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

Ayrık Sınır Koşullarında Özparametre İçeren Sturm–Liouville Probleminin Özdeğerleri

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

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
Sayfa
152–171
Yayın Tarihi
2026-07-01
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Özdeğerlere ilişkin sayısal simülasyonlar, elde edilen asimptotik tahminlerin geçerliliğini desteklemektedir.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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.
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Temel Bulgular & Veri:Daha sonra ilgili problemin özdeğerlerine ait sayısal sonuçlar, eleman bağımsız Galerkin yöntemi kullanılarak türetilmektedir.
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Sonuç & Katkı:Özdeğerlere ilişkin sayısal simülasyonlar, elde edilen asimptotik tahminlerin geçerliliğini desteklemektedir.
🔬Tür:Modelleme & Sayısal Simülasyon
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Sınır değer problemleri · Özdeğerler · Asimptotik yaklaşımlar

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.

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.

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Bu Makaleye Atıf Yap

Kabataş, A., Şengül, S. (2026). Ayrık Sınır Koşullarında Özparametre İçeren Sturm–Liouville Probleminin Özdeğerleri. Adıyaman Üniversitesi Fen Bilimleri Dergisi, Cilt: 16 Sayı: 1, 152–171. https://doi.org/10.37094/adyujsci.1899047

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