A piecewise cardinal method for a class of third-kind fractional quadratic integro-differential equations involving Caputo tempered derivative

M.H. Heydari, T. Baghban, M. Bayram and M.A. Zaky

Chaos, Solitons & Fractals, 2026, vol. 202, issue P1

Abstract: In this paper, we study a family of third-kind fractional quadratic integro-differential equations including the Caputo tempered derivative, a formulation that has received limited attention in the literature. To address the analytical and computational challenges posed by these equations, a robust numerical scheme is developed using the piecewise Chebyshev cardinal functions. To provide efficient computation, we derive corresponding matrices for the classical and tempered fractional integrals. The designed algorithm transforms the original problem into a nonlinear algebraic system, enabling accurate numerical approximation of the solution. We rigorously establish the existence of a unique solution. Convergence is verified analytically and supported by comprehensive numerical experiments, which demonstrate excellent accuracy, strong agreement with theoretical predictions, and exponential convergence with respect to the polynomial degree.

Keywords: Quadratic integro-differential equations; Tempered fractional derivative; Tempered fractional integral; Piecewise Chebyshev cardinal functions; Convergence analysis; Operational matrices (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925014791

DOI: 10.1016/j.chaos.2025.117466

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