Boussınesq tipi denklemlerin galerkın sonlu eleman yöntemi ile nümerik çözümleri
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Dosyalar
Tarih
2016
Yazarlar
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Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/embargoedAccess
Özet
Dört bölümden oluşan bu çalışmanın ilk bölümünde, lineer olmayan kısmi diferansiyel denklemler ve bu denklemlerin çözüm tiplerinden biri olan soliton ve soliter dalgalar hakkında kısaca bilgi verildi. İkinci bölümde, temel kavramlardan bahsedilirken, ele alınan model problemlerin nümerik çözümlerinin elde edilmesinde kullanılan yöntemler tanıtıldı. Model problemlerin kuadratik ve kübik B-spline bazlar kullanılarak Galerkin sonlu eleman modeli kurulduktan sonra elde edilen adi diferansiyel denklem sistemleri dördüncümertebeden Runge-Kutta yöntemi ile çözüldüğünden bu bölümde; sonlu eleman yöntemi, Galerkin sonlu eleman yöntemi, spline fonksiyonlar, B-spline fonksiyonlar, dördüncü mertebeden Runge-Kutta yöntemi ve yöntemin kararlık analizine değinildi. Üçüncü bölümde, soliter dalga üreten Boussinesq tipi denklemlerden Good Boussinesq ve Bad Boussinesq denklemleri ele alındı. Kübik B-spline bazlar yardımıyla her iki denklemin Galerkin sonlu eleman modeli oluşturuldu. Daha sonra, Good Boussinesq denklemi için dalga hareketi, iki soliter dalganın etkileşimi ve dalga ayrılması problemleri incelendi. Ayrıca etkileşim problemi içerisinde soliter dalgaların genlik seçimine göre dalga etkileşiminde karşılaşılan patlama problemlerine değinildi. Bad Boussinesq denklemi için ise soliter dalga hareketi ve iki soliter dalganın etkileşimi problemleri incelendi. Dördüncü bölümde, Boussinesq tipi denklemlerden Improved Boussinesq, Improved Boussinesq-tipi ve modifiye edilmiş Improved Boussinesq denklemleri için kuadratik B-spline bazlar kullanılarak Galerkin sonlu eleman modelleri oluşturuldu. Model problemlerden Improved Boussinesq denklemi için dalga hareketi, iki soliter dalganın etkileşimi, dalga ayrılması ve çözüm patlaması problemleri ele alındı. Improved Boussinesq tipi denklem için dalga hareketi, soliter-antisoliter dalga etkileşimi problemi, modifiye edilmiş Improved Boussinesq denklemi için ise dalga hareketi ve dalga etkileşimi problemleri incelendi.
In the first chapter of this thesis consisting of four chapters, some information are given about nonlinear partial differential equations and solitons and solitary waves which are among solution types of these equation. In the second chapter, while presenting fundamental concepts, numerical methods used to obtain numerical solutions of the model problem are explained. Since first the Galerkin finite element model of the model problems are constructed using quadratic and cubic B-spline base functions and then the obtained systems of differential equations are solved by the fourth order Runge-Kutta method, finite element method, Galerkin finite element method, spline functions, B-spline functions, the fourth order Runge-Kutta method and the stability analysis of the method are explained. In the third chapter, Good Boussinesq and Bad Boussinesq equations among Boussinesq equations producing solitary wave are considered. Galerkin finite element method models for both equations are constructed. Then, solitary wave movement, the interaction of two solitary waves and wave break-up problems for Good Boussinesq equation are considered and in interaction problem, the blow-up problems according to the choice of wave amplitudes are taken into consideration. For Bad Boussinesq equation, the solitary wave movement and the interaction of two solitary waves problems are considered. In the fourth chapter, Galerkin finite element models for Improved Boussinesq, Improved Boussinesq-type and modified Improved Boussinesq equations using quadratic B-spline bases are constructed. For improved Boussinesq equation; wave movement, the interaction of two solitary waves, wave break-up and blow-up problems are taken into consideration. For improved Boussinesq-type equation wave movement, solitary-antisolitary wave interaction problem, and for modified Improved Boussinesq equation wave movement and wave interaction problems are considered.
In the first chapter of this thesis consisting of four chapters, some information are given about nonlinear partial differential equations and solitons and solitary waves which are among solution types of these equation. In the second chapter, while presenting fundamental concepts, numerical methods used to obtain numerical solutions of the model problem are explained. Since first the Galerkin finite element model of the model problems are constructed using quadratic and cubic B-spline base functions and then the obtained systems of differential equations are solved by the fourth order Runge-Kutta method, finite element method, Galerkin finite element method, spline functions, B-spline functions, the fourth order Runge-Kutta method and the stability analysis of the method are explained. In the third chapter, Good Boussinesq and Bad Boussinesq equations among Boussinesq equations producing solitary wave are considered. Galerkin finite element method models for both equations are constructed. Then, solitary wave movement, the interaction of two solitary waves and wave break-up problems for Good Boussinesq equation are considered and in interaction problem, the blow-up problems according to the choice of wave amplitudes are taken into consideration. For Bad Boussinesq equation, the solitary wave movement and the interaction of two solitary waves problems are considered. In the fourth chapter, Galerkin finite element models for Improved Boussinesq, Improved Boussinesq-type and modified Improved Boussinesq equations using quadratic B-spline bases are constructed. For improved Boussinesq equation; wave movement, the interaction of two solitary waves, wave break-up and blow-up problems are taken into consideration. For improved Boussinesq-type equation wave movement, solitary-antisolitary wave interaction problem, and for modified Improved Boussinesq equation wave movement and wave interaction problems are considered.
Açıklama
14.08.2019 tarihine kadar kullanımı yazar tarafından kısıtlanmıştır.
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Künye
Karaağaç, B. (2016). Boussınesq tipi denklemlerin galerkın sonlu eleman yöntemi ile nümerik çözümleri. İnönü Üniversitesi Fen Bilimleri Enstitüsü. 1-160 ss.
