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Correction of BDFk for fractional Feynman-Kac equation with L\'{e}vy flight

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arxiv 2007.01563 v1 pith:X3GHEJ3R submitted 2020-07-03 math.NA cs.NA

Correction of BDFk for fractional Feynman-Kac equation with L\'{e}vy flight

classification math.NA cs.NA
keywords correctionconvergenceequationfractionalanalysisfeynman-kacflightmethod
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In this work, we present the correction formulas of the $k$-step BDF convolution quadrature at the starting $k-1$ steps for the fractional Feynman-Kac equation with L\'{e}vy flight. The desired $k$th-order convergence rate can be achieved with nonsmooth data. Based on the idea of [{\sc Jin, Li, and Zhou}, SIAM J. Sci. Comput., 39 (2017), A3129--A3152], we provide a detailed convergence analysis for the correction BDF$k$ scheme. The numerical experiments with spectral method are given to illustrate the effectiveness of the presented method. To the best of our knowledge, this is the first proof of the convergence analysis and numerical verified the sapce fractional evolution equation with correction BDF$k$.

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