The generalized Duhamel principle for fully coupled systems of fractional order

TL;DR

This paper develops a fractional Duhamel's principle for coupled systems of fractional differential equations, enabling reduction of inhomogeneous problems to homogeneous ones while capturing unique fractional effects.

Contribution

It introduces a systematic fractional Duhamel's principle for coupled systems, extending classical methods to fractional derivatives and revealing new fractional phenomena.

Findings

Provides a fractional Duhamel's principle for coupled systems

Recovers classical Duhamel's principle in the integer limit

Highlights effects of coupled fractional impulses

Abstract

Duhamel's principle reduces the Cauchy problem for an inhomogeneous partial differential equation to the corresponding homogeneous problem. In the fractional-order setting, the classical principle does not apply directly because fractional derivatives are nonlocal in time. Over the past two decades, several fractional analogues of Duhamel's principle have been developed to address this issue. In this paper, we establish a fractional version of Duhamel's principle for fully coupled systems of fractional differential-operator equations. The result provides a systematic reduction of inhomogeneous fractional problems to homogeneous ones while preserving the structure of the classical method. In the limit of integer-order derivatives, the formulation recovers the classical Duhamel principle and also reveals effects specific to coupled fractional systems, including those produced by coupled fractional impulses.