There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

A new definition of fractional derivative

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index

/pdf/theory-and-applications-of-fractional-differential-equations-4sh8ohra9h.pdf

Theory and Applications of Fractional Differential Equations

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

Theory and applications of fractional differential equations. north-holland mathematics studies

Thc paper deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. A survey of the methods and results in the theory of such ordinary fractional differential equations is given. In particular, the method based on the reduction of the Cauchy-type problem for the fractional differential equations to the Volterra integral equations is discussed, and the Laplace transform, operational calculas compositional methods for the solution of linear differential equations of fractional order are presented. Problems and new trends of research are discussed.

Juan J. Trujillo

Papers

Theory and Applications of Fractional Differential Equations

Fractional Calculus: Models and Numerical Methods

Theory and applications of fractional differential equations. north-holland mathematics studies

Differential equations of fractional order:methods results and problem —I

Existence of mild solution for evolution equation with Hilfer fractional derivative