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Relevant bibliographies by topics / Sturm-Liouville boundary conditions

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Academic literature on the topic 'Sturm-Liouville boundary conditions'

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Published: 14 March 2023

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

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Sturm-Liouville boundary conditions"

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Wintz, Nick. "Eigenvalue comparisons for an impulsive boundary value problem with Sturm-Liouville boundary conditions." Huntington, WV : [Marshall University Libraries], 2004. http://www.marshall.edu/etd/descript.asp?ref=414.

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Shlapunov, Alexander, and Nikolai Tarkhanov. "Sturm-Liouville problems in domains with non-smooth edges." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6733/.

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We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.

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Ramos, Alberto Gil Couto Pimentel. "Numerical solution of Sturm–Liouville problems via Fer streamers." Thesis, University of Cambridge, 2016. https://www.repository.cam.ac.uk/handle/1810/256997.

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The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).

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Littin, Curinao Jorge Andrés. "Quasi stationary distributions when infinity is an entrance boundary : optimal conditions for phase transition in one dimensional Ising model by Peierls argument and its consequences." Thesis, Aix-Marseille, 2013. http://www.theses.fr/2013AIXM4789/document.

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Cette thèse comporte deux chapitres principaux. Deux problèmes indépendants de Modélisation Mathématique y sont étudiés. Au chapitre 1, on étudiera le problème de l’existence et de l’unicité des distributions quasi-stationnaires (DQS) pour un mouvement Brownien avec dérive, tué en zéro dans le cas où la frontière d’entrée est l’infini et la frontière de sortie est zéro selon la classification de Feller.Ce travail est lié à l’article pionnier dans ce sujet par Cattiaux, Collet, Lambert, Martínez, Méléard, San Martín; où certaines conditions suffisantes ont été établies pour prouver l’existence et l’unicité de DQS dans le contexte d’une famille de Modèles de Dynamique des Populations.Dans ce chapitre, nous généralisons les théorèmes les plus importants de ce travail pionnier, la partie technique est basée dans la théorie de Sturm-Liouville sur la demi-droite positive. Au chapitre 2, on étudiera le problème d’obtenir des bornes inférieures optimales sur l’Hamiltonien du Modèle d’Ising avec interactions à longue portée, l’interaction entre deux spins situés à distance d décroissant comme d^(2-a), où a ϵ[0,1).Ce travail est lié à l’article publié en 2005 par Cassandro, Ferrari, Merola, Presutti où les bornes inférieures optimales sont obtenues dans le cas où a est dans [0,(log3/log2)-1) en termes de structures hiérarchiques appelées triangles et contours.Les principaux théorèmes obtenus dans cette thèse peuvent être résumés de la façon suivante:1. Il n’existe pas de borne inférieure optimale pour l’Hamiltonien en termes de triangles pour a dans ϵ[log2/log3,1). 2. Il existe une borne optimale pour l’Hamiltonien en termes de contours pour a dans a ϵ [0,1)
This thesis contains two main Chapters, where we study two independent problems of Mathematical Modelling : In Chapter 1, we study the existence and uniqueness of Quasi Stationary Distributions (QSD) for a drifted Browian Motion killed at zero, when $+infty$ is an entrance Boundary and zero is an exit Boundary according to Feller's classification. The work is related to the previous paper published in 2009 by { Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S., San Martín, where some sufficient conditions were provided to prove the existence and uniqueness of QSD in the context of a family of Population Dynamic Models. This work generalizes the most important theorems of this work, since no extra conditions are imposed to get the existence, uniqueness of QSD and the existence of a Yaglom limit. The technical part is based on the Sturm Liouville theory on the half line. In Chapter 2, we study the problem of getting quasi additive bounds on the Hamiltonian for the Long Range Ising Model when the interaction term decays according to d^{2-a}, a ϵ[0,1). This work is based on the previous paper written by Cassandro, Ferrari, Merola, Presutti, where quasi-additive bounds for the Hamiltonian were obtained for a in [0,(log3/log2)-1) in terms of hierarchical structures called triangles and Contours. The main theorems of this work can be summarized as follows: 1 There does not exist a quasi additive bound for the Hamiltonian in terms of triangles when a ϵ [0,(log3/log2)-1), 2. There exists a quasi additive bound for the Hamiltonian in terms of Contours for a in [0,1)

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Shlapunov, Alexander, and Nikolai Tarkhanov. "On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/5775/.

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We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.

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Chan, Chi-Hua, and 詹其樺. "Some eigenvalue problems for vectorial Sturm-Liouville equations with eigenparameter dependent boundary conditions." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/82373870832021424681.

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Chang, Tsorng-Hwa, and 張淙華. "Uniqueness of the potential function of the vectorial Sturm- Liouville equations with general boundary conditions." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/47758013058843078642.

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博士
淡江大學
數學學系博士班
100
Inverse spectral problems are studied for the non-self-adjoint matrix Sturm-Liouville differential equation on a finite interval. Using Weyl function, Yurko([24],2006) solved the inverse spectral problem for the matrix Sturm-Liouville operator on a finite interval with the boundary value problem L(Q(x), h, H ). At first, in this thesis, we try to solve the uniqueness theorem of the matrix-valued boundary value problem for arbitrary matrices h1 , h0 , H1 , H0 with the general boundary conditions. By the uniqueness theorem of L(Q(x),h1 , h0 , H1 , H0) described as above, our main work is to find those relations between spectra and potential Q(x) for the vectorial Sturm-Liouville differential equation. For h1 = H1 = In , we will give some characteristic functions corresponding to spectra to determine the Weyl matrix and to prove the uniqueness theorem. Furthermore, we also prove the uniqueness theorems for the vectorial Sturm-Liouville operators with real symmetric potential or real diagonal potential by given some spectra, respectively. We also obtain some results for arbitrary matrices h1 and H1.

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Yang, Ming-Chuan, and 楊名全. "Eigenvalues of Sturm-Liouville problem with periodic and related boundary condition." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/94182943963718124119.

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Book chapters on the topic "Sturm-Liouville boundary conditions"

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del Río, Rafael. "Boundary Conditions and Spectra of Sturm-Liouville Operators." In Sturm-Liouville Theory, 217–35. Basel: Birkhäuser Basel, 2005. http://dx.doi.org/10.1007/3-7643-7359-8_10.

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Jarratt, Mary. "Eigenvalue Approximations for Sturm-Liouville Differential Equations with Mixed Boundary Conditions." In Computation and Control IV, 185–202. Boston, MA: Birkhäuser Boston, 1995. http://dx.doi.org/10.1007/978-1-4612-2574-4_12.

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Aliyev, Yagub N. "Minimality Properties of Sturm-Liouville Problems with Increasing Affine Boundary Conditions." In Operator Theory, Functional Analysis and Applications, 33–49. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-51945-2_3.

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Chugunova, M. V. "Inverse Spectral problem for the Sturm-Liouville Operator with Eigenvalue Parameter Dependent Boundary Conditions." In Operator Theory, System Theory and Related Topics, 187–94. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8247-7_8.

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Ezhak, Svetlana. "On Estimates for the First Eigenvalue of the Sturm–Liouville Problem with Dirichlet Boundary Conditions and Integral Condition." In Differential and Difference Equations with Applications, 387–94. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7333-6_32.

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Karulina, Elena. "On Estimates of the First Eigenvalue for the Sturm–Liouville Problem with Symmetric Boundary Conditions and Integral Condition." In Differential and Difference Equations with Applications, 457–64. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7333-6_40.

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Behrndt, Jussi, and Friedrich Philipp. "Finite Rank Perturbations in Pontryagin Spaces and a Sturm–Liouville Problem with λ-rational Boundary Conditions." In Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations, 163–89. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-68849-7_6.

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Vladimirov, A. A., and I. A. Sheipak. "On Spectral Periodicity for the Sturm–Liouville Problem: Cantor Type Weight, Neumann and Third Type Boundary Conditions." In Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, 509–16. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0648-0_32.

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Imanbaev, Nurlan S., and Makhmud A. Sadybekov. "Regular Sturm-Liouville Operators with Integral Perturbation of Boundary Condition." In Springer Proceedings in Mathematics & Statistics, 222–34. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-67053-9_21.

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"8 Discontinuous boundary conditions." In Recent Developments in Sturm-Liouville Theory, 153–80. De Gruyter, 2021. http://dx.doi.org/10.1515/9783110719000-009.

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Conference papers on the topic "Sturm-Liouville boundary conditions"

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AKDOĞAN, Z., M. DEMIRCI, and O. SH MUKHTAROV. "STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS." In Proceedings of the International Conference (ICCMSE 2003). WORLD SCIENTIFIC, 2003. http://dx.doi.org/10.1142/9789812704658_0003.

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Klimek, Malgorzata. "Fractional Sturm-Liouville Problem and 1D Space-Time Fractional Diffusion With Mixed Boundary Conditions." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46808.

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In the paper, we show a connection between a regular fractional Sturm-Liouville problem with left and right Caputo derivatives of order in the range (1/2, 1) and a 1D space-time fractional diffusion problem in a bounded domain. Both problems include mixed boundary conditions in a finite space interval. We prove that in the case of vanishing mixed boundary conditions, the Sturm-Liouville problem can be rewritten in terms of Riesz derivatives. Then, we apply earlier results on its eigenvalues and eigenfunctions to construct a weak solution of the 1D fractional diffusion equation with variable diffusivity. Adding an assumption on the summability of the eigenvalues’ inverses series, we formulate a theorem on a strong solution of the 1D fractional diffusion problem.

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Baş, Erdal, and Ramazan Özarslan. "Spectral results of Sturm-Liouville difference equation with Dirichlet boundary conditions." In INTERNATIONAL CONFERENCE ON ADVANCES IN NATURAL AND APPLIED SCIENCES: ICANAS 2016. Author(s), 2016. http://dx.doi.org/10.1063/1.4945891.

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Klimek, Malgorzata. "Simple Case of Fractional Sturm-Liouville Problem with Homogeneous von Neumann Boundary Conditions." In 2018 23rd International Conference on Methods & Models in Automation & Robotics (MMAR). IEEE, 2018. http://dx.doi.org/10.1109/mmar.2018.8486100.

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Levitina, Tatyana V. "Free Acoustic Oscillations Inside a Triaxial Ellipsoid." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0434.

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Abstract If a Dirichlet or Neumann condition is imposed on the surface of the ellipsoid, the variables are separated in the scalar wave equation in ellipsoidal coordinates, and the problem in hand is reduced to a system of three identical ordinary differential equations, each being defined on a separate interval and subject to its own boundary conditions. Thus, the three-parameter self-adjoint Sturm-Liouville problem arises: the equations are coupled by two separation constants and the eigen frequency of the ellipsoid, i. e., the spectral parameters, which must be so chosen that all the equations of the system have simultaneously nontrivial solutions, each satisfying the corresponding boundary conditions. The effective globally converging numerical algorithm is proposed for calculating eigen frequencies and separation constants. When the modes of an ellipsoid are found, the caustic surfaces can be easily determined. The merit of the method is illustrated on the example of several calculations of the sound field and caustic surfaces in an ellipsoid.

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Mavrakos, Spyridon A., Ioannis K. Chatjigeorgiou, and Dimitra M. Lentziou. "Wave Run-Up and Second-Order Wave Forces on a Truncated Circular Cylinder Due to Monochromatic Waves." In ASME 2005 24th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2005. http://dx.doi.org/10.1115/omae2005-67104.

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The second-order diffraction potential around a truncated cylinder is considered. The solution method is based on a semi-analytical formulation for the double frequency diffraction potential. The later is properly decomposed into three components in order to satisfy all boundary conditions involved in the problem. The solution process results in a Sturm-Liouville problem for the ring-shaped outer fluid region, which is defined by the geometry of the structure. The matching of the potentials along the boundaries of neighborhood fluid regions is established with the aid of the ‘free’ wave component. The calculation of integral of the pressure distribution on the free surface is carried out using an appropriate Gauss-Legendre numerical technique. The efficiency of the method described in the present is validated through comparative numerical results.

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Shokouhmand, Hossein, Seyed Reza Mahmoudi, and Kaveh Habibi. "Analytical Solution of Hyperbolic Heat Conduction Equation for a Finite Slab With Arbitrary Boundaries, Initial Condition, and Stationary Heat Source." In ASME 2008 6th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2008. http://dx.doi.org/10.1115/icnmm2008-62058.

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This paper presents an analytical solution of the hyperbolic heat conduction equation for a finite slab that sides are subjected to arbitrary heat source, boundary, and initial conditions. In the mathematical model used in this study, the heating on both sides treated as an apparent heat source while sides of the slab assumed to be insulated. Distribution of the apparent heat source for a problem with arbitrary heating on two boundaries is solved. The solution obtained by separation of variable method using appropriate Fourier series. Being a Sturm-Liouville problem in x-direction, suitable orthogonal functions can be allocated to hyperbolic heat conduction equation depending on the type of boundary conditions. Despite ease of proposed method, very few works has been done to solve hyperbolic heat conduction problems using this method by authors. The main feature of the method is straightforward formulation. In the analysis of heat conduction involving extremely short times, the parabolic heat conduction equation breaks down. By increasing the applications of the fast heat sources such as laser pulse for annealing of semiconductors and high heat flux applications, the need for adequate model of heat conduction has arisen. The hyperbolic heat conduction equation eliminates the paradox of an infinite speed of propagation of thermal disturbances which contradicts with Einstein’s theory of relativity. Moreover, it describes the highly transient temperature distribution in a finite medium more accurately.

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Joglekar, M. M., and D. M. Joglekar. "Novel Empirical Relations for Accurately Estimating the Eigenfrequencies of Cantilever Beams With Linear Width Variation." In ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2010. http://dx.doi.org/10.1115/esda2010-24593.

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Dynamic analysis of variable cross-section beams has been the focus of numerous investigations because of its relevance to aeronautical, civil, and mechanical engineering. In this article, we analyze the case of isotropic Euler-Bernoulli cantilever beams having linearly varying width, constant thickness, and classical boundary conditions. The linear width variation is characterized by a taper parameter, which can be varied between zero and unity. The free transverse vibration problem is cast as a fourth order Sturm-Liouville eigenvalue problem, and numerically solved by using the differential transform technique. A five-parameter exponential fitting model is used to develop novel empirical relations to estimate the first five eigenfrequencies as functions of the taper parameter. The proposed empirical relations are able to predict the eigenfrequencies with an error of less than 0.1% with respect to the simulated values, which make them useful for practical design applications. Using the proposed empirical relations, we then examine the sensitivity of each eigenfrequency to the variation in taper parameter near zero and near unity.

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Chasalevris, Athanasios, and Dimitris Sfyris. "On the Analytical Evaluation of the Lubricant Pressure in the Finite Journal Bearing." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70187.

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Abstract:

The Reynolds equation for the pressure distribution of the lubricant in a journal bearing with finite length is solved analytically. Using the method of the separation of variables in an additive and in a multiplicative form, a set of particular solutions of the Reynolds equation is added in the general solution of the homogenous Reynolds equation and a closed form expression for the definition of the lubricant pressure is presented. The Reynolds equation is split in four linear ordinary differential equations of second order with non constant coefficients and together with the boundary conditions they form four Sturm-Liouville problems with the three of them to have direct forms of solution and one of them to be confronted using the method of power series. The mathematical procedure is presented up to the point that the application of the boundaries for the pressure distribution yields the final definition of the solution with the calculation of the constants. The current work gives in detail the mathematical path with which the analytical solution is derived, and it ends with the pressure evaluation and a comparison with past numerical solutions and an approximate analytical solution for a finite bearing. Also the parameters of primary interest to the bearing designer, such as load capacity, attitude angle, and stiffness and damping coefficients are evaluated and compared with numerical results.

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10

Imanbaev, Nurlan. "Stability of the basis property of system of root functions of Sturm-Liouville operator with integral boundary condition." In APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’16): Proceedings of the 42nd International Conference on Applications of Mathematics in Engineering and Economics. Author(s), 2016. http://dx.doi.org/10.1063/1.4968479.

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