Dalga dönüşümü altında indirgenebilen lineer olmayan bazı fiziksel denklemlerin yarı analitik çözümleri üzerine
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Dosyalar
Tarih
2015
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Ege Üniversitesi, Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Kısmi diferansiyel denklemler teorisi uygulamalı matematik, fizik, mühendislik ve diğer bilim alanlarındaki uygulamaları nedeniyle doğa olaylarını açıklamakta kullanılabilecek en önemli kavramlardan biridir. Lineer olmayan kısmi diferansiyel denklemlerin çözümlerini klasik yöntemler ile elde etmek her zaman mümkün değildir. Bu nedenle bu tür denklemler için çok sayıda analitik ve nümerik yöntem geliştirilmekte ve uygulanmaktadır. Bu tez çalışmasında, dalga dönüşümü altında adi diferansiyel denkleme indirgenebilen lineer olmayan bazı fiziksel denklemlerin hareketli dalga çözümleri trial (deneme) denklem yöntemi, modifiye edilmiş trial denklem yöntemi ve genişletilmiş trial denklem yöntemi kullanılarak incelenmiştir. Bu yöntemler matematiksel fiziğin bilinen denklemlerine uygulanarak bu denklemlerin hareketli dalga çözümleri elde edilmiş ve dalga tipleri belirlenmiştir. Trial denklem yöntemi ile tamsayı mertebeden lineer olmayan fiziksel denklemler için başarılı sonuçlar elde edilmiştir. Bu nedenle yöntem, uygulamalı matematikte önemli bir yere sahip olan kesirli (fractional) mertebeden kısmi diferansiyel denklemlere ve denklem sistemine de uygulanmış ve bazı yeni hareketli dalga çözümleri elde edilmiştir. Bununla birlikte, lineer olmayan fiziksel oluşum denklemlerinin tam çözümlerine ulaşmak için etkin yaklaşımlardan biri olan trial denklem yöntemi ile elde edilen çözümlerin fiziksel davranışları değerlendirilmiştir.
The theory of partial differential equations is one of the most important concepts of explaining natural phenomena because of its applications in applied mathematics, physics, engineering and other field of science. It is not always possible to obtain the solutions of nonlinear partial differential equations by classical methods. Therefore, numerous analytical and numerical methods have been developed and applied for these equations. In this thesis, we investigate the traveling wave solutions of some nonlinear physical equations reduced to the ordinary differential equations under the wave transformation by using the trial equation method, modified trial equation method and extended trial equation method. We have applied these methods to some particular equations of mathematical physics and obtained traveling wave solutions of these equations and determined the types of the waves. By means of the trial equation method, solutions of the nonlinear integer order physical equations have been successfully obtained. Therefore, modified trial equation method has been applied to the fractional partial differential equations and system of differential equation that have an important role in applied mathematics and the traveling wave solutions of these equations have been obtained. Additionally, the solutions obtained by the trial equation method which is an efficient approach to get exact solutions of the phsyical evolution equations have been evaluated.
The theory of partial differential equations is one of the most important concepts of explaining natural phenomena because of its applications in applied mathematics, physics, engineering and other field of science. It is not always possible to obtain the solutions of nonlinear partial differential equations by classical methods. Therefore, numerous analytical and numerical methods have been developed and applied for these equations. In this thesis, we investigate the traveling wave solutions of some nonlinear physical equations reduced to the ordinary differential equations under the wave transformation by using the trial equation method, modified trial equation method and extended trial equation method. We have applied these methods to some particular equations of mathematical physics and obtained traveling wave solutions of these equations and determined the types of the waves. By means of the trial equation method, solutions of the nonlinear integer order physical equations have been successfully obtained. Therefore, modified trial equation method has been applied to the fractional partial differential equations and system of differential equation that have an important role in applied mathematics and the traveling wave solutions of these equations have been obtained. Additionally, the solutions obtained by the trial equation method which is an efficient approach to get exact solutions of the phsyical evolution equations have been evaluated.
Açıklama
Anahtar Kelimeler
Lineer Olmayan Fiziksel Oluşum Denklemleri, Trial Denklem Yöntemi, Hareketli Dalga Çözümleri, Kesirli Kısmi Diferansiyel Denklemler
