Abstract
The questions of unique solvability of direct and inverse boundary value problems for a homogeneous integro-differential equation containing a quadrat of Hilfer fractional analogue for the Barenblatt-Zheltov-Kochina operator, two spatial variables and degenerate kernel are studied. The redefinition function is given in boundary condition and the additional condition has a nonlinear form. The Fourier series method based on the separation of variables is used. Sufficient coefficient conditions for unique classical solvability of the direct and inverse boundary value problems are established.