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).

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

Real and Complex Analysis

In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Exp-function method for nonlinear wave equations

The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

Partial Differential Equations- Basic Concepts- First-order Partial Differential Equations- One Dimensional Heat Flow- Higher Dimensional Heat Flow- One Dimensional Wave Equation- Higher Dimensional Wave Equation- Laplace's Equation- Nonlinear Partial Differential Equations- Linear and Nonlinear Physical Models- Numerical Applications and Pade Approximants- Solitons and Compactons- Solitray Waves Theory- Solitary Waves Theory- The Family of the KdV Equations- KdV and mKdV Equations of Higher-orders- Family of KdV-type Equations- Boussinesq, Klein-Gordon and Liouville Equations- Burgers, Fisher and Related Equations- Families of Camassa-Holm and Schrodinger Equations

Partial Differential Equations and Solitary Waves Theory

A sine-cosine method for handlingnonlinear wave equations

A reliable modification of Adomian decomposition method

This text gathers, revises and explains the newly developed Adomian decomposition method along with its modification and some traditional techniques.

Abdul-Majid Wazwaz

Papers

Partial Differential Equations and Solitary Waves Theory

A sine-cosine method for handlingnonlinear wave equations

A reliable modification of Adomian decomposition method

Partial differential equations : methods and applications

A new algorithm for calculating adomian polynomials for nonlinear operators