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A new recursive formulation of the Tau method for solving linear Abel–Volterra integral equations and its application to fractional differential equations
Calcolo ( IF 1.7 ) Pub Date : 2019-11-09 , DOI: 10.1007/s10092-019-0347-y
Y. Talaei , S. Shahmorad , P. Mokhtary

In this paper, the recursive approach of the Tau method is developed for numerical solution of Abel–Volterra type integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Müntz–Legendre polynomials as basis functions which have remarkable approximation to functions with singular behavior at origin and express Tau approximation of the exact solution based on a sequence of basis canonical polynomials that is generated by a simple recursive formula. We also provide a convergence analysis for the proposed method and obtain an exponential rate of convergence regardless of singularity behavior of the exact solution. Some examples are given to demonstrate the effectiveness of the proposed method. The results are compared with those obtained by existing numerical methods, thereby confirming the superiority of our scheme. The paper is closed by providing application of this method to approximate solution of a linear fractional integro-differential equation.

中文翻译:

Tau方法求解线性Abel–Volterra积分方程的新递推公式及其在分数阶微分方程中的应用

本文针对Abel–Volterra型积分方程的数值解,开发了Tau方法的递归方法。由于这些方程解的奇异行为,现有的频谱方法存在精度低的问题。为了克服这个缺点,我们使用Müntz–Legendre多项式作为基函数,该函数显着逼近原点具有奇异行为的函数,并基于由简单递归公式生成的一系列基础规范多项式表达精确解的Tau近似。我们还为所提出的方法提供了收敛分析,并获得了指数收敛速率,而与精确解的奇异行为无关。给出了一些例子来证明所提方法的有效性。将结果与通过现有数值方法获得的结果进行比较,从而证实了我们方案的优越性。通过提供该方法的应用来关闭线性分数阶积分微分方程的近似解。
更新日期:2019-11-09
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