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September  2020, 28(3): 1161-1189. doi: 10.3934/era.2020064

## The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions

 1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, MO 411105, China 2 Xiangtan University, Xiangtan, MO 411105, China 3 Ammosov North-Eastern Federal University, Yakutsk, MO 677000, Russia

* Corresponding author: Yin Yang

Received  March 2020 Revised  May 2020 Published  July 2020

Fund Project: Yin Yang was supported by National Natural Science Foundation of China Project (11671342, 11931003) and Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2020JJ2027); Sujuan Kang was supported by National Natural Science Foundation of China Project (11771369) and Key Project of Hunan Provincial Department of Education (17A210); Vasilev Vasilii Ivanovich was supported by Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018WK4006, 2019YZ3003)

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the $L^{\infty}$- and $L^{2}_{\omega^{\alpha, \beta}}$-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

Citation: Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064
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##### References:
The exact solution and numerical solution for $N = 10$ and $\sigma$ = 1 and 3
The exact solution and numerical solution for $N = 10$ and $\sigma$ = 6 and 9
For $N = 10$, $\sigma$ takes values of 1-9 and 28, the error of $L^\infty$ and $L^2_{\omega^{\alpha-1, 0}}$ changes as the collocation point $N$ increases
The exact solution and numerical solution for $N = 10$ and $\sigma$ = 1, 2 and 4
For $N = 10$ and $\sigma$ takes values of 1-6 and 14, the error of $L^\infty$ and $L^2_{\omega^{\alpha-1, 0}}$ changes as the collocation point $N$ increases
The exact solution and numerical solution for $N = 10$ and $\sigma$ = 1, 2 and 4
For $N = 10$ and $\sigma$ takes values of 1-6 and 8, the error of $L^\infty$ and $L^2_{\omega^{\alpha-1, 0}}$ changes as the collocation point $N$ increases
The $L^{\infty}$- and $L^2_{\omega^{\alpha-1, 0}}$-error for $N = 10$ and $\sigma$ takes values of 1-9
 $N$ $\sigma$ =1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 4.31E-03 4.31E-03 4.12E-04 5.06E-04 3.21E-05 3.14E-05 4 7.92E-04 7.07E-04 2.76E-05 2.84E-05 2.34E-08 1.87E-08 6 2.63E-04 2.32E-04 5.37E-06 4.37E-06 1.44E-11 1.05E-11 8 1.15E-04 1.01E-04 1.32E-06 1.05E-06 3.00E-15 1.76E-15 10 6.01E-05 5.26E-05 4.08E-07 3.32E-07 2.33E-15 1.20E-15 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 6.47E-03 6.21E-03 1.30E-02 1.24E-02 1.72E-02 1.65E-02 4 1.70E-05 1.29E-05 2.40E-05 1.75E-05 6.46E-09 4.82E-09 6 7.51E-07 5.35E-07 4.86E-07 3.18E-07 8.14E-10 5.29E-10 8 7.42E-08 5.85E-08 3.20E-08 2.20E-08 5.97E-12 3.60E-12 10 1.46E-08 1.05E-08 4.08E-09 2.89E-09 3.76E-14 1.94E-14 $N$ $\sigma$ = 7 $\sigma$ = 8 $\sigma$ = 9 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 1.84E-02 1.77E-02 1.68E-02 1.61E-02 1.27E-02 1.21E-02 4 0.31E-03 0.22E-03 1.03E-03 0.73E-03 2.13E-03 1.52E-03 6 5.14E-07 3.09E-07 8.26E-07 4.92E-07 3.70E-07 2.28E-07 8 1.20E-08 6.94E-09 1.11E-08 5.96E-09 9.08E-10 5.12E-10 10 7.20E-10 4.42E-10 4.26E-10 2.40E-10 3.16E-11 1.68E-11
 $N$ $\sigma$ =1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 4.31E-03 4.31E-03 4.12E-04 5.06E-04 3.21E-05 3.14E-05 4 7.92E-04 7.07E-04 2.76E-05 2.84E-05 2.34E-08 1.87E-08 6 2.63E-04 2.32E-04 5.37E-06 4.37E-06 1.44E-11 1.05E-11 8 1.15E-04 1.01E-04 1.32E-06 1.05E-06 3.00E-15 1.76E-15 10 6.01E-05 5.26E-05 4.08E-07 3.32E-07 2.33E-15 1.20E-15 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 6.47E-03 6.21E-03 1.30E-02 1.24E-02 1.72E-02 1.65E-02 4 1.70E-05 1.29E-05 2.40E-05 1.75E-05 6.46E-09 4.82E-09 6 7.51E-07 5.35E-07 4.86E-07 3.18E-07 8.14E-10 5.29E-10 8 7.42E-08 5.85E-08 3.20E-08 2.20E-08 5.97E-12 3.60E-12 10 1.46E-08 1.05E-08 4.08E-09 2.89E-09 3.76E-14 1.94E-14 $N$ $\sigma$ = 7 $\sigma$ = 8 $\sigma$ = 9 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 1.84E-02 1.77E-02 1.68E-02 1.61E-02 1.27E-02 1.21E-02 4 0.31E-03 0.22E-03 1.03E-03 0.73E-03 2.13E-03 1.52E-03 6 5.14E-07 3.09E-07 8.26E-07 4.92E-07 3.70E-07 2.28E-07 8 1.20E-08 6.94E-09 1.11E-08 5.96E-09 9.08E-10 5.12E-10 10 7.20E-10 4.42E-10 4.26E-10 2.40E-10 3.16E-11 1.68E-11
The $L^{\infty}$- and $L^2_{\omega^{\alpha-1, 0}}$-error for $N = 10$ and $\sigma$ takes values of 1-6
 $N$ $\sigma$=1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.13E-04 4.55E-04 2.79E-04 3.21E-04 1.86E-03 2.14E-03 4 2.30E-05 2.85E-05 2.11E-07 1.85E-07 3.37E-06 3.58E-06 6 4.27E-06 5.23E-06 1.62E-10 1.54E-10 5.11E-07 4.18E-07 8 1.38E-06 1.51E-06 1.01E-13 8.56E-14 6.87E-08 6.62E-08 10 5.29E-07 5.61E-07 4.77E-15 4.36E-15 1.88E-08 1.57E-08 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.94E-03 4.53E-03 5.31E-03 6.11E-03 3.86E-03 4.44E-03 4 2.70E-05 2.51E-05 1.38E-04 1.27E-04 4.51E-04 4.12E-04 6 3.23E-07 2.62E-07 8.69E-07 7.52E-07 1.09E-06 9.15E-07 8 1.44E-09 1.12E-09 5.54E-08 3.90E-08 1.67E-07 1.18E-07 10 3.14E-11 2.70E-11 2.06E-09 1.61E-09 3.68E-09 2.52E-09
 $N$ $\sigma$=1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.13E-04 4.55E-04 2.79E-04 3.21E-04 1.86E-03 2.14E-03 4 2.30E-05 2.85E-05 2.11E-07 1.85E-07 3.37E-06 3.58E-06 6 4.27E-06 5.23E-06 1.62E-10 1.54E-10 5.11E-07 4.18E-07 8 1.38E-06 1.51E-06 1.01E-13 8.56E-14 6.87E-08 6.62E-08 10 5.29E-07 5.61E-07 4.77E-15 4.36E-15 1.88E-08 1.57E-08 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.94E-03 4.53E-03 5.31E-03 6.11E-03 3.86E-03 4.44E-03 4 2.70E-05 2.51E-05 1.38E-04 1.27E-04 4.51E-04 4.12E-04 6 3.23E-07 2.62E-07 8.69E-07 7.52E-07 1.09E-06 9.15E-07 8 1.44E-09 1.12E-09 5.54E-08 3.90E-08 1.67E-07 1.18E-07 10 3.14E-11 2.70E-11 2.06E-09 1.61E-09 3.68E-09 2.52E-09
The $L^{\infty}$-and $L^2_{\omega^{\alpha-1, 0}}$-error for $N = 10$ and $\sigma$ takes values of 1-6
 $N$ $\sigma$=1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.34E-04 6.18E-04 3.65E-03 6.78E-03 3.94E-02 6.96E-02 4 5.68E-05 9.69E-05 3.54E-06 5.57E-06 1.79E-04 2.87E-04 6 1.17E-05 1.97E-05 1.77E-08 2.78E-08 5.01E-07 7.54E-07 8 3.58E-06 6.02E-06 1.66E-09 2.60E-09 3.82E-09 4.57E-09 10 1.39E-06 2.33E-06 2.68E-10 4.17E-10 2.67E-10 2.90E-10 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 7.75E-02 1.37E-01 9.07E-02 1.60E-01 8.04E-02 1.42E-01 4 7.27E-04 1.25E-03 1.02E-03 1.21E-03 2.33E-03 3.89E-03 6 1.27E-05 1.84E-05 1.00E-04 1.47E-04 3.62E-04 5.36E-04 8 6.24E-08 8.53E-08 1.10E-06 1.50E-06 8.27E-06 1.13E-05 10 2.09E-10 2.73E-10 8.63E-09 1.13E-08 1.15E-07 1.50E-07
 $N$ $\sigma$=1 $\sigma$=2 $\sigma$=3 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 3.34E-04 6.18E-04 3.65E-03 6.78E-03 3.94E-02 6.96E-02 4 5.68E-05 9.69E-05 3.54E-06 5.57E-06 1.79E-04 2.87E-04 6 1.17E-05 1.97E-05 1.77E-08 2.78E-08 5.01E-07 7.54E-07 8 3.58E-06 6.02E-06 1.66E-09 2.60E-09 3.82E-09 4.57E-09 10 1.39E-06 2.33E-06 2.68E-10 4.17E-10 2.67E-10 2.90E-10 $N$ $\sigma$ = 4 $\sigma$ = 5 $\sigma$ = 6 $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ $\|u-U\|_{L^\infty}$ $\|u-U\|_{L^2}$ 2 7.75E-02 1.37E-01 9.07E-02 1.60E-01 8.04E-02 1.42E-01 4 7.27E-04 1.25E-03 1.02E-03 1.21E-03 2.33E-03 3.89E-03 6 1.27E-05 1.84E-05 1.00E-04 1.47E-04 3.62E-04 5.36E-04 8 6.24E-08 8.53E-08 1.10E-06 1.50E-06 8.27E-06 1.13E-05 10 2.09E-10 2.73E-10 8.63E-09 1.13E-08 1.15E-07 1.50E-07
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