# American Institute of Mathematical Sciences

• Previous Article
Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model
• ERA Home
• This Issue
• Next Article
The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks
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
##### References:

show all references

##### 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
 [1] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [2] Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073 [3] Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004 [4] Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667 [5] Jean-Baptiste Burie, Ramsès Djidjou-Demasse, Arnaud Ducrot. Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2223-2243. doi: 10.3934/dcdsb.2019225 [6] Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020044 [7] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 [8] Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078 [9] Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021059 [10] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [11] Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217 [12] Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537 [13] Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129 [14] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [15] Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57 [16] Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249 [17] Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052 [18] Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011 [19] Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020080 [20] Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223

Impact Factor: 0.263