July  2021, 14(7): 2025-2039. 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  July 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

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} $
[1]

Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317

[2]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

[3]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[4]

Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080

[5]

Yuling Guo, Zhongqing Wang. A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022052

[6]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174

[7]

Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178

[8]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[9]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control and Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177

[10]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355

[11]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

[12]

Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254

[13]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073

[14]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[15]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 339-357. doi: 10.3934/dcdss.2021025

[16]

Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022021

[17]

Jiantao Jiang, Jing An, Jianwei Zhou. A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022066

[18]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[19]

Andrea L. Bertozzi, Ning Ju, Hsiang-Wei Lu. A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1367-1391. doi: 10.3934/dcds.2011.29.1367

[20]

Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4551-4574. doi: 10.3934/dcdss.2021113

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (436)
  • HTML views (465)
  • Cited by (0)

[Back to Top]