Lineer Olmayan Dalga Denkleminin Lie Grupları Analizi

Year 2018, Volume: 30 Issue: 2, 133 - 144, 30.06.2018

Saadet Özer

https://doi.org/10.7240/marufbd.405473

Abstract

Keyfi fonksiyonlar ya da parametreler içeren denklem kümesini,
denklem ailesi olarak adlandırsak, ailenin üyeleri arasında geçişi mümkün kılan
dönüşümler eşdeğerlik dönüşümleri olarak adlandırılır. Eşdeğerlik grupları,
verilen bir diferansiyel denklem ailesini değişmez bırakan dönüşüm grupları olarak
tanımlanır.  Bu nedenle diferansiyel
denklem ailelerinin eşdeğerlik grupları, aynı aileye ait, farklı denklemler
arası mümkün ilişkilerin varlığını inceler ve bu ilişkileri ortaya çıkarır. Böylelikle,
karmaşık bir denklemin kesin çözümü ya da başka deyişle davranışı, aynı aileden
daha basit bir denklem aracılığıyla belirlenebilir.  Bu çalışmada, lineer olmayan tek boyutlu dalga
denkleminin eşdeğerlik grupları, Lie gruplarının bir uygulaması çerçevesinde
incelenmiş ve bazı örnekler ile lineer ve lineer olmayan denklemler arası geçişler
sağlanmış, bazı karmaşık lineer olmayan denklemlerin çözümü belirlenmiştir. Bu
tipte dönüşümlerin varlığı için, sonsuz küçük üreteçler üzerine gelen şartlar elde
edilmiştir. Ayrıca, bu şekilde nokta dönüşümleri aracılığı ile, lineer dalga
denklemine dönüştürülebilen, lineer olmayan denklemlerin asgari fonksiyonel
bağlılıkları da belirlenmiştir.

Keywords

Eşdeğerlik Grupları, Lie Grupları, Dalga denklemi

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