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On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation

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Abstract

In this paper, an initial-boundary value problem for the nonlinear fractional Rayleigh-Stokes equation is studied in two cases, namely when the source term is globally Lipschitz or locally Lipschitz. The time-fractional derivative used in this work is the classical Riemann-Liouville derivative. Thanks to the spectral decomposition, a fixed point argument, and some useful function spaces, we establish global well-posed results for our problem. Furthermore, we demonstrate that the mild solution exists globally or blows up in finite time.

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References

  1. Brezis, H.: Functional Analysis. Springer, New York (2011)

    MATH  Google Scholar 

  2. Chen, C.M., Liu, F., Burrage, K., Chen, Y.: Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative. IMA J. Appl. Math. 78(5), 924–944 (2013)

    Article  MathSciNet  Google Scholar 

  3. Chen, C.M., Liu, F., Anh, V.: Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl. Math. Comput. 204, 340–351 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Chen, C.M., Liu, F., Anh, V.: A Fourier method and an extrapolation technique for Stokes first problem for a heated generalized second grade fluid with fractional derivative. J. Comp. Appl. Math. 223, 777–789 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid. Numer. Math. 131, 1–31 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bazhlekova, E., Bazhlekov, I.: Viscoelastic flows with fractional derivative models: computational approach by convolutional calculus of Dimovski. Fract. Calc. Appl. Anal. 17(4), 954–976 (2014)

    Article  MathSciNet  Google Scholar 

  7. Dehghan, M., Abbaszadeh, M.: A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng. Comput. 33, 587–605 (2017)

    Article  Google Scholar 

  8. Khan, M.: The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model. Nonlinear Anal. Real World Appl. 10(5), 3190–3195 (2009)

    Article  MathSciNet  Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)

    Google Scholar 

  10. Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press Inc, San Diego (1990)

    Google Scholar 

  11. Shen, F., Tan, W., Zhao, Y., Masuoka, T.: The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Anal. Real World Appl. 7(5), 1072–1080 (2006)

    Article  MathSciNet  Google Scholar 

  12. Triet, N.A., Hoan, L.V.C., Luc, N.H., Tuan, N.H., Thinh, N.V.: Identification of source term for the Rayleigh-Stokes problem with Gaussian random noise. Math. Meth. Appl. Sci. 41(14), 5593–5601 (2018)

    Article  MathSciNet  Google Scholar 

  13. Zaky, A.M.: An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid. Comput. Math. Appl. 75(7), 2243–2258 (2018)

    Article  MathSciNet  Google Scholar 

  14. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)

    Article  MathSciNet  Google Scholar 

  15. Xue, C., Nie, J.: Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space. Appl. Math. Model. 33, 524–531 (2009)

    Article  MathSciNet  Google Scholar 

  16. Zhao, C., Yang, C.: Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Appl. Math. Comput. 211, 502–509 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Fetecau, C., Jamil, M., Fetecau, C., Vieru, D.: The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid. Z. Angew. Math. Phys. 60, 921–933 (2009)

    Article  MathSciNet  Google Scholar 

  18. Akram, T., Abbas, M., Abualnaja, K.M., Iqbal, A., Majeed, A.: An efficient numerical technique based on the extended cubic B-spline functions for solving time fractional Black-Scholes model. Eng. Comput. 1–12, 5 (2021)

    Google Scholar 

  19. Majeed, A., Kamran, M., Abbas, M., Misro, M.Y.B.: An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model. Phys. Scr. 96(8), 084002 (2021)

    Article  Google Scholar 

  20. Amin, M., Abbas, M., Baleanu, D., Iqbal, M.K., Riaz, M.B.: Redefined extended cubic B-spline functions for numerical solution of time-fractional telegraph equation. CMES-Comput. Model. Eng. Sci. 127(1), 361–384 (2021)

    Google Scholar 

  21. Akram, T., Abbas, M., Ali, A., Iqbal, A., Baleanu, D.: A numerical approach of a time fractional reaction-diffusion model with a non-singular kernel. Symmetry 12(10), 1653 (2020)

    Article  Google Scholar 

  22. Majeed, A., Kamran, M., Abbas, M., Singh, J.: An efficient numerical technique for solving time-fractional generalized Fisher’s equation. Front. Phys. 8, 293 (2020)

    Article  Google Scholar 

  23. Amin, M., Abbas, M., Iqbal, M.K., Baleanu, D.: Numerical treatment of time-fractional Klein-Gordon equation using redefined extended cubic B-spline functions. Front. Phys. 8, 288 (2020)

    Article  Google Scholar 

  24. Akram, T., Muhammad, A., Azhar, I., Dumitru, B., Jihad, H.A.: Novel numerical approach based on modified extended cubic B-spline functions for solving non-linear time-fractional telegraph equation. Symmetry 12(7), 1154 (2020)

    Article  Google Scholar 

  25. Khalid, N., Abbas, M., Iqbal, M.K., Singh, J., Ismail, A.I.M.: A computational approach for solving time fractional differential equation via spline functions. Alexandr. Eng. J. 59(5), 3061–3078 (2020)

    Article  Google Scholar 

  26. Khalid, N., Abbas, M., Iqbal, M.K., Baleanu, D.: A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions. Adv. Diff. Equ. 2020(1), 1–22 (2020)

    Article  MathSciNet  Google Scholar 

  27. Akram, T., Abbas, M., Riaz, M.B., Ismail, A.I., Ali, N.M.: An efficient numerical technique for solving time fractional Burgers equation. Alexandr. Eng. J. 59(4), 2201–2220 (2020)

    Article  Google Scholar 

  28. Ngan, L.K., McLean, W., Stynes, M.: Existence, uniqueness and regularity of the solution of the time-fractional Fokker-Planck equation with general forcing. Commun. Pure Appl. Anal. 18(5), 2765–2787 (2019)

    Article  MathSciNet  Google Scholar 

  29. Khan, M., Anjum, A., Fetecau, C., Qi, H.: Exact solutions for some oscillating motions of a fractional Burgers fluid. Math. Comput. Model. 51, 682–692 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author (Do Lan) is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 101.02-2020.07

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Science Ho Chi Minh City, Ho Chi Minh City, Vietnam

    Nguyen Hoang Luc

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Hoang Luc

  3. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Nguyen Hoang Luc

  4. Department of Mathematics, Thuyloi University, 175 Tay Son Dong Da, Hanoi, Vietnam

    Do Lan

  5. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  6. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Nguyen Anh Tuan

  7. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Yong Zhou

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  1. Nguyen Hoang Luc
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  2. Do Lan
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  3. Donal O’Regan
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  4. Nguyen Anh Tuan
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  5. Yong Zhou
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Correspondence to Nguyen Anh Tuan.

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Luc, N.H., Lan, D., O’Regan, D. et al. On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation. J. Fixed Point Theory Appl. 23, 60 (2021). https://doi.org/10.1007/s11784-021-00897-7

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  • DOI: https://doi.org/10.1007/s11784-021-00897-7

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