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Zeitschrift für Analysis und ihre Anwendungen


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Volume 28, Issue 4, 2009, pp. 431–450
DOI: 10.4171/ZAA/1392

Published online: 2009-12-23

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

Sabir Umarov and Stanly Steinberg[1]

(1) University of New Mexico, Albuquerque, USA

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

Keywords: Variable order differential equations, short memory, long memory, diffusion with changing modes, Cauchy problem, Mittag–Leffler function

Umarov Sabir, Steinberg Stanly: Variable Order Differential Equations with Piecewise Constant Order-Function and Diffusion with Changing Modes. Z. Anal. Anwend. 28 (2009), 431-450. doi: 10.4171/ZAA/1392