Lineer Olmayan Dalga Denkleminin Lie Grupları Analizi

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.

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International Journal of Advances in Engineering and Pure Sciences-Cover
  • Yayın Aralığı: Yılda 4 Sayı
  • Başlangıç: 2008
  • Yayıncı: Marmara Üniversitesi