Study of Quaternion integral transforms and their applications

Abstract

Present thesis entitled Study of quaternion integral transforms and their applications newlineembodies the results of research carried out by the author. In chapter 1, mathematical definitions and properties of integral transforms and quaternions are introduced. Some basic facts of spaces are discussed with respect to integral transforms. newlineIn chapter 2, the background of integral transform is presented. The historical development of the subject is discussed. newlineIn chapter 3, the one-dimensional quaternion Mellin transform is defined. Properties like linearity, scaling, shifting, differentiation, and convolution property are demonstrated. Parseval-type property and inversion theorem are also established. To support the study, applications from mathematical physics are presented. In chapter 4, two-dimensional quaternionic fractional Mellin transform(2D-QFrMT) for a particular order is introduced. Its inversion formula is derived. Properties like linearity, Parseval s formula, and product theorem are obtained without any additional conditions. Applications are given to support the study. Generalized 2D-QFrMT is established. The testing function space and some properties are studied. The analyticity, boundedness, and inversion formula are newlineanalyzed. Operational properties and Mellin-type convolution properties are established. The applications related to mathematical physics are given in the concluding section. newlineIn chapter 5, the quaternion Hankel transform is developed. Basic operational properties and the inversion formula of quaternion Hankel transform are derived. Parseval s relation is also established. The application of the same to Cauchy s problem is demonstrated. In the concluding section, the generalized quaternion Hankel transform is defined. newline