Bir sınıf differensiyel denklemin differensiyel dönüşüm metodu ile çözülmesi
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Dosyalar
Tarih
2005-12-23
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Yayıncı
Selçuk Üniversitesi Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu tez çalışmasında; bir sınıf diferensiyel denklemin diferensiyel dönüşüm metodu ile çözümüne yer verilmiştir. Öncelikli olarak Diferensiyel Dönüşüm metodu ve metodun «-boyutlu uzaya genişletilmesinden elde edilen sonuçlar verilmiştir. Diferensiyel Dönüşüm metodu kullanılarak bazı diferensiyel denklemlerin çözümleri elde edilmiş ardından ise problemlerin çözüm aralıkları gridlere bölünerek gridli diferensiyel dönüşüm konsepti kurulmuş ve bu konsept üzerine bazı diferensiyel denklemlerin çözümleri yapılmıştır. Bu çalışma sekiz bölümden oluşmaktadır. Çalışmanın birinci bölümünde genel bir litereatür özeti verilmiştir. Metodun ilk olarak Zhou tarafından 1986 gibi yakın bir tarihte kurulmuş olması ve metot üzerine son birkaç yıla kadar çok az İİİ çalışmanın yapılmış olması; konu hakkında çıkan makalelerin çoğuna ulaşılmasına imkan tanımıştır. Bu nedenle bu ilk bölümde metodun ilk ortaya konulmasından günümüze kadar olan bütün gelişimi ayrıntılı olarak yer verilmeye çalışılmıştır. İkinci bölümde ise; çalışmanın diğer bölümlerinde sık sık kullanılan bazı tanımlar ile teoremlere yer verilmiştir. Böylelikle çalışmanın diğer bölümleri için bir anlam bütünlüğünün korunması amaçlanmıştır. Üçüncü bölümde ise; Diferensiyel Dönüşüm tanımı ilk önce bir boyutlu uzay için ardından ise n- boyutlu uzay için verilmiştir. Uygulamalarda «-boyut için verilen özellikler yeterli olacağından; metot üzerine kurulan bazı özellikler ve teoremler «-boyutlu uzaya genişletilerek bu bölümde ispatlanmışlardır. Dördüncü bölümde ise metot kullanılarak sırasıyla; beşinci mertebeden sınır değer problemlerinin çözümü, yüksek mertebeden sınır değer problemlerin çözümü ve integro-diferensiyel denklem sistemlerinin çözümlerine yer verilmiştir. Yapılan her bir çalışma; konular üzerine yapılan literatürdeki diğer çalışmalar ile karşılaştırılmış elde edilen sonuçlar tablolar halinde sunulmuştur. Beşinci bölümde; çözüm aralığının gridlere bölünerek her bir alt aralıkta çözüm bulunması esasına dayalı ve Gridli Diferensiyel Dönüşüm Metodu adı verilen metot yardımıyla bazı diferensiyel denklemlerin çözümüne yer verilmiştir. Önceki çalışmalarda sadece bir boyutlu grid yardımı ile çözümler yapılmış iken bu bölümde ayrıca iki boyutlu grid kavramı ortaya konulmuştur. Verilen bu yeni algoritma ile bazı kısmi türevli diferensiyel denklemlerin gridli diferensiyel dönüşüm ile çözümleri yapılmıştır. Altıncı bölümde ise; yapılan nümerik hesaplamalarda kullanılan ve bilgisayarda Mapple 9 paket programı altında çalıştırılan programların algoritmalarına yer verilmiştir. Yedinci bölümde de çalışmanın sonuçlan yorumlanıp öneriler sunulmuştur. Son bölümde ise yararlanılan kaynaklar sunulmuştur.
Some groups of differential equations which can be solved by employing differential transformation method were given in this study. Firstly, the differential transformation method and the solutions obtained by the extension of the method to the n-dimensional spaces were presented. Some of the differential equations were obtained by using differential transformation method the domain of the solution of the problems were divided into grids. Consequently, a differential transformation concept was established and after that, solution of some differential equations were performed with respect to this concept. This study is divided into eight chapters covering the development of the research from the back ground to the final conclusions. The different parts of this report are as follows; In the chapter one; a general literature about this method were summarized in some detail. This method was first established in 1986 by Zhou and a limited number of studies were published in the relevant literature. Therefore it was so easy to obtain the most of the articles those were related to this subject. Chapter two begins with some definitions and theorems those were used in this thesis. Thus, it was aimed to establish the relationship between the chapters of the thesis. Chapter three reviews the definition of differential transformation which was given firstly for one dimensional space and then it was given for n-dimensional spaces. In the applications, since the features of n-dimension given will be adequate, some of the features and theorems based on the method were expanded for n- dimension and they were proven. Chapter four discusses the solutions of five order boundary value problems, solution of higher order boundary value problems and solutions of integro- differential equations systems respectively. Every study, which was done, was compared one to another that was carried out on the subject in the literature and obtained results were presented in tables. Chapter five addresses the solution of some differential equation which were found by the method called as "differential transformation method with grids" based on findings in every sub-interval by dividing the solution domain to grids. Although, in the previous studies, solutions were obtained by the use of only one-dimensional grid, the concept of two dimensional grid was introduced in this research. Using the new concept, solution of some of the partial differential equations were computed by the differential transformation with grid. VI In the chapter six, the algorithms of programs which will run under the Mapple 9 software and used for numeric calculations were presented. Chapter seven summarizes the findings of this investigation interpreting and discussing of the results and conclusions and makes some recommendations for future research. Finally, the references used in this research were listed.
Some groups of differential equations which can be solved by employing differential transformation method were given in this study. Firstly, the differential transformation method and the solutions obtained by the extension of the method to the n-dimensional spaces were presented. Some of the differential equations were obtained by using differential transformation method the domain of the solution of the problems were divided into grids. Consequently, a differential transformation concept was established and after that, solution of some differential equations were performed with respect to this concept. This study is divided into eight chapters covering the development of the research from the back ground to the final conclusions. The different parts of this report are as follows; In the chapter one; a general literature about this method were summarized in some detail. This method was first established in 1986 by Zhou and a limited number of studies were published in the relevant literature. Therefore it was so easy to obtain the most of the articles those were related to this subject. Chapter two begins with some definitions and theorems those were used in this thesis. Thus, it was aimed to establish the relationship between the chapters of the thesis. Chapter three reviews the definition of differential transformation which was given firstly for one dimensional space and then it was given for n-dimensional spaces. In the applications, since the features of n-dimension given will be adequate, some of the features and theorems based on the method were expanded for n- dimension and they were proven. Chapter four discusses the solutions of five order boundary value problems, solution of higher order boundary value problems and solutions of integro- differential equations systems respectively. Every study, which was done, was compared one to another that was carried out on the subject in the literature and obtained results were presented in tables. Chapter five addresses the solution of some differential equation which were found by the method called as "differential transformation method with grids" based on findings in every sub-interval by dividing the solution domain to grids. Although, in the previous studies, solutions were obtained by the use of only one-dimensional grid, the concept of two dimensional grid was introduced in this research. Using the new concept, solution of some of the partial differential equations were computed by the differential transformation with grid. VI In the chapter six, the algorithms of programs which will run under the Mapple 9 software and used for numeric calculations were presented. Chapter seven summarizes the findings of this investigation interpreting and discussing of the results and conclusions and makes some recommendations for future research. Finally, the references used in this research were listed.
Açıklama
Anahtar Kelimeler
Diferensiyel dönüşüm metodu, Diferensiyel denklemlerin hesaplanması, Seri çözüm, Nümerik metod, Differential transformation method, Computation of differential equations, Series solution, Numerical methods
Kaynak
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Özkan, O. (2005). Bir sınıf differensiyel denklemin differensiyel dönüşüm metodu ile çözülmesi. Selçuk Üniversitesi, Yayımlanmış doktora tezi, Konya.
