Variable order differential equations with piecewise constant order-function and diffusion with changing modes

TL;DR

This paper investigates variable order diffusion equations with mode changes, proving existence and uniqueness of solutions, and analyzing memory effects relevant to heterogeneous media and biological processes.

Contribution

It introduces a new framework for variable order PDEs with piecewise constant order functions, establishing fundamental theorems and classifying memory effects.

Findings

Existence and uniqueness of solutions for the Cauchy problem.

Classification of short and long-range memory effects.

Application to diffusion in heterogeneous media and cell biology.

Abstract

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.