Zamana göre kesirli mertebeden Schrödinger denkleminin Chebyshev kollokasyon yöntemi ile nümerik çözümleri
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Dosyalar
Tarih
2020
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Be¸s bölümden olu¸san bu tez çalı¸smasının birinci bölümü Giri¸s bölümü olarak düzenlenmi¸s olup kesirli mertebeden türev ve integral kavramlarının geli¸simi ve kesirli mertebeden Schrödinger denkleminin literatür özeti bu kısımda verildi. ˙Ikinci bölümde, daha sonraki bölümlerde kullanılacak olan temel tanım ve kavramlara yer verilmi¸stir. Bu bölümde, Gama ve Beta fonksiyonları, kesirli mertebeden türev ve integral hesaplamalarında kullanılan Grünwald-Letnikov, Riemann-Liouville ve Caputo yakla¸sımları ve kesirli türevlere nümerik yakla¸sımlar ile ilgili genel bilgiler verildi. Üçüncü bölümde, çalı¸smada baz fonksiyonu olarak seçilecek olan Chebyshev polinomları ayrıntılı olarak incelenip, Chebyshev polinomlarının trigonometrik tanımları, ortogonal polinomlar ve Chebyshev polinomlarının ortogonalli˘gi ile ilgili genel bilgiler verildi. Ayrıca bu bölümde, zamana göre kesirli mertebeden Schrödinger denklemi ele alınıp dalgacıklar, Chebyshev dalgacıkları, Chebyshev dalgacıklarıyla fonksiyonlara yakla¸sım, Chebyshev dalgacıklarının integralleri ve yakınsaklık analizi ile ilgili bilgiler verildi. Dördüncü bölüm, çalı¸smanın orjinal kısımlarını içermekte olup zamana göre kesirli mertebeden Schrödinger ve ikili Schrödinger denklemleri Chebyshev dalgacık yöntemiyle ile çözüldü. Her bir kesirli model için farklı iki¸ser probleme uygulandı. Elde edilen nümerik çözümler literatürdeki mevcut bazı sonuçlarla kar¸sıla¸stırıldı, L2 ve L∞ hata normları çizelgeler halinde sunuldu. Ayrıca nümerik sonuçların grafikleri de bu bölümde verilmi¸stir. Be¸sinci bölüm, sonuç bölümü olarak düzenlenmi¸s olup çalı¸smada elde edilen sonuçlar bu bölümde de˘gerlendirilmi¸stir. Anahtar Kelimeler: Schrödinger Denklemi, Kesirli Türev, Chebyshev Dalgacık Yöntemi,Kollokasyon Yöntemi.
The first chapter of this thesis, which consists of five chapters, has been organized as an introduction, and the development of fractional order derivative and integral concepts and the literature survey of fractional order Schrödinger equation is given in this chapter. In the second chapter, fundamental definitions and concepts that will be used in the following chapters are given. In this chapter, general information about Gamma and Beta functions, Grünwald-Letnikov, Riemann-Liouville and Caputo approximations used in fractional order derivatives and integral calculations, and numerical approximations to fractional derivatives are given. In the third chapter, Chebyshev polynomials to be selected as basis functions in the study are examined in detail, and general information about the trigonometric definitions of Chebyshev polynomials, orthogonal polynomials and orthogonality of Chebyshev polynomials are given. In addition, in this chapter, information about the fractional order Schrödinger equation with respect to time is given and information on wavelets, Chebyshev wavelets, approach to functions with Chebyshev wavelets, integrals of Chebyshev wavelets and convergence analysis are given. The fourth chapter contains the original parts of the study, and the fractional order Schrödinger and coupled Schrödinger equations with respect to time were investigated by Chebyshev wavelet method. Two different problems are applied for each fractional model. The numerical methods obtained are compared with some of the results available in the literature, and the error norms L2 and L∞ are presented in tables. Also, graphs of numerical results are given in this chapter. The fifth chapter has been arranged as a conclusion section and the results obtained in the study have been evaluated in this chapter. Keywords: Schrödinger Equation, Fractional Derivation, Chebyshev Wavelet Method,Collocation Method
The first chapter of this thesis, which consists of five chapters, has been organized as an introduction, and the development of fractional order derivative and integral concepts and the literature survey of fractional order Schrödinger equation is given in this chapter. In the second chapter, fundamental definitions and concepts that will be used in the following chapters are given. In this chapter, general information about Gamma and Beta functions, Grünwald-Letnikov, Riemann-Liouville and Caputo approximations used in fractional order derivatives and integral calculations, and numerical approximations to fractional derivatives are given. In the third chapter, Chebyshev polynomials to be selected as basis functions in the study are examined in detail, and general information about the trigonometric definitions of Chebyshev polynomials, orthogonal polynomials and orthogonality of Chebyshev polynomials are given. In addition, in this chapter, information about the fractional order Schrödinger equation with respect to time is given and information on wavelets, Chebyshev wavelets, approach to functions with Chebyshev wavelets, integrals of Chebyshev wavelets and convergence analysis are given. The fourth chapter contains the original parts of the study, and the fractional order Schrödinger and coupled Schrödinger equations with respect to time were investigated by Chebyshev wavelet method. Two different problems are applied for each fractional model. The numerical methods obtained are compared with some of the results available in the literature, and the error norms L2 and L∞ are presented in tables. Also, graphs of numerical results are given in this chapter. The fifth chapter has been arranged as a conclusion section and the results obtained in the study have been evaluated in this chapter. Keywords: Schrödinger Equation, Fractional Derivation, Chebyshev Wavelet Method,Collocation Method
Açıklama
Anahtar Kelimeler
Matematik, Mathematics
Kaynak
WoS Q Değeri
Scopus Q Değeri
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Künye
Köse, G. E. (2020). Zamana göre kesirli mertebeden Schrödinger denkleminin Chebyshev kollokasyon yöntemi ile nümerik çözümleri. Yayınlanmış Doktora Tezi, İnönü Üniversitesi.
