# American Institute of Mathematical Sciences

January  2020, 25(1): 321-334. doi: 10.3934/dcdsb.2019185

## Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution

 1 School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China 2 School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China 3 Labroatory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

* Corresponding author: huangcb@csrc.ac.cn

Received  January 2018 Revised  March 2019 Published  September 2019

Fund Project: The authors are grateful to the National Natural Science Foundation of PR China (Grant Nos. 11801332, 11571002, and 11971276)

In this work, the time fractional KdV equation with Caputo time derivative of order $\alpha \in (0,1)$ is considered. The solution of this problem has a weak singularity near the initial time $t = 0$. A fully discrete discontinuous Galerkin (DG) method combining the well-known L1 discretisation in time and DG method in space is proposed to approximate the time fractional KdV equation. The unconditional stability result and O$(N^{-\min \{r\alpha,2-\alpha\}}+h^{k+1})$ convergence result for $P^k \; (k\geq 2)$ polynomials are obtained. Finally, numerical experiments are presented to illustrate the efficiency and the high order accuracy of the proposed scheme.

Citation: Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
##### References:
 [1] N. An, C. Huang and X. Yu, Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation, Appl. Math. Comput., 349 (2019), 148-157. doi: 10.1016/j.amc.2018.12.048. Google Scholar [2] W. Bu and A. Xiao, An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions, Numer. Algor., 81 (2019), 529-545. doi: 10.1007/s11075-018-0559-2. Google Scholar [3] H. Chen and T. Sun, A Petrov-Galerkin spectral method for the linearized time fractional KdV equation, Int. J. Comput. Math., 95 (2018), 1292-1307. doi: 10.1080/00207160.2017.1410544. Google Scholar [4] Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp., 77 (2008), 699-730. doi: 10.1090/S0025-5718-07-02045-5. Google Scholar [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Google Scholar [6] B. Cockburn and K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math., 130 (2015), 293-314. doi: 10.1007/s00211-014-0661-x. Google Scholar [7] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, volume 16 of Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2000. Google Scholar [8] M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796. Google Scholar [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. Google Scholar [10] R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar [11] C. Huang, N. An and X. Yu, A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Anal., 97 (2018), 659-675. doi: 10.1080/00036811.2017.1281407. Google Scholar [12] C. Huang, M. Stynes and N. An, Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem, BIT. Numer. Math, 58 (2018), 661-690. doi: 10.1007/s10543-018-0707-z. Google Scholar [13] C. Huang, X. Yu, C. Wang, Z. Li and N. An, A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation, Appl. Math. Comput., 264 (2015), 483-492. doi: 10.1016/j.amc.2015.04.093. Google Scholar [14] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739. Google Scholar [15] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001. Google Scholar [16] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138. doi: 10.1017/S1446181111000617. Google Scholar [17] S. Momani and A. Yıldı rım, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J. Comput. Math., 87 (2010), 1057-1065. doi: 10.1080/00207160903023581. Google Scholar [18] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145. doi: 10.1016/j.camwa.2008.02.015. Google Scholar [19] K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995. doi: 10.1090/S0025-5718-09-02234-0. Google Scholar [20] K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal., 32 (2012), 906-925. doi: 10.1093/imanum/drr027. Google Scholar [21] K. Mustapha, M. Nour and B. Cockburn, Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems, Adv. Comput. Math., 42 (2016), 377-393. doi: 10.1007/s10444-015-9428-x. Google Scholar [22] I. Podlubny, Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Google Scholar [23] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386. Dedicated to the 60th anniversary of Prof. Francesco Mainardi. Google Scholar [24] J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray.Google Scholar [25] M. Stynes, E. O'Riordan and J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. doi: 10.1137/16M1082329. Google Scholar [26] I. Turner, F. Liu, V. Anh and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245. doi: 10.1007/BF02936089. Google Scholar [27] L. Wei, Y. He, A. Yildirim and S. Kumar, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, ZAMM Z. Angew. Math. Mech., 93 (2013), 14-28. doi: 10.1002/zamm.201200003. Google Scholar [28] G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008], J. Stat. Phys., 135 (2009), 389-391. doi: 10.1007/s10955-009-9713-5. Google Scholar [29] G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Google Scholar [30] N. Zabusky and M. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240. Google Scholar [31] Q. Zhang, J. Zhang, S. Jiang and Z. Zhang, Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comput., 87 (2018), 693-719. doi: 10.1090/mcom/3229. Google Scholar

show all references

##### References:
 [1] N. An, C. Huang and X. Yu, Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation, Appl. Math. Comput., 349 (2019), 148-157. doi: 10.1016/j.amc.2018.12.048. Google Scholar [2] W. Bu and A. Xiao, An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions, Numer. Algor., 81 (2019), 529-545. doi: 10.1007/s11075-018-0559-2. Google Scholar [3] H. Chen and T. Sun, A Petrov-Galerkin spectral method for the linearized time fractional KdV equation, Int. J. Comput. Math., 95 (2018), 1292-1307. doi: 10.1080/00207160.2017.1410544. Google Scholar [4] Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp., 77 (2008), 699-730. doi: 10.1090/S0025-5718-07-02045-5. Google Scholar [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Google Scholar [6] B. Cockburn and K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math., 130 (2015), 293-314. doi: 10.1007/s00211-014-0661-x. Google Scholar [7] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, volume 16 of Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2000. Google Scholar [8] M. Fung, Kdv equation as an euler-poincare equation, Chinese J. Phys., 35 (1997), 789-796. Google Scholar [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. Google Scholar [10] R. Hilfer, editor., Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar [11] C. Huang, N. An and X. Yu, A fully discrete direct discontinuous Galerkin method for the fractional diffusion-wave equation, Appl. Anal., 97 (2018), 659-675. doi: 10.1080/00036811.2017.1281407. Google Scholar [12] C. Huang, M. Stynes and N. An, Optimal ${L}^\infty ({L}^2)$ error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem, BIT. Numer. Math, 58 (2018), 661-690. doi: 10.1007/s10543-018-0707-z. Google Scholar [13] C. Huang, X. Yu, C. Wang, Z. Li and N. An, A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation, Appl. Math. Comput., 264 (2015), 483-492. doi: 10.1016/j.amc.2015.04.093. Google Scholar [14] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739. Google Scholar [15] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001. Google Scholar [16] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), 123-138. doi: 10.1017/S1446181111000617. Google Scholar [17] S. Momani and A. Yıldı rım, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method, Int. J. Comput. Math., 87 (2010), 1057-1065. doi: 10.1080/00207160903023581. Google Scholar [18] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145. doi: 10.1016/j.camwa.2008.02.015. Google Scholar [19] K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995. doi: 10.1090/S0025-5718-09-02234-0. Google Scholar [20] K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J. Numer. Anal., 32 (2012), 906-925. doi: 10.1093/imanum/drr027. Google Scholar [21] K. Mustapha, M. Nour and B. Cockburn, Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems, Adv. Comput. Math., 42 (2016), 377-393. doi: 10.1007/s10444-015-9428-x. Google Scholar [22] I. Podlubny, Fractional Differential Equations, volume 198 of Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Google Scholar [23] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5 (2002), 367–386. Dedicated to the 60th anniversary of Prof. Francesco Mainardi. Google Scholar [24] J. Russell, Report of the committee on waves, Rep. Meet. Brit. Assoc. Adv. Sci., 7th Liverpool, 1837, London, John Murray.Google Scholar [25] M. Stynes, E. O'Riordan and J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. doi: 10.1137/16M1082329. Google Scholar [26] I. Turner, F. Liu, V. Anh and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245. doi: 10.1007/BF02936089. Google Scholar [27] L. Wei, Y. He, A. Yildirim and S. Kumar, Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, ZAMM Z. Angew. Math. Mech., 93 (2013), 14-28. doi: 10.1002/zamm.201200003. Google Scholar [28] G. H. Weiss, R. Klages, G. Radons and I. M. Sokolov (eds.), Anomalous transport: Foundations and applications [book review of WILEY-VCH Verlag GmbH & Co., Weinheim, 2008], J. Stat. Phys., 135 (2009), 389-391. doi: 10.1007/s10955-009-9713-5. Google Scholar [29] G. B. Witham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Google Scholar [30] N. Zabusky and M. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240-243. doi: 10.1103/PhysRevLett.15.240. Google Scholar [31] Q. Zhang, J. Zhang, S. Jiang and Z. Zhang, Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comput., 87 (2018), 693-719. doi: 10.1090/mcom/3229. Google Scholar
The numerical solution for Example 4.1 with $\alpha = 0.4$
$L^\infty(L^2)$ errors and orders of convergence on temporal direction for Example 4.1 with $r = (2-\alpha)/\alpha$
 N = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 3.0496E-2 1.1110E-2 3.9235E-3 1.3578E-3 4.6379E-4 1.5729E-4 1.4567 1.5016 1.5307 1.5498 1.5600 $\alpha = 0.6$ 3.8341E-2 1.5127E-2 5.8825E-3 2.2665E-3 8.6831E-4 3.3157E-4 1.3417 1.3626 1.3759 1.3842 1.3888 $\alpha = 0.8$ 5.9953E-2 2.6607E-2 1.1728E-2 5.1485E-3 2.2540E-3 9.8512E-4 1.1720 1.1817 1.1878 1.1916 1.1941
 N = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 3.0496E-2 1.1110E-2 3.9235E-3 1.3578E-3 4.6379E-4 1.5729E-4 1.4567 1.5016 1.5307 1.5498 1.5600 $\alpha = 0.6$ 3.8341E-2 1.5127E-2 5.8825E-3 2.2665E-3 8.6831E-4 3.3157E-4 1.3417 1.3626 1.3759 1.3842 1.3888 $\alpha = 0.8$ 5.9953E-2 2.6607E-2 1.1728E-2 5.1485E-3 2.2540E-3 9.8512E-4 1.1720 1.1817 1.1878 1.1916 1.1941
Errors and orders of convergence on space direction for Example 4.1 with $\alpha = 0.4$
 Polynomial M $\|u-u_h\|_{L^2}$ Order $\|u-u_h\|_{L^{\infty}}$ Order $P^2$ 5 5.3831E-01 - 3.2328E-01 - 10 7.8579E-02 2.7762 4.7729E-02 2.7598 20 9.9319E-03 2.9840 6.2124E-03 2.9416 40 1.1426E-04 3.1196 7.5845E-04 3.0340 $P^3$ 5 1.7236E-02 - 1.3819E-02 - 10 1.1399E-03 3.9184 8.7589E-04 3.9798 15 2.2712E-04 3.9406 1.7695E-04 3.9667 20 7.2979E-05 3.9418 6.1408E-04 3.9070
 Polynomial M $\|u-u_h\|_{L^2}$ Order $\|u-u_h\|_{L^{\infty}}$ Order $P^2$ 5 5.3831E-01 - 3.2328E-01 - 10 7.8579E-02 2.7762 4.7729E-02 2.7598 20 9.9319E-03 2.9840 6.2124E-03 2.9416 40 1.1426E-04 3.1196 7.5845E-04 3.0340 $P^3$ 5 1.7236E-02 - 1.3819E-02 - 10 1.1399E-03 3.9184 8.7589E-04 3.9798 15 2.2712E-04 3.9406 1.7695E-04 3.9667 20 7.2979E-05 3.9418 6.1408E-04 3.9070
$L^\infty(L^2)$ errors and orders of convergence on temporal direction for Example 4.2 with $r = (2-\alpha)/\alpha$
 N = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 2.6605E-2 9.8042E-3 3.4860E-3 1.2119E-3 4.1549E-4 1.4179E-4 1.4402 1.4918 1.5243 1.5444 1.5510 $\alpha = 0.6$ 3.0086E-2 1.2002E-2 4.6980E-3 1.8177E-3 6.9850E-4 2.6770E-4 1.3258 1.3531 1.3699 1.3797 1.3836 $\alpha = 0.8$ 4.1374E-2 1.8226E-2 7.9818E-3 3.4836E-3 1.5178E-3 6.6104E-4 1.1827 1.1912 1.1961 1.1985 1.1992
 N = 32 N = 64 N = 128 N = 256 $N = 512$ N = 1024 $\alpha = 0.4$ 2.6605E-2 9.8042E-3 3.4860E-3 1.2119E-3 4.1549E-4 1.4179E-4 1.4402 1.4918 1.5243 1.5444 1.5510 $\alpha = 0.6$ 3.0086E-2 1.2002E-2 4.6980E-3 1.8177E-3 6.9850E-4 2.6770E-4 1.3258 1.3531 1.3699 1.3797 1.3836 $\alpha = 0.8$ 4.1374E-2 1.8226E-2 7.9818E-3 3.4836E-3 1.5178E-3 6.6104E-4 1.1827 1.1912 1.1961 1.1985 1.1992
Errors and orders of convergence on space direction for Example 4.2 with $\alpha = 0.4$
 Polynomial M $\|u-u_h\|_{L^2}$ Order $\|u-u_h\|_{L^{\infty}}$ Order $P^2$ 5 3.8931E-01 - 2.3483E-01 - 10 5.6563E-02 2.7829 3.4418E-02 2.7704 20 7.1139E-03 2.9911 4.4696E-03 2.9449 40 7.7979E-04 3.1894 5.4210E-04 3.0435 $P^3$ 5 1.2812E-02 - 1.0564E-02 - 10 8.3809E-03 3.9342 6.7270E-04 3.9731 15 1.6615E-04 3.9552 1.2940E-04 4.0071 20 5.3162E-05 3.9564 4.3749E-05 3.9578
 Polynomial M $\|u-u_h\|_{L^2}$ Order $\|u-u_h\|_{L^{\infty}}$ Order $P^2$ 5 3.8931E-01 - 2.3483E-01 - 10 5.6563E-02 2.7829 3.4418E-02 2.7704 20 7.1139E-03 2.9911 4.4696E-03 2.9449 40 7.7979E-04 3.1894 5.4210E-04 3.0435 $P^3$ 5 1.2812E-02 - 1.0564E-02 - 10 8.3809E-03 3.9342 6.7270E-04 3.9731 15 1.6615E-04 3.9552 1.2940E-04 4.0071 20 5.3162E-05 3.9564 4.3749E-05 3.9578
$L^2$ errors and orders of convergence on temporal direction for Example 4.3 with $r = (2-\alpha)/\alpha$
 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 2.4959E-3 8.4989E-4 2.8741E-4 9.6635E-5 3.2367E-5 1.5542 1.5641 1.5720 1.5784 $\alpha = 0.6$ 6.0125E-3 2.3383E-3 8.9915E-4 3.4367E-4 1.3092E-4 1.3624 1.3788 1.3875 1.3923 $\alpha = 0.8$ 1.0359E-2 4.6808E-3 2.0899E-3 1.2645E-4 4.0879E-4 1.1460 1.1633 1.1736 1.1803
 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 2.4959E-3 8.4989E-4 2.8741E-4 9.6635E-5 3.2367E-5 1.5542 1.5641 1.5720 1.5784 $\alpha = 0.6$ 6.0125E-3 2.3383E-3 8.9915E-4 3.4367E-4 1.3092E-4 1.3624 1.3788 1.3875 1.3923 $\alpha = 0.8$ 1.0359E-2 4.6808E-3 2.0899E-3 1.2645E-4 4.0879E-4 1.1460 1.1633 1.1736 1.1803
$L^2$ errors and orders of convergence on temporal direction for Example 4.3 with $r = 2(2-\alpha)/\alpha$
 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 6.0380E-3 2.2285E-3 7.2999E-4 2.4866E-4 8.4023E-5 1.5110 1.5371 1.5536 1.5653 $\alpha = 0.6$ 1.1214E-2 4.4769E-3 1.7480E-3 6.7438E-4 2.5839E-4 1.3248 1.3567 1.3741 1.3839 $\alpha = 0.8$ 1.5602E-2 7.0291E-3 3.1157E-3 1.3694E-3 5.9926E-4 1.1503 1.1737 1.1859 1.1923
 N = 64 N = 128 N = 256 N = 512 N = 1024 $\alpha = 0.4$ 6.0380E-3 2.2285E-3 7.2999E-4 2.4866E-4 8.4023E-5 1.5110 1.5371 1.5536 1.5653 $\alpha = 0.6$ 1.1214E-2 4.4769E-3 1.7480E-3 6.7438E-4 2.5839E-4 1.3248 1.3567 1.3741 1.3839 $\alpha = 0.8$ 1.5602E-2 7.0291E-3 3.1157E-3 1.3694E-3 5.9926E-4 1.1503 1.1737 1.1859 1.1923
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