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

1
Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, Tokat 60250, Turkey
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Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku 370141, Azerbaijan
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Osmancık Ö.D. Vocational School, Hitit University, Osmancık/Çorum 19500, Turkey
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(3), 415; https://doi.org/10.3390/math8030415 (registering DOI)
Received: 4 February 2020 / Revised: 6 March 2020 / Accepted: 10 March 2020 / Published: 13 March 2020
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. View Full-Text
Keywords: two-interval problems; Sturm–Liouville equation; transmission conditions; eigenvalues; eigenfunctions; adomian decomposition method two-interval problems; Sturm–Liouville equation; transmission conditions; eigenvalues; eigenfunctions; adomian decomposition method
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Mukhtarov, O.S.; Yücel, M. A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique. Mathematics 2020, 8, 415.

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