In this paper, we consider the multi-order fractional differential equation and recast it into an integral equation. Based on the integral equation, we develop an hp-version Legendre spectral collocation method and the integral terms with the weakly singular kernels are calculated precisely according to the properties of Legendre and Jacobi polynomials. The hp-version error bounds under the L²-norm and the \(L^{\infty }\)-norm are derived rigorously. Numerical experiments are included to illustrate the efficiency of the proposed method and the rationality of the theoretical results.

中文翻译：

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高阶分数阶微分方程的hp版本Legendre光谱配置方法

在本文中，我们考虑了多阶分数阶微分方程，并将其重铸为一个积分方程。基于积分方程，我们开发了hp版本的Legendre光谱配置方法，并根据Legendre和Jacobi多项式的性质精确计算了具有弱奇异核的积分项。的马力下-version误差界限大号²范数和\（L ^ {\ infty} \）范数是严格的。数值实验表明了该方法的有效性和理论结果的合理性。