Group analysis of solutions of 2-dimensional differential equations


Год издания: 2009

Аннотация: It is well known [4, 9] that if a system of differential equations admits the Lie group of point transformations (point symmetry), then any solution of the system is transformed to a solution of this system. This property permits the construction of new solutions without integrating the given system of partial differential equationПоказать полностьюs (PDEs), by means of group transformations alone under known solutions. This is an effective method if we have a sufficiently rich group of point transformations. By applying point transformations to exact solution, a family of so-called S-solutions can be constructed, i.e., obtained by means of symmetries. This family of S-solutions is dependent on the group parameter. If this parameter is equal to zero, then we have an initial solution. This procedure is called the production [9] or repro-duction of solutions [4]. Moreover, it is easy to show that under a group transformation characteristic curves of the system of PDEs of the hyperbolic type are transformed to the characteristics curves. The evolution of characteristic curves permits to find out the boundary condi-tions for new S-solutions. In the present chapter authors will show some applications of this procedure for the system of the theory of ideal plane plasticity, developing results obtained in [12]. In particular, we shall use an infinite subgroup of the group of symmetries for deformation of characteristics curves of the considered hyperbolic system of PDEs to construct a new analytical solutions. From the system of PDEs an automorphic system will be deduced, which permits find out some relations between different solutions by means of group transformations.

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Журнал: Lie Groups: New Research

Номера страниц: 123-138

Издатель: Nova Science Publishers, Inc.


  • Senashov S.I. (Univ. de Guadalajara,Univ. Centre of Sciences and Ingeneira)
  • Yakhno A. (Univ. de Guadalajara,Univ. Centre of Sciences and Ingeneira)

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