Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica

Evolution of the Central Indian tectonic zone: Geochemical and isotope-geochronological data

In the framework of the Russian-Indian joint research projects geochemical and geochronological study of granitoid rocks across the Central Indian Tectonic Zone has been carried out. Geochronological data suggest that the Central Indian Tectonic Zone is composed primarily of Proterozoic rocks, formed as a result of several stages of granitoid magmatism: at 2.43, 2.34-2.31, 1.73-1.72 and 1.53-1.51 Ga. Metamorphic transformations reflected by Sm–Nd and Rb–Sr systems of rocks and minerals occurred 1.37-1.1 Ga ago that allows comparing the final processes in the Central Indian Tectonic Zone with the Grenville orogeny and it can be used for the reconstruction of Rodini