Approximate analytical solutions to conformable modified Burgers equation using homotopy analysis method
Abstract
In this paper the authors aspire to obtain the approximate analytical solution of Modified Burgers Equation with newly defined conformable derivative by employing homotopy analysis method (HAM).
Keywords
modified Burgers equation; conformable derivative; homotopy analysis method
References
1. Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57–66.
2. Atangana A., Baleanu D., Alsaedi A., New properties of conformable derivative, Open Math. 13 (2015), 889–898.
3. Çenesiz Y., Baleanu D., Kurt A., Tasbozan O., New exact solutions of Burgers’ type equations with conformable derivative, Waves Random Complex Media 27 (2017), no. 1, 103–116.
4. Eslami M., Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Appl. Math. Comput. 285 (2016), 141–148.
5. He S., Sun K., Mei X., Yan B., Xu S., Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative, Eur. Phys. J. Plus 132 (2017), no. 1, Art. 36, 11 pp.
6. Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
7. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science, San Diego, 2006.
8. Kurt A., Çenesiz Y., Tasbozan O., On the solution of Burgers’ equation with the new fractional derivative, Open Phys. 13 (2015), 355–360.
9. Liao S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
10. Liao S.J., Beyond Perturbation. Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, 2004.
11. Liao S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 4, 983–997.
12. Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
13. Nicolescu B.N., Macarie T., Petrescu T.C., Application of homotopy analysis method for solving equation of vehicle’s move in the linear case, Applied Mechanics and Materials 822 (2016), 3–11.
14. Oruç Ö., Bulut F., Esen A., A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation, J. Math. Chem. 53 (2015), no. 7, 1592–1607.
15. Podlubny I., Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, 1999.
16. Tasbozan O., Kurt A., Approximate analytical solution of ZK-BBM equation, Sohag J. Math. 2 (2015), no. 2, 57–60.
17. Ünal E., Gökdoğan A., Solution of conformable fractional ordinary differential equations via differential transform method, Optik 128 (2017), 264–273.
2. Atangana A., Baleanu D., Alsaedi A., New properties of conformable derivative, Open Math. 13 (2015), 889–898.
3. Çenesiz Y., Baleanu D., Kurt A., Tasbozan O., New exact solutions of Burgers’ type equations with conformable derivative, Waves Random Complex Media 27 (2017), no. 1, 103–116.
4. Eslami M., Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Appl. Math. Comput. 285 (2016), 141–148.
5. He S., Sun K., Mei X., Yan B., Xu S., Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative, Eur. Phys. J. Plus 132 (2017), no. 1, Art. 36, 11 pp.
6. Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
7. Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science, San Diego, 2006.
8. Kurt A., Çenesiz Y., Tasbozan O., On the solution of Burgers’ equation with the new fractional derivative, Open Phys. 13 (2015), 355–360.
9. Liao S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
10. Liao S.J., Beyond Perturbation. Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, 2004.
11. Liao S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 4, 983–997.
12. Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
13. Nicolescu B.N., Macarie T., Petrescu T.C., Application of homotopy analysis method for solving equation of vehicle’s move in the linear case, Applied Mechanics and Materials 822 (2016), 3–11.
14. Oruç Ö., Bulut F., Esen A., A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation, J. Math. Chem. 53 (2015), no. 7, 1592–1607.
15. Podlubny I., Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, 1999.
16. Tasbozan O., Kurt A., Approximate analytical solution of ZK-BBM equation, Sohag J. Math. 2 (2015), no. 2, 57–60.
17. Ünal E., Gökdoğan A., Solution of conformable fractional ordinary differential equations via differential transform method, Optik 128 (2017), 264–273.
KurtA., & TasbozanO. (2018). Approximate analytical solutions to conformable modified Burgers equation using homotopy analysis method. Annales Mathematicae Silesianae, 33, 159-167. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13666
Ali Kurt
Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Turkey Turkey
Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Turkey Turkey
Orkun Tasbozan
otasbozan@mku.edu.tr
Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Turkey Turkey
Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Turkey Turkey
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
