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We study a Sturm–Liouville problem in the cross-section of the ocean waveguide covered by pack ice. We prove the basis properties of the eigenfunctions, the convergence of the corresponding Fourier...

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

In this study, we recover potential function and separable boundary conditions for the inverse Sturm-Liouville problem in normal form by using two partial subsets of the data which consist of its one spectrum and sequence of endpoints of eigenfunctions.

Critical point results for a 2nth-order differential equation with Sturm-liouville operator is exploited in order to prove that a suitable class possesses at least one solution under an asymptotica...

In this paper, two classical inverse spectral problems are investigated, namely, the inverse second-order Sturm-Liouville problem and the inverse fourth-order Sturm-Liouville problem. Based on Lidskii’s theorem, we derive trace formulas showing relations between the unknown coefficients and eigenvalues explicitly for both problems. According to those trace formulas, two efficient algorithms are proposed to recover the symmetric potential from one spectrum for second-order Sturm-Liouville problem and two coefficients simultaneously from three spectra for fourth-order Sturm-Liouville problem, respectively. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithms.

The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

In this paper, we construct numerical schemes based on the Lagrange polynomial interpolation to solve Fractional Sturm–Liouville problems (FSLPs) in which the fractional derivatives are considered in the Caputo sense. First, we convert the differential equation with boundary conditions into integral form and discretize the fractional integral to generate a system of algebraic equations in the matrix form. Next, we calculate the set of approximate eigenvalues and corresponding eigenvectors. The eigenfunctions are approximated and some of their properties are investigated. The experimental rate of convergence of numerical calculations for the eigenvalues is reported and the order convergence of the numerical method is obtained. Finally, some examples are presented to illustrate the efficiency and accuracy of the numerical method.

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions., Comment: 2 figures

A collocation method based on Bernoulli polynomial is developed to compute the eigenvalues and eigenfunctions of some known fourth-order Sturm-Liouville problems. Properties of Bernoulli matrix method are presented to convert the problem into a system of linear algebraic equations. Error estimation is introduced. The eigenfunctions are calculated for the test problems. A comparison is made with other relevant studies.

In this paper we consider the Sturm-Liouville transform ℱ(f) on ℝ+. We analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.

In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions.

Let I⊆R be an interval and β:I→I a strictly increasing and continuous function with a unique fixed point s0∈I that satisfies (s0−t)(β(t)−t)≥ 0 for all t∈I, where the equality holds only when t=s0. ...

In this study, we solve a Sturm-Liouville problem on time scales with constant graininess by using Laplace transform which is one of the finest representatives of integral transformation used in applied mathematics. Eigenfunctions on the time scale were obtained in different cases with the Laplace transform. Thus, it was seen that the Laplace transform is an effective method on time scales. The results that will contribute to the spectral theory were obtained on the time scale with the examples discussed. It is very interesting that the results obtained differ as the time scale changes and this transformation can be applied to other types of problems. The problems that were established and solved enabled the subject to be understood on the time scale.

In this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , where ${}_{0}^{C}D_{t}^{\alpha }$ D t α 0 C , ${}_{t}D_{T}^{\alpha }$ D T α t are the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha \in ( {\frac{1}{2},1} ]$ α ∈ ( 1 2 , 1 ] , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.

In the present paper, a boundary value problem consisting of a Sturm–Liouville equation with boundary conditions dependent on the eigenparameter and discontinuous conditions inside the interval is investigated. We present new Prufer substitutions and obtain the asymptotic form of eigenvalues, nodal points and nodal lengths. Then, we prove the uniqueness theorem for the solution of the inverse nodal problem, present a constructive procedure for the potential function by using nodal lengths. Finally, we study Lipschitz stability for the inverse problem.

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.

In this paper, we establish new Lyapunov-type inequalities for a Hadamard fractional differential equation under Sturm-Liouville boundary conditions. Our conclusions cover many results in the literature.

In this paper, we consider bifurcation from zero or infinity of nontrivial solutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontrivial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line $\mathbb{R} \times \{\infty\}$ (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

We prove that if the Neumann eigenvalues of the impulsive Sturm–Liouville operator - D 2 + q {-D^{2}+q} in L 2 ⁢ ( 0 , π ) {L^{2}(0,\pi)} coincide with those of the Neumann Laplacian, then q = 0 {q=0} .

The paper is concerned with the Neumann eigenvalues for second-order Sturm–Liouville difference equations. By analyzing the new discriminant function, we show the interlacing properties between the periodic, antiperiodic, and Neumann eigenvalues. Moreover, when the potential sequence is symmetric and symmetric monotonic, we show the order relation between the first Dirichlet eigenvalue and the second Neumann eigenvalue, and prove that the minimum of the first Neumann eigenvalue gap is attained at the constant potential sequence.

We extend the classical boundary values (0.1) g ( a ) = − W ( u a ( λ 0 , ⋅ ) , g ) ( a ) = lim x ↓ a ⁡ g ( x ) u ˆ a ( λ 0 , x ) , g [ 1 ] ( a ) = ( p g ′ ) ( a ) = W ( u ˆ a ( λ 0 , ⋅ ) , g ) ( a ) = lim x ↓ a ⁡ g ( x ) − g ( a ) u ˆ a ( λ 0 , x ) u a ( λ 0 , x ) , for regular Sturm–Liouville operators associated with differential expressions of the type τ = r ( x ) − 1 [ − ( d / d x ) p ( x ) ( d / d x ) + q ( x ) ] for a.e. x ∈ ( a , b ) ⊂ R , to the case where τ is singular on ( a , b ) ⊆ R and the associated minimal operator T m i n is bounded from below. Here u a ( λ 0 , ⋅ ) and u ˆ a ( λ 0 , ⋅ ) denote suitably normalized principal and nonprincipal solutions of τ u = λ 0 u for appropriate λ 0 ∈ R , respectively. Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of τ u = λ 0 u is closely related to a seminal 1992 paper by Niessen and Zettl [58] . We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of T m i n in the singular case in a manner familiar from the regular case. We briefly discuss the singular Weyl–Titchmarsh–Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.

We study the inverse spectral problem of reconstructing nonsplitting boundary conditions for the Sturm–Liouville operator from a minimal amount of spectral data. Necessary and sufficient conditions are derived for the existence of solutions of the problem under consideration and for these solutions to be isolated or nonisolated. We propose a simple analytical algorithm for reconstructing boundary conditions. (Explicit formulas are provided for the coefficients of the boundary conditions.) It is shown that the condition determining the properties of the problem is the dimension of the linear span of matrices of fundamental solution systems corresponding to the given spectral data.

The Sturm–Liouville equation is one of classical famous differential equations which has been studied from different of views in the literature. In this work, we are going to review its fractional hybrid inclusion version. In this way, we investigate two fractional hybrid Sturm–Liouville differential inclusions with multipoint and integral hybrid boundary conditions. Also, we provide two examples to illustrate our main results.

In this paper, we look for local minima for the Euler functional corresponding to a dynamic Sturm–Liouville boundary value problem on time scales which turns out as an optimization problem. In fact, applying variational methods we obtain the existence of infinitely many solutions for our dynamic problem. An example is also given to illustrate the main results.

Purpose This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions. Design/methodology/approach This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error. Findings Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance. Originality/value The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

In this paper, we consider a class of fractional Sturm-Liouville problems, in which the second order derivative is replaced by the Caputo fractional derivative. The Laplace transform method is applied to obtain algebraic equations. Then, the eigenvalues and the eigenfunctions of the fractional Sturm-Liouville problems are obtained numerically. We provide a convergence analysis for given method. Finally, the simplicity and efficiency of the numerical method is shown by some examples.

Local perturbation analysis and oscillation of eigenvalues of the two parameter Sturm–Liouville system 1 $$\begin{aligned} -y_{1}^{\prime \prime } + q_{1} y_{1}\,\, \,= \,\,\, ( \lambda r_{11} + \mu r_{12} ) y_{1} \,\,\, \text{ on } \,\, [0,1] \end{aligned}$$ with the boundary conditions $$\begin{aligned} \frac{y_{1}' (0)}{y_{1} (0) } = \cot \alpha _{1} \,\, \text{ and } \,\,\, \frac{y_{1}' (1) }{y_{1} (1) } = \frac{a_{1} \lambda + b_{1} }{c_{1} \lambda + d_{1} } \end{aligned}$$ and 2 $$\begin{aligned} -y_{2}^{\prime \prime } + q_{2} y_{2} \,\, \,= \,\,\, ( \lambda r_{21} + \mu r_{22}) y_{2} \end{aligned}$$ with the boundary conditions $$\begin{aligned} \frac{y_{2}' (0) }{y_{2} (0)} = \cot \alpha _{2} \,\, \text{ and } \,\,\, \frac{y_{2}' (1) }{y_{2} (1)} = \frac{ a_{2} \mu + b_{2} }{c_{2} \mu + d_{2} }, \end{aligned}$$ are studied on the compact interval [0,1]. Here $$q_{i}$$ and $$r_{ij}$$ are continuous real valued functions on [0, 1], the angle $$\alpha _{i}$$ is in $$[0,\pi )$$ and $$a_{i} , b_{i} , c_{i}, d_{i}$$ are real numbers with $$\delta _{i} = a_{i} d_{i} - b_{i}c_{i} > 0$$ and $$c_{i} \ne 0$$ for $$i,j=1,2$$ . Results on local analysis, algebraic multiplicity and oscillation of eigenvalues are given. Analysis on the presence of complex eigenvalues is conducted and an upper bound for the number of nonreal eigenvalues is obtained.

Inverse nodal problems for the Sturm–Liouville operator are to reconstruct this operator from the given nodal points(zeros) of its eigenfunctions. In this paper, the potential q ( x ) up to its mea...

This paper presents a study of the performance of the collocation methods using sinc and Bernstein bases functions to determine the eigenvalues of Sturm–Liouville problems with nonlocal boundary conditions. The two methods have the linear systems solved by the Q-Z method. This study shows that Bernstein-collocation method performs better than the sinc-collocation method for the cases considered.

It is well known that the Sturm–Liouville equation has many applications in different areas of science. Thus, it is important to review different versions of the well-known equation. The technique of α-admissible α-ψ-contractions was introduced by Samet et al. in (Nonlinear Anal. 75:2154–2165, 2012). Our aim in this work is to study a fractional hybrid version of the Sturm–Liouville equation by mixing the technique of Samet. In fact, by using the technique of α-admissible α-ψ-contractions, we investigate the existence of solutions for the fractional hybrid Sturm–Liouville equation by using the multi-point boundary value conditions. Also, we review the existence of solutions for a fractional hybrid version of the problem under the integral boundary value conditions. Finally, we provide two examples to illustrate our main results.

In this work, we consider the inverse nodal problem for the Sturm-Liouville problem with a weight and the jump condition at the middle point. It is shown that the dense nodes of the eigenfunctions can uniquely determine the potential on the whole interval and some parameters.

We find necessary and sufficient conditions for the uniform approximation of continuous functions by Lagrange–Sturm–Liouville processes in an arbitrary interval and obtain estimates for approximation errors.

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vector-functional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.

In this article, we consider a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions. By constructing some new Lagrange-type identities and two fundamental functions, we obtain not only the existence, the simplicity, and the interlacing properties of the real eigenvalues, but also the oscillation properties, orthogonality of the eigenfunctions, and the expansion theorem. Finally, we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems.

We give a new formula for the zeta determinant of a Sturm-Liouville operator on a line segment.

An inverse spectral problem is studied for the matrix Sturm–Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the ...

We establish a Sturm–Liouville theorem for quadratic operator pencils with matrix-valued potentials counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.