Helmholtz denklemi ve onbir koordinat sisteminde çözümü
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Tarih
2007
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info:eu-repo/semantics/openAccess
Özet
Doğadaki olayları açıklamak için en etkin ve sistematik yol Diferansiyel Denklem dilini kullanmaktır. Fizik, Kimya, Biyoloji, Astroloji, Mühendislik, Ekonomi ve diğer pek çok Uygulamalı Bilimler, Diferansiyel Denklemlerin önemli uygulama alanlarıdır. Bunun dışında, matematiğin kendi içinde de diferansiyel denklemlerin önemli bir yeri vardır. Diferansiyel Denklemler ve koordinat sistemleri birbirleri ile yakından ilişkilidirler. Özellikle denklemlerin çözümlerinin bulunması denklemlerin koordinat sistemlerinde uygun ifade edilmelerine bağlıdır. Çalışmanın I. Bölümünde Eğrisel Koordinatlar ve Ortogonal Koordinat Sistemleri hakkında genel kavramlar ile Gradyent, Diverjans, Rotasyonel ve Laplasyen ifadeleri verilmiştir. II. Bölümde Özel Ortogonal Koordinat Sistemleri tanıtılarak özellikleri irdelenmiştir. III. Bölümde Helmholtz Denklemi tanıtılmış, Stackel Matris ve Helmholtz Denkleminin Basit Ayrıştırması irdelenmiştir. IV. Bölümde Helmholtz Denkleminin Özel Koordinat Sistemlerinde Çözümü verilmiştir. Anahtar Kelimeler: Eğrisel Koordinatlar, Helmholtz Denklemi, Ayrıştırma.
In order to explain the events in the nature, the most effective and systematic way is to use the language of Differential Equation. Physics, Chemistry, Biology, Astrnomy, Engineering, Economics and many other practical Applied Sciences are the important fields for application of Differential Equation. A part from these, differential equation have an important place in mathematics itself. Differential Equations and coordinate systems are closely related to each other. Especialy, finding the solutions of equations depens on the appropriate expression of the equations in coordinate systems. In the first chapter of this study, the general concepts about Curvilinear Coordinates ant Orthogonal Coordinate Sysstems are given and the terms Gradient, Divergence, Rotational and Laplacian are determined. In the second chapter, Special Orthogonal Coordinate Systems are given and their characteristics are studied. In the third chapter, Helmholtz Equations is given and the Basic Separation of Helmholtz Equations and Stackel Matrix are studied. In the fourth chapter, The Solution of the Helmholtz Equation in Special Coordinate Systems are given. Key words: Curvilinear Coordinates, Helmholtz Equation, Separation.
In order to explain the events in the nature, the most effective and systematic way is to use the language of Differential Equation. Physics, Chemistry, Biology, Astrnomy, Engineering, Economics and many other practical Applied Sciences are the important fields for application of Differential Equation. A part from these, differential equation have an important place in mathematics itself. Differential Equations and coordinate systems are closely related to each other. Especialy, finding the solutions of equations depens on the appropriate expression of the equations in coordinate systems. In the first chapter of this study, the general concepts about Curvilinear Coordinates ant Orthogonal Coordinate Sysstems are given and the terms Gradient, Divergence, Rotational and Laplacian are determined. In the second chapter, Special Orthogonal Coordinate Systems are given and their characteristics are studied. In the third chapter, Helmholtz Equations is given and the Basic Separation of Helmholtz Equations and Stackel Matrix are studied. In the fourth chapter, The Solution of the Helmholtz Equation in Special Coordinate Systems are given. Key words: Curvilinear Coordinates, Helmholtz Equation, Separation.
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Anahtar Kelimeler
Matematik, Mathematics
