Finding Self-Similar Solutions by Means of Power Geometry

Here ideas and algorithms of Power Geometry are applied for a study of one partial differential equation. To each differential monomial we put in correspondence a point in <b>R</b>ⁿ that is it's vector power exponent. To the differential equation there corresponds it's support that is a set of the vector power exponents of it's monomials. The affine hull of the support is called a supersupport and it's dimension is called the dimension of the equation. If the dimension is less than n then the equation is quasihomogeneous and has the quasihomogeneous (self-similar) solutions. Such a solution is defined by a function of less number of independent variables. Here it is shown how to calculate all kinds of self-similar solutions of the equation by means of the methods of linear algebra using the support of the equation. Equations of the combustion process without a source and with a source are considered as examples. Generalizations for a system of equations are formulated as well.