigenfunctions of the given problems. We show that the eigenfunctions corresponding to same eigenvalues are unique modulo a constant multiplicative factor and reality of the eigenvalues of multiplicative regular S-L problems. Finally, we present some examples to illustrate our main results.

Spectrum completion and inverse Sturm–Liouville problems

Given a finite set of eigenvalues of a regular Sturm–Liouville problem for the equation , the potential of which is unknown. We show the possibility to compute more eigenvalues without any additional information on the potential . Moreover, considering the Sturm–Liouville problem with the boundary conditions and , where are some constants, we complete its spectrum without additional information neither on the potential nor on the constants and . The eigenvalues are computed with a uniform absolute accuracy. Based on this result, we propose a new method for numerical solution of the inverse Sturm–Liouville problem of recovering the potential from two spectra. The method includes the completion of the spectra in the first step and reduction to a system of linear algebraic equations in the second. The potential is recovered from the first component of the solution vector. The approach is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations possessing remarkable properties and leads to an efficient numerical algorithm for solving inverse Sturm–Liouville problems.

Dependence of Eigenvalues of Sturm–Liouville Problems with Eigenparameter-Dependent Boundary Conditions and Interface Conditions

In this paper, regular Sturm–Liouville problems with eigenparameter-dependent boundary conditions and interface conditions are investigated. We obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, the interface conditions, as well as the endpoints. Moreover, we find the differential expressions for each parameters, respectively.

Eigenvalues of Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions

In this paper, regular Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions are investigated. We obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, and the endpoints, in particular, the eigenparameter-dependent boundary condition matrix. Moreover, we find the differential expressions for each parameters.

Dynamic identification of pretensile forces in a spider orb-web

In this paper we propose a constructive procedure to determine the radial pretension in an axially-symmetric orb-web, on the basis of one eigenfrequency and its corresponding vibration mode. The method can be applied both to axisymmetric modes, corresponding to the solution of a regular Sturm–Liouville problem, and to non-axisymmetric modes, corresponding to the solution of a singular Sturm–Liouville problem. In the absence of measurement errors of the eigenfunction, the identification of the pretension field is almost perfect. On the contrary, if the measurement presents some noise, then identification is imprecise. In this case, a regularization technique applied to the measurement of the eigenfunction, or to its derivative, is proposed, as well as a filtering process, which significantly improves the reconstruction of the pretension field. The fundamental mode of vibration, sampled with a suitable number of points, is the one that provides the lowest identification error.

General characteristics of a fractal Sturm–Liouville problem

In this paper, we consider a regular fractal Sturm-Liouville boundary value problem. We prove the self-adjointness of the differential operator which is generated by the $F^\alpha$-derivative introduced in [32]. We obtained the $F^\alpha$-analogue of Liouville's theorem, and we show some properties of eigenvalues and eigenfunctions. We present examples to demonstrate the efficiency and applicability of the obtained results. The findings of this paper can be regarded as a contribution to an emerging field.

Existence of solutions for infinite systems of regular fractional Sturm-Liouville problems in the spaces of tempered sequences

In this paper we investigate the existence of solutions for infinite systems of regular fractional Sturm-Liouville problems by an application of Meir-Keeler fixed point theorem in the tempered sequence spaces. We provide examples to check validity of our obtained results.

Quantitative projections in the Sturm Oscillation Theorem

There is c>0 such that for all f∈C[0,π] with at most d−1 roots inside (0,π)∑1≤n≤d|〈f,sin⁡(nx)〉|≥κ−κ2log⁡κ‖f‖L2whereκ=c‖∇f‖L2‖f‖L2. This quantifies the Sturm-Hurwitz Theorem and connects a purely topological condition (number of roots) to the Fourier spectrum. It is also one of few estimates on Fourier coefficients from below. The result holds more generally for eigenfunctions of regular Sturm-Liouville problems−(p(x)y′(x))′+q(x)y(x)=λw(x)y(x)on(a,b). Sturm-Liouville theory shows the existence of a sequence of solutions (ϕn)n=1∞ that form an orthogonal basis of L2(a,b) with respect to w(x)dx. Sturm himself proved that if f:(a,b)→R is a finite linear combinations of ϕn having d−1 roots inside (a,b), then f cannot be orthogonal to A=span{ϕ1,…,ϕd}. We prove a lower bound on the size of the projection ‖πAf‖.

Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions

This paper is concerned with regular Sturm–Liouville problems with eigenparameter dependent boundary conditions. We obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, in particular, eigenparameter-dependent boundary condition matrix. Moreover the differential expression of the eigenvalues with respect to the data is given. The dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by Pru¨fer transformation and the inequalities of eigenvalues.

Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

It is purpose of this paper to investigate Sturm-Liouville equation on many-interval with the eigenvalue parameter appearing linearly in the boundary conditions and with two supplementary transmission conditions. The classical Sturmian theory did not cover such type of many-interval boundary value transmission problems. For the classical Sturm-Liouville problems it is guaranteed that the problem is self-adjoint with compact resolvent, the spectrum is disctrete and consist of eigenvalues and the corresponding eigenfunctions form an orthogonal basis in the well-known Hilbert space . But the boundary-value-transmission problems are not self-adjoint and the system of eigenfunctions did not form a basis in the classical Hilbert space in general. Taking in view this fact we suggest a new approach for self-adjoint realization of such type transmission problems. Moreover, we define some new Hilbert spaces to establish positiveness of corresponding operator-pencil. At first we define a concept of generalized eigenfunctions for this kind of spectral problems. In particular it is shown that if the potential is continuous then the generalized eigenfunctions satisfies the considered problem is the classical sense. Then we introduce to the consideration some compact operators such a way that the considered boundary-value-transmission problem can be reduced to the appropriate operator-pencil equation. Finally, we prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. It is important to note that the obtained results extends classical results associated with regular Sturm-Liouville problems.

Bounds for distance between eigenvalues of boundary value problems with retarded argument

Abstract In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1, the transmission coefficients γ 1 = δ 1, γ 2 = δ 2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

Wave equations & energy

The focus of this work is apply Fourier analytic methods based on Parseval's equality to the computation of kinetic and potential energy of solutions of initial boundary value problems for general wave type equations on a finite interval. As a consequence, an energy equipartion principle for the solution is obtained. Within our methods are some new results regarding eigenfunction expansions arising from regular Sturm-Liouville problems in Sobolev spaces.

Existence of positive solutions for regular fractional Sturm-Liouville problems

Existence of positive solutions for regular fractional Sturm-Liouville problems

Uniform local Lipschitz continuity of eigenvalues with respect to the potential in L^1[a,b

The present paper shows that the eigenvalue sequence $\{\lambda_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.

Sufficient Condition for Convergence of Lagrange–Sturm–Liouville Processes in Terms of One-Sided Modulus of Continuity

A sufficient condition for the uniform convergence in the interval (0, π) of interpolation processes based on the eigenfunctions of a regular Sturm–Liouville problem with a continuous bounded variation potential is obtained. The condition is formulated in terms of a one-sided modulus of continuity of a function.

A Criterion of Convergence of Lagrange–Sturm–Liouville Processes in Terms of One-Sided Module of Variation

We obtain a criterion of uniform convergence inside the interval (0, π) of interpolation processes determined by eigenfunctions of the regular Sturm–Liouville problem with a continuous potential of bounded variation. The criterion is formulated in terms of one-sided modulus of variation.

Exact and numerical solutions of the fractional Sturm–Liouville problem

Abstract In the paper, we discuss the regular fractional Sturm-Liouville problem in a bounded domain, subjected to the homogeneous mixed boundary conditions. The results on exact and numerical solutions are based on transformation of the differential fractional Sturm-Liouville problem into the integral one. First, we prove the existence of a purely discrete, countable spectrum and the orthogonal system of eigenfunctions by using the tools of Hilbert-Schmidt operators theory. Then, we construct a new variant of the numerical method which produces eigenvalues and approximate eigenfunctions. The convergence of the procedure is controlled by using the experimental rate of convergence approach and the orthogonality of eigenfunctions is preserved at each step of approximation. In the final part, the illustrative examples of calculations and estimation of the experimental rate of convergence are presented.

Dependence of eigenvalues on the boundary conditions of Sturm–Liouville problems with one singular endpoint

This paper is concerned with continuous and differentiable dependence of isolated eigenvalues on the boundary conditions of self-adjoint Sturm–Liouville problems with one singular endpoint. Locally continuous dependence of eigenvalues on the boundary conditions is proved. Especially, in the limit point case, the continuous dependence of isolated eigenvalues is shown by the relationships between Weyl–Titchmarsh m(λ)-function and the spectrum of the singular problems. Then continuous eigenvalue branches through each isolated eigenvalue over the space of boundary conditions are formed. It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound in Lw2 norm. Its proof is different from that in the regular Sturm–Liouville problems. The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.