Oluşum türü denklemlerin simetri indirgemeleri, korunum kanunları ve tam çözümleri
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Date
2017-12-12
Authors
Giresunlu, İlker Burak
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Publisher
Uludağ Üniversitesi
Abstract
Bu doktora tezinde, tam ve kesirli mertebeli oluşum türü denklemlerin simetri indirgemeleri, korunum kanunları ve tam çözümleri araştırılarak uygulamaları yapılmıştır. Diferensiyel denklemlerin incelenmesinde oldukça önemli bir yere sahip olan Lie simetri grupları yöntemi varyant Boussinesq sistemine, Schamel-Korteweg-de Vries denklemine, Konopelchencho-Dubrovski sistemine, logaritmik KdV-benzeri ve logaritmik KP-benzeri denklemlerine ve zaman kesirli Schamel-Korteweg-de Vries denklemine uygulandı. Gözönüne alınan denklem veya sistemlerin simetri indirgemeleri, tam çözümleri ve korunum kanunlarına ulaşıldı. Bu tezde sadece Lie nokta simetrileri ele alındı. Bunun yanında tezde Lie nokta simetri ve korunum vektörleri arasındaki ilişkiler araştırıldı. Korunum vektörlerini sistematik olarak elde etmek için üç tip farklı yöntem ele alındı. Bunlar sırasıyla çarpan yöntemi, yerel olmayan korunum yöntemi ve eşlenik simetri yaklaşımdır. Bu üç yöntem arasındaki ilişkiler tartışılmış ve logaritmik KdV-benzeri ve KP-benzeri denklemlerine uygulandı. Bununla birlikte elde edilen korunum kanunları ve elde edilen simetriler yardımıyla denklemin hem mertebesi hem de değişken sayısında indirgemeye olanak sağlayan "çift indirgeme" yöntemi kullanılarak kapalı çözüm formlarına ulaşıldı. Lie simetri grupları yönteminin kesirli mertebeli diferensiyel denklemlere uyarlanması ele alındı ve bu yeni yaklaşım kullanılarak zaman kesirli mertebeli Schamel-Korteweg-de Vries denkleminin Lie simetri grupları ve korunum kanunları elde edildi. Elde edilen simetri üreteçlerinin orijinal tam mertebeli denkleme göre daha az üreteç kabul etmesine rağmen elde edilen simetri indirgemesinin özel integral operatörlerini içeren kesirli mertebeden adi diferensiyel denklemlere ulaşıldığı gözlemlendi. Lie simetri indirgemelerinin ilerleyen dalga tipindeki çözümlerine, bazı tam çözüm bulma algoritmaları kullanılarak ulaşıldı. Bu doktora tezinde elde edilen sonuçlar gözönüne alınan modellerin arkasındaki fiziksel olgunun açıklanmasında kullanılabilir. Bununla birlikte elde edilen tam çözümler kullanılarak sayısal simülasyonlar yapılabilir ve sayısal çözüm bulma şemalarında test fonksiyonu olarak kullanılabilir.
In this doctoral thesis, we study symmetry reductions, conservation laws and exact solutions of integer and fractional order evolution differential equations. The Lie symmetry group method, which has a very important role in the study of the differential equations, is applied to variant Boussinesq system, Schamel-Korteweg-de Vries equation, Konopelchencho-Dubrovski system, logarithmic KdV-like and KP-like equations and time fractional Schamel-Korteweg-de Vries equation. The symmetry reductions, exact solutions and conservation laws of the considered equations or systems have been reached. We have restricted ourselves to only Lie point symmetry. In addition, relations between Lie point symmetry and conservation vectors were investigated. Three types of different methods have been dealt with in order to systematically obtain conservation laws. These are multiplier method, non-local conservation method and adjoint symmetry approaches respectively. Relations between these three methods are discussed and applied to KdV-like and KP-like equations with logarithmic structure. If the considered equation or system has relationship between symmetries and conservation laws one can construct closed solution forms exploiting by the "double reduction" approach which allows the equation to be reduced both in the order and in the variable number. The adaptation of the Lie symmetry groups method to the fractional order differential equations was studied and the Lie symmetry groups and conservation laws of the time-fractional Schamel-Kortewegde Vries equation were obtained. Though the obtained symmetry generators are less than the original integer-order equation, we have yield fractional order ordinary differential equation including special integral operators. In this thesis, traveling wave type solutions are reached by using some powerful algorithms. The results obtained in this thesis can be used to explain the physical phenomenas behind the models considered. Numerical simulations can be made using the exact solutions and can be used as test functions in numerical solution finding schemes.
In this doctoral thesis, we study symmetry reductions, conservation laws and exact solutions of integer and fractional order evolution differential equations. The Lie symmetry group method, which has a very important role in the study of the differential equations, is applied to variant Boussinesq system, Schamel-Korteweg-de Vries equation, Konopelchencho-Dubrovski system, logarithmic KdV-like and KP-like equations and time fractional Schamel-Korteweg-de Vries equation. The symmetry reductions, exact solutions and conservation laws of the considered equations or systems have been reached. We have restricted ourselves to only Lie point symmetry. In addition, relations between Lie point symmetry and conservation vectors were investigated. Three types of different methods have been dealt with in order to systematically obtain conservation laws. These are multiplier method, non-local conservation method and adjoint symmetry approaches respectively. Relations between these three methods are discussed and applied to KdV-like and KP-like equations with logarithmic structure. If the considered equation or system has relationship between symmetries and conservation laws one can construct closed solution forms exploiting by the "double reduction" approach which allows the equation to be reduced both in the order and in the variable number. The adaptation of the Lie symmetry groups method to the fractional order differential equations was studied and the Lie symmetry groups and conservation laws of the time-fractional Schamel-Kortewegde Vries equation were obtained. Though the obtained symmetry generators are less than the original integer-order equation, we have yield fractional order ordinary differential equation including special integral operators. In this thesis, traveling wave type solutions are reached by using some powerful algorithms. The results obtained in this thesis can be used to explain the physical phenomenas behind the models considered. Numerical simulations can be made using the exact solutions and can be used as test functions in numerical solution finding schemes.
Description
Keywords
Lie simetrileri, Korunum kanunları, Eşlenik denklem, Eşlenik simetri, Çarpan yöntemi, Lineer olmayan kendi eşleniklik, Çift indirgeme yöntemi, Simetri indirgemeleri, Özyineleme formülü, Lie symmetries, Conservation laws, Adjoint equation, Adjoint symmetry, Multiplier method, Nonlinear self-adjointness, Double reduction method, Symmetry reductions, Recursion formula
Citation
Giresunlu, İ. B. (2017). Oluşum türü denklemlerin simetri indirgemeleri, korunum kanunları ve tam çözümleri. Yayınlanmamış doktora tezi. Uludağ Üniversitesi Fen Bilimleri Enstitüsü.