Variable Order Differential Equations with Piecewise Constant Order-Function and Diffusion with Changing Modes

In this paper diffusion processes with changing modes are studied involvingthe variable order partial differential equations. We prove the existence anduniqueness theorem of a solution of the Cauchy problem for fractional variableorder (with respect to the time derivative) pseudo-differential equations.Depending on the parameters of variable order derivatives short or long rangememories may appear when diffusion modes change. These memory effects areclassified and studied in detail. Processes that have distinctive regimes ofdifferent types of diffusion depending on time are ubiquitous in the nature.Examples include diffusion in a heterogeneous media and protein movement incell biology.