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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
References:
[1]

W. M. Ahmad and R. EL-Khazali, Fractional-order dynamical models of love, Chaos Solitons Fractals, 33 (2007), 1367-1375.  doi: 10.1016/j.chaos.2006.01.098.  Google Scholar

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A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40 (2009), 521-529.  doi: 10.1016/j.chaos.2007.08.001.  Google Scholar

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A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Appl. Math. Model., 40 (2016), 832-845.  doi: 10.1016/j.apm.2015.06.012.  Google Scholar

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H. Brunner, Theory and Numerical Solution of Volterra Functional Integral Equations, HIT Summer Seminar, 2010. Google Scholar

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H. Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernel, SIAM J. Numer. Anal., 20 (1983), 1106-1119.  doi: 10.1137/0720080.  Google Scholar

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P. Baratella and A. P. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 163 (2004), 401-418.  doi: 10.1016/j.cam.2003.08.047.  Google Scholar

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Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equationS with a weakly singular kernel, Math. Comp., 79 (2010), 147-167.  doi: 10.1090/S0025-5718-09-02269-8.  Google Scholar

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K. Du, On well-conditioned spectral collocation and spectral methods by the integral reformulation, SIAM J. Sci. Comput., 38 (2016), A3247–A3263. doi: 10.1137/15M1046629.  Google Scholar

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F. Ghoreishi and P. Mokhtary, Spectral collocation method for multi-order fractional differential equations, Int. J. Comput. Methods, 11 (2014), 1350072, 23 pp. doi: 10.1142/s0219876213500722.  Google Scholar

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J. H. He, Nonlinear Oscillation with Fractional Derivative and its Cahpinpalications, International Conference on Vibrating Engineering98, 1998. Google Scholar

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L. HuangX. F. LiY. L. Zhao and X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl., 62 (2011), 1127-1134.  doi: 10.1016/j.camwa.2011.03.037.  Google Scholar

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Y. J. JiaoL. L. Wang and C. Huang, Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis, J. Comput. Phys., 305 (2016), 1-28.  doi: 10.1016/j.jcp.2015.10.029.  Google Scholar

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M. Kolk and A. Pedas, Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integro-differential equations with weakly singular kernels, WSEAS Trans. Math., 6 (2007), 537–544. http://www.crm.umontreal.ca/AARMS07/pdf/pedas.pdf  Google Scholar

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X. Li and T. Tang, Convergence analysis of Jacobi spectral collocation methods for Abel–CVolterra integral equations of second kind, Front. Math. China, 7 (2012), 69-84.  doi: 10.1007/s11464-012-0170-0.  Google Scholar

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X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942.  Google Scholar

[22]

H. Liang and M. Stynes, Collocation methods for general caputo two-point boundary value problems, J. Sci. Comput., 76 (2018), 390-425.  doi: 10.1007/s10915-017-0622-5.  Google Scholar

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Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp., 41 (1983), 87-102.  doi: 10.1090/S0025-5718-1983-0701626-6.  Google Scholar

[24]

G. Mastroianni and D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey., J. Comput. Appl. Math., 134 (2001), 325-341.  doi: 10.1016/S0377-0427(00)00557-4.  Google Scholar

[25]

S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differential equations, Comput. Math. Appl., 52 (2006), 459-470.  doi: 10.1016/j.camwa.2006.02.011.  Google Scholar

[26]

G. Monegato and L. Scuderi, High order methods for weakly singular integral equations with nonsmooth input functions, Math. Comp., 67 (1998), 1493-1515.  doi: 10.1090/S0025-5718-98-01005-9.  Google Scholar

[27]

A. PedasE. Tamme and M. Vikerpuur, Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems, J. Comput. Appl. Math., 317 (2017), 1-16.  doi: 10.1016/j.cam.2016.11.022.  Google Scholar

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J. Shen, T. Tang and L. L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

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Y. YangY. Chen and Y. Huang, Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51 (2014), 203-224.  doi: 10.4134/JKMS.2014.51.1.203.  Google Scholar

[31]

Y. YangY. ChenY. Huang and H. Wei, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, J. Comput. Appl. Math., 73 (2017), 1218-1232.  doi: 10.1016/j.camwa.2016.08.017.  Google Scholar

[32]

Y. YangW. QiaoJ. Wang and S. Zhang, Spectral collocation methods for nonlinear coupled time fractional Nernest-Planck equations in two dimensions and its convergence analysis, Comput. Math. Appl., 78 (2019), 1431-1449.  doi: 10.1016/j.camwa.2018.12.018.  Google Scholar

[33]

Y. YangY. Chen and Y. Huang, Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta Math. Sci. Ser. B (Engl. Ed.), 34 (2014), 673-690.  doi: 10.1016/S0252-9602(14)60039-4.  Google Scholar

[34]

C. Yang and J. Hou, Numerical solution of Volterra integro-differential equations of fractional order by Laplace decomposition method, International Journal of Mathematical, Computational, Natural and Physical Engineering, 7 (2013), 549-553.  doi: 10.5281/zenodo.1087866.  Google Scholar

[35]

Y. YangY. Huang and Y. Zhou, Numerical solutions for solving time fractional Fokker-Planck equations based on spectral collocation methods, J. Comput. Appl. Math., 339 (2018), 389-404.  doi: 10.1016/j.cam.2017.04.003.  Google Scholar

[36]

Y. YangY. Huang and Y. Zhou, Numerical simulation of time fractional Cable equations and convergence analysis, Numer. Methods Partial Differential Equations, 34 (2018), 1556-1576.  doi: 10.1002/num.22225.  Google Scholar

[37]

Y. Yang and E. Tohidi, Numerical solution of multi-Pantograph delay boundary value problems via an efficient approach with the convergence analysis, Comput. Appl. Math., 38 (2019), Paper No. 127, 14 pp. doi: 10.1007/s40314-019-0896-3.  Google Scholar

[38]

M. A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357 (2019), 103-122.  doi: 10.1016/j.cam.2019.01.046.  Google Scholar

show all references

References:
[1]

W. M. Ahmad and R. EL-Khazali, Fractional-order dynamical models of love, Chaos Solitons Fractals, 33 (2007), 1367-1375.  doi: 10.1016/j.chaos.2006.01.098.  Google Scholar

[2]

A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40 (2009), 521-529.  doi: 10.1016/j.chaos.2007.08.001.  Google Scholar

[3]

A. H. Bhrawy and M. A. Zaky, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Appl. Math. Model., 40 (2016), 832-845.  doi: 10.1016/j.apm.2015.06.012.  Google Scholar

[4]

H. Brunner, Theory and Numerical Solution of Volterra Functional Integral Equations, HIT Summer Seminar, 2010. Google Scholar

[5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511543234.  Google Scholar
[6]

H. Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernel, SIAM J. Numer. Anal., 20 (1983), 1106-1119.  doi: 10.1137/0720080.  Google Scholar

[7]

P. Baratella and A. P. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 163 (2004), 401-418.  doi: 10.1016/j.cam.2003.08.047.  Google Scholar

[8]

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, 1997. Google Scholar

[9]

Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equationS with a weakly singular kernel, Math. Comp., 79 (2010), 147-167.  doi: 10.1090/S0025-5718-09-02269-8.  Google Scholar

[10]

K. Du, On well-conditioned spectral collocation and spectral methods by the integral reformulation, SIAM J. Sci. Comput., 38 (2016), A3247–A3263. doi: 10.1137/15M1046629.  Google Scholar

[11]

F. Ghoreishi and P. Mokhtary, Spectral collocation method for multi-order fractional differential equations, Int. J. Comput. Methods, 11 (2014), 1350072, 23 pp. doi: 10.1142/s0219876213500722.  Google Scholar

[12]

Z. Hao and W. Cao, An improved algorithm based on finite difference schemes for fractional Boundary Value Problems with nonsmooth solution, J. Sci. Comput., 73 (2017), 395-415.  doi: 10.1007/s10915-017-0417-8.  Google Scholar

[13]

J. H. He, Nonlinear Oscillation with Fractional Derivative and its Cahpinpalications, International Conference on Vibrating Engineering98, 1998. Google Scholar

[14]

J. H. He, Some applications of nonlinear fractional differential equations and therir approximations, Bulletin of Science Technology and Society, 15 (1999), 86-90.   Google Scholar

[15]

C. HuangY. J. JiaoL. L. Wang and Z. M. Zhang, Optimal fractional integration preconditioning and error analysis of fractional collocation method using nodal generalized Jacobi functions, SIAM J. Numer. Anal., 54 (2016), 3357-3387.  doi: 10.1137/16M1059278.  Google Scholar

[16]

L. HuangX. F. LiY. L. Zhao and X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl., 62 (2011), 1127-1134.  doi: 10.1016/j.camwa.2011.03.037.  Google Scholar

[17]

M. JaniD. Bhatta and S. Javadi, Numerical solution of fractional integro-differential equations with nonlocal conditions, Appl. Appl. Math., 12 (2017), 98-111.  doi: 10.1007/s40314-019-0896-3.  Google Scholar

[18]

Y. J. JiaoL. L. Wang and C. Huang, Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis, J. Comput. Phys., 305 (2016), 1-28.  doi: 10.1016/j.jcp.2015.10.029.  Google Scholar

[19]

M. Kolk and A. Pedas, Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integro-differential equations with weakly singular kernels, WSEAS Trans. Math., 6 (2007), 537–544. http://www.crm.umontreal.ca/AARMS07/pdf/pedas.pdf  Google Scholar

[20]

X. Li and T. Tang, Convergence analysis of Jacobi spectral collocation methods for Abel–CVolterra integral equations of second kind, Front. Math. China, 7 (2012), 69-84.  doi: 10.1007/s11464-012-0170-0.  Google Scholar

[21]

X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131.  doi: 10.1137/080718942.  Google Scholar

[22]

H. Liang and M. Stynes, Collocation methods for general caputo two-point boundary value problems, J. Sci. Comput., 76 (2018), 390-425.  doi: 10.1007/s10915-017-0622-5.  Google Scholar

[23]

Ch. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp., 41 (1983), 87-102.  doi: 10.1090/S0025-5718-1983-0701626-6.  Google Scholar

[24]

G. Mastroianni and D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey., J. Comput. Appl. Math., 134 (2001), 325-341.  doi: 10.1016/S0377-0427(00)00557-4.  Google Scholar

[25]

S. Momani and R. Qaralleh, An efficient method for solving systems of fractional integro-differential equations, Comput. Math. Appl., 52 (2006), 459-470.  doi: 10.1016/j.camwa.2006.02.011.  Google Scholar

[26]

G. Monegato and L. Scuderi, High order methods for weakly singular integral equations with nonsmooth input functions, Math. Comp., 67 (1998), 1493-1515.  doi: 10.1090/S0025-5718-98-01005-9.  Google Scholar

[27]

A. PedasE. Tamme and M. Vikerpuur, Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems, J. Comput. Appl. Math., 317 (2017), 1-16.  doi: 10.1016/j.cam.2016.11.022.  Google Scholar

[28]

I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[29]

J. Shen, T. Tang and L. L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[30]

Y. YangY. Chen and Y. Huang, Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51 (2014), 203-224.  doi: 10.4134/JKMS.2014.51.1.203.  Google Scholar

[31]

Y. YangY. ChenY. Huang and H. Wei, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, J. Comput. Appl. Math., 73 (2017), 1218-1232.  doi: 10.1016/j.camwa.2016.08.017.  Google Scholar

[32]

Y. YangW. QiaoJ. Wang and S. Zhang, Spectral collocation methods for nonlinear coupled time fractional Nernest-Planck equations in two dimensions and its convergence analysis, Comput. Math. Appl., 78 (2019), 1431-1449.  doi: 10.1016/j.camwa.2018.12.018.  Google Scholar

[33]

Y. YangY. Chen and Y. Huang, Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta Math. Sci. Ser. B (Engl. Ed.), 34 (2014), 673-690.  doi: 10.1016/S0252-9602(14)60039-4.  Google Scholar

[34]

C. Yang and J. Hou, Numerical solution of Volterra integro-differential equations of fractional order by Laplace decomposition method, International Journal of Mathematical, Computational, Natural and Physical Engineering, 7 (2013), 549-553.  doi: 10.5281/zenodo.1087866.  Google Scholar

[35]

Y. YangY. Huang and Y. Zhou, Numerical solutions for solving time fractional Fokker-Planck equations based on spectral collocation methods, J. Comput. Appl. Math., 339 (2018), 389-404.  doi: 10.1016/j.cam.2017.04.003.  Google Scholar

[36]

Y. YangY. Huang and Y. Zhou, Numerical simulation of time fractional Cable equations and convergence analysis, Numer. Methods Partial Differential Equations, 34 (2018), 1556-1576.  doi: 10.1002/num.22225.  Google Scholar

[37]

Y. Yang and E. Tohidi, Numerical solution of multi-Pantograph delay boundary value problems via an efficient approach with the convergence analysis, Comput. Appl. Math., 38 (2019), Paper No. 127, 14 pp. doi: 10.1007/s40314-019-0896-3.  Google Scholar

[38]

M. A. Zaky, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357 (2019), 103-122.  doi: 10.1016/j.cam.2019.01.046.  Google Scholar

Figure 1.  The exact solution and numerical solution for $ N = 10 $ and $ \sigma $ = 1 and 3
Figure 2.  The exact solution and numerical solution for $ N = 10 $ and $ \sigma $ = 6 and 9
Figure 3.  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
Figure 4.  The exact solution and numerical solution for $ N = 10 $ and $ \sigma $ = 1, 2 and 4
Figure 5.  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
Figure 6.  The exact solution and numerical solution for $ N = 10 $ and $ \sigma $ = 1, 2 and 4
Figure 7.  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
Table 1.  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} $
26.47E-036.21E-031.30E-021.24E-021.72E-021.65E-02
41.70E-051.29E-052.40E-051.75E-056.46E-094.82E-09
67.51E-075.35E-074.86E-073.18E-078.14E-105.29E-10
87.42E-085.85E-083.20E-082.20E-085.97E-123.60E-12
101.46E-081.05E-084.08E-092.89E-093.76E-141.94E-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} $
21.84E-021.77E-021.68E-021.61E-021.27E-021.21E-02
40.31E-030.22E-031.03E-030.73E-032.13E-031.52E-03
65.14E-073.09E-078.26E-074.92E-073.70E-072.28E-07
81.20E-086.94E-091.11E-085.96E-099.08E-105.12E-10
107.20E-104.42E-104.26E-102.40E-103.16E-111.68E-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} $
26.47E-036.21E-031.30E-021.24E-021.72E-021.65E-02
41.70E-051.29E-052.40E-051.75E-056.46E-094.82E-09
67.51E-075.35E-074.86E-073.18E-078.14E-105.29E-10
87.42E-085.85E-083.20E-082.20E-085.97E-123.60E-12
101.46E-081.05E-084.08E-092.89E-093.76E-141.94E-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} $
21.84E-021.77E-021.68E-021.61E-021.27E-021.21E-02
40.31E-030.22E-031.03E-030.73E-032.13E-031.52E-03
65.14E-073.09E-078.26E-074.92E-073.70E-072.28E-07
81.20E-086.94E-091.11E-085.96E-099.08E-105.12E-10
107.20E-104.42E-104.26E-102.40E-103.16E-111.68E-11
Table 2.  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} $
23.94E-034.53E-035.31E-036.11E-033.86E-034.44E-03
42.70E-052.51E-051.38E-041.27E-044.51E-044.12E-04
63.23E-072.62E-078.69E-077.52E-071.09E-069.15E-07
81.44E-091.12E-095.54E-083.90E-081.67E-071.18E-07
103.14E-112.70E-112.06E-091.61E-093.68E-092.52E-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} $
23.94E-034.53E-035.31E-036.11E-033.86E-034.44E-03
42.70E-052.51E-051.38E-041.27E-044.51E-044.12E-04
63.23E-072.62E-078.69E-077.52E-071.09E-069.15E-07
81.44E-091.12E-095.54E-083.90E-081.67E-071.18E-07
103.14E-112.70E-112.06E-091.61E-093.68E-092.52E-09
Table 3.  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} $
27.75E-021.37E-019.07E-021.60E-018.04E-021.42E-01
47.27E-041.25E-031.02E-031.21E-032.33E-033.89E-03
61.27E-051.84E-051.00E-041.47E-043.62E-045.36E-04
86.24E-088.53E-081.10E-061.50E-068.27E-061.13E-05
102.09E-102.73E-108.63E-091.13E-081.15E-071.50E-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} $
27.75E-021.37E-019.07E-021.60E-018.04E-021.42E-01
47.27E-041.25E-031.02E-031.21E-032.33E-033.89E-03
61.27E-051.84E-051.00E-041.47E-043.62E-045.36E-04
86.24E-088.53E-081.10E-061.50E-068.27E-061.13E-05
102.09E-102.73E-108.63E-091.13E-081.15E-071.50E-07
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