On a singular time-fractional order wave equation with Bessel operator and Caputo derivative

This paper deals with the study of the well-posedness of a mixed fractional problem for the wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. We prove the existence and uniqueness of solution as well as its dependence on the given data. Our results develop and show the efficiency and effectiveness of the functional analysis method when we deal with fractional partial differential equations instead of the nonfractional equations which have been extensively studied by many authors during the last three decades.