2 research outputs found
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 () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with 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 . 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
Nonlinear Hyperbolic Equations and Linear Heat Conduction with Memory
The model of a rigid heat conductor with memory is considered. Specifically, in the one-dimensional case, a connection, via Cole-Hopf Transformation, between the linear integro-differential evolution equation which describes heat conduction with memory and a nonlinear partial integro-differential equation of hyperbolic type is established. Notably, when the heat conductor is homogeneous, as well as when the homogeneity hypothesis is removed, the differential operator of the transformed nonlinear partial differential equation is of hyperbolic type