Preliminary symmetry analysis of nonlinear heat equations

Abstract

This thesis presents a preliminary study of symmetry analysis of a class of nonlinear heat equations of the formH : ut = f (x,ux)uxx+g(x,ux). We first review the notions related to symmetry analysis of differential equations and then followed by examples that are relevant in the appli cations. The preliminary study of the class of nonlinear heat equations includes the computation of the equivalence group, the equivalence al gebra and the analysis of the determining equations of equation from the class under study. Using the direct method, all equivalence transformations connecting two equations from the class H were obtained. These transformations preserve the equations of the class and they are projectable to the space of variables and arbitrary elements. The class under study is not normalized. The infinitesimal counterparts of the one parameter group for this class was computed and the commuta tion relations of the vector fields spanning the maximal Lie invariance Lie algebra was presented. The investigation and analysis of the de termining equations for Lie symmetries of an equation from the class H lead to the expression of the maximal kernel invariance Lie algebra for equations from the class H. It is shown there that this Lie algebra is finite-dimensional, whereas the dimension of the linear span of the maximal Lie invariance algebra is infinite.