The subject of fractional calculus and its applications (that is, convolution-type pseudo-differential operators including integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, mainly due to its applications in diverse fields of science and engineering. These operators have been used to model problems with anomalous dynamics, however, they also are an effective tool as filters and controllers, and they can be applied to write complicated functions in terms of fractional integrals or derivatives of elementary functions, and so on. 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 was written with a trade-off in mind 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. Numerical code is also provided.

Description-Table Of Contents

1. Preliminaries. 1.1. Fourier and Laplace transforms. 1.2. Special functions and their properties. 1.3. Fractional operators -- 2. A survey of numerical methods for the solution of ordinary and partial fractional differential equations. 2.1. Approximation of fractional operators. 2.2. Direct methods for fractional ODEs. 2.3. Indirect methods for fractional ODEs. 2.4. Linear multistep methods. 2.5. Other methods. 2.6. Methods for terminal value problems. 2.7. Methods for multi-term FDE and multi-order FDS. 2.8. Extension to fractional PDEs -- 3. Efficient numerical methods. 3.1. Methods for ordinary differential equations. 3.2. Methods for partial differential equations -- 4. Generalized Stirling numbers and applications. 4.1. Introduction. 4.2. Stirling functions s([symbol],k), [symbol]. 4.3. General Stirling functions s([symbol], [symbol]) with complex arguments. 4.4. Stirling functions of the second kind S([symbol],k). 4.5. Generalized Stirling functions S(n,[symbol]), [symbol]. 4.6. Generalized Stirling functions S([symbol],[symbol]), [symbol]. 4.7. Connections between s([symbol],[symbol]) and S([symbol],k) -- 5. Fractional variational principles. 5.1. Fractional Euler-Lagrange equations. 5.2. Fractional Hamiltonian dynamics -- 6. CTRW and fractional diffusion models. 6.1. Introduction. 6.2. The definition of continuous-time random walks. 6.3. Fractional diffusion and limit theorems -- 7. Applications of CTRW to finance and economics. 7.1. Introduction. 7.2. Models of price fluctuations in financial markets. 7.3. Simulation. 7.4. Option pricing. 7.5. Other applications.