Abstract:
This article is devoted to the study of the Cauchy problem for a second-order differential equation with a non-invertible operator at the highest derivative, as a result of which, the solution exists not for every initial value. This operator is Fredholm with a zero index. The cascade splitting method is used to solve the problem. This method splits the equation and conditions into the corresponding equation and conditions in subspaces of smaller dimensions. The case of invertibility of some operator constructed by using the operator coefficients of the equation is investigated. The conditions under which a solution to the problem exists and is unique are determined; it is found in the analytical form.
Keywords:Cauchy problem, degenerate second order differential equation, Fredholm operator, cascade splitting
Fulltext:
PDF file (569 kB)