Publication: Kesirli hesaplar ve uygulamaları
Abstract
KESİRLİ HESAPLAR VE UYGULAMALARI İntegral ve türev kavramları klasik analizin en temel kavramlarıdır. Herhangi bir fonksiyonun tamsayı mertebeden türev ve integralleri kolaylıkla hesaplanabilmektedir. Fakat kesirli mertebeden türev ve integral hesaplamaları klasik anlamda türev ve integral almak kadar kolay değildir. Bu tezin ikinci bölümünde kesirli analiz incelenmiştir. Kesirli analizin tarihçesi, kesirli integral tanımı, kesirli integral için örnekler, kuvvet kuralı, Laplace dönüşümü ve Leibniz formülü verilmiştir. Ayrıca fonksiyon sınıfları ele alınmıştır Kesirli türev tanımı verilmiş ve kuvvet kuralı uygulanmıştır. Kesirli türev için Laplace dönüşümü ve Leibniz formülü incelenmiştir. Üçüncü bölümde Bessel ve hipergeometrik fonksiyonlar sınıfına ait bazı özel fonksiyonların kesirli türev ve integralleri incelenmiştir. Dördüncü bölümde Weyl kesirli integrali ve türevi üzerinde durulmuş ve örnekler verilmiştir. Son olarak beşinci bölümde Krätzel fonksiyonu üzerinde çalışmalar yapılmıştır. Birinci kısımda fonksiyona ait çeşitli özellikler ispatlandıktan sonra ikinci kısımda Krätzel fonksiyonunun Weyl kesirli türev ve integrali incelenmiştir. Buna bağlı olarak çeşitli bağıntılar elde edilmiştir. Üçüncü kısımda genelleştirilmiş Krätzel fonksiyonunun Weyl kesirli türev ve integrali incelenmiştir. İkinci tür genelleştirilmiş fonksiyon tanımlanarak yeni bir indirgeme bağıntısı elde edilmiştir. Son olarak Krätzel fonksiyonunun monotonluğu ve konveksliği incelenmiş ve genelleştirilmiş fonksiyonlar için yeni sonuçlar elde edilmiştir. Böylece, uygulamalı matematikte karşılaşılan bir çok problemin çözümü için kesirli analiz ve uygulamalarının geniş yer tuttuğu açıkça görülmüştür
FRACTIONAL CALCULUS AND APPLICATIONS Integral and derivative concepts are the most basic concepts of the classical analysis. Integer order derivative and integrals of any function can be calculated easily. However, the fractional-order derivative and integral is not as easy as taking derivative and integral in the classical sense. In the second part of this thesis, fractional analysis is examined. Fractional analysis history, the fractional integral definition, examples for fractional integral, the law of exponents, the Leibniz formula and the Laplace transform are given. Furthermore, the function classes are focused on. The definition of fractional derivative is given and the law of exponents is applied. Leibniz formula and Laplace transform are evaluated for fractional derivative. In the third part, fractional derivatives and integrals of special functions belonging to the class of hypergeometric and Bessel functions are issued. The fourth part focuses on the Weyl fractional integral and derivative. Moreover, examples are given. Last but not least, Krätzel function studies are done on the fifth chapter. After proving various features of Krätzel function in the first section, Weyl fractional derivative and integral of the Krätzel function is issued in the second section. Accordingly, various relations are obtained. In the third section, generalized Krätzel function’ s Weyl fractional derivative and integral are examined. A new recurrence formula is obtained by defining second type of generalized function. Finally, the monotonicity and convexity of Krätzel function is issued and new applications for generalized funcitons are developed. Thus, for solutions of many problems encountered in applied mathematics, it is clearly observed that fractional analysis and applications play a significant role.
FRACTIONAL CALCULUS AND APPLICATIONS Integral and derivative concepts are the most basic concepts of the classical analysis. Integer order derivative and integrals of any function can be calculated easily. However, the fractional-order derivative and integral is not as easy as taking derivative and integral in the classical sense. In the second part of this thesis, fractional analysis is examined. Fractional analysis history, the fractional integral definition, examples for fractional integral, the law of exponents, the Leibniz formula and the Laplace transform are given. Furthermore, the function classes are focused on. The definition of fractional derivative is given and the law of exponents is applied. Leibniz formula and Laplace transform are evaluated for fractional derivative. In the third part, fractional derivatives and integrals of special functions belonging to the class of hypergeometric and Bessel functions are issued. The fourth part focuses on the Weyl fractional integral and derivative. Moreover, examples are given. Last but not least, Krätzel function studies are done on the fifth chapter. After proving various features of Krätzel function in the first section, Weyl fractional derivative and integral of the Krätzel function is issued in the second section. Accordingly, various relations are obtained. In the third section, generalized Krätzel function’ s Weyl fractional derivative and integral are examined. A new recurrence formula is obtained by defining second type of generalized function. Finally, the monotonicity and convexity of Krätzel function is issued and new applications for generalized funcitons are developed. Thus, for solutions of many problems encountered in applied mathematics, it is clearly observed that fractional analysis and applications play a significant role.