Implicit operators for networked mechanical and thermal systems with integer-order components

TL;DR

This paper introduces implicit operators that generalize fractional derivatives, describing the complex dynamic behavior of infinite networked systems composed of integer-order components, which are crucial for modeling such systems.

Contribution

It demonstrates that certain network configurations lead to implicit integro-differential operators governing system dynamics, extending the concept of fractional derivatives.

Findings

Equivalent operators can be implicit and expressed as operator equations.

Special cases yield fractional-order derivatives.

Implicit operators are essential for analyzing complex networked systems.

Abstract

Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an implicit integro-differential operator even though the individual components themselves satisfy equations that use explicit operators of integer order. The networks considered here are infinite trees and ladders, and each is composed only of two types of integer-order components with potential-driven flows that are repeated ad infinitum. In special cases the equivalent operator for the system is a fractional-order derivative, but in general it is implicit and can only be expressed as a solution of an operator equation. These implicit operators, which are a generalization of fractional-order derivatives, play an important role in the analysis and modeling of complex systems.