Homotopy Analysis Method for the Time-Fractional Boussinesq Equation

He Yang

doi:10.32323/ujma.634491

EN

Homotopy Analysis Method for the Time-Fractional Boussinesq Equation

Abstract

In this paper, the exact and approximate analytical solutions to the time-fractional Boussinesq equation are constructed using the homotopy analysis method. Several examples about the fourth-order and sixth-order time-fractional Boussinesq equations show the flexibility and efficiency of the method. Furthermore, by choosing an appropriate value for the auxiliary parameter $h$, we can obtain the $N$-term approximate solution with improved accuracy.

Keywords

Homotopy analysis method,time-fractional Boussinesq equation

References

[1] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424-438.
[2] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.
[3] A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937-954.
[4] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Frac. Differ. App., 1 (2015), 73-85.
[5] H. Hosseini, R. Ansari, New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Wave Random Complex, 27 (2017), 628-636.
[6] H. Jafari, N. Kadkhoda, D. Baleanu, Fractional Lie group method of the time-fractional Boussinesq equation, Nonlinear Dyn., 81 (2015), 1569-1574.
[7] J. Jia, H. Wang, A fast finite volume method for conservative space-fractional diffusion equations in convex domains, J. Comput. Phys., 310 (2016), 63-84.
[8] J. Jia, H. Wang, Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions, J. Comput. Phys., 293 (2015), 359-369.
[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[10] S-J Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC, Boca Raton, 2004.

[11] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
[12] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28.
[13] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Frac. Calc. Appl. Anal., 4 (2001), 153-192.
[14] M. Meerschaert, Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56 (2006), 80-90.
[15] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solutions and Some of their Applications, Academic Press, San Diego, 1999.
[16] F. Xu, Y. Gao, W. Zhang, Construction of analytic solution for time-fractional Boussinesq equation using iterative method, Adv. Math. Phys., 2015, Article ID 506140, 7 pages.
[17] Q. Xu, J.S. Hesthaven, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM J. Numer. Anal., 52 (2014), 405-423.
[18] M. Yavuz, N. Özdemir, European vanilla option pricing model of fractional order without singular kernel, Fractal Fractional, 2 (2018), 1.
[19] M. Yavuz, N. Özdemir, On the solutions of fractional Cauchy problem featuring conformable derivative, Proceedings of ITM Web of Conferences, EDP Sciences, (2018), 01045.
[20] M. Yavuz, B. Yaşkiran, Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator, Balikesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20 (2018), 75-89.
[21] H. Yang, A new high-order method for the time-fractional diffusion equation with a source, J. Frac. Calc. Appl., 11 (2020), 111-129.
[22] H. Yang, J. Guo, J.-H. Jung, Schwartz duality of the Dirac delta function for the Chebyshev collocation approximation to the fractional advection equation, Appl. Math. Lett., 64 (2017), 205-212.
[23] F. Zeng, C. Li, F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976-A3000.
[24] H. Zhang, X. Jiang, M. Zhao, R. Zheng, Spectral method for solving the time fractional Boussinesq equation, Appl. Math. Lett., 85 (2018), 164-170.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

He Yang ^*
0000-0001-9608-4920
United States

Publication Date

March 25, 2020

Submission Date

October 18, 2019

Acceptance Date

February 10, 2020

Published in Issue

Year 2020 Volume: 3 Number: 1

DOI

https://doi.org/10.32323/ujma.634491

IZ

https://izlik.org/JA69TG85EP

APA

Yang, H. (2020). Homotopy Analysis Method for the Time-Fractional Boussinesq Equation. Universal Journal of Mathematics and Applications, 3(1), 12-18. https://doi.org/10.32323/ujma.634491

AMA

1.Yang H. Homotopy Analysis Method for the Time-Fractional Boussinesq Equation. Univ. J. Math. Appl. 2020;3(1):12-18. doi:10.32323/ujma.634491

Chicago

Yang, He. 2020. “Homotopy Analysis Method for the Time-Fractional Boussinesq Equation”. Universal Journal of Mathematics and Applications 3 (1): 12-18. https://doi.org/10.32323/ujma.634491.

EndNote

Yang H (March 1, 2020) Homotopy Analysis Method for the Time-Fractional Boussinesq Equation. Universal Journal of Mathematics and Applications 3 1 12–18.

IEEE

[1]H. Yang, “Homotopy Analysis Method for the Time-Fractional Boussinesq Equation”, Univ. J. Math. Appl., vol. 3, no. 1, pp. 12–18, Mar. 2020, doi: 10.32323/ujma.634491.

ISNAD

Yang, He. “Homotopy Analysis Method for the Time-Fractional Boussinesq Equation”. Universal Journal of Mathematics and Applications 3/1 (March 1, 2020): 12-18. https://doi.org/10.32323/ujma.634491.

JAMA

1.Yang H. Homotopy Analysis Method for the Time-Fractional Boussinesq Equation. Univ. J. Math. Appl. 2020;3:12–18.

MLA

Yang, He. “Homotopy Analysis Method for the Time-Fractional Boussinesq Equation”. Universal Journal of Mathematics and Applications, vol. 3, no. 1, Mar. 2020, pp. 12-18, doi:10.32323/ujma.634491.

Vancouver

1.He Yang. Homotopy Analysis Method for the Time-Fractional Boussinesq Equation. Univ. J. Math. Appl. 2020 Mar. 1;3(1):12-8. doi:10.32323/ujma.634491