Solutions of some differential equations in the distributions space

The concept of solution for differential equations with generalized coefficients is not determined in classical distributions theory because the product of arbitrary distributions can not be defined.

A number of approaches for solving the problem of distributions multiplication were suggested by different mathematicians such as V. Ivanov, J.-F. Colombeau, Yu. Egorov, E. E. Rosinger and others. The general idea of these approaches is to consider new objects which preserve some properties of distributions and form an algebra at the same time. Such objects are called new generalized functions or mnemofunctions.

The simplest linear differential equation of the form

$u'+qu=0$

where $q$ is a distribution is under consideration in this paper. We analyze the solvability conditions for such equation when $q=b\delta$, $q=b\delta'$ and $q=P(\frac{1}{x})+c\delta$. The theory of mnemofunctions provides a great tool for solving such equation in the space of distributions. According to mnemofunctions theory approach we change the generalized coefficient $q$ by its approximation $q_\varepsilon(x)$ by smooth functions depending on small parameter $\varepsilon$. Then we consider the family of equations of the form $u'_\varepsilon(x)+q_\varepsilon(x)u_\varepsilon(x)=0$ and investigate the behaviour of it’s solutions $u_\varepsilon(x)$ as $\varepsilon$ tends to zero. If $u_\varepsilon(x)$ have a limit in the space of distributions we declare it as a solution of initial equation.

Examples considered in this paper show that there are no common statements about solvability for differential equations with generalized coefficient and each case has it’s own conditions of solvability which depend on singularity type of coefficient.

Keywords: differential equation with generalized coefficient, mnemofunction, solvability condition, analytical representation of distribution.