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Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers’ Equations

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Abstract

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers’ equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi–Gauss–Lobatto collocation points. By means of the simplified Jacobi operational matrix, we produce the differentiation matrix and transfer the space-fractional Burgers’ equation into a system of ordinary differential equations that can be solved by the fourth-order Runge–Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers’ equation.

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References

  1. Afsane, S., Mahmoud, B.: Application Jacobi spectral method for solving the time-fractional differential equation. J. Comput. Appl. Math. 399, 49–68 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Bhrawy, A.H., Taha, M.T., Machado, J.A.T.: A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn. 81, 1023–1052 (2015)

    Article  MathSciNet  Google Scholar 

  3. Biler, P., Funaki, T., Woyczynski, W.A.: Fractal Burgers equations. J. Differ. Equ. 148(1), 9–46 (1998)

    Article  MathSciNet  Google Scholar 

  4. Canuto, C.G., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, New York (2010)

    MATH  Google Scholar 

  5. Chester, W.N.: Resonant ocillations in closed tubes. J. Fluid Mech. 18(1), 44–64 (1964)

    Article  Google Scholar 

  6. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)

    Article  MathSciNet  Google Scholar 

  7. El-Shahed, M.: Adomian decomposition method for solving Burgers equation with fractional derivative. J. Fract. Calc. 24, 23–28 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Esen, A., Bulut, F., Oruç, Ö.: A unified approach for the numerical solution of time fractional Burgers’ type equations. Eur. Phys. J. Plus 131(4), 1–13 (2016)

    Article  Google Scholar 

  9. Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)

    Book  Google Scholar 

  10. Guo, B.Y., Wang, L.L.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227–276 (2001)

    Article  MathSciNet  Google Scholar 

  11. Guo, B.Y., Wang, L.L.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 128, 1–41 (2004)

    Article  MathSciNet  Google Scholar 

  12. Inc, M.: The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345(1), 476–484 (2008)

    Article  MathSciNet  Google Scholar 

  13. Kakutanil, T., Matsuuchi, K.: Effect of viscosity on long gravity waves. J. Phys. Soc. Jpn. 39, 237–246 (1975)

    Article  MathSciNet  Google Scholar 

  14. Keller, J.J.: Propagation of simple non-linear waves in gas filled tubes with friction. Z. Angew. Math. Phys. 32(2), 170–181 (1982)

    Article  Google Scholar 

  15. Khatera, A.H., Temsaha, R.S., Hassanb, M.M.: A Chebyshev spectral collocation method for solving Burgers’-type equations. J. Comput. Appl. Math. 222, 333–350 (2008)

    Article  MathSciNet  Google Scholar 

  16. Li, D., Zhang, C., Ran, M.: A linear finite difference scheme for generalized time fractional Burgers equation. Appl. Math. Model. 40(11), 6069–6081 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ma, H.P., Sun, W.W.: Optimal error estimates of the Legendre–Petrov–Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. Anal. 39, 1380–1394 (2001)

    Article  MathSciNet  Google Scholar 

  18. Miksis, M.J., Ting, L.: Effective equations for multiphase flows-waves in a bubbly liquid. Adv. Appl. Mech. 28, 141–260 (1991)

    Article  MathSciNet  Google Scholar 

  19. Momani, S.: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos Soliton. Fract. 28(4), 930–937 (2006)

    Article  MathSciNet  Google Scholar 

  20. Podulubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  21. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)

    MATH  Google Scholar 

  22. Shen, J., Tang, T., Wang, L.L.: Spectral Methods Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, Berlin (2011)

    Google Scholar 

  23. Sugimoto, N.: Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 225, 631–653 (1991)

    Article  MathSciNet  Google Scholar 

  24. Wu, H., Ma, H.P., Li, H.Y.: Optimal error estimates of the Chebyshev–Legendre method for solving the generalized Burgers equation. SIAM J. Numer. Anal. 41, 659–672 (2003)

    Article  MathSciNet  Google Scholar 

  25. Yang, Y.B., Ma, H.P.: The Legendre Galerkin–Chebyshev collocation method for generalized space-fractional Burgers equations. J. Numer. Meth. Comput. Appl. 38(3), 236–244 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40–A62 (2014)

    Article  MathSciNet  Google Scholar 

  27. Zhao, T.G., Wu, Y.J., Ma, H.P.: Error analysis of Chebyshev–Legendre pseudo-spectral method for a class of nonclassical parabolic equation. J. Sci. Comput. 52, 588–602 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11701358, 11774218). The authors wish to thank Professor Heping Ma and Professor Changpin Li for their valuable discussions.

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Qingqing Wu & Xiaoyan Zeng

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  1. Qingqing Wu
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  2. Xiaoyan Zeng
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Correspondence to Xiaoyan Zeng.

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Wu, Q., Zeng, X. Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers’ Equations. Commun. Appl. Math. Comput. 2, 305–318 (2020). https://doi.org/10.1007/s42967-019-00053-6

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  • DOI: https://doi.org/10.1007/s42967-019-00053-6

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