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Title:	Meijer's G-Functions In Univalent Function Theory And Its Applications
Authors:	Amir Pishkoo
Supervisor:	Maslina Darus, Prof. Dr.
Keywords:	Meijer's G-functions (MGFs)
Issue Date:	30-Aug-2013
Description:	This study focuses on applying the Meijer's G-functions (MGFs) in univalent function theory and its applications in physical problems. MGFs are defined as Mellin-Barnes contour integrals which have been in existence for over 60 years. The main objective of this thesis is to study analytic function theory and particularly the geometric properties of univalent functions through path integral representation. This is a new version of representation for univalent functions besides the usual series representation. First of all, some basic materials which are required in subsequent chapters, will be presented. These materials include the classical fractional calculus and the generalised fractional calculus operators, the partial differential equations (PDEs) in physics and the process of making Micro- and Nano- structures. The aim of using generalised fractional calculus operators is to obtain the image of MGFs which will be implemented by applying certain fractional integral, differential or differ-integral transformations. These transformations are the single and double Eldélyi-Kober operators, preserving the univalence of MGFs. All MGFs are analytic, but some of them are univalent. Further, the three basic univalent MGFs; will be considered, from which a number of univalent MGFs can be obtained. An interesting discovery is that the Koebe function is an MGF. In fact, due to the path representation for MGFs, images of some univalent MGFs which are identified by three operators will be discussed. The images are described by fractional differ-integral transformations of MGFs. The operators studied here include the Biernacki, Libera, and Ruscheweyh operators. To study these operators, it is enough to convert the form of some well-known operators into generalised fractional calculus operators (Eldélyi-Kober), which shall be discussed more in the thesis. MGFs are path integrals on the complex plane in which the position of poles and zeroes can be easily obtained. Moreover, the operation on how each of these operators translate poles (and zeroes), remove poles (and zeroes) and create poles (and zeroes) on the complex plane will be comprehensively shown. In addition, some new integral theorems in terms of MGFs are obtained. Finally, MGF will be shown as a tool in solving physical problems. These problems include solving the Schrödinger equation, Laplace equation, Diffusion equation, and the Reaction-Diffusion equation which all of them are PDEs. The last equation is related to the process of making Micro- and Nano- structures. The main attention here is to concentrate on G-function's ordinary linear differential equation (OLDE), and to equate G-function's OLDE and ordinary differential equations obtained from each of their problems. This shows that G-functions are the solutions of the PDEs. A modified separation of variables method will be implemented in the studies. The PDEs mentioned previously are solved in both Cartesian and cylindrical coordinates systems.,PHD
Pages:	111
Publisher:	UKM, Bangi
Appears in Collections:	Faculty of Science and Technology / Fakulti Sains dan Teknologi

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