On two backward problems with Dzherbashian-Nersesian operator

We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired results here are more general than those obtained by making use of conventional fractional operators such as fractional Riemann-Liouville derivative (FRLD), fractional Caputo derivative (FCD) and fractional Hilfer derivative (FHD).