广义Abel-Volterra积分方程组的一种新的递归谱Tau方法
A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations
摘要：
本文给出了谱Tau方法逼近广义Abel-Volterra积分方程组解的一种有效的递推方法。在这方面，我们首先研究的存在性，唯一性以及光滑性的假设下，给定的数据。接下来，从数值的角度来看，我们表示为一个线性组合的合适的规范多项式，这是由一个易于使用的递归公式构造的近似解。大多数情况下，通过求解与逼近阶无关的低维代数系统来计算未知参数，从而避免了高的计算成本。显然，由于精确解的奇异性，用经典多项式构造规范多项式，会导致精度不高。在这方面，我们利用Müntz-勒让德多项式构造了一个新的分数阶标准多项式，它与原问题的解具有相同的渐近性质.讨论了收敛性分析，并在$L^{\infty}$范数下达到了常见的谱精度。最后，该方法的可靠性进行评估，使用各种问题。
英文摘要：
This paper provides an efficient recursive approach of the spectral Tau method to approximate the solution of system of generalized Abel-Volterra integral equations. In this regards, we first investigate the existence, uniqueness as well as smoothness of the solutions under assumption 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 a low dimensional algebraic systems independent of degree of approximation which prevent from high computational costs. Obviously, due to singular behavior of the exact solutions, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regards, we develop a new fractional order canonical polynomials using Müntz-Legendre polynomials which have a same asymptotic behavior with the solution of 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 problems.
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