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Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic

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Abstract

A finite difference/spectral scheme is proposed for the time fractional Ito equation. The mass conservation and stability of the numerical solution are deduced by the energy method in the \(L^2\) norm form. To reduce the computation costs, the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations. The effectiveness of the proposed algorithm is verified by the first numerical example. The mass conservation property and stability statement are confirmed by two other numerical examples.

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References

  1. Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral methods in fluid dynamics. Springer-Verlag, New York (1988)

    Book  MATH  Google Scholar 

  2. Cao, W., Xu, Y., Zheng, Z.: Finite difference/collocation method for a generalized time-fractional KDV equation. Appl. Sci. 8, 42 (2018)

    Article  Google Scholar 

  3. Cen, D., Wang, Z.: Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett. 129, 107919 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cen, D., Wang, Z., Mo, Y.: Second order difference schemes for time-fractional KdV-Burgers’ equation with initial singularity. Appl. Math. Lett. 112, 106829 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gupta, A., Ray, S.: Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method. Appl. Math. Model. 39, 5121–5130 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heydari, M., Avazzadeh, Z., Atangana, A.: Shifted Vieta-Fibonacci polynomials for the fractal-fractional fifth-order KdV equation. Math. Method. Appl. Sci. 44, 6716–6730 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ito, M.: An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders. J. Phys. Soc. Jpn. 49, 771–778 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Iyiola, O.: A numerical study of Ito equation and Sawada-Kotera equation both of time-fractional type. Adv. Math. Sci. J. 2, 71–79 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Li, L., Li, D.: Exact solutions and numerical study of time fractional Burgers equations. Appl. Math. Lett. 100, 106011 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Luo, D., Huang, W., Qiu, J.: A hybrid LDG-HWENO scheme for KdV-type equations. J. Comput. Phys. 313, 754–774 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Poochinapan, K., Wongsaijai, B.: A novel convenient finite difference method for shallow water waves derived by fifth-order Kortweg and De-Vries-type equation. Numer. Methods Partial Differential Equations (2022). https://doi.org/10.1002/num.22875

    Article  Google Scholar 

  12. Sun, Z., Ji, C., Du, R.: A new analytical technique of the L-type difference schemes for time fractional mixed sub-diffusion and diffusion-wave equations. Appl. Math. Lett. 102, 106115 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, J., Xu, T., Wang, G.: Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials. Appl. Math. Comput. 338, 1–11 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wazwaz, A.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184, 1002–1014 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ye, X., Cheng, X.: The Fourier spectral method for the Cahn-Hilliard equation. Appl. Math. Comput. 171, 345–357 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and reviewers for their constructive comments and suggestions, which helped the authors to improve the quality of the paper significantly.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, China

    Dakang Cen & Zhibo Wang

  2. Department of Mathematics, University of Macau, Macau, China

    Dakang Cen & Seakweng Vong

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  1. Dakang Cen
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  2. Zhibo Wang
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  3. Seakweng Vong
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Correspondence to Zhibo Wang.

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The authors declare that they have no conflict of interest.

Additional information

This research was partly supported by the National Natural Science Foundation of China (No. 11701103), the Young Top-notch Talent Program of Guangdong Province of China (No. 2017GC010379), the Natural Science Foundation of Guangdong Province of China (No. 2022A1515012147), the Project of Science and Technology of Guangzhou of China (No. 202102020704), the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University of China (2021023), the Science and Technology Development Fund, Macau SAR (File No. 0005/2019/A), and the University of Macau of China (File Nos. MYRG2020-00035-FST, MYRG2018-00047-FST).

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Cen, D., Wang, Z. & Vong, S. Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic. Commun. Appl. Math. Comput. 5, 1591–1600 (2023). https://doi.org/10.1007/s42967-022-00223-z

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  • DOI: https://doi.org/10.1007/s42967-022-00223-z

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