Spectral Collocation Method for Solving Nonlinear Riesz Distributed-Order Fractional Differential Equations

In this article, we present an efficient and highly accurate numerical scheme that achieves exponential convergence for solving nonlinear Riesz distributed-order fractional differential equations (RDFDEs) in one- and two-dimensional initial–boundary value problems. The proposed method is based on a two-stage collocation framework. In the first stage, spatial discretization is performed using the shifted Legendre–Gauss–Lobatto (SL-G-L) collocation method, where the approximate solutions and spatial derivatives are expressed in terms of shifted Legendre polynomial expansions. This reduces the original problem to a system of fractional differential equations (FDEs) for the expansion coefficients. Then, the temporal discretization is achieved in the second stage via Romanovski–Gauss–Radau collocation approach, which converts the system into a system of algebraic equations that can be solved efficiently. The method is applied to one- and two-dimensional nonlinear RDFDEs, and numerical experiments confirm its spectral accuracy, computational efficiency, and reliability. Existing numerical approaches to distributed-order fractional models often suffer from poor accuracy, instability in nonlinear settings, and high computational costs. By combining the efficiency of Legendre polynomials for bounded spatial domains with the stability of Romanovski polynomials for temporal discretization, the proposed two-stage framework effectively overcomes these limitations and achieves superior accuracy and stability.