Fractional Heat Conduction in Infinite One-Dimensional Composite Medium

The problem of fractional heat conduction in a composite medium consisting of two semi-infinite regions being in perfect thermal contact is considered. The heat conduction in each region is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α and β, respectively. The solution is obtained using the Laplace transform with respect to time and is expressed in terms of the Mittag–Leffler function and Mainardi function. Numerical results are illustrated graphically.