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

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.

New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors. The methods of integral transforms via local fractional calculus have been used to solve various local fractional ordinary and local fractional partial differential equations and also to figure out the presence of the fractal phenomenon. The book presents the basics of the local fractional derivative operators and investigates some new results in the area of local integral transforms.Provides applications of local fractional Fourier SeriesDiscusses definitions for local fractional Laplace transformsExplains local fractional Laplace transforms coupled with analytical methods

Local Fractional Integral Transforms and Their Applications

This paper points out a novel local fractional variational iteration method for processing the local fractional heat conduction equation arising in fractal heat transfer.

/pdf/fractal-heat-conduction-problem-solved-by-local-fractional-5fjj1j2p89.pdf

Fractal heat conduction problem solved by local fractional variation iteration method

In this paper, a family of local fractional two-dimensional Burgers-type equations (2DBEs) is investigated. The local fractional Riccati differential equation method is proposed here for the first time. The travelling wave transformation of the non-differentiable type is presented. The non-differentiable exact travelling wave solutions for the problems are obtained. The present methodology is shown to provide a useful approach to solve the local fractional nonlinear partial differential equations (LFNPDEs) in mathematical physics.

Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations

In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.

A new fractional operator of variable order: Application in the description of anomalous diffusion

In this article we propose a new fractional derivative without singular
 kernel. We consider the potential application for modeling the steady
 heat-conduction problem. The analytical solution of the fractional-order heat
 flow is also obtained by means of the Laplace transform.

/pdf/a-new-fractional-derivative-without-singular-kernel-9et0fa2lhw.pdf

Xiao-Jun Yang

Papers

Local Fractional Integral Transforms and Their Applications

Fractal heat conduction problem solved by local fractional variation iteration method

Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations

A new fractional operator of variable order: Application in the description of anomalous diffusion

A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL Application to the Modelling of the Steady Heat Flow