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Table Of Integrals Series And Products

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).

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

Stability of Solitary Waves

In this paper, a new generalized exponential rational function method is employed to extract new solitary wave solutions for the Zakharov–Kuznetsov equation (ZKE). The ZKE exhibits the behavior of weakly nonlinear ion-acoustic waves in incorporated hot isothermal electrons and cold ions in the presence of a uniform magnetic field. Furthermore, the stability for the governing equations is investigated via the aspect of linear stability analysis. Numerical simulations are made to shed light on the characteristics of the obtained solutions.

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The new exact solitary wave solutions and stability analysis for the ( 2 + 1 ) $(2+1)$ -dimensional Zakharov–Kuznetsov equation

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y) = 0 with respect to the standard Lagrangian L = xny′2/2 — xn ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αyr , where r ≠ 0,1. The case r = 5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r = 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.

https://www.tandfonline.com/doi/pdf/10.2991/jnmp.2008.15.2.3

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation

In this paper we discuss the integrability of the generalized Lane–Emden equations of the first and second kinds. We carry out their Noether symmetry classification. Various cases for the arbitrary functions in the equations are obtained for which the equations have Noether point symmetries. First integrals of such cases are obtained and also reduction to quadrature of the corresponding Lane–Emden equations are presented. New cases are found.

Exact solutions of the generalized Lane–Emden equations of the first and second kind

Lagrangian approach to a generalized coupled Lane–Emden system: Symmetries and first integrals

We classify a generalized coupled Lane–Emden system t u + n u + t f ( t ) v q = 0 , t v + n v + t f ( t ) u p = 0 with respect to the standard first-order Lagrangian according to the Noether point symmetries it admits. First integrals of the various cases which admit Noether point symmetries are then obtained for the generalized coupled Lane–Emden system under consideration.

Ben Muatjetjeja

Papers

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation

Exact solutions of the generalized Lane–Emden equations of the first and second kind

Lagrangian approach to a generalized coupled Lane–Emden system: Symmetries and first integrals

First integrals for a generalized coupled Lane–Emden system