On Nonlinear Distributed Order Fractional Differential Equations for Star-Sun Graphs Clique Polynomials

This paper introduces a graph-theoretic strategy for solving Nonlinear Distributed-Order Fractional Differential Equa- tions (NDOFDEs) by utilizing the Caputo fractional derivative and the clique polynomial of Star-Sun graphs. By encoding weight functions via clique polynomials, we achieve 85–92% error reduction versus Legendre-wavelet methods [1, 2] and 40% faster computation by transforming complex integration into summations. The Caputo derivative leverages our method's compatibility with initial conditions, while Python-implemented clique computations validate scalability for massive graphs. Theoretical novelty lies in bridging clique polynomials with distributed-order calculus—a paradigm shift for multi-scale modeling in viscoelasticity, signal processing, and anomalous diffusion.