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Category: General Natural Science(6/8)

E-books | Detail

Title: Fractional Calculus: A New Approach from an Operator-Based Formulation

Author: Takahiro INOUE

Category: General Natural Science
Number of pages: 162
Size: A5

e-Book compatible devices: pc
Viewer: ActiBook
Language: English



Book Summary

This book presents a new approach to fractional calculus. The theory constructed here is based on a function sequence called the lambda-sequence in this book. This function sequence is generated from a repeated operation of a linear operator satisfying Leibniz's rule to a function which belongs to a sufficiently broad function space. By using the generating function of the lambda-sequence, the integral representation of the lambda-sequence is derived. The differentiation and integration to arbitrary complex order of such a function are defined by extending this integral representation by analytic continuation. In this book, many theorems, corollaries, lemmas, and remarks are presented to reveal the interesting properties(including linearity and continuity) of fractional differentiation and integration, and also to show the relationship to the classical fractional calculus. Various special functions of arbitrary order are derived inclusively as special cases of the lambda-sequence for different pairs of an operator and an operand function. Applications of the theory are given to an order-reduction of ordinary differential equations of a certain type, solutions of basic differ-integral equations, infinite continued square root, diffusion equations, Schroedinger's equation, the WKB approximation, electric circuits, and electromagnetic wave propagation problems.

Author Profile

Takahiro INOUE is a professor emeritus, Kumamoto University, a Doctor of Engineering, and a Fellow of IEICE(The Institute of Electronics, Information and Communication Engineers).
Up to now, he has published 175 journal papers and 108 proceedings papers of international conferences. As for books, he is a writer of Chapters 21 to 24 of “Electronic Circuit Handbook" published in 2006 by Asakura Publishing Co., Japan, a main author of the text book titled “Analog Electronic Circuits Learning by Examples" published in 2009 by Morikita Publishing Co., Japan, and an author of “A Unified Theory of Generalized Differentiation and Integration" published in 2016 by BookWay, Japan.

From the Author

This book is an extended version of my previous books published under the title "A Unified Theory of Generalized Differentiation and Integration." The content of the previous books is extended and enhanced especially about the spaces on which the proposed theory can be constructed. The subject of this book is the fractional calculus that treats differentiation and integration to arbitrary complex order. This book presents a new approach where the fractional calculus is constructed from an operator-based formulation. In this theory, linear operators which satisfy the Leibniz rule are extended to operators to arbitrary complex order. Many theorems, corollaries, and lemmas are presented in this book to reveal the properties (including linearity, continuity, etc.) of fractional differ-integration to arbitrary complex order. Various special functions of arbitrary order are derived as applications to show the usefulness of the proposed theory. In this book, readers can find other interesting applications to miscellaneous problems in mathematics and physics.