İkili Burgers denkleminin trigonometrik B-spline kollokasyon yöntemi ile nümerik çözümleri
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2021
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info:eu-repo/semantics/openAccess
Özet
Bu yüksek lisans tezi beş bölümden oluşmaktadır. Birinci bölümde, tezin temel amacından kısaca bahsedildi ve sonlu eleman yöntemi hakkında bilgi verildi. İkinci bölümde, bu tez çalışmasında kullandığımız temel kavramlar olan sonlu eleman yöntemi, kollokasyon yöntemi ve trigonometrik B-spline fonksiyonlar hakkında bilgi verildi. Ayrıca bu bölümde tezde göz önüne alınacak olan sonlu eleman yönteminin tarihsel gelişiminden bahsedildi. Üçüncü bölümde ikili (coupled) Burgers denklemi ile ilgili literatürdeki çalışmalar sunuldu ve nümerik çözümü yapılacak olan ikili Burgers denkleminin farklı başlangıç ve sınır ¸sartları ile verilen üç model problemi tanıtıldı. Dördüncü bölümde, ikili Burgers denklemindeki lineer olmayan terimler yerine LİN-1 ve LİN-2 olmak üzere iki farklı lineerleştirme tekniği kullanılarak kübik ve kuintik trigonometrik B-spline kollokasyon sonlu eleman yöntemi ile nümerik ¸semaları elde edildi. Bu nümerik ¸semaların önceki bölümde verilen üç model probleme uygulanmasıyla elde edilen nümerik sonuçlar mevcut tam çözümle ve/veya literatürdeki farklı çalışmalardaki sonuçlarla çizelgeler halinde kar¸sıla¸stırıldı. Her iki lineerleştirme tekniğinde elde edilen nümerik ¸semaların kararlılık analizleri benzer olacagından LİN-1 ve LİN-2 için kübik trigonometrik B-spline kollokasyon yöntemleriyle elde edilen ¸semaların kararlılık analizi von Neumann yöntemiyle incelendi. Beşinci ve son bölümde, uygulanan her iki lineerleştirme tekniğinden elde edilen sonuçlar her bir model problem için kendi içerisinde çizelgeler halinde karşılaştırıldı.
This master thesis consists of five chapters. In the first chapter, the main purpose of the thesis was briefly mentioned and some information about the finite element method was given. In the second chapter, information was given about the basic concepts that we used in this thesis, the finite element method, the collocation method and trigonometric B-spline basis functions. In addition, in this chapter, the historical development of the finite element method, which is going to be considered in the thesis, is mentioned. In the third chapter, the studies related to the coupled Burgers equation in the literature are presented and three model problems of the coupled Burgers equation to be solved numerically with different boundary and initial conditions are introduced. In the fourth chapter, using two different linearization techniques, LIN-1 and LIN-2, instead of the non-linear terms in the coupled Burgers equation, numerical schemes were obtained by cubic and quintic trigonometric B-spline collocation finite element method. The numerical results obtained by applying these schemes to the three models given in the previous section were compared with the available exact solution and/or the results from different studies in the literature in tabular form. Since the stability analysis of the numerical schemes obtained in both linearization techniques are going to be similar, the stability analysis of the schemes obtained by the cubic trigonometric B-spline collocation methods for LIN-1 and LIN-2 was examined by the von Neumann method. In the fifth and last part, the results obtained from both applied linearization techniques were compared for each model problem in the form of tables.
This master thesis consists of five chapters. In the first chapter, the main purpose of the thesis was briefly mentioned and some information about the finite element method was given. In the second chapter, information was given about the basic concepts that we used in this thesis, the finite element method, the collocation method and trigonometric B-spline basis functions. In addition, in this chapter, the historical development of the finite element method, which is going to be considered in the thesis, is mentioned. In the third chapter, the studies related to the coupled Burgers equation in the literature are presented and three model problems of the coupled Burgers equation to be solved numerically with different boundary and initial conditions are introduced. In the fourth chapter, using two different linearization techniques, LIN-1 and LIN-2, instead of the non-linear terms in the coupled Burgers equation, numerical schemes were obtained by cubic and quintic trigonometric B-spline collocation finite element method. The numerical results obtained by applying these schemes to the three models given in the previous section were compared with the available exact solution and/or the results from different studies in the literature in tabular form. Since the stability analysis of the numerical schemes obtained in both linearization techniques are going to be similar, the stability analysis of the schemes obtained by the cubic trigonometric B-spline collocation methods for LIN-1 and LIN-2 was examined by the von Neumann method. In the fifth and last part, the results obtained from both applied linearization techniques were compared for each model problem in the form of tables.
Açıklama
Anahtar Kelimeler
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İnönü Üniversitesi Fen Bilimleri Enstitüsü
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Künye
YİĞİT, M. K. (2021). İkili Burgers denkleminin trigonometrik B-spline kollokasyon yöntemi ile nümerik çözümleri. Yüksek Lisans Tezi, İnönü Üniversitesi.
