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.