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

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Published: 14 March 2023

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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 (May 21, 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 (December 31, 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

Klimek, Malgorzata. "Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions." Symmetry 13, no. 12 (November 28, 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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6

Klimek, Malgorzata. "Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem." Fractional Calculus and Applied Analysis 22, no. 1 (February 25, 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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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

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 (February 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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10

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 (October 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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11

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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12

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 (March 13, 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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13

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 (February 27, 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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14

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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15

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 (January 18, 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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16

Buterin, Sergey. "An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions." Tamkang Journal of Mathematics 50, no. 3 (September 2, 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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17

Kravchenko, Vladislav V., Kira V. Khmelnytskaya, and Fatma Ayça Çetinkaya. "Recovery of Inhomogeneity from Output Boundary Data." Mathematics 10, no. 22 (November 19, 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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18

Ibrahim, Sobhy El-Sayed. "On the boundary conditions for products of Sturm-Liouville differential operators." Tamkang Journal of Mathematics 32, no. 3 (September 30, 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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19

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 (April 2004): 241–48. http://dx.doi.org/10.1017/s030821050000319x.

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20

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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21

Bondarenko, Natalia Pavlovna. "Inverse Sturm-Liouville problem with analytical functions in the boundary condition." Open Mathematics 18, no. 1 (June 10, 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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22

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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23

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 (June 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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24

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

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25

Garbuza, Tatjana. "EXPRESSIONS FOR FUČIK SPECTRA FOR STURM‐LIOUVILLE BVP." Mathematical Modelling and Analysis 12, no. 1 (March 31, 2007): 51–60. http://dx.doi.org/10.3846/1392-6292.2007.12.51-60.

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26

Perera, Upeksha, and Christine Böckmann. "Solutions of Sturm-Liouville Problems." Mathematics 8, no. 11 (November 20, 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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27

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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28

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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29

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 (June 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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30

Harutyunyan, Tigran. "The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems." Tamkang Journal of Mathematics 50, no. 3 (September 2, 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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31

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 (November 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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32

Binding, P. A., and P. J. Browne. "Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 2 (April 2000): 239–47. http://dx.doi.org/10.1017/s0308210500000135.

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Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

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33

Dehghan, Mohammad, and Angelo B. Mingarelli. "Fractional Sturm–Liouville Eigenvalue Problems, II." Fractal and Fractional 6, no. 9 (August 30, 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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34

Akhtyamov, A. M. "On degenerate boundary conditions in the Sturm–Liouville problem." Differential Equations 52, no. 8 (August 2016): 1085–87. http://dx.doi.org/10.1134/s0012266116080140.

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35

Rodríguez, Jesús, and Zachary Abernathy. "Nonlinear discrete Sturm–Liouville problems with global boundary conditions." Journal of Difference Equations and Applications 18, no. 3 (March 2012): 431–45. http://dx.doi.org/10.1080/10236198.2010.505237.

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36

Korotyaev, E., and D. Chelkak. "The inverse Sturm–Liouville problem with mixed boundary conditions." St. Petersburg Mathematical Journal 21, no. 5 (October 1, 2010): 761. http://dx.doi.org/10.1090/s1061-0022-2010-01116-6.

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37

Sadovnichii, V. A., Ya T. Sultanaev, and A. M. Akhtyamov. "Inverse Sturm-Liouville problem with generalized periodic boundary conditions." Differential Equations 45, no. 4 (April 2009): 526–38. http://dx.doi.org/10.1134/s0012266109040065.

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38

Akdoĝan, Z., M. Demirci, and O. Sh Mukhtarov. "STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS." Acta Mathematica Scientia 25, no. 4 (October 2005): 731–40. http://dx.doi.org/10.1016/s0252-9602(17)30213-8.

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Wang, Aiping, and Anton Zettl. "Self-adjoint Sturm-Liouville problems with discontinuous boundary conditions." Methods and Applications of Analysis 22, no. 1 (2015): 37–66. http://dx.doi.org/10.4310/maa.2015.v22.n1.a2.

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del rio Castillo, Rafael. "On Boundary Conditions of an Inverse Sturm–Liouville Problem." SIAM Journal on Applied Mathematics 50, no. 6 (December 1990): 1745–51. http://dx.doi.org/10.1137/0150103.

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Bhattacharyya, T., P. A. Binding, and K. Seddighi. "Multiparameter Sturm–Liouville Problems with Eigenparameter Dependent Boundary Conditions." Journal of Mathematical Analysis and Applications 264, no. 2 (December 2001): 560–76. http://dx.doi.org/10.1006/jmaa.2001.7695.

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del Rio, R., and Olga Tchebotareva. "Boundary conditions of Sturm–Liouville operators with mixed spectra." Journal of Mathematical Analysis and Applications 288, no. 2 (December 2003): 518–29. http://dx.doi.org/10.1016/j.jmaa.2003.09.008.

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Eberhard, Walter, Gerhard Freiling, and Anton Zettl. "Sturm-Liouville problems with singular non-selfadjoint boundary conditions." Mathematische Nachrichten 278, no. 12-13 (October 2005): 1509–23. http://dx.doi.org/10.1002/mana.200310318.

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Kong, Lingju, Qingkai Kong, Man K. Kwong, and James S. W. Wong. "Linear Sturm-Liouville problems with multi-point boundary conditions." Mathematische Nachrichten 286, no. 11-12 (February 28, 2013): 1167–79. http://dx.doi.org/10.1002/mana.201200187.

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Castro, Hernán, and Hui Wang. "A singular Sturm–Liouville equation under homogeneous boundary conditions." Journal of Functional Analysis 261, no. 6 (September 2011): 1542–90. http://dx.doi.org/10.1016/j.jfa.2011.05.012.

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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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Abstract:

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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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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Adalar, İbrahim. "Spectral problems for Sturm-Liouville operator with eigenparameter boundary conditions on time scales." Filomat 36, no. 8 (2022): 2519–29. http://dx.doi.org/10.2298/fil2208519a.

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In this paper, we consider the inverse problem for Sturm-Liouville operators with eigenparameter dependent boundary conditions on time scales. We give new uniqueness theorems and investigate its some special cases.

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Binding, Paul A., Patrick J. Browne, Warren J. Code, and Bruce A. Watson. "TRANSFORMATION OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 3 (October 2004): 533–52. http://dx.doi.org/10.1017/s0013091504000197.

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AbstractWe consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

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Tharwat, M. M., A. H. Bhrawy, and A. S. Alofi. "Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation." Abstract and Applied Analysis 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/498457.

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The eigenvalues of discontinuous Sturm-Liouville problems which contain an eigenparameter appearing linearly in two boundary conditions and an internal point of discontinuity are computed using the derivative sampling theorem and Hermite interpolations methods. We use recently derived estimates for the truncation and amplitude errors to investigate the error analysis of the proposed methods for computing the eigenvalues of discontinuous Sturm-Liouville problems. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Moreover, it is shown that the proposed methods are significantly more accurate than those based on the classical sinc method.

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