Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion

The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0 < r < R) and a matrix (R < r < infinity) being in perfect thermal contact at r = R is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order 0 < alpha <= 2 and 0 < beta < 2, respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the Mittag-Leffler, Wright, and Mainardi functions.