Poisson Denklemi ve Çözümleri
Loading...
Date
2011
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Trakya Üniversitesi Fen Bilimleri Enstitüsü
Access Rights
info:eu-repo/semantics/openAccess
Abstract
Kısmi Türevli Diferansiyel Denklemler, Uygulamalı Matematiğin bir dalı olup Temel Bilimlerden Mühendisliğin tüm alanlarında geniş uygulaması vardır. Fizik ve mühendislik alanında karşılaşılan diferansiyel denklemler, Laplace, Poisson, Helmholtz veya dalga, Schrödinger gibi denklemlerdir. Bu tip denklemlerin ortak özellikleri; doğrusal, ikinci mertebeden kısmi diferansiyel denklemler olmalarıdır. Bu denklemlerin çözümlerinde seriler, değişkenlerin ayrılması, Green fonksiyonları ve integral dönüşümler sıkça kullanılır. Analitik tekniklerin yetersiz olduğu durumlarda sayısal yöntemlere başvurulur. Bu çalışmanın I. Bölümünde Diferansiyel Denklemler ile ilgili genel kısa bilgiler verilmiş, Kısmi Diferansiyel Denklemlerle ilgili genel kavramların yanısıra, Laplace, Poisson, Difüzyon, Helmholtz, Dalga Denklemleri kısaca tanıtılmıştır. II. Bölümde Genel Koordinatlar, Ortogonal Koordinat Sistemleri, Özel Ortogonal Koordinatlar tanıtılarak, Gradyenti, Diverjans, Rotasyonel ve Laplasyen ifadeleri verilmiş, Bessel ve Legendre Fonksiyonlarının temel özellikleri kısaca tanıtılmıştır. III. Bölümde Poisson Denklemi tanıtılarak Silindirik ve Küresel Koordinatlarda yapılan bazı özel çözümleri gözden geçirilmiştir. Anahtar Kelimeler: Özel Ortogonal Koordinatlar, Poisson Denklemi, Green Fonksiyonu.
Abstract
Partial Differential Equations, a branch of applied Mathematics, have many applications in every branch of engineering in basic science. Differential Equations that faced in Physics and Engineering are equations as Laplace, Poisson, Helmholtz or wave, Schrödinger equations. Common features of this kind of equations are linear and partial differential equations from second degree. Series, differentiation of variables, Green Functions and integral transformations are often used in solving these equations. In some situations, analytical techniques are inadequate so numerical methods are used. In this study, in Part I, general and short information about Differential Equations are given, and in addition to general concepts about Partial Differential Equations, Laplace, Poisson, Diffusion, Helmholtz, Wave Equations are shortly introduced. In Part II, general coordinates, Orthogonal Coordinate Systems, Special orthogonal coordinates are introduced and Gradient, Divergence, Rotational and Laplacian expressions are given and basic features of Bessel and Legendre Functions are shortly introduced. In Part III, Poisson Equation is introduced and special solutions which are applied in Cylinder and Spherical Coordinates are looked over.
Abstract
Partial Differential Equations, a branch of applied Mathematics, have many applications in every branch of engineering in basic science. Differential Equations that faced in Physics and Engineering are equations as Laplace, Poisson, Helmholtz or wave, Schrödinger equations. Common features of this kind of equations are linear and partial differential equations from second degree. Series, differentiation of variables, Green Functions and integral transformations are often used in solving these equations. In some situations, analytical techniques are inadequate so numerical methods are used. In this study, in Part I, general and short information about Differential Equations are given, and in addition to general concepts about Partial Differential Equations, Laplace, Poisson, Diffusion, Helmholtz, Wave Equations are shortly introduced. In Part II, general coordinates, Orthogonal Coordinate Systems, Special orthogonal coordinates are introduced and Gradient, Divergence, Rotational and Laplacian expressions are given and basic features of Bessel and Legendre Functions are shortly introduced. In Part III, Poisson Equation is introduced and special solutions which are applied in Cylinder and Spherical Coordinates are looked over.
Description
Yüksek Lisans Tezi
Keywords
Matematik, Mathematics, Poisson Denklemleri, Poisson Equations
