Properties and applications of some fractional derivatives and integrals obtained by global derivative

In the first part of the study, the purpose of the study, its importance and research hypotheses are detailed. The aim of the study is to compile and analyze current knowledge and applications on fractional derivatives and integrals. The importance of the study covers a wide range of fields in science and engineering, as well as its contribution to mathematical modeling and applications. The research hypotheses are aimed at testing the accuracy of innovative approaches provided by fractional calculations. In the second part of the study, the contributions made by important mathematicians involved in the definition and development of fractional derivatives are discussed. The mathematics of fractional derivatives, their properties, especially linearity, Leibniz's rule and derivatives of composite functions are emphasized. In addition, Laplace transforms and related properties are explained, Gamma, Beta and Mittag-Leffler functions are detailed. Application areas are examined in various fields such as physics, control theory, electrical engineering, biomedical engineering, finance and geophysics. In the third part of the study, the definition of fractional integrals, fractional derivatives and differentiation concepts are explained. The details of Riemann-Liouville fractional integration, its kernel structures and Caputo's definition are emphasized. In addition, various examples are presented regarding the applicability of fractional integrals. The fourth section of the study is related to the definitions of global derivatives and their sample applications. Initially developed to solve physical problems, this calculus eventually became a fundamental tool in mathematics, engineering, economics, and many other fields. While differential calculus examines rates of change, integral calculus deals with the summation of these changes. In summary, fractional derivatives and integrals have emerged as an important field of mathematics, developing since the 18th century with fundamental concepts such as the Riemann-Liouville and Caputo definitions. Fractional analysis serves as a bridge between theoretical mathematics and practical applications, offering innovative solutions across various disciplines