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This multi-volume handbook is the most up-to-date and comprehensive
reference work in the field of fractional calculus and its numerous
applications. This second volume collects authoritative chapters
covering the mathematical theory of fractional calculus, including
ordinary and partial differential equations of fractional order,
inverse problems, and evolution equations.
This multi-volume handbook is the most up-to-date and comprehensive
reference work in the field of fractional calculus and its numerous
applications. This first volume collects authoritative chapters
covering the mathematical theory of fractional calculus, including
fractional-order operators, integral transforms and equations,
special functions, calculus of variations, and probabilistic and
other aspects.
The aim of this book is to develop a new approach which we called
the hyper geometric one to the theory of various integral
transforms, convolutions, and their applications to solutions of
integro-differential equations, operational calculus, and
evaluation of integrals. We hope that this simple approach, which
will be explained below, allows students, post graduates in
mathematics, physicists and technicians, and serious mathematicians
and researchers to find in this book new interesting results in the
theory of integral transforms, special functions, and convolutions.
The idea of this approach can be found in various papers of many
authors, but systematic discussion and development is realized in
this book for the first time. Let us explain briefly the basic
points of this approach. As it is known, in the theory of special
functions and its applications, the hypergeometric functions play
the main role. Besides known elementary functions, this class
includes the Gauss's, Bessel's, Kummer's, functions et c. In
general case, the hypergeometric functions are defined as a linear
combinations of the Mellin-Barnes integrals. These ques tions are
extensively discussed in Chapter 1. Moreover, the Mellin-Barnes
type integrals can be understood as an inversion Mellin transform
from the quotient of products of Euler's gamma-functions. Thus we
are led to the general construc tions like the Meijer's G-function
and the Fox's H-function."
The aim of this book is to develop a new approach which we called
the hyper geometric one to the theory of various integral
transforms, convolutions, and their applications to solutions of
integro-differential equations, operational calculus, and
evaluation of integrals. We hope that this simple approach, which
will be explained below, allows students, post graduates in
mathematics, physicists and technicians, and serious mathematicians
and researchers to find in this book new interesting results in the
theory of integral transforms, special functions, and convolutions.
The idea of this approach can be found in various papers of many
authors, but systematic discussion and development is realized in
this book for the first time. Let us explain briefly the basic
points of this approach. As it is known, in the theory of special
functions and its applications, the hypergeometric functions play
the main role. Besides known elementary functions, this class
includes the Gauss's, Bessel's, Kummer's, functions et c. In
general case, the hypergeometric functions are defined as a linear
combinations of the Mellin-Barnes integrals. These ques tions are
extensively discussed in Chapter 1. Moreover, the Mellin-Barnes
type integrals can be understood as an inversion Mellin transform
from the quotient of products of Euler's gamma-functions. Thus we
are led to the general construc tions like the Meijer's G-function
and the Fox's H-function."
Dieses Lehrbuch beinhaltet eine Einfuhrung in die vielfaltige und
faszinierende Welt der mathematischen Modellierung und eignet sich
ideal fur alle, die auf diesem Gebiet noch keine grossen
Erfahrungen sammeln konnten. Insbesondere wurde dabei an die
Studierenden im Bachelor-Studium gedacht, die beim Durcharbeiten
des Buchs das noetige Rustzeug bekommen, um sich selbststandig an
die mathematische Modellierung von realen Anwendungen zu wagen und
die in der Spezialliteratur beschriebenen Modelle kreativ
anzupassen und einzusetzen. Wahrend der erste Teil des Buchs sich
der Methodik des Modellierens und den Aktivitaten im
Modellierungszyklus widmet, halt der zweite Teil einen
Werkzeugkasten fur die einzelnen Modellierungsschritte parat. Die
dritte Saule des Buchs bilden einige Fallstudien, die nach der
vorgestellten Methodik und mit den Techniken aus dem Werkzeugkasten
bearbeitet werden. Das Modellieren beschrankt sich dabei nicht -
und das ist das Besondere an diesem Buch - auf die Modellentwurfe,
sondern beinhaltet auch ihre Analyse, numerische Behandlung,
Implementierung von Algorithmen, Rechnungen, Visualisierung und
Analyse der Ergebnisse. Fur die Implementierung der Berechnungen
und die Visualisierung der Ergebnisse wird dabei das Softwarepaket
MATLAB (R) eingesetzt, alle Beispiele sind jedoch ebenso in Octave
lauffahig. Die vorliegende zweite Auflage wurde in einigen Teilen
wesentlich erweitert, um die Bedeutung der mathematischen
Modellierung in aktuellen Anwendungen noch deutlicher zu machen.
Insbesondere werden jetzt auch wichtige Modellansatze aus dem
Bereich des maschinellen Lernens vorgestellt und eine neue
Fallstudie uber Computertomographie behandelt die Modellierung von
inversen schlecht gestellten Problemen.
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