Volume 65, pp. 372-386, 2026.

An operational Jacobi collocation framework for solving linear systems of distributed-order fractional differential equations

Z. Saki and P. Mokhtary

Abstract

This paper introduces a novel collocation-based framework for the analysis and numerical solution of linear systems of distributed-order fractional differential equations (DOFDEs), formulated in the Caputo sense. Under suitable conditions we establish the analytic form of the exact solution and investigate its smoothness properties. The proposed numerical methodology is structured into two main steps. First, a high-order Gauss–Legendre quadrature rule is employed to accurately discretize the distributed-order integral, thereby reducing the original system to an equivalent multi-term fractional differential system. Second, an operational spectral collocation scheme is developed, based on a special family of generalized Jacobi polynomials as basis functions and Gauss–Legendre nodes as collocation points to further transform the problem into a well-structured block system of algebraic equations. A comprehensive convergence analysis is presented, demonstrating the spectral accuracy of the method. The efficiency of the proposed approach is validated through several illustrative examples, confirming its potential for solving complex fractional systems with distributed-order dynamics.

Full Text (PDF) [331 KB], BibTeX , DOI: 10.1553/etna_vol65s372

Key words

linear system of distributed-order fractional differential equations (DOFDEs), collocation method, generalized Jacobi polynomials, convergence analysis

AMS subject classifications

26A33, 34A08, 39A33, 65D05, 37N30, 93C27