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Approximate Solution of Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Equations by Residual Power Series Method (RPSM)

Year 2021, Volume 5, Issue 2, 78 - 91, 30.12.2021
https://doi.org/10.29002/asujse.992420

Abstract

Drinfeld–Sokolov–Satsuma–Hirota (DSSH) equation approximation solution is acquired by using the residual power series method (RPSM) in this study. The conformable sense is used to characterize the fractional derivatives and the approximate solutions of the RPSM technique are compared to the exact solutions by using various tables and graphs for different fractional operators. The findings demonstrate that the method introduced is simple to use, very successful, consistent; and provides a considerable advance in this area compared to prior approaches.

References

  • [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (The Netherlands: Elsevier, Amsterdam, 2006).
  • [2] S. S. Ray, S. Giri, New soliton solutions of the time fractional Drinfeld–Sokolov–Satsuma–Hirota system in dispersive water waves, Mathematical Methods in the Applied Sciences, Early View (2021) https://doi.org/10.1002/mma.7691.
  • [3] I. Podlubny, Fractional Differential Equations (NY: Academic Press, New York, 1999).
  • [4] S.S. Ray, and R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167(1) (2005) 561-571.
  • [5] S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics, 207(1) (2007) 96-110.
  • [6] L. Kexue, and P. Jigen, Laplace transform and fractional differential equations, Applied Mathematics Letters, 24(12) (2011) 2019-2023.
  • [7] M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34(1) (2010) 24-35.
  • [8] XB. Yin, S. Kumar, D. Kumar, Modified homotopy analysis method for solution of fractional wave equations, Advances in Mechanical Engineering, 7(12) (2015) 1-8.
  • [9] A. Yildirim, An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(4) (2009) 445-450.
  • [10] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36(1) (2008) 167-174.
  • [11] M. Senol, O. Tasbozan, A. Kurt, Comparison of two reliable methods to solve fractional Rosenau-Hyman equation, Mathematical Methods in the Applied Sciences, 44(10) (2021) 7904-7914.
  • [12] M. Şenol, and I.T. Dolapci, On the Perturbation–Iteration Algorithm for fractional differential equations, Journal of King Saud University-Science, 28(1) (2016) 69-74.
  • [13] A. Arafa, G. Elmahdy, Application of residual power series method to fractional coupled physical equations arising in fluids flow, International Journal of Differential Equations, 2018 (2018) 1- 10.
  • [14] O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, Journal of Advance Research in Applied Mathematics, 5 (2013) 31-52.
  • [15] H. M. Jaradat, I. Jaradat, M. Alquran, et al, Approximate solutions to the generalized time-fractional Ito system, Italian Journal of Pure and Applied Mathematics, 37 (2017) 699-710.
  • [16] H.M. Jaradat, S. Al-Shara, Q. J. Khan, M. Alquran, and K. Al-Khaled, Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method, IAENG International Journal of Applied Mathematics, 46(1) (2016) 64-70.
  • [17] V. G. Drinfeld, V. Sokolov, Equations of Korteweg–de Vries type and simple Lie algebras, Doklady Akademii Nauk SSSR, 258(1) (1981) 11–16.
  • [18] J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, Journal of the Physical Society of Japan, 51(10) (1982) 3390–3397.
  • [19] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014) 65-70.
  • [20] M. Senol, Analytical and approximate solutions of (2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation, Communications in Theoretical Physics, 72(5) (2020) 1-11.
  • [21] A. Kurt, H. Rezazadeh, M. Senol, A. Neirameh, O. Tasbozan, M.E.M. Mirzazadeh, Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves, Journal of Ocean Engineering and Science, 4(1) (2019) 24-32.

Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü

Year 2021, Volume 5, Issue 2, 78 - 91, 30.12.2021
https://doi.org/10.29002/asujse.992420

Abstract

Bu çalışmada, Drinfeld–Sokolov–Satsuma–Hirota (DSSH) denkleminin yaklaşık çözümü rezidual kuvvet seri metodu (RKSM) kullanılarak elde edildi. Kesirli türevleri karakterize etmek için uyumlu türev yaklaşımı kullanıldı ve farklı operatör değerleri için RKSM yaklaşık çözümleri kesin çözümlerle çeşitli tablo ve grafikler kullanılarak karşılaştırıldı. Sonuçlar; uygulanan tekniğin kullanımının kolay, çok başarılı ve tutarlı olduğunu göstermekle birlikte daha önceki yapılmış çalışmalara kıyasla da önemli bir ilerleme getirdiğini göstermektedir.

References

  • [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (The Netherlands: Elsevier, Amsterdam, 2006).
  • [2] S. S. Ray, S. Giri, New soliton solutions of the time fractional Drinfeld–Sokolov–Satsuma–Hirota system in dispersive water waves, Mathematical Methods in the Applied Sciences, Early View (2021) https://doi.org/10.1002/mma.7691.
  • [3] I. Podlubny, Fractional Differential Equations (NY: Academic Press, New York, 1999).
  • [4] S.S. Ray, and R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167(1) (2005) 561-571.
  • [5] S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics, 207(1) (2007) 96-110.
  • [6] L. Kexue, and P. Jigen, Laplace transform and fractional differential equations, Applied Mathematics Letters, 24(12) (2011) 2019-2023.
  • [7] M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34(1) (2010) 24-35.
  • [8] XB. Yin, S. Kumar, D. Kumar, Modified homotopy analysis method for solution of fractional wave equations, Advances in Mechanical Engineering, 7(12) (2015) 1-8.
  • [9] A. Yildirim, An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(4) (2009) 445-450.
  • [10] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36(1) (2008) 167-174.
  • [11] M. Senol, O. Tasbozan, A. Kurt, Comparison of two reliable methods to solve fractional Rosenau-Hyman equation, Mathematical Methods in the Applied Sciences, 44(10) (2021) 7904-7914.
  • [12] M. Şenol, and I.T. Dolapci, On the Perturbation–Iteration Algorithm for fractional differential equations, Journal of King Saud University-Science, 28(1) (2016) 69-74.
  • [13] A. Arafa, G. Elmahdy, Application of residual power series method to fractional coupled physical equations arising in fluids flow, International Journal of Differential Equations, 2018 (2018) 1- 10.
  • [14] O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, Journal of Advance Research in Applied Mathematics, 5 (2013) 31-52.
  • [15] H. M. Jaradat, I. Jaradat, M. Alquran, et al, Approximate solutions to the generalized time-fractional Ito system, Italian Journal of Pure and Applied Mathematics, 37 (2017) 699-710.
  • [16] H.M. Jaradat, S. Al-Shara, Q. J. Khan, M. Alquran, and K. Al-Khaled, Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method, IAENG International Journal of Applied Mathematics, 46(1) (2016) 64-70.
  • [17] V. G. Drinfeld, V. Sokolov, Equations of Korteweg–de Vries type and simple Lie algebras, Doklady Akademii Nauk SSSR, 258(1) (1981) 11–16.
  • [18] J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, Journal of the Physical Society of Japan, 51(10) (1982) 3390–3397.
  • [19] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014) 65-70.
  • [20] M. Senol, Analytical and approximate solutions of (2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation, Communications in Theoretical Physics, 72(5) (2020) 1-11.
  • [21] A. Kurt, H. Rezazadeh, M. Senol, A. Neirameh, O. Tasbozan, M.E.M. Mirzazadeh, Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves, Journal of Ocean Engineering and Science, 4(1) (2019) 24-32.

Details

Primary Language Turkish
Subjects Basic Sciences
Journal Section Research Article
Authors

Ahmet KAYA (Primary Author)
NEVŞEHİR HACI BEKTAŞ VELİ ÜNİVERSİTESİ
0000-0001-5109-8130
Türkiye

Publication Date December 30, 2021
Application Date September 8, 2021
Acceptance Date September 22, 2021
Published in Issue Year 2021, Volume 5, Issue 2

Cite

Bibtex @research article { asujse992420, journal = {Aksaray University Journal of Science and Engineering}, issn = {}, eissn = {2587-1277}, address = {Aksaray Üniversitesi, Fen Bilimleri Enstitüsü, Merkez Kampüs, 68100 Aksaray}, publisher = {Aksaray University}, year = {2021}, volume = {5}, pages = {78 - 91}, doi = {10.29002/asujse.992420}, title = {Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü}, key = {cite}, author = {Kaya, Ahmet} }
APA Kaya, A. (2021). Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü . Aksaray University Journal of Science and Engineering , 5 (2) , 78-91 . DOI: 10.29002/asujse.992420
MLA Kaya, A. "Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü" . Aksaray University Journal of Science and Engineering 5 (2021 ): 78-91 <http://asujse.aksaray.edu.tr/en/pub/issue/63258/992420>
Chicago Kaya, A. "Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü". Aksaray University Journal of Science and Engineering 5 (2021 ): 78-91
RIS TY - JOUR T1 - Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü AU - Ahmet Kaya Y1 - 2021 PY - 2021 N1 - doi: 10.29002/asujse.992420 DO - 10.29002/asujse.992420 T2 - Aksaray University Journal of Science and Engineering JF - Journal JO - JOR SP - 78 EP - 91 VL - 5 IS - 2 SN - -2587-1277 M3 - doi: 10.29002/asujse.992420 UR - https://doi.org/10.29002/asujse.992420 Y2 - 2021 ER -
EndNote %0 Aksaray University Journal of Science and Engineering Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü %A Ahmet Kaya %T Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü %D 2021 %J Aksaray University Journal of Science and Engineering %P -2587-1277 %V 5 %N 2 %R doi: 10.29002/asujse.992420 %U 10.29002/asujse.992420
ISNAD Kaya, Ahmet . "Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü". Aksaray University Journal of Science and Engineering 5 / 2 (December 2021): 78-91 . https://doi.org/10.29002/asujse.992420
AMA Kaya A. Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü. Aksaray J. Sci. Eng.. 2021; 5(2): 78-91.
Vancouver Kaya A. Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü. Aksaray University Journal of Science and Engineering. 2021; 5(2): 78-91.
IEEE A. Kaya , "Drinfeld–Sokolov–Satsuma–Hirota (DSSH) Denkleminin Rezidual Kuvvet Seri Metodu (RKSM) Yardımıyla Yaklaşık Çözümü", Aksaray University Journal of Science and Engineering, vol. 5, no. 2, pp. 78-91, Dec. 2021, doi:10.29002/asujse.992420

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