Computing eigenvalues of periodic sturm-liouville problems using shooting technique and direct integration method

Malathi, V.; Suleiman, Mohamed; Taib, Bachok M.

doi:10.1080/00207169808804682

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…The first approximation y 0 (x) can be obtained by using initial and boundary conditions. Thus we have recurrence formulas (8) and (9) for obtaining other components y 1 (x), y 2 (x), ... of the decomposition (7). Convergence of this decomposition and rapidity of this convergence have been established by Y. Cherrualt [25].…”

Section: Outline Of the Decomposition Methods For Linear And Nonlinearmentioning

confidence: 98%

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A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Mukhtarov

Yücel

2020

Mathematics

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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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Section: Outline Of the Decomposition Methods For Linear And Nonlinearmentioning

confidence: 98%

“…By using the shooting technique and the direct integrating method, Malathi et al [9] computed eigenvalues of periodic Sturm-Liouville problems.…”

Section: Introductionmentioning

confidence: 99%

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Mukhtarov

Yücel

2020

Mathematics

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“…Ji and Wong used Prüfer transformation and the shooting method in their works [10,11,23]. Malathi et al [15] used the shooting method and direct integration method for computing eigenvalues of the periodic Sturm-Liouville problems. One of the interesting approaches was given by Dinibütün and Veliev [7].…”

Section: Introduction and Preliminary Factsmentioning

confidence: 99%

“…where the terms in this equation are defined in (14) and (15). Nevertheless, even if we have obtained this equation from [6], the method of investigation is absolutely different.…”

Section: Introduction and Preliminary Factsmentioning

confidence: 99%

On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

Nur

2018

Bound Value Probl

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“…Ji and Wong used Prüfer transformation and shooting method in their studies [7][8][9]. Malathi et al [10] used shooting technique and direct integration method for computing eigenvalues of periodic Sturm-Liouville problems.…”

Section: Introductionmentioning

confidence: 99%

On the Estimations of the Small Periodic Eigenvalues

Dinibutun

Veliev

2013

Abstract and Applied Analysis

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We estimate the small periodic and semiperiodic eigenvalues of Hill's operator with sufficiently differentiable potential by two different methods. Then using it we give the high precision approximations for the length of th gap in the spectrum of Hill-Sehrodinger operator and for the length of th instability interval of Hill's equation for small values of Finally we illustrate and compare the results obtained by two different ways for some examples.

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Computing eigenvalues of periodic sturm-liouville problems using shooting technique and direct integration method

Cited by 7 publications

References 8 publications

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

On the Estimations of the Small Periodic Eigenvalues

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