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doi: 10.3934/dcdss.2020402

Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme

1. 

Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran, Iran

2. 

Department of Mathematics, University of Mazandaran, Babolsar, Iran, Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa

3. 

Department of Mathematics, Cankaya University, Ankara, Turkey, Institute of Space Sciences, Magurele-Bucharest, Romania

* Corresponding author: Hamid Safdari

Received  December 2019 Revised  January 2020 Published  June 2020

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order $ \mathcal{O}(\delta \tau ^{2}) $ is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.

Citation: Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020402
References:
[1]

M. AbdelhakemH. MoussaD. Baleanu and M. El-Kady, Shifted Chebyshev schemes for solving fractional optimal control problems, Journal of Vibration and Control, 25 (2019), 2143-2150.  doi: 10.1177/1077546319852218.  Google Scholar

[2]

M. BadrA. Yazdani and H. Jafari, Stability of a finite volume element method for the time–fractional advection–diffusion equation, Numerical Methods for Partial Differential Equations, 34 (2018), 1459-1471.  doi: 10.1002/num.22243.  Google Scholar

[3]

V. O. Bohaienko, A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative, Comput. Appl. Math., 38 (2019), 105, 21 pp. doi: 10.1007/s40314-019-0878-5.  Google Scholar

[4]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition. Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[5]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to complex geometries and applications to fluid dynamics. Scientific Computation. Springer, Berlin, 2007.  Google Scholar

[6]

M. Caputo, Linear models of dissipation whose {Q} is almost frequency independent–ii, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x.  Google Scholar

[7]

V. J. ErvinN. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), 572-591.  doi: 10.1137/050642757.  Google Scholar

[8]

R. M. Ganji and H. Jafari, Numerical solution of variable order integro-differential equations, Advanced Mathematical Models & Applications, 4 (2019), 64-69.   Google Scholar

[9]

M. M. GhalibA. A. ZafarZ. HammouchM. B. Riaz and K. Shabbir, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Discrete & Continuous Dynamical Systems - S, 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.  Google Scholar

[10]

A. Golbabai, O. Nikan and T. Nikazad, Numerical analysis of time fractional {B}lack–{S}choles european option pricing model arising in financial market, Comput. Appl. Math., 38 (2019), Paper No. 173, 24 pp. doi: 10.1007/s40314-019-0957-7.  Google Scholar

[11]

A. GoswamiJ. Singh and D. Kumar et al, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Phys. A, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.  Google Scholar

[12]

H. HassaniJ. A. Tenreiro Machado and E. Naraghirad, Generalized shifted Chebyshev polynomials for fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 75 (2019), 50-61.  doi: 10.1016/j.cnsns.2019.03.013.  Google Scholar

[13]

B. I. Henry and S. L. Wearne, Existence of turing instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math., 62 (2001/02), 870-887.  doi: 10.1137/S0036139900375227.  Google Scholar

[14]

M. H. Heydari, A. Atangana and Z. Avazzadeh, Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation, Engineering with Computers, (2019), 1–12. doi: 10.1007/s00366-019-00889-9.  Google Scholar

[15]

M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2535-2542.  doi: 10.1016/j.cnsns.2010.09.007.  Google Scholar

[16]

M. A. Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis e via the {C}aputo–{F}abrizio derivative, Math. Model. Nat. Phenom., 14 (2019), Paper No. 311, 19 pp. doi: 10.1051/mmnp/2018074.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[19]

D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2020), 443-457.  doi: 10.1002/mma.5903.  Google Scholar

[20]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[21]

O. Nikan, A. Golbabai, J. A. Tenreiro Machado and T. Nikazad, Numerical solution of the fractional {R}ayleigh–{S}tokes model arising in a heated generalized second-grade fluid, Engineering with Computers, (2020), 1–14. doi: 10.1007/s00366-019-00913-y.  Google Scholar

[22]

O. Nikan, J. A. Tenreiro Machado, A. Golbabai and T. Nikazad, Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media, International Communications in Heat and Mass Transfer, 111 (2020), 104443. doi: 10.1016/j.icheatmasstransfer.2019.104443.  Google Scholar

[23]

K. M. Owolabi and A. Atangana, High-order solvers for space-fractional differential equations with Riesz derivative, Discrete & Continuous Dynamical Systems-S, 12 (2019), 567-590.  doi: 10.3934/dcdss.2019037.  Google Scholar

[24]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[25]

L. Ren and L. Liu, A high-order compact difference method for time fractional Fokker–Planck equations with variable coefficients, Comput. Appl. Math., 38 (2019), Paper No. 101, 16 pp. doi: 10.1007/s40314-019-0865-x.  Google Scholar

[26]

E. Reyes-MeloJ. Martinez-VegaC. Guerrero-Salazar and U. Ortiz-Mendez, Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, Journal of Applied Polymer Science, 98 (2005), 923-935.  doi: 10.1002/app.22057.  Google Scholar

[27]

A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1135-1142.  doi: 10.1016/j.camwa.2011.04.014.  Google Scholar

[28]

J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas–Milovic model with Mittag-Leffler law, Math. Model. Nat. Phenom., 14 (2019), Paper No. 303, 23 pp. doi: 10.1051/mmnp/2018068.  Google Scholar

[29]

E. Sousa, Numerical approximations for fractional diffusion equations via splines, Comput. Math. Appl., 62 (2011), 938-944.  doi: 10.1016/j.camwa.2011.04.015.  Google Scholar

[30]

N. H. SweilamA. M. Nagy and A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Solitons Fractals, 73 (2015), 141-147.  doi: 10.1016/j.chaos.2015.01.010.  Google Scholar

[31]

N. SweilamA. Nagy and A. A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, Journal of King Saud University-Science, 28 (2016), 41-47.   Google Scholar

[32]

C. TadjeranM. M. Meerschaert and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205-213.  doi: 10.1016/j.jcp.2005.08.008.  Google Scholar

[33]

S. UllahM. Altaf Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Solitons Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001.  Google Scholar

[34]

S. Ullah, M. Altaf Khan and M. Farooq, A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 237. doi: 10.1016/j.chaos.2018.09.001.  Google Scholar

[35]

S. UllahM. Altaf KhanM. FarooqZ. Hammouch and D. Baleanu, A fractional model for the dynamics of tuberculosis infection using {C}aputo-{F}abrizio derivative, Discrete & Continuous Dynamical Systems-S, 13 (2020), 975-993.  doi: 10.3934/dcdss.2020057.  Google Scholar

show all references

References:
[1]

M. AbdelhakemH. MoussaD. Baleanu and M. El-Kady, Shifted Chebyshev schemes for solving fractional optimal control problems, Journal of Vibration and Control, 25 (2019), 2143-2150.  doi: 10.1177/1077546319852218.  Google Scholar

[2]

M. BadrA. Yazdani and H. Jafari, Stability of a finite volume element method for the time–fractional advection–diffusion equation, Numerical Methods for Partial Differential Equations, 34 (2018), 1459-1471.  doi: 10.1002/num.22243.  Google Scholar

[3]

V. O. Bohaienko, A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative, Comput. Appl. Math., 38 (2019), 105, 21 pp. doi: 10.1007/s40314-019-0878-5.  Google Scholar

[4]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition. Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[5]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to complex geometries and applications to fluid dynamics. Scientific Computation. Springer, Berlin, 2007.  Google Scholar

[6]

M. Caputo, Linear models of dissipation whose {Q} is almost frequency independent–ii, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x.  Google Scholar

[7]

V. J. ErvinN. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal., 45 (2007), 572-591.  doi: 10.1137/050642757.  Google Scholar

[8]

R. M. Ganji and H. Jafari, Numerical solution of variable order integro-differential equations, Advanced Mathematical Models & Applications, 4 (2019), 64-69.   Google Scholar

[9]

M. M. GhalibA. A. ZafarZ. HammouchM. B. Riaz and K. Shabbir, Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary, Discrete & Continuous Dynamical Systems - S, 13 (2020), 683-693.  doi: 10.3934/dcdss.2020037.  Google Scholar

[10]

A. Golbabai, O. Nikan and T. Nikazad, Numerical analysis of time fractional {B}lack–{S}choles european option pricing model arising in financial market, Comput. Appl. Math., 38 (2019), Paper No. 173, 24 pp. doi: 10.1007/s40314-019-0957-7.  Google Scholar

[11]

A. GoswamiJ. Singh and D. Kumar et al, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Phys. A, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.  Google Scholar

[12]

H. HassaniJ. A. Tenreiro Machado and E. Naraghirad, Generalized shifted Chebyshev polynomials for fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 75 (2019), 50-61.  doi: 10.1016/j.cnsns.2019.03.013.  Google Scholar

[13]

B. I. Henry and S. L. Wearne, Existence of turing instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math., 62 (2001/02), 870-887.  doi: 10.1137/S0036139900375227.  Google Scholar

[14]

M. H. Heydari, A. Atangana and Z. Avazzadeh, Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation, Engineering with Computers, (2019), 1–12. doi: 10.1007/s00366-019-00889-9.  Google Scholar

[15]

M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2535-2542.  doi: 10.1016/j.cnsns.2010.09.007.  Google Scholar

[16]

M. A. Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis e via the {C}aputo–{F}abrizio derivative, Math. Model. Nat. Phenom., 14 (2019), Paper No. 311, 19 pp. doi: 10.1051/mmnp/2018074.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[19]

D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods Appl. Sci., 43 (2020), 443-457.  doi: 10.1002/mma.5903.  Google Scholar

[20]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[21]

O. Nikan, A. Golbabai, J. A. Tenreiro Machado and T. Nikazad, Numerical solution of the fractional {R}ayleigh–{S}tokes model arising in a heated generalized second-grade fluid, Engineering with Computers, (2020), 1–14. doi: 10.1007/s00366-019-00913-y.  Google Scholar

[22]

O. Nikan, J. A. Tenreiro Machado, A. Golbabai and T. Nikazad, Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media, International Communications in Heat and Mass Transfer, 111 (2020), 104443. doi: 10.1016/j.icheatmasstransfer.2019.104443.  Google Scholar

[23]

K. M. Owolabi and A. Atangana, High-order solvers for space-fractional differential equations with Riesz derivative, Discrete & Continuous Dynamical Systems-S, 12 (2019), 567-590.  doi: 10.3934/dcdss.2019037.  Google Scholar

[24]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[25]

L. Ren and L. Liu, A high-order compact difference method for time fractional Fokker–Planck equations with variable coefficients, Comput. Appl. Math., 38 (2019), Paper No. 101, 16 pp. doi: 10.1007/s40314-019-0865-x.  Google Scholar

[26]

E. Reyes-MeloJ. Martinez-VegaC. Guerrero-Salazar and U. Ortiz-Mendez, Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, Journal of Applied Polymer Science, 98 (2005), 923-935.  doi: 10.1002/app.22057.  Google Scholar

[27]

A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1135-1142.  doi: 10.1016/j.camwa.2011.04.014.  Google Scholar

[28]

J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas–Milovic model with Mittag-Leffler law, Math. Model. Nat. Phenom., 14 (2019), Paper No. 303, 23 pp. doi: 10.1051/mmnp/2018068.  Google Scholar

[29]

E. Sousa, Numerical approximations for fractional diffusion equations via splines, Comput. Math. Appl., 62 (2011), 938-944.  doi: 10.1016/j.camwa.2011.04.015.  Google Scholar

[30]

N. H. SweilamA. M. Nagy and A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos Solitons Fractals, 73 (2015), 141-147.  doi: 10.1016/j.chaos.2015.01.010.  Google Scholar

[31]

N. SweilamA. Nagy and A. A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, Journal of King Saud University-Science, 28 (2016), 41-47.   Google Scholar

[32]

C. TadjeranM. M. Meerschaert and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205-213.  doi: 10.1016/j.jcp.2005.08.008.  Google Scholar

[33]

S. UllahM. Altaf Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Solitons Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001.  Google Scholar

[34]

S. Ullah, M. Altaf Khan and M. Farooq, A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 237. doi: 10.1016/j.chaos.2018.09.001.  Google Scholar

[35]

S. UllahM. Altaf KhanM. FarooqZ. Hammouch and D. Baleanu, A fractional model for the dynamics of tuberculosis infection using {C}aputo-{F}abrizio derivative, Discrete & Continuous Dynamical Systems-S, 13 (2020), 975-993.  doi: 10.3934/dcdss.2020057.  Google Scholar

Figure 1.  Plots of the approximate solution (left side) and absolute error (right side) of Example 5.1 at $ T = 1 $, $ M = 400 $ and $ N = 5 $
Figure 2.  The maximum absolute error and error norm $ L_{2} $ of Example 5.1 at $ T = 1 $, $ N = 5 $ and $ M = 200,400,600, \ldots, 3000 $
Figure 3.  Error histories of Example 5.1 at $ T = 1 $, $ N = 5 $ and $ M = 100,200,400,800, 1600 $
Figure 4.  Error histories of Example 5.1 at $ T = 1 $, $ M = 400 $ and $ N = 3, 5, 7, 9 $
Figure 5.  Error histories of Example 5.2 at $ T = 1 $, with $ M = 100,200,400,800, 1600, $ $ N = 5 $ (left side) and $ N = 7 $ (right side)
Table 1.  The absolute error of Example 5.1 at $ T = 1 $
$ x $ with $N=7$ in with $N=7$ in with $N=3$ in our method with $N=3$
[15] [27] [30]
$ 0 $ $ 2.81\times 10^{-5} $ $ 0 $ $ 0 $ $ 4.77\times 10^{-17} $
$ 0.1 $ $ 4.26\times 10^{-5} $ $ 4.66\times 10^{-5} $ $ 5.46\times 10^{-6} $ $ 3.17\times 10^{-9} $
$ 0.2 $ $ 5.39\times 10^{-5} $ $ 7.74\times 10^{-5} $ $ 8.51\times 10^{-6} $ $ 5.85\times 10^{-9} $
$ 0.3 $ $ 6.12\times 10^{-5} $ $ 5.00\times 10^{-5} $ $ 9.60\times 10^{-6} $ $ 7.97\times 10^{-9} $
$ 0.4 $ $ 6.48\times 10^{-5} $ $ 2.30\times 10^{-5} $ $ 9.18\times 10^{-6} $ $ 9.44\times 10^{-9} $
$ 0.5 $ $ 6.45\times 10^{-5} $ $ 2.74\times 10^{-5} $ $ 7.69\times 10^{-6} $ $ 1.02\times 10^{-8} $
$ 0.6 $ $ 5.98\times 10^{-5} $ $ 4.38\times 10^{-5} $ $ 5.60\times 10^{-6} $ $ 1.01\times 10^{-8} $
$ 0.7 $ $ 5.23\times 10^{-5} $ $ 3.87\times 10^{-5} $ $ 3.33\times 10^{-6} $ $ 9.12\times 10^{-9} $
$ 0.8 $ $ 4.48\times 10^{-5} $ $ 1.01\times 10^{-5} $ $ 1.34\times 10^{-6} $ $ 7.17\times 10^{-9} $
$ 0.9 $ $ 3.91\times 10^{-5} $ $ 3.35\times 10^{-5} $ $ 8.39\times 10^{-8} $ $ 4.16\times 10^{-9} $
$ 1.0 $ $ 2.81\times 10^{-5} $ $ 0 $ $ 0 $ $ 7.55\times 10^{-17} $
$ x $ with $N=7$ in with $N=7$ in with $N=3$ in our method with $N=3$
[15] [27] [30]
$ 0 $ $ 2.81\times 10^{-5} $ $ 0 $ $ 0 $ $ 4.77\times 10^{-17} $
$ 0.1 $ $ 4.26\times 10^{-5} $ $ 4.66\times 10^{-5} $ $ 5.46\times 10^{-6} $ $ 3.17\times 10^{-9} $
$ 0.2 $ $ 5.39\times 10^{-5} $ $ 7.74\times 10^{-5} $ $ 8.51\times 10^{-6} $ $ 5.85\times 10^{-9} $
$ 0.3 $ $ 6.12\times 10^{-5} $ $ 5.00\times 10^{-5} $ $ 9.60\times 10^{-6} $ $ 7.97\times 10^{-9} $
$ 0.4 $ $ 6.48\times 10^{-5} $ $ 2.30\times 10^{-5} $ $ 9.18\times 10^{-6} $ $ 9.44\times 10^{-9} $
$ 0.5 $ $ 6.45\times 10^{-5} $ $ 2.74\times 10^{-5} $ $ 7.69\times 10^{-6} $ $ 1.02\times 10^{-8} $
$ 0.6 $ $ 5.98\times 10^{-5} $ $ 4.38\times 10^{-5} $ $ 5.60\times 10^{-6} $ $ 1.01\times 10^{-8} $
$ 0.7 $ $ 5.23\times 10^{-5} $ $ 3.87\times 10^{-5} $ $ 3.33\times 10^{-6} $ $ 9.12\times 10^{-9} $
$ 0.8 $ $ 4.48\times 10^{-5} $ $ 1.01\times 10^{-5} $ $ 1.34\times 10^{-6} $ $ 7.17\times 10^{-9} $
$ 0.9 $ $ 3.91\times 10^{-5} $ $ 3.35\times 10^{-5} $ $ 8.39\times 10^{-8} $ $ 4.16\times 10^{-9} $
$ 1.0 $ $ 2.81\times 10^{-5} $ $ 0 $ $ 0 $ $ 7.55\times 10^{-17} $
Table 2.  The absolute error of Example 5.1 at $ T = 2 $
$ x $ with $N=5$ in with $N=5$ in with $N=3$ in our method with $N=3$
$ $ [15] [27] [30]
$ 0 $ $ 2.74\times 10^{-5} $ $ 0 $ $ 0 $ $ 1.86\times 10^{-17} $
$ 0.1 $ $ 4.20\times 10^{-5} $ $ 4.47\times 10^{-6} $ $ 3.33\times 10^{-6} $ $ 1.28\times 10^{-8} $
$ 0.2 $ $ 3.76\times 10^{-5} $ $ 2.78\times 10^{-7} $ $ 5.65\times 10^{-6} $ $ 2.05\times 10^{-8} $
$ 0.3 $ $ 8.44\times 10^{-5} $ $ 5.81\times 10^{-6} $ $ 7.05\times 10^{-6} $ $ 2.40\times 10^{-8} $
$ 0.4 $ $ 3.27\times 10^{-5} $ $ 1.02\times 10^{-5} $ $ 7.64\times 10^{-6} $ $ 2.40\times 10^{-8} $
$ 0.5 $ $ 3.61\times 10^{-5} $ $ 1.17\times 10^{-5} $ $ 7.52\times 10^{-6} $ $ 2.15\times 10^{-8} $
$ 0.6 $ $ 1.94\times 10^{-5} $ $ 1.08\times 10^{-5} $ $ 6.80\times 10^{-6} $ $ 1.72\times 10^{-8} $
$ 0.7 $ $ 2.95\times 10^{-5} $ $ 8.54\times 10^{-6} $ $ 5.59\times 10^{-6} $ $ 1.21\times 10^{-8} $
$ 0.8 $ $ 4.92\times 10^{-5} $ $ 6.06\times 10^{-6} $ $ 3.98\times 10^{-6} $ $ 6.93\times 10^{-9} $
$ 0.9 $ $ 2.83\times 10^{-5} $ $ 3.67\times 10^{-6} $ $ 2.08\times 10^{-6} $ $ 2.62\times 10^{-9} $
$ 1.0 $ $ 7.73\times 10^{-5} $ $ 0 $ $ 0 $ $ 8.24\times 10^{-18} $
$ x $ with $N=5$ in with $N=5$ in with $N=3$ in our method with $N=3$
$ $ [15] [27] [30]
$ 0 $ $ 2.74\times 10^{-5} $ $ 0 $ $ 0 $ $ 1.86\times 10^{-17} $
$ 0.1 $ $ 4.20\times 10^{-5} $ $ 4.47\times 10^{-6} $ $ 3.33\times 10^{-6} $ $ 1.28\times 10^{-8} $
$ 0.2 $ $ 3.76\times 10^{-5} $ $ 2.78\times 10^{-7} $ $ 5.65\times 10^{-6} $ $ 2.05\times 10^{-8} $
$ 0.3 $ $ 8.44\times 10^{-5} $ $ 5.81\times 10^{-6} $ $ 7.05\times 10^{-6} $ $ 2.40\times 10^{-8} $
$ 0.4 $ $ 3.27\times 10^{-5} $ $ 1.02\times 10^{-5} $ $ 7.64\times 10^{-6} $ $ 2.40\times 10^{-8} $
$ 0.5 $ $ 3.61\times 10^{-5} $ $ 1.17\times 10^{-5} $ $ 7.52\times 10^{-6} $ $ 2.15\times 10^{-8} $
$ 0.6 $ $ 1.94\times 10^{-5} $ $ 1.08\times 10^{-5} $ $ 6.80\times 10^{-6} $ $ 1.72\times 10^{-8} $
$ 0.7 $ $ 2.95\times 10^{-5} $ $ 8.54\times 10^{-6} $ $ 5.59\times 10^{-6} $ $ 1.21\times 10^{-8} $
$ 0.8 $ $ 4.92\times 10^{-5} $ $ 6.06\times 10^{-6} $ $ 3.98\times 10^{-6} $ $ 6.93\times 10^{-9} $
$ 0.9 $ $ 2.83\times 10^{-5} $ $ 3.67\times 10^{-6} $ $ 2.08\times 10^{-6} $ $ 2.62\times 10^{-9} $
$ 1.0 $ $ 7.73\times 10^{-5} $ $ 0 $ $ 0 $ $ 8.24\times 10^{-18} $
Table 3.  The absolute error of Example 5.1 at $ T = 10 $
$ x $ $ N=3 $ $ N=5 $ $ N=7 $
$ 0 $ $ 5.82\times 10^{-21} $ $ 5.93\times 10^{-22} $ $ 4.43\times 10^{-21} $
$ 0.2 $ $ 1.01\times 10^{-9} $ $ 4.74\times 10^{-9} $ $ 2.28\times 10^{-9} $
$ 0.4 $ $ 8.21\times 10^{-9} $ $ 8.11\times 10^{-9} $ $ 4.21\times 10^{-9} $
$ 0.6 $ $ 1.28\times 10^{-9} $ $ 1.17\times 10^{-9} $ $ 1.15\times 10^{-9} $
$ 0.8 $ $ 3.76\times 10^{-9} $ $ 7.93\times 10^{-10} $ $ 2.71\times 10^{-10} $
$ 1.0 $ $ 4.34\times 10^{-21} $ $ 3.78\times 10^{-21} $ $ 1.14\times 10^{-22} $
$ x $ $ N=3 $ $ N=5 $ $ N=7 $
$ 0 $ $ 5.82\times 10^{-21} $ $ 5.93\times 10^{-22} $ $ 4.43\times 10^{-21} $
$ 0.2 $ $ 1.01\times 10^{-9} $ $ 4.74\times 10^{-9} $ $ 2.28\times 10^{-9} $
$ 0.4 $ $ 8.21\times 10^{-9} $ $ 8.11\times 10^{-9} $ $ 4.21\times 10^{-9} $
$ 0.6 $ $ 1.28\times 10^{-9} $ $ 1.17\times 10^{-9} $ $ 1.15\times 10^{-9} $
$ 0.8 $ $ 3.76\times 10^{-9} $ $ 7.93\times 10^{-10} $ $ 2.71\times 10^{-10} $
$ 1.0 $ $ 4.34\times 10^{-21} $ $ 3.78\times 10^{-21} $ $ 1.14\times 10^{-22} $
Table 4.  The convergence order, the errors $ L_{2} $ and $ L_{\infty} $ for Example 5.1 with $ T = 1 $ and $ N = 3 $
$ \delta \tau $ $ L_{\infty} $ $ C_{\delta \tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.62773\times 10^{-7} $ $ 3.76647\times 10^{-7} $
$ \frac{1}{200} $ $ 4.06928\times 10^{-8} $ $ 2.00002 $ $ 9.41607\times 10^{-8} $ $ 2.00002 $
$ \frac{1}{400} $ $ 1.01732\times 10^{-8} $ $ 2.00000 $ $ 2.35401\times 10^{-8} $ $ 2.00000 $
$ \frac{1}{800} $ $ 2.54329\times 10^{-9} $ $ 2.00000 $ $ 5.88503\times 10^{-9} $ $ 2.00000 $
$ \frac{1}{1600} $ $ 6.35828\times 10^{-10} $ $ 1.99999 $ $ 1.47127\times 10^{-9} $ $ 1.99999 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
$ \delta \tau $ $ L_{\infty} $ $ C_{\delta \tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.62773\times 10^{-7} $ $ 3.76647\times 10^{-7} $
$ \frac{1}{200} $ $ 4.06928\times 10^{-8} $ $ 2.00002 $ $ 9.41607\times 10^{-8} $ $ 2.00002 $
$ \frac{1}{400} $ $ 1.01732\times 10^{-8} $ $ 2.00000 $ $ 2.35401\times 10^{-8} $ $ 2.00000 $
$ \frac{1}{800} $ $ 2.54329\times 10^{-9} $ $ 2.00000 $ $ 5.88503\times 10^{-9} $ $ 2.00000 $
$ \frac{1}{1600} $ $ 6.35828\times 10^{-10} $ $ 1.99999 $ $ 1.47127\times 10^{-9} $ $ 1.99999 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
Table 5.  The convergence order, the errors $ L_{2} $ and $ L_{\infty} $ for Example 5.1 with $ T = 10 $ and $ N = 3 $
$ \delta \tau $ $ L_{\infty} $ $ C_{\delta\tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.63402\times 10^{-7} $ $ 3.10926\times 10^{-7} $
$ \frac{1}{200} $ $ 4.08673\times 10^{-8} $ $ 1.99941 $ $ 7.77632\times 10^{-8} $ $ 1.99941 $
$ \frac{1}{400} $ $ 1.02179\times 10^{-8} $ $ 1.99985 $ $ 1.94428\times 10^{-8} $ $ 1.99985 $
$ \frac{1}{800} $ $ 2.55453\times 10^{-9} $ $ 1.99996 $ $ 4.86082\times 10^{-9} $ $ 1.99996 $
$ \frac{1}{1600} $ $ 6.38636\times 10^{-10} $ $ 1.99999 $ $ 1.21521\times 10^{-9} $ $ 1.99999 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
$ \delta \tau $ $ L_{\infty} $ $ C_{\delta\tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.63402\times 10^{-7} $ $ 3.10926\times 10^{-7} $
$ \frac{1}{200} $ $ 4.08673\times 10^{-8} $ $ 1.99941 $ $ 7.77632\times 10^{-8} $ $ 1.99941 $
$ \frac{1}{400} $ $ 1.02179\times 10^{-8} $ $ 1.99985 $ $ 1.94428\times 10^{-8} $ $ 1.99985 $
$ \frac{1}{800} $ $ 2.55453\times 10^{-9} $ $ 1.99996 $ $ 4.86082\times 10^{-9} $ $ 1.99996 $
$ \frac{1}{1600} $ $ 6.38636\times 10^{-10} $ $ 1.99999 $ $ 1.21521\times 10^{-9} $ $ 1.99999 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
Table 6.  The convergence order, the errors $ L_{2} $ and $ L_{\infty} $ for Example 5.2 with $ N = 7 $ at $ T = 1 $
$ \delta\tau $ $ L_{\infty} $ $ C_{\delta\tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.71816\times 10^{-6} $ $ 3.73349\times 10^{-6} $
$ \frac{1}{200} $ $ 4.29538\times 10^{-7} $ $ 2.00000 $ $ 9.33372\times 10^{-7} $ $ 2.00000 $
$ \frac{1}{400} $ $ 1.07384\times 10^{-7} $ $ 2.00000 $ $ 2.33343\times 10^{-7} $ $ 2.00000 $
$ \frac{1}{800} $ $ 2.68460\times 10^{-8} $ $ 2.00000 $ $ 5.83360\times 10^{-8} $ $ 2.00000 $
$ \frac{1}{1600} $ $ 6.71143\times 10^{-9} $ $ 2.00002 $ $ 1.45842\times 10^{-8} $ $ 1.99998 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
$ \delta\tau $ $ L_{\infty} $ $ C_{\delta\tau} $ $ L_{2} $ $ C_{\delta \tau} $
$ \frac{1}{100} $ $ 1.71816\times 10^{-6} $ $ 3.73349\times 10^{-6} $
$ \frac{1}{200} $ $ 4.29538\times 10^{-7} $ $ 2.00000 $ $ 9.33372\times 10^{-7} $ $ 2.00000 $
$ \frac{1}{400} $ $ 1.07384\times 10^{-7} $ $ 2.00000 $ $ 2.33343\times 10^{-7} $ $ 2.00000 $
$ \frac{1}{800} $ $ 2.68460\times 10^{-8} $ $ 2.00000 $ $ 5.83360\times 10^{-8} $ $ 2.00000 $
$ \frac{1}{1600} $ $ 6.71143\times 10^{-9} $ $ 2.00002 $ $ 1.45842\times 10^{-8} $ $ 1.99998 $
$ \mathrm{TCO} $ $ 2 $ $ 2 $
Table 7.  The comparison of maximum error of our proposed method and [32] for Example 5.2, at $ T = 1 $
Max error-CN [32] Max error-ext CN [32] the present method with N=3
$ 6.84895\times 10^{-4} $ $ 2.82750 \times 10^{-5} $ $ 9.95930\times 10^{-8} $
Max error-CN [32] Max error-ext CN [32] the present method with N=3
$ 6.84895\times 10^{-4} $ $ 2.82750 \times 10^{-5} $ $ 9.95930\times 10^{-8} $
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