Publication: Sonlu fourıer sıne, fourıer cosıne, laplace dönüşümlerinin genelleştirilmesi ve uygulamaları
Abstract
İntegral dönüşümleri sadece matematiğin konusu olmayıp temel bilimlerin diğer önemli disiplinleri olan fizik ve mühendislik alanlarında da sıkça kullanılmaktadır. Sınır değer, başlangıç değer problemleri, diferansiyel denklemler ve integral denklemlerinin çözümüne ulaşmak için integral dönüşümlerinden faydalanılmaktadır. Bu tezin 1. Bölümü’nde tez çalışmasında sıkça kullanılan Laplace, Sonlu Laplace, Sonlu Fourier Sinüs ve Kosinüs ve Mellin dönüşümlerinin tanımları, özellikleri, önemli örnekleriyle birlikte verilmiştir.2. Bölüm’de, çalışmanın konusunu oluşturan dönüşümlere kaynak olacak makaleler taranmış ve bu makalelerde yer alan Laplace ve L_2 dönüşümlerini içeren Parseval-Goldstein türü teoremler ve sonuçları incelenmiştir. Ayrıca Mellin dönüşümü ve genelleştirilmiş fonksiyonların dönüşümleri de incelenmiştir. 3. bölümde, ilk olarak sonlu Laplace dönüşümünü içeren Parseval tipi teoremler ve sonuçları incelenmiştir. Ardından Genelleştirilmiş Sonlu Laplace dönüşümü tanımlanıp, sonlu Laplace dönüşümü ile birlikte yer aldığı Parseval tipi teoremler ve sonuçları ortaya konmuştur. Ortaya çıkan bu bağıntılar sayesinde bazı fonksiyonların integral hesabını kolaylaştıran yöntemlerin uygulaması gösterilmiştir. Bu çalışmalar “Sonlu Laplace Dönüşümü ve Uygulamaları Üzerine Bağıntılar” isimli bildiri ile “ICOME-2017” sunulmuş ve “Genelleştirilmiş Sonlu Laplace Dönüşümü Üzerine Sonuçlar” adlı makale olarak “Communications in Mathematics and Applications-CMA” adlı uluslararası dergide yayınlanmıştır. Son olarak Genelleştirilmiş Mellin dönüşümü tanımlanıp bu dönüşümün sağladığı özelliklere ve elementer fonksiyonların dönüşümlerine yer verilmiştir. Genelleştirilmiş Mellin dönüşümü kullanılarak çözülen sınır değer problemleri de incelenmiştir. Yapılan bu çalışma “ICAA-2016” konferansında “Genelleştirilmiş Mellin Dönüşümü ve Uygulamaları Üzerine Sonuçlar” isimli bildiri ile sunulmuştur.4. Bölüm’de, tez çalışması süresince ulaşılan sonuçlar sıralanmıştır.
Integral transforms are not only included in mathematics, but also they are frequently used in different disciplines such as physics, engineering, etc. In order to solve boundary value and initial value problems, differential equations as well as integral equations, integral transfroms are generally applied to the problems. By thanks to defining new integral transforms, the existing integral transform tables can be extended.In the first chapter of the thesis, especially Laplace, Finite Laplace, Finite Fourier Sine and Cosine as well as Mellin transforms are mentioned by using their definitions, properties and important examples.In the second chapter, referenced articles for the study are investigated and Parseval-Goldstein type theorems and conclusions, which consists of Laplace transform are examined. Moreover, Mellin transform and the transform of generalized functions are also analyzed in this part. In the third chapter, firstly Parseval type theorems and conclusions that includes finite Laplace transform are examined. After that, generalized finite Laplace transform is defined and then Parseval type relations including both these transforms together are given. By thanks to these methods, integrals of some certain functions can be evaluated more easily. These findings are presented with the name of “Some Relations On The Finite Laplace Transform And Applications” in the “ICOME-2017”, and also the article with the name of “Some Results on Generalized Finite Laplace Transform” is published in “Communications in Mathematics and Applications-CMA”. Finally Generalized Mellin transforms are defined and the basic operational properties of the transforms are given. And also, boundary value problems that are solved by using Generalized Mellin transforms are shown. This study which is called “Some Results On The Generalized Mellin Transforms And Applications” is presented on “ICAA-2016”. In the fourth and last chapter, the whole results obtained in the thesis are shown.
Integral transforms are not only included in mathematics, but also they are frequently used in different disciplines such as physics, engineering, etc. In order to solve boundary value and initial value problems, differential equations as well as integral equations, integral transfroms are generally applied to the problems. By thanks to defining new integral transforms, the existing integral transform tables can be extended.In the first chapter of the thesis, especially Laplace, Finite Laplace, Finite Fourier Sine and Cosine as well as Mellin transforms are mentioned by using their definitions, properties and important examples.In the second chapter, referenced articles for the study are investigated and Parseval-Goldstein type theorems and conclusions, which consists of Laplace transform are examined. Moreover, Mellin transform and the transform of generalized functions are also analyzed in this part. In the third chapter, firstly Parseval type theorems and conclusions that includes finite Laplace transform are examined. After that, generalized finite Laplace transform is defined and then Parseval type relations including both these transforms together are given. By thanks to these methods, integrals of some certain functions can be evaluated more easily. These findings are presented with the name of “Some Relations On The Finite Laplace Transform And Applications” in the “ICOME-2017”, and also the article with the name of “Some Results on Generalized Finite Laplace Transform” is published in “Communications in Mathematics and Applications-CMA”. Finally Generalized Mellin transforms are defined and the basic operational properties of the transforms are given. And also, boundary value problems that are solved by using Generalized Mellin transforms are shown. This study which is called “Some Results On The Generalized Mellin Transforms And Applications” is presented on “ICAA-2016”. In the fourth and last chapter, the whole results obtained in the thesis are shown.