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We study a model that intermediates among the wave, heat, and transportequations. The approach considers the propagation of initial disturbances in aone-dimensional medium that can vibrate. The medium is nonlinear in such a formthat nonlocal differential expressions are required to describe thetime-evolution of solutions. Non-locality is modeled with a space-timefractional differential equation of order $1 \leq \alpha \leq 2$ in time, andorder $1\leq \beta \leq 2$ in space. We adopt the notion of Caputo fortime-derivative and the Riesz pseudo-differential operator forspace-derivative. The corresponding Cauchy problem is solved for zero initialvelocity and initial disturbance represented by either the Dirac delta or theGaussian distributions. Well known results for the partial differentialequations of wave propagation, diffusion and (modified) transport processes arerecovered as particular cases. In addition, regular solutions are found for thepartial differential equation that arises from $\alpha=2$ and $\beta=1$. Unlikethe above cases, the latter equation permits the presence of nodes in itssolutions.

Transition from the wave equation to either the heat or the transport equations through fractional differential expressions