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Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations

机译：分数阶弱奇异积分-微分方程的运算Jacobi Tau方法的数值分析

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摘要

The main concern of this paper is to develop and analyze an operational Tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. This strategy is an application of the matrix-vector-product approach in Tau formulation of the problem. We first study the regularity of the exact solution and show that some derivatives of the exact solution have a singularity at origin dependence on both order of fractional derivative and weakly singular kernel function which makes poor convergence results for the Tau discretization of the problem. In order to recover high-order of convergence, we propose a new variable transformation to regularize the given functions and then to approximate the solution via a satisfactory order of convergence using an operational Tau method. Convergence analysis of this novel method is presented and the expected spectral rate of convergence for the proposed method is established. Numerical results are given which confirm both the theoretical predictions obtained and efficiency of the proposed method.
机译：本文的主要关注点是开发和分析可操作的Tau方法，以便在将Jacobi多项式用作自然基函数时获得分数阶弱奇异积分-微分方程的数值解。该策略是矩阵向量乘积方法在问题的Tau公式化中的应用。我们首先研究精确解的正则性，并证明精确解的某些导数在原点上具有奇异性，既依赖于分数导数的阶数又具有弱奇异核函数，这对于该问题的Tau离散化而言，收敛效果较差。为了恢复高阶收敛性，我们提出了一种新的变量变换来对给定函数进行正则化，然后使用可操作的Tau方法通过令人满意的收敛阶数来逼近解。提出了这种新方法的收敛性分析，并建立了该方法的预期频谱收敛速率。数值结果证实了所获得的理论预测和所提方法的效率。

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