B-Spline sonlu eleman yöntemleri ile coupled diferansiyel denklemlerin nümerik çözümleri
Yükleniyor...
Dosyalar
Tarih
2011
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu tez dört bölümden oluşmaktadır. Birinci bölümde, tezde kullanılacak olan sonlu eleman yöntemleri hakkında bazı genel bilgiler verildikten sonra, Galerkin, Petrov-Galerkin, subdomain ve kollokasyon yöntemleri ile birlikte spline fonksiyonlar ve B-spline fonksiyonlar hakkında temel kavramlar verildi. İkinci, üçüncü ve dördüncü bölümler bu tezin orijinal kısımlarını oluşturmaktadır. İkinci bölümde, coupled Burgers denklemi, farklı dereceden B-spline fonksiyonlar yardımıyla Galerkin, Petrov-Galerkin, subdomain ve kollokasyon sonlu eleman yöntemleri ile çözüldü. Bu yöntemler göz önüne alınan üç model probleme uygulandı. Elde edilen nümerik sonuçlar literatürdeki mevcut sonuçlar ile karşılaştırılarak L_{2 } ve L_{? }hata normları tablolar halinde verildi. Ayrıca her bir yöntemin uygulanmasıyla elde edilen sonlu eleman yaklaşımının kararlılık analizi incelendi. Üçüncü bölümde, coupled Korteweg-de Vries (KdV) denklemi hakkında genel bilgiler verildikten sonra farklı dereceden B-spline fonksiyonlar kullanılarak Galerkin, Petrov-Galerkin, subdomain ve kollokasyon sonlu eleman yöntemleri ile üç model problem çözüldü. Elde edilen nümerik sonuçlar literatürdeki mevcut sonuçlar ile karşılaştırılarak L_{2 }ve L_{? } hata normları ile korunum sabitleri tablolar halinde verildi. Ayrıca her bir yöntemin uygulanmasıyla elde edilen yaklaşımın kararlılık analizi incelendi. Dördüncü bölümde, coupled modified Korteweg-de Vries (mKdV) denklemi farklı dereceden B-spline fonksiyonlar kullanılarak Galerkin, Petrov-Galerkin, subdomain ve kollokasyon sonlu eleman yöntemleri ile çözüldü. Bu yöntemler göz önüne alınan beş model probleme uygulandı. Elde edilen nümerik sonuçlar literatürdeki mevcut sonuçlar ile karşılaştırılarak L_{2 } ve L_{? } hata normları ile korunum sabitleri tablolar halinde verildi. Ayrıca her bir yöntemin uygulanmasıyla elde edilen yaklaşımın kararlılık analizi yapıldı.
This thesis consists of four chapters. In the first chapter, after giving some general information about the finite element methods which will be used in the thesis, fundamental concepts about Galerkin, Petrov-Galerkin, subdomain and collocation methods together with spline functions and B-spline functions are presented. The second, third and fourth chapters of this thesis make up its original parts. In the second chapter, coupled Burgers' equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods with different degrees B-spline functions. These methods are applied to three model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, the error norms L_{2 } and L_{? }are given in the form of tables. The stability analysis of the finite element approximation obtained by applying each method is also investigated. In the third chapter, after giving general information about the coupled Korteweg-de Vries (KdV) equation, the three test problems are solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? }and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated. In the fourth chapter, the coupled modified Korteweg-de Vries (mKdV) equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. These methods are applied to five model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? } and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated.
This thesis consists of four chapters. In the first chapter, after giving some general information about the finite element methods which will be used in the thesis, fundamental concepts about Galerkin, Petrov-Galerkin, subdomain and collocation methods together with spline functions and B-spline functions are presented. The second, third and fourth chapters of this thesis make up its original parts. In the second chapter, coupled Burgers' equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods with different degrees B-spline functions. These methods are applied to three model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, the error norms L_{2 } and L_{? }are given in the form of tables. The stability analysis of the finite element approximation obtained by applying each method is also investigated. In the third chapter, after giving general information about the coupled Korteweg-de Vries (KdV) equation, the three test problems are solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? }and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated. In the fourth chapter, the coupled modified Korteweg-de Vries (mKdV) equation is solved by Galerkin, Petrov-Galerkin, subdomain and collocation finite element methods by using different degrees B-spline functions. These methods are applied to five model problems which are taken into consideration in the thesis. The obtained numerical results are compared with existing results in the literature, and they are given with the error norms L_{2 }, L_{? } and the invariants in the form of tables. The stability analysis of the approximation obtained by applying each method is also investigated.
Açıklama
Anahtar Kelimeler
Coupled Burgers Denklemi, Coupled KdV Denklemi, Coupled mKdV Denklemi, Sonlu Eleman Yöntemleri, B-Spline Fonksiyonlar, Galerkin Yöntemi, Petrov-Galerkin Yöntemi, Subdomain Yöntemi, Kollokasyon Yöntemi, Kararlılık Analizi, Coupled Burgers' Equation, Coupled KdV Equation, Coupled mKdV Equation, Finite Element Methods, B-Spline Functions, Galerkin Method, Petrov-Galerkin Method, Subdomain Method, Collocation Method, Stability Analysis
Kaynak
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Uçar, Y. (2011). B-Spline sonlu eleman yöntemleri ile coupled diferansiyel denklemlerin nümerik çözümleri. İnönü Üniversitesi Fen Bilimleri Enstitüsü. 1-266 ss.
