The Connection of the Continuity, the Diffusion and the Wave Equation by the Fractional Derivative

In the present work, we discuss the fractional continuity, diffusion and wave equation and their interconnection. The motivation for this kind of approach is the possibility to derive a solution in analytical representation, which solves the three equations for the simple case where the velocity field in the continuity equation is a constant and the diffusion parameter is isotropic and constant. For the unified operator for fractional differentiation and integration we employ the Riemann-Liouville definition and apply it to the three aforementioned equations. Further we show by a taxonomy scheme, that the three equations may be understood as similar equations and the only difference is the specific choice of the fractional orders for the space and time description of the differential equation. The solution is derived upon separating the space and time part in close analogy to a spectral method and solve each fractional differential equation by the use of three parametric Mittag-Leffler type functions, which upon introducing a suitable truncation in the series represent approximate solutions in analytical representation for the generic fractional derivative differential equation in 1⊕1 space-time dimensions.