Regularization of generalized functions in convolution operator algebra

A problem of Cauchy integral calculation with the aid of n-fold multiple integral by integer-order operators is investigated. This n-fold multiple differentiation results in n-order uniform and non-uniform systems of ordinary differential equations. The solution of the first system is equal to the convolution of the second system’s solution with arbitrary function forming heterogeneity of the first system. This is the necessary condition of existence of the given problem’s solution. The convolution is a sufficient condition for establishing of the fraction order operator algebra that is equivalent to convolution operator algebra. Besides that, existence of the ordinary differential equation defining stability of time, is important, too. Subalgebra of fractional order less than 1 defines convolution operators as parametric generalized functions, their asymptotic values and unity operators. Both algebras determine identity of ordinary differential equations after substituting n-fold multiple operator integral in these equations. Vladimirov’s regularization according to Horsthemke-Saichev theorem corresponds to Bogolubov’s regularization for superfluidity. Time stability of superfluidity is described by the Newton equation. Parametric generalized functions and their symmetry are stable.