Diferansiyel denklem çözümlerinde nokta dönüşümleri
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Tarih
2004
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info:eu-repo/semantics/openAccess
Özet
Uygulamalı Bilimlerde bir olayı açıklamak ya da bir problemi çözmek çoğu kez bir diferansiyel denklem kurmayı ve onu çözmeyi gerektirir. Bundan dolayı diferansiyel denklemleri sınıflandırmak, çözülebilirlik durumlarını araştırmak çözüm yöntemleri geliştirmek diferansiyel denklemler teorisinin başlıca konulandır. Adi Diferansiyel Denklemler için geliştirilmiş değişik çözüm yöntemleri vardır. Bunlardan biride simetriler yardımı ile çözümlerin bulunmasıdır. Bayağı Diferansiyel Denklemler ve çözümleri ile ilgili olan bu çalışma 6 Bölümden oluşmaktadır. I, II. Bölümlerde; nokta dönüşümleri, dönüşüm kuralları, dönüşüm ve üreteçlerin normal formları, dönüşümlerin çok parametreli grupları ve üreteçleri, bayağı diferansiyel denklemlerin lie nokta simetrileri ve bu simetrilerin bulunması, simetrilerle ilgili tanımlar, bir ve ikinci mertebeden diferansiyel denklemler ile ilgili bilgiler verilmiştir. III. ve IV. Bölümlerde; bir simetrisi bilinen diferansiyel denklemlerin çözümleri, lie cebrinin bazı temel özellikleri, lie nokta simetrilerinin kullanımı, integrasyon teknikleri incelenmiştir. V. Bölümde; belli türden non-lineer bir denklem sınıfının belli koşullarda yapılabilen çözümleri ile, aynı denklem türünün simetriler yardımı ile yapılan bir çözümü incelenerek karşılaştmlmıştır. VI. Bölümde; çalışılan konunun tartışması yapılmıştır. Ek-A, Ek-B, Ek-C, Ek-D de ise lie nokta simetrilerinin bulunması, simetrileri bilinen diferansiyel denklemlerin çözümleri ile ilgili örnekler verilmiştir.
In Applied Sciences, the construction of a differential equation and its solution is often required to explain an event, or to solve a problem. Therefore, the differential equations theory mainly focuses on the classification of the differential equations, the research on resolvable conditions and improving the solution methods. There exist different solution methods developed for the ordinary differential equations. One of the most important methods is to find solutions by the help of the symmetries. This study, which is considered the ordinary differential equations and their solutions consists of six chapters. In the first two chapters; the point transformations, the transformation rules, the normal forms of the transformations and generators, the groups with polyparameter and the generators of the transformations, Lie point symmetries of the ordinary differantial equations and finding these symmetries, the definitions related with symmetries, the information about the differential equations of \th and 2th order are given. In the third and fourth chapters, the solutions of the differential equations with known symmetry, the certain fundamental properties of the Lie Algebra, the use of Lie point symmetries, the integration techniques are studied. In the fifth chapter, the solutions which can be found at obvious conditions of the obvious type of a nonlinear equation class are compared with the solution of the same type of the equation carried out by the help of the symmetries. In sixth chapter discusses the overall subject. In the appendiceses A-B-C-D, finding Lie point symmetries and the examples related with the solutions of the differential equations whose symmetries are known are presented.
In Applied Sciences, the construction of a differential equation and its solution is often required to explain an event, or to solve a problem. Therefore, the differential equations theory mainly focuses on the classification of the differential equations, the research on resolvable conditions and improving the solution methods. There exist different solution methods developed for the ordinary differential equations. One of the most important methods is to find solutions by the help of the symmetries. This study, which is considered the ordinary differential equations and their solutions consists of six chapters. In the first two chapters; the point transformations, the transformation rules, the normal forms of the transformations and generators, the groups with polyparameter and the generators of the transformations, Lie point symmetries of the ordinary differantial equations and finding these symmetries, the definitions related with symmetries, the information about the differential equations of \th and 2th order are given. In the third and fourth chapters, the solutions of the differential equations with known symmetry, the certain fundamental properties of the Lie Algebra, the use of Lie point symmetries, the integration techniques are studied. In the fifth chapter, the solutions which can be found at obvious conditions of the obvious type of a nonlinear equation class are compared with the solution of the same type of the equation carried out by the help of the symmetries. In sixth chapter discusses the overall subject. In the appendiceses A-B-C-D, finding Lie point symmetries and the examples related with the solutions of the differential equations whose symmetries are known are presented.
