The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. & Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489.) is combined with a formula for the convolution of two Legendre series (Hale, N. & Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 36, A1207–A1220.) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.

中文翻译：

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卷积型线性Fredholm和Volterra积分-微分方程的超球谱方法

常微分方程基于勒让德-超球谱方法（奥尔弗，S。＆汤森，A.（2013）的快速和良好的空调谱方法。SIAM启。，55，462-489。）与式为组合两个勒让德勒级数的卷积（Hale，N.＆Townsend，A.（2014a）一个勒让德勒级数卷积的算法。SIAMJ. Sci。Comput 。，36，A1207–A1220。）产生一种求解新技术卷积型核的线性Fredholm和Volterra积分微分方程。当核函数和系数函数足够平滑时，则该方法在频谱上是准确的，并且可以用线性复杂度求解所得的几乎带状线性系统。