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).

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Finite element methods for Maxwell's equations.

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Purpose – The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The space and time‐fractional derivatives are described in the Caputo sense. The paper witnesses the extension of HAM for fractional KdV equations.Design/methodology/approach – This paper presents numerical solutions for the space‐ and time‐fractional KdV equation using HAM. The space and time‐fractional derivatives are described in the Caputo sense.Findings – In this paper, the application of homotopy analysis method was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The homotopy analysis method was clearly a very efficient and powerful technique in finding the solutions of the proposed equations.Originality/value – In this paper, the application of HAM was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation w...

Homotopy analysis method for space‐ and time‐fractional KdV equation

We extend He's variational iteration method (VIM) to find the approximate solutions for nonlinear differential-difference equation. Simple but typical examples are applied to illustrate the validity and great potential of the generalized variational iteration method in solving nonlinear differential-difference equation. The results reveal that the method is very effective and simple. We find the extended method for nonlinear differential-difference equation is of good accuracy.

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Applying He's Variational Iteration Method for Solving Differential-Difference Equation

A collocation approach for solving high-order linear Fredholm–Volterra integro-differential equations

Exploring collision-free path planning by using homotopy continuation methods

Although the significance of the vibration equation has recently attracted the researchers because of the experimental, empirical or numerical analyses, there is a lack of modern fractional analytic approaches. The main aim of this investigation is to analyze the dual treatment of vibration equation for large membrane through the modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional operators. In order to analyze the fractional model of vibration equation for large membrane, an analytic study is carried out by using Laplace and Hankel transforms satisfying the imposed conditions. A comparative analysis of vibration equation is addressed by newly presented non-integer-order derivatives with and without singular kernel, namely Michele Caputo–Mauro Fabrizio and Atangana–Baleanu fractional derivatives. The analytical solutions are obtained via both fractional approaches and then separately expressed in terms of newly presented Wiman special function $$\varvec{E}_{\eta ,\xi } \left( z \right)$$
 . The present fractional methods performed extremely well in terms of reliabilities and computational efficiencies. For the accuracy and validations of analytical treatment of fractional model of vibration equation for large membrane, a graphical comparison is made between Caputo–Fabrizio and Atangana–Baleanu fractional derivatives, which results in various similarities and differences on pertinent parameters involved in the vibration equation.

Ahmet Yildirim

Papers

Homotopy analysis method for space‐ and time‐fractional KdV equation

Applying He's Variational Iteration Method for Solving Differential-Difference Equation

A collocation approach for solving high-order linear Fredholm–Volterra integro-differential equations

Exploring collision-free path planning by using homotopy continuation methods

Fractional Treatment of Vibration Equation Through Modern Analogy of Fractional Differentiations Using Integral Transforms