A new collection of real world applications of fractional calculus in science and engineering

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

The role of fractional calculus in modeling biological phenomena: A review

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.

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A Review of Definitions for Fractional Derivatives and Integral

Abstract Recently, the conformable derivative and its properties have been introduced. In this work we have investigated in more detail some new properties of this derivative and we have proved some useful related theorems. Also, some new definitions have been introduced.

New properties of conformable derivative

Survey of Numerical Methods to Solve Ordinary and Partial Fractional Differential Equations Specific and Efficient Methods to Solve Ordinary and Partial Fractional Differential Equations Fractional Variational Principles Continuous-Time Random Walks (CTRWs) Applications to Finance and Economics Generalized Stirling Numbers of First and Second Kind in the Framework of Fractional Calculus.

Fractional Calculus: Models and Numerical Methods

This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. Moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models.

All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications. This book will keep in mind the trade-off between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice.

The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.

Contents:

Preface
Preface to the Second Edition
Preliminaries
A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations
Specific Efficient Methods for the Solution of Ordinary and Partial Fractional Differential Equations
Generalized Stirling Numbers and Applications
Fractional Variational Principles
Continuous-Time Random Walks and Fractional Diffusion Models
Applications of Continuous-Time Random Walks
Selected Applications of Fractional Calculus Models in the Life Sciences
Appendix: Source Codes
Bibliography
Index

Readership: Undergraduate and graduate students in mathematics as well as researchers in applied mathematics and applied sciences whose interests involve differential equations, fractional calculus, special functions, variational methods, numerical methods and their applications in physics, mechanics, engineering, finance, economics and other applied fields including life science and social science.

Kai Diethelm

Papers

Fractional Calculus: Models and Numerical Methods

Fractional calculus: models and numerical methods (2nd edition)

Continuous-Time Random Walks and Fractional Diffusion Models

Applications of Continuous-Time Random Walks to Finance and Economics

A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations