NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2017, 8 (3), P. 323–333
On a problem for the loaded degenerating mixed type equation involving integral-differential operators
B. I. Islomov -National University of Uzbekistan, Universitetskaya-4, VUZgorodog, Tashkent, 100174, Uzbekistan; islomovbozor@yandex.ru
O. Kh. Abdullayev – National University of Uzbekistan, Universitetskaya-4, VUZgorodog, Tashkent, 100174, Uzbekistan; obidjon.mth@gmail.com
N. K. Ochilova – Tashkent financial institute, st. Amir Timur. 57, Tashkent, 100000, Uzbekistan; nargiz.ochilova@gmail.com
This work aims to study the existence and uniqueness of a solution for a problem of the loaded degenerating mixed type equation. We consider the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals. The uniqueness of solution is proved by using the method of integral energy applying an extremum principle. Based on the statement of equivalence between “The existence and uniqueness of solution” and “Solvability of the respectively Fredholm type integral equations”, the existence of a solution was proved.
Keywords: loaded degenerating equation, parabolic-hyperbolic type, integral operators, Caputo fractional derivative, existence and uniqueness
of solution, integral equations.
PACS 2010 MSC-35M10
DOI 10.17586/2220-8054-2017-8-3-323-333