July  2021, 26(7): 3921-3942. doi: 10.3934/dcdsb.2020269

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

* Corresponding author: Xiaozhong Yang

Received  December 2019 Revised  July 2020 Published  July 2021 Early access  September 2020

Fund Project: The work was supported in part by the Subproject of Major Science and Technology Program of China (No.2017ZX07101001-01) and the National Natural Science Foundation of China (No.11371135)

Fractional telegraph equations are an important class of evolution equations and have widely applications in signal analysis such as transmission and propagation of electrical signals. Aiming at the one-dimensional time-fractional telegraph equation, a class of explicit-implicit (E-I) difference methods and implicit-explicit (I-E) difference methods are proposed. The two methods are based on a combination of the classical implicit difference method and the classical explicit difference method. Under the premise of smooth solution, theoretical analysis and numerical experiments show that the E-I and I-E difference schemes are unconditionally stable, with 2nd order spatial accuracy, $ 2-\alpha $ order time accuracy, and have significant time-saving, their calculation efficiency is higher than the classical implicit scheme. The research shows that the E-I and I-E difference methods constructed in this paper are effective for solving the time-fractional telegraph equation.

Citation: Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269
References:
[1]

T. Akram, M. Abbas, A. I. Ismail, N. Hj. M. Ali and D. Baleanu, Extended cubic B-splines in the numerical solution of time fractional telegraph equation, Adv. Differ. Equ., (2019), Paper No. 365, 20 pp. doi: 10.1186/s13662-019-2296-9.

[2]

D. J. Arrigo and S. G. Krantz, Analytical Techniques for Solving Nonlinear Partial Differential Equations, Morgan & Claypool Publishers, 2019. doi: 10.2200/S00907ED1V01Y201903MAS025.

[3]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114.  doi: 10.1016/j.jcp.2014.12.043.

[4]

R. C. CascavalE. C. EcksteinC. L. Frota and J. A. Goldstein, Fractional telegraph equations, J. Math. Anal. Appl., 276 (2002), 145-159.  doi: 10.1016/S0022-247X(02)00394-3.

[5]

J. ChenF. Liu and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377.  doi: 10.1016/j.jmaa.2007.06.023.

[6]

S. DasK. VishalP. K. Gupta and A. Yildirim, An approximate analytical solution of time-fractional telegraph equation, Appl. Math. Comput., 217 (2011), 7405-7411.  doi: 10.1016/j.amc.2011.02.030.

[7]

W. Deng and Z. Zhang, High Accuracy Algorithm for the Differentail Equations Governing Anomalous Diffusion, Algorithm and Models for Anomalous Diffusion, World Scientific, Singapore, 2019.

[8]

K. Diethelm, The Analysis of Fraction Differential Equations, Springer, 2010. doi: 10.1007/978-3-642-14574-2.

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M. Ferreira, M. M. Rodrigues and N. Vieira, First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac oprators, Adv. Appl. Clifford Algebr, 28 (2018), Art. 42, 14 pp. doi: 10.1007/s00006-018-0858-7.

[10]

N. J. FordM. M. RodriguesJ. Xiao and Y. Yan, Numerical analysis of a two-parameter fractional telegraph equation, J. Comput. Appl. Math., 249 (2013), 95-106.  doi: 10.1016/j.cam.2013.02.009.

[11] B. GuoX. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, 2011.  doi: 10.1142/9543.
[12]

L. Hervé and L. Brigitte, Partial Differential Equations: Modeling, Analysis and Numerical Approximation, Springer International Publishing, Switzerland, 2016.

[13]

M. H. HeydariM. R. Hooshmandasl and F. Mohammadi, Two-Dimensional legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6 (2014), 247-260.  doi: 10.4208/aamm.12-m12132.

[14]

V. R. HosseiniW. Chen and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem, 38 (2014), 31-39.  doi: 10.1016/j.enganabound.2013.10.009.

[15]

W. Jiang and Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3639-3645.  doi: 10.1016/j.cnsns.2010.12.019.

[16]

K. Kumar, R. K. Pandey and S. Yadav, Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms, Physica A, 535 (2019), Art. 122271, 15 pp. doi: 10.1016/j.physa.2019.122271.

[17] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015. 
[18]

C. LiZ. Zhao and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.

[19]

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.

[20]

F. LiuP. ZhuangV. AnhI. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191 (2007), 17-20.  doi: 10.1016/j.amc.2006.08.162.

[21] F. LiuP. Zhuang and Q. Liu, Numerical Solutions of Fractional Order Partial Differential Equations and its Applications, Science Press, Beijing, 2015. 
[22]

M. O. Mamchuev, Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method, Fract. Calc. Appl. Anal., 20 (2017), 190-211.  doi: 10.1515/fca-2017-0010.

[23]

S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170 (2005), 1126-1134.  doi: 10.1016/j.amc.2005.01.009.

[24]

E. Orsingher and L. Beghin, Time-fractional telegraph equations and telegraph processes with brownian time, Probab. Theory Relat. Fields, 128 (2004), 141-160.  doi: 10.1007/s00440-003-0309-8.

[25]

A. Saadatmandi and M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155-166. 

[26]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, World Book Incorporated Beijing, 2014. doi: 10.1007/978-1-4020-6042-7.

[27]

M. StynesE. O' Riordan and J. L. 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.

[28] Z. Sun and G. Gao, Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015. 
[29]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Volume II Applications, Higher Education Press; Springer, Heidelberg, Beijing, 2013. doi: 10.1007/978-3-642-33911-0.

[30]

S. VongP. Lyu and Z. Wang, A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under neumann boundary conditions, J. Sci. Comput., 66 (2016), 725-739.  doi: 10.1007/s10915-015-0040-5.

[31]

H. WangA. Cheng and K. Wang, Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 1427-1441.  doi: 10.3934/dcdsb.2015.20.1427.

[32]

L. WeiH. DaiD. Zhang and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.  doi: 10.1007/s10092-013-0084-6.

[33]

P. Xanthoulos and G. E. Zouraris, A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 239-263.  doi: 10.3934/dcdsb.2008.10.239.

[34]

X. Yang, L. Wu, S. Sun and X. Zhang, A universal difference method for time-space fractional Black-Scholes equation, Adv. Differ. Equ., (2016), Paper No. 71, 14 pp. doi: 10.1186/s13662-016-0792-8.

[35]

A. Yildirim, He's homtopy perturbation method for solving the space and time fractional telegraph equations, Int. J. Comput. Math., 87 (2010), 2998-3006.  doi: 10.1080/00207160902874653.

[36]

Z. Zhao and C. Li, Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975-2988.  doi: 10.1016/j.amc.2012.09.022.

show all references

References:
[1]

T. Akram, M. Abbas, A. I. Ismail, N. Hj. M. Ali and D. Baleanu, Extended cubic B-splines in the numerical solution of time fractional telegraph equation, Adv. Differ. Equ., (2019), Paper No. 365, 20 pp. doi: 10.1186/s13662-019-2296-9.

[2]

D. J. Arrigo and S. G. Krantz, Analytical Techniques for Solving Nonlinear Partial Differential Equations, Morgan & Claypool Publishers, 2019. doi: 10.2200/S00907ED1V01Y201903MAS025.

[3]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104-114.  doi: 10.1016/j.jcp.2014.12.043.

[4]

R. C. CascavalE. C. EcksteinC. L. Frota and J. A. Goldstein, Fractional telegraph equations, J. Math. Anal. Appl., 276 (2002), 145-159.  doi: 10.1016/S0022-247X(02)00394-3.

[5]

J. ChenF. Liu and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377.  doi: 10.1016/j.jmaa.2007.06.023.

[6]

S. DasK. VishalP. K. Gupta and A. Yildirim, An approximate analytical solution of time-fractional telegraph equation, Appl. Math. Comput., 217 (2011), 7405-7411.  doi: 10.1016/j.amc.2011.02.030.

[7]

W. Deng and Z. Zhang, High Accuracy Algorithm for the Differentail Equations Governing Anomalous Diffusion, Algorithm and Models for Anomalous Diffusion, World Scientific, Singapore, 2019.

[8]

K. Diethelm, The Analysis of Fraction Differential Equations, Springer, 2010. doi: 10.1007/978-3-642-14574-2.

[9]

M. Ferreira, M. M. Rodrigues and N. Vieira, First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac oprators, Adv. Appl. Clifford Algebr, 28 (2018), Art. 42, 14 pp. doi: 10.1007/s00006-018-0858-7.

[10]

N. J. FordM. M. RodriguesJ. Xiao and Y. Yan, Numerical analysis of a two-parameter fractional telegraph equation, J. Comput. Appl. Math., 249 (2013), 95-106.  doi: 10.1016/j.cam.2013.02.009.

[11] B. GuoX. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, 2011.  doi: 10.1142/9543.
[12]

L. Hervé and L. Brigitte, Partial Differential Equations: Modeling, Analysis and Numerical Approximation, Springer International Publishing, Switzerland, 2016.

[13]

M. H. HeydariM. R. Hooshmandasl and F. Mohammadi, Two-Dimensional legendre wavelets for solving time-fractional telegraph equation, Adv. Appl. Math. Mech., 6 (2014), 247-260.  doi: 10.4208/aamm.12-m12132.

[14]

V. R. HosseiniW. Chen and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem, 38 (2014), 31-39.  doi: 10.1016/j.enganabound.2013.10.009.

[15]

W. Jiang and Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3639-3645.  doi: 10.1016/j.cnsns.2010.12.019.

[16]

K. Kumar, R. K. Pandey and S. Yadav, Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms, Physica A, 535 (2019), Art. 122271, 15 pp. doi: 10.1016/j.physa.2019.122271.

[17] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, 2015. 
[18]

C. LiZ. Zhao and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.

[19]

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.

[20]

F. LiuP. ZhuangV. AnhI. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput, 191 (2007), 17-20.  doi: 10.1016/j.amc.2006.08.162.

[21] F. LiuP. Zhuang and Q. Liu, Numerical Solutions of Fractional Order Partial Differential Equations and its Applications, Science Press, Beijing, 2015. 
[22]

M. O. Mamchuev, Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method, Fract. Calc. Appl. Anal., 20 (2017), 190-211.  doi: 10.1515/fca-2017-0010.

[23]

S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170 (2005), 1126-1134.  doi: 10.1016/j.amc.2005.01.009.

[24]

E. Orsingher and L. Beghin, Time-fractional telegraph equations and telegraph processes with brownian time, Probab. Theory Relat. Fields, 128 (2004), 141-160.  doi: 10.1007/s00440-003-0309-8.

[25]

A. Saadatmandi and M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155-166. 

[26]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado (Editors), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, World Book Incorporated Beijing, 2014. doi: 10.1007/978-1-4020-6042-7.

[27]

M. StynesE. O' Riordan and J. L. 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.

[28] Z. Sun and G. Gao, Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015. 
[29]

V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers: Volume II Applications, Higher Education Press; Springer, Heidelberg, Beijing, 2013. doi: 10.1007/978-3-642-33911-0.

[30]

S. VongP. Lyu and Z. Wang, A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under neumann boundary conditions, J. Sci. Comput., 66 (2016), 725-739.  doi: 10.1007/s10915-015-0040-5.

[31]

H. WangA. Cheng and K. Wang, Fast finite volume methods for space-fractional diffusion equations, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 1427-1441.  doi: 10.3934/dcdsb.2015.20.1427.

[32]

L. WeiH. DaiD. Zhang and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.  doi: 10.1007/s10092-013-0084-6.

[33]

P. Xanthoulos and G. E. Zouraris, A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves, Discrete Contin. Dyn. Syst. Ser. B., 10 (2008), 239-263.  doi: 10.3934/dcdsb.2008.10.239.

[34]

X. Yang, L. Wu, S. Sun and X. Zhang, A universal difference method for time-space fractional Black-Scholes equation, Adv. Differ. Equ., (2016), Paper No. 71, 14 pp. doi: 10.1186/s13662-016-0792-8.

[35]

A. Yildirim, He's homtopy perturbation method for solving the space and time fractional telegraph equations, Int. J. Comput. Math., 87 (2010), 2998-3006.  doi: 10.1080/00207160902874653.

[36]

Z. Zhao and C. Li, Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219 (2012), 2975-2988.  doi: 10.1016/j.amc.2012.09.022.

Figure 1.  Curved surface of exact solution
Figure 2.  Curved surface of implicit scheme solution
Figure 3.  Curved surface of E-I scheme solution
Figure 4.  Curved surface of I-E scheme solution
Figure 5.  SRET of numerical solutions of three schemes
Figure 6.  The distribution of DTE in spatial grid
Figure 7.  Comparison of computing time among three schemes
Figure 8.  Curved surface of exact solution
Figure 9.  Curved surface of implicit scheme solution
Figure 10.  Curved surface of E-I scheme solution
Figure 11.  Curved surface of I-E scheme solution
Figure 12.  SRET of numerical solutions of two schemes
Figure 13.  The distribution of DTE in spatial grid
Table 1.  Comparison of numerical and exact solutions
21 41 61 81
Exact solution 0.032816 0.038767 0.045800 0.054104
Implicit scheme solution 0.032727 0.038662 0.045673 0.053957
E-I scheme solution 0.032238 0.037960 0.044992 0.053152
I-E scheme solution 0.032168 0.038002 0.044895 0.053037
21 41 61 81
Exact solution 0.032816 0.038767 0.045800 0.054104
Implicit scheme solution 0.032727 0.038662 0.045673 0.053957
E-I scheme solution 0.032238 0.037960 0.044992 0.053152
I-E scheme solution 0.032168 0.038002 0.044895 0.053037
Table 2.  Spatial convergence orders and numerical errors of implicit scheme, E-I scheme and I-E scheme($ \tau = h^{2}) $
$ \alpha $ M Implicit scheme E-I scheme I-E scheme
$ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1
0.6 6 0.601261438 0.670908875 0.676867797
12 0.171331703 1.811200301 0.185496548 1.854724481 0.185742355 1.865571243
24 0.027626239 1.861354661 0.020497637 1.880049324 0.056380674 1.888092124
0.7 6 0.572754131 0.625245364 0.632403830
12 0.154051855 1.894499914 0.167207909 1.902779364 0.167706812 1.914904807
24 0.043125187 1.968989813 0.042627245 1.954591537 0.042269721 1.953396613
0.8 6 0.331546455 0.384936527 0.392207783
12 0.093281787 1.829543269 0.000168779 1.942187657 0.000592926 1.963089311
24 0.025790914 1.887189262 0.042627245 1.954591537 0.042269721 1.953396613
0.9 6 0.312856198 0.351172503 0.354347245
12 0.079460269 1.977194091 0.084117921 2.061694794 0.084447694 2.069033890
24 0.020221416 1.978009907 0.021169310 2.057999271 0.021427164 2.056110578
$ \alpha $ M Implicit scheme E-I scheme I-E scheme
$ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1
0.6 6 0.601261438 0.670908875 0.676867797
12 0.171331703 1.811200301 0.185496548 1.854724481 0.185742355 1.865571243
24 0.027626239 1.861354661 0.020497637 1.880049324 0.056380674 1.888092124
0.7 6 0.572754131 0.625245364 0.632403830
12 0.154051855 1.894499914 0.167207909 1.902779364 0.167706812 1.914904807
24 0.043125187 1.968989813 0.042627245 1.954591537 0.042269721 1.953396613
0.8 6 0.331546455 0.384936527 0.392207783
12 0.093281787 1.829543269 0.000168779 1.942187657 0.000592926 1.963089311
24 0.025790914 1.887189262 0.042627245 1.954591537 0.042269721 1.953396613
0.9 6 0.312856198 0.351172503 0.354347245
12 0.079460269 1.977194091 0.084117921 2.061694794 0.084447694 2.069033890
24 0.020221416 1.978009907 0.021169310 2.057999271 0.021427164 2.056110578
Table 3.  Time convergence orders and numerical errors of implicit scheme, E-I scheme and I-E scheme($ h = \frac{1}{40} $)
$ \alpha $ N Implicit scheme E-I scheme I-E scheme
$ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2
0.6 300 0.001748225 0.001257729 0.001257729
600 0.000647496 1.432945906 0.000769857 1.408158851 0.000795404 1.415571243
1200 0.000238734 1.437531875 0.000300146 1.406435732 0.000300015 1.406473635
0.7 300 0.003765612 0.002220764 0.002081502
600 0.001462733 1.364217584 0.000902026 1.299814239 0.000902020 1.206383164
1200 0.000662609 1.342435234 0.000608222 1.325685702 0.000608231 1.322349016
0.8 300 0.002921061 0.001592343 0.001453911
600 0.001224089 1.254076687 0.000647496 1.298205933 0.000647496 1.236914092
1200 0.000532251 1.205163582 0.000267776 1.296425166 0.000264108 1.293815168
0.9 300 0.003178454 0.002285670 0.002149714
600 0.001481610 1.101159256 0.000973810 1.230904340 0.000910832 1.238886806
1200 0.000791962 1.052349016 0.000348563 1.116311302 0.000451147 1.116552467
$ \alpha $ N Implicit scheme E-I scheme I-E scheme
$ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2
0.6 300 0.001748225 0.001257729 0.001257729
600 0.000647496 1.432945906 0.000769857 1.408158851 0.000795404 1.415571243
1200 0.000238734 1.437531875 0.000300146 1.406435732 0.000300015 1.406473635
0.7 300 0.003765612 0.002220764 0.002081502
600 0.001462733 1.364217584 0.000902026 1.299814239 0.000902020 1.206383164
1200 0.000662609 1.342435234 0.000608222 1.325685702 0.000608231 1.322349016
0.8 300 0.002921061 0.001592343 0.001453911
600 0.001224089 1.254076687 0.000647496 1.298205933 0.000647496 1.236914092
1200 0.000532251 1.205163582 0.000267776 1.296425166 0.000264108 1.293815168
0.9 300 0.003178454 0.002285670 0.002149714
600 0.001481610 1.101159256 0.000973810 1.230904340 0.000910832 1.238886806
1200 0.000791962 1.052349016 0.000348563 1.116311302 0.000451147 1.116552467
Table 4.  Comparison of numerical and exact solutions
21 41 61 81
Exact solution -0.03079 -0.04618 -0.04618 -0.03079
Implicit scheme solution -0.03008 -0.04505 -0.04505 -0.03008
E-I scheme solution -0.03001 -0.04492 -0.04492 -0.03001
I-E scheme solution -0.03001 -0.04492 -0.04492 -0.03001
21 41 61 81
Exact solution -0.03079 -0.04618 -0.04618 -0.03079
Implicit scheme solution -0.03008 -0.04505 -0.04505 -0.03008
E-I scheme solution -0.03001 -0.04492 -0.04492 -0.03001
I-E scheme solution -0.03001 -0.04492 -0.04492 -0.03001
Table 5.  Spatial convergence orders and numerical errors of implicit scheme and E-I scheme ($ \tau = h^{2}) $
$ \alpha $ M Implicit scheme E-I scheme
$ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1
8 0.007395715 0.006928145
0.2 16 0.001926582 1.940537064 0.001786161 1.955606884
32 0.000486619 1.985178414 0.000447580 1.996642455
64 0.000121970 1.996305831 0.000110261 2.004338515
8 0.007399299 0.006730926
0.3 16 0.001926975 1.944911724 0.001748227 1.944911722
32 0.000486657 1.996642453 0.000440517 1.988623313
64 0.000121970 1.996374655 0.000110261 1.998266445
8 0.007406310 0.006655592
0.4 16 0.001927758 1.941830430 0.001736738 1.938186701
32 0.000486747 1.985359045 0.000438647 1.988623313
64 1.988623313 1.996518415 0.000109876 1.996547989
8 0.007418404 0.006638599
0.5 16 0.001929358 1.942987829 0.001735124 1.935839557
32 0.000486961 1.986241089 0.000438424 1.984638177
64 0.000122001 1.996814720 0.000109876 1.996450948
$ \alpha $ M Implicit scheme E-I scheme
$ E_{2}(h, \tau) $ Order1 $ E_{2}(h, \tau) $ Order1
8 0.007395715 0.006928145
0.2 16 0.001926582 1.940537064 0.001786161 1.955606884
32 0.000486619 1.985178414 0.000447580 1.996642455
64 0.000121970 1.996305831 0.000110261 2.004338515
8 0.007399299 0.006730926
0.3 16 0.001926975 1.944911724 0.001748227 1.944911722
32 0.000486657 1.996642453 0.000440517 1.988623313
64 0.000121970 1.996374655 0.000110261 1.998266445
8 0.007406310 0.006655592
0.4 16 0.001927758 1.941830430 0.001736738 1.938186701
32 0.000486747 1.985359045 0.000438647 1.988623313
64 1.988623313 1.996518415 0.000109876 1.996547989
8 0.007418404 0.006638599
0.5 16 0.001929358 1.942987829 0.001735124 1.935839557
32 0.000486961 1.986241089 0.000438424 1.984638177
64 0.000122001 1.996814720 0.000109876 1.996450948
Table 6.  Time convergence orders and numerical errors of implicit scheme and E-I scheme ($ h = \frac{1}{40} $)
$ \alpha $ N Implicit scheme E-I scheme
$ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2
150 0.003247111 0.003127877
0.2 300 0.0038902781 0.260726667 0.003593341 0.200141963
600 0.0046878951 0.269069826 0.000447580 0.211580117
150 0.003053496 0.002880601
0.3 300 0.003751789 0.297116513 0.003603101 0.322869023
600 0.004664256 0.314069826 0.004535791 0.332115801
150 0.002847062 0.002645836
0.4 300 0.003604515 0.340331208 0.003490906 0.399877794
600 0.004639253 0.359191478 0.004623847 0.405493914
150 0.002620013 0.002391991
0.5 300 0.003442873 0.394038915 0.003330367 0.477464645
600 0.004561314 0.405837085 0.004690116 0.493943176
$ \alpha $ N Implicit scheme E-I scheme
$ E_{2}(h, \tau) $ Order2 $ E_{2}(h, \tau) $ Order2
150 0.003247111 0.003127877
0.2 300 0.0038902781 0.260726667 0.003593341 0.200141963
600 0.0046878951 0.269069826 0.000447580 0.211580117
150 0.003053496 0.002880601
0.3 300 0.003751789 0.297116513 0.003603101 0.322869023
600 0.004664256 0.314069826 0.004535791 0.332115801
150 0.002847062 0.002645836
0.4 300 0.003604515 0.340331208 0.003490906 0.399877794
600 0.004639253 0.359191478 0.004623847 0.405493914
150 0.002620013 0.002391991
0.5 300 0.003442873 0.394038915 0.003330367 0.477464645
600 0.004561314 0.405837085 0.004690116 0.493943176
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