Transition from the Wave Equation to Either the Heat or the Transport Equations through Fractional Differential Expressions

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

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