Rosenau-Kawahara denkleminin sayısal çözümü üzerine bir çalışma
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2021
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info:eu-repo/semantics/openAccess
Özet
Dört bölümden oluşan bu yüksek lisans tezinin birinci bölümünde, tezde göz önüne alınan Rosenau-Kawahara denklemi hakkında kısa bir literatür taraması verildikten sonra yaklaşık çözümleri bulunacak olan Problem 1 ve Problem 2 olarak isimlendirilen iki model problem tanıtıldı. İkinci bölümde ise öncelikle bu tez çalışmasında kullanılacak olan Crank-Nicolson korunumlu kapalı sonlu fark yönteminden bahsedildi. Sonra Rosenau-Kawahara denklemindeki lineer olmayan terim yerine Lin-I, Lin-II ve Lin-III ile verilen üç farklı lineerleştirme tekniği kullanılarak Problem 1'in Crank-Nicolson korunumlu kapalı sonlu fark yöntemi yardımıyla nümerik şemaları elde edildi. Ayrıca Lin-I ile elde edilen nümerik şemanın kararlılık analizi von-Neumann yöntemi ile incelendi. Daha sonra şemalardan hesaplanan L₂ ve L∞ hata normları, doğruluk mertebesi ve korunum sabitleri çizelgeler halinde sunuldu ve aynı zamanda elde edilen ayrık çözümlerin sürekliliği ile birlikte problemin fiziksel davranışını ne kadar iyi sergilediğini göstermek için dalga grafikleri verildi. Üçüncü bölümde, Rosenau-Kawahara denklemi önce konuma göre parçalanarak Problem 2 olarak adlandırılan ikili (coupled) diferansiyel denklem sistemine dönüştürüldü. Sonra ikili denklem sistemindeki lineer olmayan terim yerine ikinci bölümde göz önüne alınan üç farklı lineerleştirme tekniği kullanılarak Problem 2'nin Crank-Nicolson korunumlu kapalı sonlu fark yöntemi ile nümerik şemaları verildi. Ayrıca Lin-I için elde edilen nümerik şemanın kararlılık analizi yine ikinci bölümde olduğu gibi von-Neumann yöntemi ile incelendi. Daha sonra nümerik şemalardan elde edilen L₂ ve L∞ hata normları, doğruluk mertebesi ve korunum sabitleri çizelgeler ve grafikler halinde sunuldu. Tezin son bölümü olan dördüncü bölümde ise önerilen şemalarla birlikte gelecekteki araştırmalar için bir sonuç verildi.
In the first chapter of this thesis, consisting of four chapters, after a brief literature survey about the Rosenau-Kawahara equation considered in the thesis, two model problems called Problem 1 and Problem 2, of which approximate solutions are going to be found, are introduced. In the second chapter, first of all, the Crank-Nicolson conservative implicit finite difference method, which is going to be used in this thesis, is mentioned. Then, instead of the nonlinear term in the Rosenau-Kawahara equation, using three different linearization techniques represented by Lin-I, Lin-II and Lin-III, numerical schemes of Problem 1 are obtained with the help of Crank-Nicolson conservative implicit finite difference method. In addition, the stability analysis of the numerical scheme obtained with Lin-I is examined by the von-Neumann method. Then the error norms L₂ and L∞ calculated from the schemes, the order of accuracy and the conservation constants are presented in graphs, and at the same time, wave graphs are given to show how well the problem exhibits the physical behavior with the continuity of the discrete solutions obtained. In the third chapter, the Rosenau-Kawahara equation is first split with respect to the spatial variable and converted into a coupled differential equation system called as Problem 2. Then, instead of the non-linear term in the coupled equation system, numerical schemes of Problem 2 are given using the Crank-Nicolson conservative implicit finite difference method using three different linearization techniques considered in the second chapter. In addition, the stability analysis of the numerical scheme obtained for Lin-I is examined by the von-Neumann method, as in the second chapter. Then, error norms L₂ and L∞, the order of accuracy and conservation constants obtained from the numerical schemes are presented in tables and graphs. In the fourth chapter, which is the last chapter of the thesis, a brief conclusion for future researcies is given along with the proposed schemes.
In the first chapter of this thesis, consisting of four chapters, after a brief literature survey about the Rosenau-Kawahara equation considered in the thesis, two model problems called Problem 1 and Problem 2, of which approximate solutions are going to be found, are introduced. In the second chapter, first of all, the Crank-Nicolson conservative implicit finite difference method, which is going to be used in this thesis, is mentioned. Then, instead of the nonlinear term in the Rosenau-Kawahara equation, using three different linearization techniques represented by Lin-I, Lin-II and Lin-III, numerical schemes of Problem 1 are obtained with the help of Crank-Nicolson conservative implicit finite difference method. In addition, the stability analysis of the numerical scheme obtained with Lin-I is examined by the von-Neumann method. Then the error norms L₂ and L∞ calculated from the schemes, the order of accuracy and the conservation constants are presented in graphs, and at the same time, wave graphs are given to show how well the problem exhibits the physical behavior with the continuity of the discrete solutions obtained. In the third chapter, the Rosenau-Kawahara equation is first split with respect to the spatial variable and converted into a coupled differential equation system called as Problem 2. Then, instead of the non-linear term in the coupled equation system, numerical schemes of Problem 2 are given using the Crank-Nicolson conservative implicit finite difference method using three different linearization techniques considered in the second chapter. In addition, the stability analysis of the numerical scheme obtained for Lin-I is examined by the von-Neumann method, as in the second chapter. Then, error norms L₂ and L∞, the order of accuracy and conservation constants obtained from the numerical schemes are presented in tables and graphs. In the fourth chapter, which is the last chapter of the thesis, a brief conclusion for future researcies is given along with the proposed schemes.
Açıklama
Anahtar Kelimeler
Kaynak
İnönü Üniversitesi Fen Bilimleri Enstitüsü
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Künye
ÖZDEMİR EKİCİ, H. (2021).Rosenau-Kawahara denkleminin sayısal çözümü üzerine bir çalışma. Yüksek Lisans Tezi, İnönü Üniversitesi.
