Fractional heat conduction in a semi-infinite composite body

A medium consisting of a region 0 < x < L and a region L < x < infinity is considered. Heat conduction in one region is described by the equation with the Caputo time-fractional derivative of order ?, whereas heat conduction in another region is described by the equation with the time derivative of the order ?. The problem is solved under conditions of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The solution valid for small values of time is expressed in terms of the Mittag-Leffler function and the Mainardi function. Several particular cases are considered and illustrated graphically.