Document Type : Research Article
Authors
1
Department of Mathematics, Faculty of Science, Mansoura University, Dakahlia 35516, Egypt.
2
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.
10.22067/ijnao.2026.100588.1955
Abstract
This study proposes a numerical scheme based on Jacobi spectral collocation to approximate solutions for a coupled system of nonlinear Volterra integro-differential equations, which are utilized to model the interaction between two biological species. The novelty of this work lies in the unified treatment of multiple Jacobi families within a single framework, the rigorous residual-based error analysis tailored to this nonlinear coupled system with memory terms, and the consistent use of Gauss–Legendre quadrature to avoid incorrect weighting in the memory integrals. This unified framework correctly handles unweighted memory integrals, avoiding a common pitfall in spectral methods, and is supported by concrete actual error estimation to validate the theoretical bounds. The unknown functions are approximated by shifted Jacobi polynomials, and the governing equations are enforced at Jacobi–Gauss collocation nodes. The memory integrals are treated by a change of variable and evaluated using Gauss–Legendre quadrature, which avoids imposing an incorrect Jacobi weight on the original unweighted integral terms. The resulting nonlinear algebraic system is solved by a Picard-warm-started Newton method. An error analysis is developed based on Jacobi truncation estimates. In particular, bounds for the two residuals of the integro-differential system are derived, and it is shown that, under sufficient smoothness, the residuals decay algebraically with the collocation degree. An algorithmic description of the implementation is included. Four test examples are solved using several Jacobi families, including first-, second-, third-, and fourth-kind Chebyshev polynomials, Legendre polynomials, and a specific Jacobi polynomial family (e.g., parameters $a = 1, b = 1$). The numerical results are discussed and compared with the theoretical error bounds. The computed errors are significantly smaller than the theoretical estimates and confirm the high accuracy and efficiency of the proposed method.
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