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Journals / Adıyaman Üniversitesi Fen Bilimleri Dergisi / 2026 / Cilt: 16 Sayı: 1

Eigenvalues of Sturm-Liouville Problem Containing an Eigenparameter in Seperated Boundary Conditions

Ayrık Sınır Koşullarında Özparametre İçeren Sturm–Liouville Probleminin Özdeğerleri

Ayşe Kabataş(KARADENIZ TECHNICAL UNIVERSITY, FACULTY OF SCIENCE, DEPARTMENT OF MATHEMATICS)0000-0003-3273-3666
Süleyman Şengül(RECEP TAYYIP ERDOGAN UNIVERSITY, FACULTY OF ARTS AND SCIENCES, DEPARTMENT OF MATHEMATICS)0000-0002-0359-4137
Pages
152–171
Publication Date
2026-07-01
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Numerical simulations of the eigenvalues support the validity of the obtained asymptotic estimates.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The present paper is concerned with the asymptotic behavior and numerical computation of the eigenvalues of Sturm–Liouville problems in which the eigenvalue parameter appears both in the differential equation and in the separated boundary conditions.
📊
Key Findings & Data:The potential function is assumed to be absolutely continuous and symmetric with a single-well structure.
💡
Conclusion & Impact:Numerical simulations of the eigenvalues support the validity of the obtained asymptotic estimates.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Sınır değer problemleri · Özdeğerler · Asimptotik yaklaşımlar

Abstract

The present paper is concerned with the asymptotic behavior and numerical computation of the eigenvalues of Sturm–Liouville problems in which the eigenvalue parameter appears both in the differential equation and in the separated boundary conditions. The potential function is assumed to be absolutely continuous and symmetric with a single-well structure. As the eigenparameter becomes sufficiently large, asymptotic approximations are obtained using the relationship between the first-order Riccati equation and the second-order Sturm–Liouville equation. Numerical results for the eigenvalues of the related problem are then derived using the element-free Galerkin method. Numerical simulations of the eigenvalues support the validity of the obtained asymptotic estimates.

Özet

Bu makale, özdeğer parametresinin hem diferansiyel denklemde hem de ayrık sınır koşullarında yer aldığı Sturm–Liouville problemlerinin özdeğerlerinin asimptotik davranışı ve sayısal hesaplanması ile ilgilenmektedir. Potansiyel fonksiyonun mutlak sürekli ve simetrik tek kuyulu yapıya sahip olduğu varsayılmaktadır. Özparametre yeterince büyük olduğunda, birinci mertebeden Riccati denklemi ile ikinci mertebeden Sturm–Liouville denklemi arasındaki ilişki kullanılarak asimptotik yaklaşımlar elde edilmektedir. Daha sonra ilgili problemin özdeğerlerine ait sayısal sonuçlar, eleman bağımsız Galerkin yöntemi kullanılarak türetilmektedir. Özdeğerlere ilişkin sayısal simülasyonlar, elde edilen asimptotik tahminlerin geçerliliğini desteklemektedir.

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Cite This Article

Kabataş, A., Şengül, S. (2026). Eigenvalues of Sturm-Liouville Problem Containing an Eigenparameter in Seperated Boundary Conditions. Adıyaman Üniversitesi Fen Bilimleri Dergisi, Cilt: 16 Sayı: 1, 152–171. https://doi.org/10.37094/adyujsci.1899047

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