The generalized Duhamel principle for fully coupled systems of fractional order
Sabir Umarov
TL;DR
The paper develops a fractional Duhamel principle for fully coupled systems of distributed-order differential-operator equations of the form $A \circ \mathfrak{D}^{\mathcal{A}} U(t) = F U(t) + H(t)$ with $U(0)=\Phi$, where $A=(a_{ij})$, $\mathfrak{D}^{\mathcal{A}}=(D^{\alpha_{ij}})$ and $0<\alpha_{ij}\le 1$. It shows that the nonhomogeneous problem can be reduced to a homogeneous one via a memory integral $U(t)=\int_0^t V(t,\tau)\,d\tau$, where $V$ solves the homogeneous system with a coupled initial impulse $V(t,\tau^+)= I \circ \mathfrak{D}_{+ ,\tau}^{I-\Lambda}(H(\tau)-G(\tau))$, $\Lambda=I\circ\mathcal{A}$, and $G$ solves an auxiliary Volterra-type system. The existence proof relies on a Volterra-type operator $\mathcal{K}$ with a convergent Neumann series for $(I+\mathcal{K})^{-1}$, and the framework covers both Caputo and Riemann–Liouville derivatives, revealing how coupling induces collective memory via fractional impulses. In the limit $\alpha_{ij}\to 1$, the classical Duhamel principle is recovered, while the results illuminate memory entanglement in fully coupled fractional dynamics and the phenomenon of memory decoupling when off-diagonal coupling vanishes. These contributions provide a rigorous, implementable method to reduce inhomogeneous fractional problems to homogeneous ones while preserving the Duhamel philosophy in nonlocal-in-time settings.
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.