TY - JOUR
AU - A. M. Samoilenko
PY - 2019/04/25
Y2 - 2020/12/03
TI - Singular integral equation equivalent in the space of smooth functions
to an ordinary differential equation, method of successive approximations
for the construction of its smooth solutions and its nonsmooth solutions
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 71
IS - 4
SE - Research articles
DO -
UR - http://umj.imath.kiev.ua/index.php/umj/article/view/1456
AB - We propose a singular integral equation whose definition is extended to a singular point by additional conditions. In thespace of smooth functions, this equation becomes equivalent, by the indicated extended definition, to an ordinary differentialequation, whereas in the space of piecewise discontinuous functions, it becomes equivalent to an impulsive differentialequation. For smooth solutions of the singular equation, we substantiate the method of successive approximations. For theordinary differential equation, this method turns into a new algorithm for the construction of successive approximations. Forthe investigated equation, we specify a solution of new type, which is equivalent, for the impulsive differential equation,to a solution with discontinuity of the second kind (a “solution with needle”). We propose an algorithmic formula for thegeneral solution of the initial-value problem for the impulsive differential equation.
ER -