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Correction of High-Order \(L_k\) Approximation for Subdiffusion

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Abstract

The subdiffusion equations with a Caputo fractional derivative of order \(\alpha \in (0,1)\) arise in a wide variety of practical problems, which describe the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree k (\(k\le 6\)) convolution quadrature, called \(L_k\) approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of \(L_k\) approximation by the polylogarithm function or Bose-Einstein integral. To construct a \(\tau _8\) approximation of Bose-Einstein integral, the desired \((k+1-\alpha )\)th-order convergence rate can be proved for the correction \(L_k\) scheme with nonsmooth data, which is higher than kth-order BDFk method in [**, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

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Funding

This work was supported by NSFC 11601206, 11901266 and Natural Science Foundation of Gansu Province (No. 21JR7RA253).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jiankang Shi & Minghua Chen

  2. Department of Physical, Mathematical and Engineering, Faculty of Science and Engineering, University of Chester, Parkgate Road, Chester, Ch14BJ, UK

    Yubin Yan

  3. School of Sciences, Lanzhou University of Technology, Lanzhou, 730000, People’s Republic of China

    Jianxiong Cao

Authors
  1. Jiankang Shi
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  2. Minghua Chen
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  3. Yubin Yan
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Correspondence to Minghua Chen.

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Appendix

Appendix

The coefficients \(\omega ^{(k)}_{j}\) of \(L_k\) approximation in (2.4) are given explicitly by the following

  • \(L_1\) approximation

$$\begin{aligned}&\omega ^{(1)}_{0}=\frac{1}{\varGamma (2-\alpha )}, \quad \omega ^{(1)}_{j}=\frac{(j+1)^{1-\alpha }-2j^{1-\alpha }+(j-1)^{1-\alpha }}{\varGamma (2-\alpha )},~~j\ge 1. \end{aligned}$$

  • \(L_2\) approximation

    $$\begin{aligned} \omega ^{(2)}_{0}=&\frac{1}{\varGamma (3-\alpha )} +\frac{1}{2} \frac{1}{\varGamma (2-\alpha )}, \qquad \omega ^{(2)}_{1}= \frac{\left( 2^{2-\alpha }-3\right) }{\varGamma (3-\alpha )} +\frac{1}{2}\frac{\left( 2^{1-\alpha }-3\right) }{\varGamma (2-\alpha )},\\ \omega ^{(2)}_{j} =&\frac{\left( (j+1)^{2-\alpha }-3j^{2-\alpha }+3(j-1)^{2-\alpha }-(j-2)^{2-\alpha }\right) }{\varGamma (3-\alpha )} \\&+\frac{1}{2}\frac{\left( (j+1)^{1-\alpha }-3j^{1-\alpha }+3(j-1)^{1-\alpha }-(j-2)^{1-\alpha }\right) }{\varGamma (2-\alpha )},~~j\ge 2. \end{aligned}$$

  • \(L_3\) approximation

    $$\begin{aligned} \omega ^{(3)}_{0}=&\frac{1}{\varGamma (4-\alpha )} + \frac{1}{\varGamma (3-\alpha )}+\frac{1}{3}\frac{1}{\varGamma (2-\alpha )}, \\ \omega ^{(3)}_{1}=&\frac{ 2^{3-\alpha }-4 }{\varGamma (4-\alpha )} + \frac{ 2^{2-\alpha }-4 }{\varGamma (3-\alpha )} +\frac{1}{3}\frac{ 2^{1-\alpha }-4 }{\varGamma (2-\alpha )},\\ \omega ^{(3)}_{2}=&\frac{ 3^{3-\alpha }-4\times 2^{3-\alpha }+6 }{\varGamma (4-\alpha )} + \frac{ 3^{2-\alpha }-4\times 2^{2-\alpha }+6 }{\varGamma (3-\alpha )} +\frac{1}{3}\frac{ 3^{1-\alpha }-4\times 2^{1-\alpha }+6 }{\varGamma (2-\alpha )},\\ \omega ^{(3)}_{j} =&\frac{ (j+1)^{3-\alpha }-4j^{4-\alpha }+6(j-1)^{3-\alpha }-4(j-2)^{3-\alpha } +(j-3)^{3-\alpha } }{\varGamma (4-\alpha )} \\&+ \frac{ (j+1)^{2-\alpha }-4j^{2-\alpha }+6(j-1)^{2-\alpha }-4(j-2)^{2-\alpha } +(j-3)^{2-\alpha } }{\varGamma (3-\alpha )} \\&+\frac{1}{3}\frac{ (j+1)^{1-\alpha }-4j^{1-\alpha }+6(j-1)^{1-\alpha }-4(j-2)^{1-\alpha } +(j-3)^{1-\alpha } }{\varGamma (2-\alpha )},~j\ge 3. \end{aligned}$$

  • \(L_4\) approximation

$$\begin{aligned} \omega ^{(4)}_{0}=&\frac{1}{\varGamma (5-\alpha )} +\frac{3}{2} \frac{1}{\varGamma (4-\alpha )}+\frac{11}{12}\frac{1}{\varGamma (3-\alpha )} +\frac{1}{4}\frac{1}{\varGamma (2-\alpha )},\\ \omega ^{(4)}_{1}=&\frac{ 2^{4-\alpha }-5 }{\varGamma (5-\alpha )} +\frac{3}{2}\frac{ 2^{3-\alpha }-5 }{\varGamma (4-\alpha )} +\frac{11}{12}\frac{ 2^{2-\alpha }-5 }{\varGamma (3-\alpha )}+\frac{1}{4}\frac{ 2^{1-\alpha }-5 }{\varGamma (2-\alpha )},\\ \omega ^{(4)}_{2} =&\frac{ 3^{4-\alpha }-5\times 2^{4-\alpha }+10 }{\varGamma (5-\alpha )} +\frac{3}{2}\frac{ 3^{3-\alpha }-5\times 2^{3-\alpha }+10 }{\varGamma (4-\alpha )} +\frac{11}{12}\frac{ 3^{2-\alpha }-5\times 2^{2-\alpha }+10 }{\varGamma (3-\alpha )} \\&+\frac{1}{4}\frac{ 3^{1-\alpha }-5\times 2^{1-\alpha }+10 }{\varGamma (2-\alpha )},\\ \omega ^{(4)}_{3} =&\frac{ 4^{4-\alpha }-5\times 3^{4-\alpha }+10\times 2^{4-\alpha } -10 }{\varGamma (5-\alpha )} + \frac{3}{2}\frac{ 4^{3-\alpha }-5\times 3^{3-\alpha }+10\times 2^{3-\alpha } -10 }{\varGamma (4-\alpha )} \\&+ \frac{11}{12}\frac{ 4^{2-\alpha }-5\times 3^{2-\alpha }+10\times 2^{2-\alpha }-10 }{\varGamma (3-\alpha )} +\frac{1}{4}\frac{ 4^{1-\alpha }-5\times 3^{1-\alpha }+10\times 2^{1-\alpha }-10 }{\varGamma (2-\alpha )},\\ \omega ^{(4)}_{j} =&\frac{ (j+1)^{4-\alpha }-5j^{4-\alpha }+10(j-1)^{4-\alpha } -10(j-2)^{4-\alpha } +5(j-3)^{4-\alpha }-(j-4)^{4-\alpha } }{\varGamma (5-\alpha )} \\&+\frac{3}{2}\frac{ (j+1)^{3-\alpha }-5j^{3-\alpha }+10(j-1)^{3-\alpha }-10(j-2)^{3-\alpha } +5(j-3)^{3-\alpha }-(j-4)^{3-\alpha } }{\varGamma (4-\alpha )} \\&+\frac{11}{12}\frac{ (j+1)^{2-\alpha }-5j^{2-\alpha }+10(j-1)^{2-\alpha } -10(j-2)^{2-\alpha } +5(j-3)^{2-\alpha }-(j-4)^{2-\alpha } }{\varGamma (3-\alpha )} \\&+ \! \frac{1}{4}\frac{(j \! + \! 1)^{1-\alpha } \! - \! 5j^{1-\alpha } \! + \! 10(j \! - \! 1)^{1-\alpha } \! - \! 10(j \! - \! 2)^{1-\alpha } \! + \! 5(j \! - \! 3)^{1-\alpha } \! - \! (j\!-\!4)^{1-\alpha } }{\varGamma (2-\alpha )},~j\ge 4.\\ \end{aligned}$$

  • \(L_5\) approximation

    $$\begin{aligned} \omega ^{(5)}_{0}=&\frac{1}{\varGamma (6-\alpha )} + 2 \frac{1}{\varGamma (5-\alpha )} + \frac{7}{4}\frac{1}{\varGamma (4-\alpha )} + \frac{5}{6}\frac{1}{\varGamma (3-\alpha )} + \frac{1}{5}\frac{1}{\varGamma (2-\alpha )},\\ \omega ^{(5)}_{1}=&\frac{ 2^{5-\alpha }-6 }{\varGamma (6-\alpha )} + 2\frac{ 2^{4-\alpha }-6 }{\varGamma (5-\alpha )} + \frac{7}{4}\frac{ 2^{3-\alpha }-6 }{\varGamma (4-\alpha )}+\frac{5}{6}\frac{ 2^{2-\alpha }-6 }{\varGamma (3-\alpha )} + \frac{1}{5}\frac{ 2^{1-\alpha }-6 }{\varGamma (2-\alpha )},\\ \omega ^{(5)}_{2} =&\frac{ 3^{5-\alpha }-6\times 2^{5-\alpha }+15 }{\varGamma (6-\alpha )} + 2\frac{ 3^{4-\alpha }-6\times 2^{4-\alpha }+15 }{\varGamma (5-\alpha )} +\frac{7}{4}\frac{ 3^{3-\alpha }-6\times 2^{3-\alpha }+15 }{\varGamma (4-\alpha )} \\&+\frac{5}{6}\frac{ 3^{2-\alpha }-6\times 2^{2-\alpha }+15 }{\varGamma (3-\alpha )} + \frac{1}{5}\frac{ 3^{1-\alpha }-6\times 2^{1-\alpha }+15 }{\varGamma (2-\alpha )},\\ \omega ^{(5)}_{3} =&\frac{ 4^{5-\alpha }-6\times 3^{5-\alpha }+15\times 2^{5-\alpha } -20 }{\varGamma (6-\alpha )} +2\frac{ 4^{4-\alpha }-6\times 3^{4-\alpha }+15\times 2^{4-\alpha } -20 }{\varGamma (5-\alpha )} \\&+\frac{7}{4}\frac{ 4^{3-\alpha }-6\times 3^{3-\alpha }+15\times 2^{3-\alpha }-20 }{\varGamma (4-\alpha )} +\frac{5}{6}\frac{ 4^{2-\alpha }-6\times 3^{2-\alpha }+15\times 2^{2-\alpha }-20 }{\varGamma (3-\alpha )} \\&+\frac{1}{5}\frac{ 4^{1-\alpha }-6\times 3^{1-\alpha }+15\times 2^{1-\alpha } -20 }{\varGamma (2-\alpha )},\\ \omega ^{(5)}_{4} =&\frac{5^{5-\alpha }-6\times 4^{5-\alpha }+15\times 3^{5-\alpha } -20 \times 2^{5-\alpha } +15 }{\varGamma (6-\alpha )} +2\frac{5^{4-\alpha }-6\times 4^{4-\alpha }+15\times 3^{4-\alpha } }{\varGamma (5-\alpha )} \\&+2\frac{-20 \times 2^{4-\alpha } +15}{\varGamma (5-\alpha )} +\frac{7}{4}\frac{5^{3-\alpha }-6\times 4^{3-\alpha }+15\times 3^{3-\alpha } -20 \times 2^{3-\alpha } +15 }{\varGamma (4-\alpha )} \\&+\frac{5}{6}\frac{5^{2-\alpha }-6\times 4^{2-\alpha }+15\times 3^{2-\alpha } -20 \times 2^{2-\alpha } +15}{\varGamma (3-\alpha )} \\&+\frac{1}{5}\frac{5^{1-\alpha }-6\times 4^{1-\alpha } +15\times 3^{1-\alpha } -20 \times 2^{1-\alpha } +15 }{\varGamma (2-\alpha )},\\ \omega ^{(5)}_{j} =&\frac{(j+1)^{5-\alpha }-6j^{5-\alpha }+15(j-1)^{5-\alpha } -20(j-2)^{5-\alpha } + 15(j-3)^{5-\alpha }-6(j-4)^{5-\alpha } }{\varGamma (6-\alpha )} \\&+2\frac{(j+1)^{4-\alpha }-6j^{4-\alpha }+15(j-1)^{4-\alpha } -20(j-2)^{4-\alpha }+ 15(j-3)^{4-\alpha }-6(j-4)^{4-\alpha } }{\varGamma (5-\alpha )} \\&+\frac{7}{4}\frac{(j+1)^{3-\alpha }-6j^{3-\alpha }+15(j-1)^{3-\alpha }-20(j-2)^{3-\alpha } + 15(j-3)^{3-\alpha }-6(j-4)^{3-\alpha } }{\varGamma (4-\alpha )}\\&+\frac{5}{6}\frac{(j+1)^{2-\alpha }-6j^{2-\alpha }+15(j-1)^{2-\alpha }-20(j-2)^{2-\alpha } + 15(j-3)^{2-\alpha }-6(j-4)^{2-\alpha }}{\varGamma (3-\alpha )} \\&+\frac{1}{5}\frac{(j+1)^{1-\alpha }-6j^{1-\alpha }+15(j-1)^{1-\alpha } -20(j-2)^{1-\alpha } + 15(j-3)^{1-\alpha }-6(j-4)^{1-\alpha }}{\varGamma (2-\alpha )} \\&+ \frac{(j-5)^{5-\alpha }}{\varGamma (6-\alpha )} +2\frac{(j-5)^{4-\alpha }}{\varGamma (5-\alpha )} +\frac{7}{4}\frac{(j-5)^{3-\alpha }}{\varGamma (4-\alpha )} +\frac{5}{6}\frac{(j-5)^{2-\alpha } }{\varGamma (3-\alpha )} +\frac{1}{5}\frac{(j-5)^{1-\alpha } }{\varGamma (2-\alpha )},~j\ge 5. \end{aligned}$$

  • \(L_6\) approximation

$$\begin{aligned} \omega ^{(6)}_{0} =&\frac{1}{\varGamma (7 \! - \! \alpha )} \! + \! \frac{5}{2}\frac{1}{\varGamma (6 \! - \! \alpha )} \! +\! \frac{17}{6}\frac{1}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{1}{\varGamma (4 \! - \! \alpha )} \! + \! \frac{137}{180}\frac{1}{\varGamma (3 \! - \! \alpha )} \! + \! \frac{1}{6}\frac{1}{\varGamma (2 \! - \! \alpha )},\\ \omega ^{(6)}_{1} =&\frac{2^{6-\alpha } \! - \! 7}{\varGamma (7 \! - \! \alpha )} \! + \! \frac{5}{2}\frac{2^{5-\alpha } \! - \! 7}{\varGamma (6 \! - \! \alpha )} \! + \! \frac{17}{6}\frac{ 2^{4-\alpha } \! - \! 7}{\varGamma (5 \! - \! \alpha )} \! + \! \frac{15}{8}\frac{2^{3-\alpha } \! - \! 7}{\varGamma (4 \! - \! \alpha )} \! + \! \frac{137}{180}\frac{2^{2-\alpha } \! - \! 7}{\varGamma (3 \! - \! \alpha )} \! + \! \frac{1}{6}\frac{ 2^{1-\alpha } \! - \! 7}{\varGamma (2 \! - \! \alpha )},\\ \omega ^{(6)}_{2}=&\frac{ 3^{6-\alpha }-7\times 2^{6-\alpha }+21 }{\varGamma (7-\alpha )} +\frac{5}{2}\frac{ 3^{5-\alpha }-7\times 2^{5-\alpha }+21 }{\varGamma (6-\alpha )} +\frac{17}{6}\frac{ 3^{4-\alpha }-7\times 2^{4-\alpha }+21 }{\varGamma (5-\alpha )} \\ +&\frac{15}{8}\frac{3^{3-\alpha } \! - \! 7\times 2^{3-\alpha } \! + \! 21 }{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ 3^{2-\alpha }-7\times 2^{2-\alpha }+21 }{\varGamma (3-\alpha )} \! + \! \frac{1}{5}\frac{ 3^{1-\alpha }-7\times 2^{1-\alpha }+21 }{\varGamma (2-\alpha )},\\ \omega ^{(6)}_{3}=&\frac{ 4^{6-\alpha }-7\times 3^{6-\alpha }+21\times 2^{6-\alpha }-35 }{\varGamma (7-\alpha )} + \frac{5}{2}\frac{ 4^{5-\alpha }-7\times 3^{5-\alpha }+21\times 2^{5-\alpha } -35 }{\varGamma (6-\alpha )} \\ +&\frac{17}{6}\frac{ 4^{4-\alpha }-7\times 3^{4-\alpha }+21\times 2^{4-\alpha }-35 }{\varGamma (5-\alpha )} +\frac{15}{8}\frac{ 4^{3-\alpha }-7\times 3^{3-\alpha } +21\times 2^{3-\alpha } -35 }{\varGamma (4-\alpha )} \\&+\frac{137}{180}\frac{ 4^{2-\alpha }-7\times 3^{2-\alpha }+21\times 2^{2-\alpha } -35 }{\varGamma (3-\alpha )} +\frac{1}{6}\frac{ 4^{1-\alpha }-7\times 3^{1-\alpha } +21\times 2^{1-\alpha } -35 }{\varGamma (2-\alpha )},\\ \omega ^{(6)}_{4}&=\frac{ 5^{6-\alpha }-7\times 4^{6-\alpha }+21\times 3^{6-\alpha } -35 \times 2^{6-\alpha } +35 }{\varGamma (7-\alpha )} +\frac{5}{2}\frac{ 5^{5-\alpha }-7\times 4^{5-\alpha }+21\times 3^{5-\alpha } }{\varGamma (6-\alpha )} \\&+\frac{5}{2}\frac{ -35 \times 2^{5-\alpha } +35}{\varGamma (6-\alpha )} +\frac{17}{6}\frac{ 5^{4-\alpha }-7\times 4^{4-\alpha }+21\times 3^{4-\alpha } -35 \times 2^{4-\alpha } +35 }{\varGamma (5-\alpha )} \\&\! + \! \frac{15}{8}\frac{5^{3-\alpha } \! - \! 7\times 4^{3-\alpha } \! + \! 21\times 3^{3-\alpha } \! - \! 35 \times 2^{3-\alpha } \! + \! 35 }{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{5^{2-\alpha } \! - \! 7\times 4^{2-\alpha } \! + \! 21\times 3^{2-\alpha }}{\varGamma (3-\alpha )} \\&+\frac{137}{180}\frac{ -35 \times 2^{2-\alpha } +35 }{\varGamma (3-\alpha )} +\frac{1}{6}\frac{ 5^{1-\alpha }-7\times 4^{1-\alpha }+21\times 3^{1-\alpha } -35 \times 2^{1-\alpha } +35 }{\varGamma (2-\alpha )}, \end{aligned}$$
$$\begin{aligned} \omega ^{(6)}_{5} =&\frac{ 6^{6-\alpha }-7\times 5^{6-\alpha }+21\times 4^{6-\alpha } -35 \times 3^{6-\alpha } +35\times 2^{6-\alpha }-21 }{\varGamma (7-\alpha )} +\frac{5}{2}\frac{ 6^{5-\alpha }-7\times 5^{5-\alpha } }{\varGamma (6-\alpha )} \\&+\frac{5}{2}\frac{ 21\times 4^{5-\alpha } -35\times 3^{5-\alpha } +35\times 2^{5-\alpha }-21 }{\varGamma (6-\alpha )} +\frac{17}{6}\frac{ 6^{4-\alpha }-7\times 5^{4-\alpha }+21\times 4^{4-\alpha }}{\varGamma (5-\alpha )}\\&+\frac{17}{6}\frac{-35\times 3^{4-\alpha } + 35\times 2^{4-\alpha }-21 }{\varGamma (5-\alpha )} +\frac{15}{8}\frac{ 6^{3-\alpha }-7\times 5^{3-\alpha }+21\times 4^{3-\alpha }-35\times 3^{3-\alpha } }{\varGamma (4-\alpha )} \\&+\frac{15}{8}\frac{ 35\times 2^{3-\alpha }-21 }{\varGamma (4-\alpha )} +\frac{137}{180}\frac{ 6^{2-\alpha }-7\times 5^{2-\alpha }+21\times 4^{2-\alpha }-35\times 3^{2-\alpha }+35\times 2^{2-\alpha }}{\varGamma (3-\alpha )} \\&+\frac{137}{180}\frac{ -21 }{\varGamma (3-\alpha )} +\frac{1}{6}\frac{ 6^{1-\alpha }-7\times 5^{1-\alpha }+21\times 4^{1-\alpha }-35\times 3^{1-\alpha }+35\times 2^{1-\alpha }-21 }{\varGamma (2-\alpha )},\\ \omega ^{(6)}_{j} =&\frac{(j+1)^{6-\alpha }}{\varGamma (7-\alpha )}+\frac{5}{2}\frac{(j+1)^{5-\alpha }}{\varGamma (6-\alpha )} +\frac{17}{6}\frac{(j+1)^{4-\alpha }}{\varGamma (5-\alpha )} + \frac{15}{8}\frac{(j+1)^{3-\alpha }}{\varGamma (4-\alpha )} + \frac{137}{180}\frac{ (j+1)^{2-\alpha } }{\varGamma (3-\alpha )} \\&-7\left( \frac{j^{6-\alpha }}{\varGamma (7-\alpha )}+\frac{5}{2}\frac{j^{5-\alpha }}{\varGamma (6-\alpha )} +\frac{17}{6}\frac{j^{4-\alpha }}{\varGamma (5-\alpha )} + \frac{15}{8}\frac{j^{3-\alpha }}{\varGamma (4-\alpha )} + \frac{137}{180}\frac{ j^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&+21\left( \frac{(j-1)^{6-\alpha }}{\varGamma (7-\alpha )} \! + \! \frac{5}{2}\frac{(j-1)^{5-\alpha }}{\varGamma (6-\alpha )} \! + \! \frac{17}{6}\frac{(j-1)^{4-\alpha }}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{(j-1)^{3-\alpha }}{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ (j-1)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&-35\left( \frac{(j-2)^{6-\alpha }}{\varGamma (7-\alpha )} \! + \! \frac{5}{2}\frac{(j-2)^{5-\alpha }}{\varGamma (6-\alpha )} \! + \! \frac{17}{6}\frac{(j-2)^{4-\alpha }}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{(j-2)^{3-\alpha }}{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ (j-2)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&+ 35\left( \frac{(j-3)^{6-\alpha }}{\varGamma (7-\alpha )} \! + \! \frac{5}{2}\frac{(j-3)^{5-\alpha }}{\varGamma (6-\alpha )} \! + \! \frac{17}{6}\frac{(j-3)^{4-\alpha }}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{(j-3)^{3-\alpha }}{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ (j-3)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&-21\left( \frac{(j-4)^{6-\alpha }}{\varGamma (7-\alpha )} \! + \! \frac{5}{2}\frac{(j-4)^{5-\alpha }}{\varGamma (6-\alpha )} \! + \! \frac{17}{6}\frac{(j-4)^{4-\alpha }}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{(j-4)^{3-\alpha }}{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ (j-4)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&+7\left( \frac{(j-5)^{6-\alpha }}{\varGamma (7-\alpha )} \! + \! \frac{5}{2}\frac{(j-5)^{5-\alpha }}{\varGamma (6-\alpha )} \! + \! \frac{17}{6}\frac{(j-5)^{4-\alpha }}{\varGamma (5-\alpha )} \! + \! \frac{15}{8}\frac{(j-5)^{3-\alpha }}{\varGamma (4-\alpha )} \! + \! \frac{137}{180}\frac{ (j-5)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&-\left( \frac{(j-6)^{6-\alpha }}{\varGamma (7-\alpha )}+\frac{5}{2}\frac{(j-6)^{5-\alpha }}{\varGamma (6-\alpha )} +\frac{17}{6}\frac{(j-6)^{4-\alpha }}{\varGamma (5-\alpha )} + \frac{15}{8}\frac{(j-6)^{3-\alpha }}{\varGamma (4-\alpha )} + \frac{137}{180}\frac{ (j-6)^{2-\alpha } }{\varGamma (3-\alpha )} \right) \\&+ \frac{1}{6}\frac{ (j+1)^{1-\alpha } \! - \! 7j^{1-\alpha } \! + \! 21(j-1)^{1-\alpha } \! - \! 35(j-2)^{1-\alpha } \! + \! 35(j-3)^{1-\alpha } \! - \! 21(j-4)^{1-\alpha } }{\varGamma (2-\alpha )} \\&+ \frac{1}{6}\frac{ 7(j-5)^{1-\alpha } - (j-6)^{1-\alpha } }{\varGamma (2-\alpha )},~j\ge 6. \end{aligned}$$

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Shi, J., Chen, M., Yan, Y. et al. Correction of High-Order \(L_k\) Approximation for Subdiffusion. J Sci Comput 93, 31 (2022). https://doi.org/10.1007/s10915-022-01984-8

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