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Journal articles on the topic 'Sturm-Liouville boundary conditions'

Author: Grafiati

Published: 14 March 2023

Last updated: 19 July 2025

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1

Sadovnichy, V. A., Ya T. Sultanaev, and A. M. Akhtyamov. "Degenerate boundary conditions on a geometric graph." Доклады Академии наук 485, no. 3 (2019): 272–75. http://dx.doi.org/10.31857/s0869-56524853272-275.

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The boundary conditions of the Sturm-Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that if the lengths of the edges are different, then the Sturm-Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are the same, then the characteristic determinant of the Sturm-Liouville problem can not be equal to a constant different from zero. But the set of Sturm-Liouville problems for which the characteristic determinant is identically equal to zero is an infinite (continuum). In this way, in contrast to the Sturm-Liouville problem defined on an interval, the set of boundary-value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor A124 for matrix of coefficients is nonzero, it does not consist of two problems, as in the case of the Sturm-Liouville problem given on an interval, but of 18 classes, each containing two to four arbitrary constants.

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2

Karahan, D., and K. R. Mamedov. "ON A q-BOUNDARY VALUE PROBLEM WITH DISCONTINUITY CONDITIONS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 4 (2021): 5–12. http://dx.doi.org/10.14529/mmph210401.

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In this paper, we studied q-analogue of Sturm–Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the q-Sturm–Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of q-Sturm–Liouville boundary value problem. We shown that eigenfunctions of q-Sturm–Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson’s type.

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3

Akhtyamov, Azamat M., and Khanlar R. Mamedov. "Inverse Sturm–Liouville problems with polynomials in nonseparated boundary conditions." Baku Mathematical Journal 1, no. 2 (2022): 179–94. http://dx.doi.org/10.32010/j.bmj.2022.19.

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An nonself-adjoint Sturm–Liouville problem with two polynomials in nonseparated boundary conditions are considered. It is shown that this problem have an infinite countable spectrum. The corresponding inverse problems is solved. Criterions for unique reconstruction of the nonself-adjoint Sturm-Liouville problem by eigenvalues of this problem and the spectral data of an additional problem with separated boundary conditions are proved. Schemes for unique reconstruction of the Sturm-Liouville problems with polynomials in nonseparated boundary conditions and corresponding examples are given

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4

Vitkauskas, Jonas, and Artūras Štikonas. "Relations between spectrum curves of discrete Sturm-Liouville problem with nonlocal boundary conditions and graph theory." Lietuvos matematikos rinkinys 61 (February 18, 2021): 1–6. http://dx.doi.org/10.15388/lmr.2020.22474.

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Sturm-Liouville problem with nonlocal boundary conditions arises in many scientific fields such as chemistry, physics, or biology. There could be found some references to graph theory in a discrete Sturm-Liouville problem, especially in investigation of spectrum curves. In this paper, relations between discrete Sturm-Liouville problem with nonlocal boundary conditions characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found.

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5

Zhang, Yunyang, Shaojie Chen, and Jing Li. "New Results on a Nonlocal Sturm–Liouville Eigenvalue Problem with Fractional Integrals and Fractional Derivatives." Fractal and Fractional 9, no. 2 (2025): 70. https://doi.org/10.3390/fractalfract9020070.

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In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary condition. Furthermore, we discussed the continuous dependence of the eigenvalues on the potential function for a nonlocal Sturm–Liouville equation under a Dirichlet boundary condition.

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6

Klimek, Malgorzata. "Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions." Symmetry 13, no. 12 (2021): 2265. http://dx.doi.org/10.3390/sym13122265.

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In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

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7

Vitkauskas, Jonas, and Artūras Štikonas. "Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II." Lietuvos matematikos rinkinys 62 (December 15, 2021): 1–8. http://dx.doi.org/10.15388/lmr.2021.25128.

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In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.

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8

Şen, Erdoğan. "A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition." Physics Research International 2013 (September 1, 2013): 1–9. http://dx.doi.org/10.1155/2013/159243.

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We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.

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9

De la Sen, Manuel. "Loss of the Sturm–Liouville Property of Time-Varying Second-Order Differential Equations in the Presence of Delayed Dynamics." Mathematical and Computational Applications 29, no. 5 (2024): 89. http://dx.doi.org/10.3390/mca29050089.

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This paper considers a nominal undelayed and time-varying second-order Sturm–Liouville differential equation on a finite time interval which is a nominal version of another perturbed differential equation subject to a delay in its dynamics. The nominal delay-free differential equation is a Sturm–Liouville system in the sense that it is subject to prescribed two-point boundary conditions. However, the perturbed differential system is not a Sturm–Liouville system, in general, due to the presence of delayed dynamics. The main objective of the paper is to investigate the loss of the boundary values of the Sturm–Liouville nominal undelayed system in the presence of the delayed dynamics. Such a delayed dynamics is considered a perturbation of the nominal dynamics related to the Sturm–Liouville system with given two-point boundary values. In particular, this loss of the Sturm–Liouville exact tracking of the prescribed two-point boundary values might happen because the nominal boundary values may become lost by the state trajectory solution in the presence of delays, related to the undelayed case, due to the presence of the delayed dynamics. The worst-case error description of the deviation of the two-point boundary values of the current perturbed differential with respect to those of the nominal Sturm–Liouville system is characterized in terms of error norms related to the nominal system. Under sufficiently small deviations of the parameterization of the perturbed system with respect to the nominal one, such a worst-error characterization makes the current perturbed system an approximate Sturm–Liouville system of the nominal undelayed one.

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10

Klimek, Malgorzata. "Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem." Fractional Calculus and Applied Analysis 22, no. 1 (2019): 78–94. http://dx.doi.org/10.1515/fca-2019-0005.

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Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

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11

Alama, Hameda Mohamed. "On some spectral properties of nonlocal boundary-value problems for nonlinear differential inclusion." Ukrains’kyi Matematychnyi Zhurnal 76, no. 10 (2024): 1427–43. http://dx.doi.org/10.3842/umzh.v76i10.7772.

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UDC 517.9 We study the solutions to the Sturm–Liouville boundary-value problem for a nonlinear differential inclusion with nonlocal conditions. The maximal and minimal solutions are demonstrated. The analysis of eigenvalues and eigenfunctions is performed. It is discussed whether multiple solutions may exist for the inhomogeneous Sturm–Liouville boundary-value problem for differential equation with nonlocal conditions.

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12

Sun, Fu, and Kun Li. "Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems." Applied Mathematics-A Journal of Chinese Universities 38, no. 1 (2023): 100–110. http://dx.doi.org/10.1007/s11766-023-3991-6.

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AbstractThe present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems. The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

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13

Li, Shuang, Jinming Cai, and Kun Li. "Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity." Axioms 12, no. 5 (2023): 479. http://dx.doi.org/10.3390/axioms12050479.

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Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. It has been proven that each Sturm–Liouville problem is equivalent to the given matrix eigenvalue problem.

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14

Isayev, Hamlet, and Bilender Allahverdiev. "Self-adjoint and non-self-adjoint extensions of symmetric q-Sturm-Liouville operators." Filomat 37, no. 24 (2023): 8057–66. http://dx.doi.org/10.2298/fil2324057i.

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Aspace of boundary values is constructed for minimal symmetric regular and singular q-Sturm- Liouville operators in limit-point and limit-circle cases. A description of all maximal dissipative, maximal accumulative, self-adjoint, and other extensions of such symmetric q-Sturm-Liouville operators is given in terms of boundary conditions.

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15

Mukhtarov, Oktay Sh, and Merve Yücel. "A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique." Mathematics 8, no. 3 (2020): 415. http://dx.doi.org/10.3390/math8030415.

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The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

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16

Binding, P. A., P. J. Browne, and K. Seddighi. "Sturm–Liouville problems with eigenparameter dependent boundary conditions." Proceedings of the Edinburgh Mathematical Society 37, no. 1 (1994): 57–72. http://dx.doi.org/10.1017/s0013091500018691.

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Sturm theory is extended to the equationfor 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditionsandOscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

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17

Pivovarchik, Vyacheslav. "Direct and inverse three‐point Sturm–Liouville problem with parameter‐dependent boundary conditions." Asymptotic Analysis 26, no. 3-4 (2001): 219–38. https://doi.org/10.3233/asy-2001-445.

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The problem of small vibrations of a smooth inhomogeneous string damped at an interior point and fixed at the endpoints is reduced to a three‐point Sturm–Liouville boundary problem. This problem is considered as an eigenvalue problem for a nonmonic quadratic operator pencil of a special type with the spectrum located in the upper half‐plane of the spectral parameter. Concerning the corresponding inverse problem it is shown that the spectrum does not determine the potential of the Sturm–Liouville problem (and consequently the density of the string) uniquely. The conditions are given sufficient for a sequence of complex numbers to be the spectrum of the considered Sturm–Liouville problem with real‐valued potential which belongs to L2(0,a). In order to recover the potential uniquely the spectrum of the corresponding so to say truncated Sturm–Liouville problem is chosen as an additional information. Then the problem of recovering of the potential turns out to be overdetermined. The self‐consistency of the two spectra is discussed.

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18

Pečiulytė, S., and A. Štikonas. "Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions." Nonlinear Analysis: Modelling and Control 11, no. 1 (2006): 47–78. http://dx.doi.org/10.15388/na.2006.11.1.14764.

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The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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19

El-Sayed, Ahmed M. A., Eman M. A. Hamdallah, and Hameda M. A. Alama. "Multiple solutions of a Sturm-Liouville boundary value problem of nonlinear differential inclusion with nonlocal integral conditions." AIMS Mathematics 7, no. 6 (2022): 11150–64. http://dx.doi.org/10.3934/math.2022624.

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<abstract><p>The existence of solutions for a Sturm-Liouville boundary value problem of a nonlinear differential inclusion with nonlocal integral condition is studied. The maximal and minimal solutions will be studied. The existence of multiple solutions of the nonhomogeneous Sturm-Liouville boundary value problem of differential equation with nonlocal integral condition is considered. The eigenvalues and eigenfunctions are investigated.</p></abstract>

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20

Freiling, G., and V. Yurko. "Boundary value problems with regular singularities and singular boundary conditions." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1481–95. http://dx.doi.org/10.1155/ijmms.2005.1481.

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Singular boundary conditions are formulated for nonselfadjoint Sturm-Liouville operators with singularities and turning points. For boundary value problems with singular boundary conditions, properties of the spectrum are studied and the completeness of the system of root functions is proved.

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21

Buterin, Sergey. "An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions." Tamkang Journal of Mathematics 50, no. 3 (2019): 207–21. http://dx.doi.org/10.5556/j.tkjm.50.2019.3347.

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The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

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22

Binding, Paul A., Patrick J. Browne, and Bruce A. Watson. "STURM–LIOUVILLE PROBLEMS WITH REDUCIBLE BOUNDARY CONDITIONS." Proceedings of the Edinburgh Mathematical Society 49, no. 3 (2006): 593–608. http://dx.doi.org/10.1017/s0013091505000131.

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AbstractThe regular Sturm–Liouville problem$$ \tau y:=-y''+qy=\lambda y\quad\text{on }[0,1],\ \lambda\in\CC, $$is studied subject to boundary conditions$$ P_j(\lambda)y'(j)=Q_j(\lambda)y(j),\quad j=0,1, $$where $q\in L^1(0,1)$ and $P_j$ and $Q_j$ are polynomials with real coefficients. A comparison is made between this problem and the corresponding ‘reduced’ one where all common factors are removed from the boundary conditions. Topics treated include Jordan chain structure, eigenvalue asymptotics and eigenfunction oscillation.

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23

Kravchenko, Vladislav V., Kira V. Khmelnytskaya, and Fatma Ayça Çetinkaya. "Recovery of Inhomogeneity from Output Boundary Data." Mathematics 10, no. 22 (2022): 4349. http://dx.doi.org/10.3390/math10224349.

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We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter. Special cases of this problem include the recovery of the potential from the Weyl function, the inverse two-spectra Sturm–Liouville problem, as well as the recovery of the potential from the output boundary values of a plane wave that interacted with the potential. The method is based on the special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid, the problem is reduced to the classical inverse Sturm–Liouville problem of recovering the potential from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

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24

Klimek, Malgorzata, Mariusz Ciesielski, and Tomasz Blaszczyk. "Exact and Numerical Solution of the Fractional Sturm–Liouville Problem with Neumann Boundary Conditions." Entropy 24, no. 2 (2022): 143. http://dx.doi.org/10.3390/e24020143.

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In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. Finally, we present the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule.

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25

Chanchlani, Lata, Pratibha Manohar, Ajay Sharma, and Sangeeta Choudhary. "Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives." Indian Journal Of Science And Technology 17, no. 16 (2024): 1702–12. http://dx.doi.org/10.17485/ijst/v17i16.2514.

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Objectives: The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. Methods: Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. Findings: Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. Novelty: The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. Keywords: Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem

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26

Ao, Ji-Jun, and Juan Wang. "Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales." Filomat 33, no. 6 (2019): 1747–57. http://dx.doi.org/10.2298/fil1906747a.

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The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

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27

Ibrahim, Sobhy El-Sayed. "On the boundary conditions for products of Sturm-Liouville differential operators." Tamkang Journal of Mathematics 32, no. 3 (2001): 187–99. http://dx.doi.org/10.5556/j.tkjm.32.2001.374.

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In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

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28

Bondarenko, Natalia Pavlovna. "Inverse Sturm-Liouville problem with analytical functions in the boundary condition." Open Mathematics 18, no. 1 (2020): 512–28. http://dx.doi.org/10.1515/math-2020-0188.

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Abstract The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

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29

Binding, Paul, and Branko Ćurgus. "A counterexample in Sturm–Liouville completeness theory." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 2 (2004): 241–48. http://dx.doi.org/10.1017/s030821050000319x.

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30

Garbuza, Tatjana. "EXPRESSIONS FOR FUČIK SPECTRA FOR STURM‐LIOUVILLE BVP." Mathematical Modelling and Analysis 12, no. 1 (2007): 51–60. http://dx.doi.org/10.3846/1392-6292.2007.12.51-60.

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31

Everitt, W. N., M. Möller, and A. Zettl. "Sturm—Liouville problems and discontinuous eigenvalues." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 4 (1999): 707–16. http://dx.doi.org/10.1017/s0308210500013093.

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If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

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32

Lata, Chanchlani, Manohar Pratibha, Sharma Ajay, and Choudhary Sangeeta. "Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives." Indian Journal of Science and Technology 17, no. 16 (2024): 1702–12. https://doi.org/10.17485/IJST/v17i16.2514.

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Abstract <strong>Objectives:</strong> The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. <strong>Methods:</strong> Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. <strong>Findings:</strong> Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. <strong>Novelty:</strong> The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. <strong>Keywords:</strong> Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem

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33

Kong, Q., H. Wu, A. Zettl, and M. Möller. "Indefinite Sturm–Liouville problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 3 (2003): 639–52. http://dx.doi.org/10.1017/s0308210500002584.

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We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.

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34

Gan, Yu, Zhaowen Zheng, and Kun Li. "Asymptotic estimations of eigenvalues and eigenfunctions for a multi-point nonlocal boundary value problem." Filomat 38, no. 30 (2024): 10555–66. https://doi.org/10.2298/fil2430555g.

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In this paper, the multi-point nonlocal second order linear Sturm-Liouville problem is considered consisting of the equation-u?? (t) + q(t)u(t) = ?u(t) on [0, 1] and multi-point boundary value conditions. The geometric multiplicity of eigenvalues, the asymptotic formulas for eigenvalues and eigenfunctions are expressed explicitly under certain mild conditions, these results can be used to investigate the inverse spectral problems of certain Sturm-Liouville operators.

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35

Al-Refai, Mohammed, and Thabet Abdeljawad. "Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems." Complexity 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/3720471.

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We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

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36

Perera, Upeksha, and Christine Böckmann. "Solutions of Sturm-Liouville Problems." Mathematics 8, no. 11 (2020): 2074. http://dx.doi.org/10.3390/math8112074.

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This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.

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37

Çöl, Aynur. "Spectral Characteristics of the Sturm-Liouville Problem with Spectral Parameter-Dependent Boundary Conditions." Journal of New Theory, no. 48 (September 30, 2024): 1–10. http://dx.doi.org/10.53570/jnt.1501326.

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38

Kong, Q., H. Wu, and A. Zettl. "Geometric aspects of Sturm—Liouville problems I. Structures on spaces of boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 3 (2000): 561–89. http://dx.doi.org/10.1017/s0308210500000305.

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We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

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Olǧar, H., F. Muhtarov, and O. Mukhtarov. "Operator-pencil treatment of multi-interval Sturm-Liouville equation with boundary-transmission conditions." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 115, no. 3 (2024): 126–36. http://dx.doi.org/10.31489/2024m3/126-136.

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This paper is devoted to a new type of boundary-value problems for Sturm-Liouville equations defined on three disjoint intervals (−π,−π+d),(−π+d,π−d) and (π−d,π) together with eigenparameter dependent boundary conditions and with additional transmission conditions specified at the common end points −π+d and π−d, where 0<d<π. The considered problem cannot be treated by known techniques within the usual framework of classical Sturm-Liouville theory. To establish some important spectral characteristics we introduced the polynomial-operator formulation of the problem. Moreover, we develop a new modification of the Rayleigh method to obtain lower bound of eigenvalues.

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Novickij, Jurij, and Artūras Štikonas. "On the equivalence of discrete Sturm–Liouville problem with nonlocal boundary conditions to the algebraic eigenvalue problem." Lietuvos matematikos rinkinys 56 (December 23, 2015): 66–71. http://dx.doi.org/10.15388/lmr.a.2015.12.

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We consider the ﬁnite diﬀerence approximation of the second order Sturm–Liouville equation with nonlocal boundary conditions (NBC). We investigate the condition when the discrete Sturm–Liouville problem can be transformed to an algebraic eigenvalue problem and denote this condition as solvability condition. The examples of the solvability for the most popular NBCs are provided. The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/ 2014).

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Harutyunyan, Tigran. "The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems." Tamkang Journal of Mathematics 50, no. 3 (2019): 233–52. http://dx.doi.org/10.5556/j.tkjm.50.2019.3352.

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We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.

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Allahverdiev, B. P. "Dissipative Sturm-Liouville Operators with Nonseparated Boundary Conditions." Monatshefte f�r Mathematik 140, no. 1 (2003): 1–17. http://dx.doi.org/10.1007/s00605-003-0035-4.

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Allahverdiev, Bilender, Hamlet Isayev, and Hüseyin Tuna. "Impulsive q-Sturm-Liouville problems." Applicable Analysis and Discrete Mathematics, no. 00 (2024): 9. http://dx.doi.org/10.2298/aadm230428009a.

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In this study, impulsive q-Sturm-Liouville problems are considered. First, symmetry is obtained with the help of boundary conditions. Then, the existence and uniqueness problem for such equations is discussed. Finally, eigenfunction expansion was obtained with the help of characteristic determinant and Green?s function.

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ZHANG, Liang, Jijun AO, and Wenyan LÜ. "Inverse Sturm-Liouville Problems with a Class of Non-Self-Adjoint Boundary Conditions Containing the Spectral Parameter." Wuhan University Journal of Natural Sciences 29, no. 6 (2024): 508–16. https://doi.org/10.1051/wujns/2024296508.

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The inverse spectral theory of a class of Atkinson-type Sturm-Liouville problems with non-self-adjoint boundary conditions containing the spectral parameter is investigated. Based on the so-called matrix representations of such problems and a special class of inverse matrix eigenvalue problems, some of the coefficient functions of the corresponding Sturm-Liouville problems are constructed by using priori known two sets of complex numbers satisfying certain conditions. To best understand the result, an algorithm and some examples are posted.

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Basak, Partha. "Application of Lagrange Multiplier for Solving Non – Homogenous Differential Equations." American Journal of Applied Mathematics 12, no. 5 (2024): 111–17. http://dx.doi.org/10.11648/j.ajam.20241205.11.

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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.

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Liu, Dan, Xuejun Zhang, and Mingliang Song. "Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero." Journal of Mathematics 2021 (September 14, 2021): 1–10. http://dx.doi.org/10.1155/2021/4221459.

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We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

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47

ÖZTÜRK, Sevda Nur, Oktay MUKHTAROV, and Kadriye AYDEMİR. "Non-classical periodic boundary value problems with impulsive conditions." Journal of New Results in Science 12, no. 1 (2023): 1–8. http://dx.doi.org/10.54187/jnrs.1201577.

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This study investigates some spectral properties of a new type of periodic Sturm-Liouville problem. The problem under consideration differs from the classical ones in that the differential equation is given on two disjoint segments that have a common end, and two additional interaction conditions are imposed on this common end (such interaction conditions are called various names, including transmission conditions, jump conditions, interface conditions, impulsive conditions, etc.). At first, we proved that all eigenvalues are real and there is a corresponding real-valued eigenfunction for each eigenvalue. Then we showed that two eigenfunctions corresponding to different eigenvalues are orthogonal. We also defined some left and right-hand solutions, in terms of which we constructed a new transfer characteristic function. Finally, we have defined asymptotic formulas for the transfer characteristic functions and also for the eigenvalues. The results obtained are a generalization of similar results of the classical Sturm-Liouville theory.

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BINDING, PAUL A., PATRICK J. BROWNE, and BRUCE A. WATSON. "TRANSFORMATIONS BETWEEN STURM–LIOUVILLE PROBLEMS WITH EIGENVALUE DEPENDENT AND INDEPENDENT BOUNDARY CONDITIONS." Bulletin of the London Mathematical Society 33, no. 6 (2001): 749–57. http://dx.doi.org/10.1112/s0024609301008177.

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Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.

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49

Metin, Turk, and Erdal Bas. "Energy-dependent fractional Sturm-Liouville impulsive problem." Thermal Science 23, Suppl. 1 (2019): 139–52. http://dx.doi.org/10.2298/tsci171017338m.

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In study, we show the existence and integral representation of solution for energy-dependent fractional Sturm-Liouville impulsive problem of order with ? ? (1,2] impulsive and boundary conditions. An existence theorem is proved for energy-dependent fractional Sturm-Liouville impulsive problem by using Schaefer fixed point theorem. Furthermore, in the last part of the article, an application is given for the problem and visual results are shown by figures.

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50

Dehghan, Mohammad, and Angelo B. Mingarelli. "Fractional Sturm–Liouville Eigenvalue Problems, II." Fractal and Fractional 6, no. 9 (2022): 487. http://dx.doi.org/10.3390/fractalfract6090487.

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We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0<α<1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α→1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation.

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