Bazı lineer ve lineer olmayan diferansiyel denklemlerin homotopi pertürbasyon ve homotopi analiz metotları ile çözümlerinin analizi üzerine
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Dosyalar
Tarih
2009
Yazarlar
Dergi Başlığı
Dergi ISSN
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Yayıncı
Trakya Üniversitesi Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu çalışmanın amacı, homotopi pertürbasyon metodu ve homotopi analiz metodunu kullanarak bazı lineer ve nonlineer problemlerin çözümlerini elde etmek ve çözümlerin karşılaştırmasını yapmaktır. Üç bölümden oluşan bu çalışmanın I. bölümünde, homotopi kavramı ve pertürbasyon teorisi kısaca verilmiş, homotopi pertürbasyon metodu tanıtılmış ve bu metot bir ve iki-boyutlu homojen, üç-boyutlu homojen olmayan değişken katsayılı ısı-tipi ve dalga-tipi başlangıç ve sınır değer problemlerine uygulanarak çözümler elde edilmiş ve bulunan çözümlerin yakınsaklık analizi verilmiştir. II. Bölümde homotopi analiz metodu tanıtılmış ve bu metot kullanılarak lineer backward ve forward Kolmogorov ve nonlineer Fokker-Planck denklemlerinin -yakınsaklık kontrol parametresine bağlı seri çözümleri elde edilmiş ve çözümlerin analizi yapılmıştır. III. Bölümünde her iki metotla bulunan çözümler karşılaştırılmıştır. Son olarak elde edilen sonuçlar değerlendirilmiştir.
The purpose of this study is obtaining the solutions of certain linear and nonlinear problems by using homotopy perturbation method and homotopy analysis method and making the comparison of these solutions. This study consists three chapters in total and in the first chapter the concept of homotopy and perturbation theory are given briefly, the homotopy perturbation method is introduced and solutions are obtained by applying this method to one and two-dimensional homogeneous, three-dimensional nonhomogeneous heat-like and wave-like initial and boundary value problems with variable coefficients. Then convergence analysis of the found solutions is given. In the second chapter, homotopy analysis method is introduced and by using this method the series solutions according to the -convergence control parameter of linear backward and forward Kolmogorov and nonlinear Fokker-Planck equations are obtained and analysis of the solutions is made. In the third chapter, the solutions found by both of the methods are compared. In conclusion, the results obtained are evaluated.
The purpose of this study is obtaining the solutions of certain linear and nonlinear problems by using homotopy perturbation method and homotopy analysis method and making the comparison of these solutions. This study consists three chapters in total and in the first chapter the concept of homotopy and perturbation theory are given briefly, the homotopy perturbation method is introduced and solutions are obtained by applying this method to one and two-dimensional homogeneous, three-dimensional nonhomogeneous heat-like and wave-like initial and boundary value problems with variable coefficients. Then convergence analysis of the found solutions is given. In the second chapter, homotopy analysis method is introduced and by using this method the series solutions according to the -convergence control parameter of linear backward and forward Kolmogorov and nonlinear Fokker-Planck equations are obtained and analysis of the solutions is made. In the third chapter, the solutions found by both of the methods are compared. In conclusion, the results obtained are evaluated.
Açıklama
Doktora Tezi
Anahtar Kelimeler
Matematik, Mathematics, Diferensiyel Denklem Yöntemi, Differential Equation Method, Doğrusal Olmayan Denklemler, Nonlinear Equations
