An application of Power Geometry to Finding Self-Similar Solutions

A. Bruno

Abstract


Here ideas and algorithms of Power Geometry are applied to a study of a partial differential equation. We put in correspondence to each differential monomial a point in JR:n, that is, its vector power exponent. To the differential equation there corresponds its support, that is, a set of the vector power exponents of its monomials. The affine hull of the support is called a super-support, and its dimension is called the dimension of the equation. If the dimension is smaller than n, then the equation is quasihomogeneous, and it has quasihomogeneous (self-similar) solutions. Such a solution is defined by a function of a smaller 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.

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