Abstract
In this paper, we consider new classes of differential equations of fractional order related to Hadamard derivatives. These equations generalize the well-known heat conduction equation for the fractional exponent of the time derivative. For the equations under consideration, mixed problems with Dirichlet and Neumann boundary conditions are studied. The Fourier method is used to solve these problems. Two auxiliary problems are obtained for ordinary differential equations of fractional order and ordinary differential equations with involution. The spectral properties of ordinary differential operators with involution are studied. For the main problems, theorems on the existence and uniqueness of solutions are proved.