This paper provides an efficient recursive approach of the spectral Tau method, to approximate the solution of a system of generalized Abel-Volterra integral equations. In this regard, we first investigate the existence, uniqueness as well as smoothness of the solutions under various assumptions on the given data. Next, from a numerical perspective, we express approximated solution as a linear combination of suitable canonical polynomials which are constructed by an easy-to-use recursive formula. Mostly, the unknown parameters are calculated by solving low-dimensional algebraic systems independent of the degree of approximation which prevents high computational costs. Obviously, due to the singular behavior of the exact solution, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regard, we develop new fractional-order canonical polynomials using Müntz-Legendre polynomials which have the same asymptotic behavior with the solution of the underlying problem. The convergence analysis is discussed, and the familiar spectral accuracy is achieved in \(L^{\infty }\)-norm. Finally, the reliability of the method is evaluated using various examples and applying it in solving a class of fractional differential equations systems.

本文提供了一种谱 Tau 方法的有效递归方法，以逼近广义 Abel-Volterra 积分方程组的解。在这方面，我们首先在给定数据的各种假设下研究解的存在性、唯一性和平滑性。接下来，从数值的角度来看，我们将近似解表示为合适的规范多项式的线性组合，这些规范多项式由易于使用的递归公式构造而成。大多数情况下，未知参数是通过求解与近似程度无关的低维代数系统来计算的，从而避免了高计算成本。显然，由于精确解的奇异行为，使用经典多项式构造规范多项式会导致结果精度低。在这方面，我们使用 Müntz-Legendre 多项式开发新的分数阶规范多项式，这些多项式具有与潜在问题的解相同的渐近行为。讨论了收敛性分析，并在\(L^{\infty }\) -范数。最后，通过各种实例评估了该方法的可靠性，并将其应用于求解一类分数阶微分方程组。