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

Abstract This paper introduces a graph-theoretic strategy for solving nonlinear distributed-order fractional differential equations (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 and 40% faster computation by transforming complex integration into summations. This novel weighting scheme offers a new way to represent distributed-order kernels. The Caputo derivative leverages our method's compatibility with initial conditions, while python-implemented clique computations validate scalability for massive graphs. Error behavior and convergence analysis confirm that the clique polynomial collocation method (CCM) remains stable compared to classical collocation techniques. Theoretical novelty lies in bridging clique polynomials with distributed-order calculus—a paradigm shift for multiscale modeling in viscoelasticity, signal processing, and anomalous diffusion. Future directions include orthogonalized bases and broader applications to complex solution profiles. Some of the highlights are as follows: (1) Developed a clique polynomial-based method for solving nonlinear distributed-order fractional differential equations. (2) Utilized the Caputo derivative to effectively manage initial conditions. (3) Demonstrated computational advantages by transforming integral equations into summation expressions. (4) Validated the proposed method through numerical comparison with existing schemes. (5) Implemented a python program for computing clique polynomials, facilitating further research in graph-theoretic applications. (6) Bridged graph theory and fractional calculus, fostering interdisciplinary applications in diffusion modeling, viscoelasticity, and signal analysis.