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

Öz

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.

Anahtar Kelimeler

• Boundary value problems
• Eigenvalues
• Asymptotic approximations
• Element free Galerkin method

Ayrık Sınır Koşullarında Özparametre İçeren Sturm–Liouville Probleminin Özdeğerleri

Öz

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.

Anahtar Kelimeler

• Sınır değer problemleri
• Özdeğerler
• Asimptotik yaklaşımlar
• Eleman bağımsız Galerkin yöntemi

Kaynakça

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