Rapidly convergent infinite state representation for fractional calculus and applications to fractional rheology

Fractional calculus, as a core tool for modeling complex systems, plays a critical role in numerous fields. However, its prominent nonlocal characteristics introduce a series of challenging problems for numerical analysis. The infinite-state approach has demonstrated great potential in the fast computation of fractional calculus. Nevertheless, the weak singularity and weak algebraic decay properties of infinite-state representations lead to generally low convergence orders of the infinite-state approach, which even fail completely in some cases. To address these issues, this paper proposes two strategies: I) Innovatively developing a local hybrid interpolation quadrature method to achieve high-precision computation of weakly singular integrals; II) Adopting an interval segmentation strategy to overcome the weak algebraic decay of infinite-state representations and realize their exponential decay rate. By combining these strategies, a high-precision infinite-state approach with linearly increasing computational complexity O(n)) is intended to be constructed. Moreover, the proposed algorithm fundamentally eliminates the need for solving high-dimensional ordinary differential equation systems and circumvents stiff problem challenges that plague conventional infinite-state approach. The convergence order of the method is rigorously established through theoretical analysis and subsequently validated via numerical experiments on benchmark problems. In addition, the method is applied to the fast solution of fractional viscoelastic constitutive equations, achieving high-precision fitting to experimental data.