On some direct and inverse problems for an integro-differential equation

The direct and two inverse problems defined for an integro-differential equation on a bounded domain have been considered. The spectral problem of the integro-differential equation constitutes the Legendre differential equation in space variable. Finding a space-dependent source term whenever the data at some time, say T, as over-specified condition, constitutes the Ist inverse problem. The 2nd inverse problem consists of recovering a time-dependent coefficient in the source term from an integral type over-specified condition. The Fourier approach is used to have the analytical series solution of the problems. The existence and uniqueness results for the direct and inverse problems under certain regularity conditions on the data are presented.