Generalized integral transforms and applications to some problems involving fractional derivatives

In this thesis, generalized forms of integral transforms, which have important application areas in various branches of science, are investigated. Firstly, generalized Fourier, Fourier sine, Fourier cosine, Laplace and Mellin transforms and their inverse transforms are defined, their basic properties are examined and their applications to elementary functions are illustrated with examples. Then, solutions of various ordinary, partial and fractional differential equations are obtained using generalized integral transforms. Finally, tables of generalized integral transforms are given and some graphs are presented to visualise how these transforms behave under different parameters.