Skip to main content
Erschienen in: Journal of Applied Mathematics and Computing 1-2/2017

06.10.2016 | Original Research

Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems

verfasst von: Hira Tariq, Ghazala Akram

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2017

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, a recent analytic iterative technique, named as residual power series method is implemented to find the approximate solution of the nonlinear time-space-fractional Benney-Lin equation. The convergence analysis of the proposed scheme is also discussed. To test the validity, potentiality, and practical usefulness of the proposed method in solving such a complicated equation, several numerical examples with various initial conditions are considered. The analysis of the obtained approximate solution results reveal that the proposed method is a significant addition for exploring nonlinear fractional models in fractional theory and its computations.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATH Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATH
2.
Zurück zum Zitat Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)MATH Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)MATH
3.
Zurück zum Zitat McBrid, A., Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer Publishing Company, New York (2007)MATH McBrid, A., Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer Publishing Company, New York (2007)MATH
4.
Zurück zum Zitat Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATH Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATH
5.
Zurück zum Zitat Alquran, M., Al-Khaled, K., Chattopadhyay, J.: Analytical solutions of fractional population diffusion model: residual power series. Nonlinear Stud. 22(1), 31–39 (2015)MathSciNetMATH Alquran, M., Al-Khaled, K., Chattopadhyay, J.: Analytical solutions of fractional population diffusion model: residual power series. Nonlinear Stud. 22(1), 31–39 (2015)MathSciNetMATH
6.
Zurück zum Zitat Alquran, M.: Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J. Appl. Anal. Comput. 5(4), 589–599 (2015)MathSciNet Alquran, M.: Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J. Appl. Anal. Comput. 5(4), 589–599 (2015)MathSciNet
7.
Zurück zum Zitat Alquran, M., Al-Khaled, K., Sardar, T., Chattopadhyay, J.: Revisited Fisher’s equation in a new outlook: a fractional derivative approach. Phys. A 438, 81–93 (2015)MathSciNetCrossRef Alquran, M., Al-Khaled, K., Sardar, T., Chattopadhyay, J.: Revisited Fisher’s equation in a new outlook: a fractional derivative approach. Phys. A 438, 81–93 (2015)MathSciNetCrossRef
8.
Zurück zum Zitat Jaradat, H.M., Awawdeh, F., Al-Shara, S., Alquran, M., Momani, S.: Controllable dynamical behaviors and the analysis of fractal burgers hierarchy with the full effects of inhomogeneities of media. Rom. J. Phys. 60, 324–343 (2015) Jaradat, H.M., Awawdeh, F., Al-Shara, S., Alquran, M., Momani, S.: Controllable dynamical behaviors and the analysis of fractal burgers hierarchy with the full effects of inhomogeneities of media. Rom. J. Phys. 60, 324–343 (2015)
9.
Zurück zum Zitat Alquran, M., Jaradat, H.M., Al-Shara, S., Awawdeh, F.: A new simplified bilinear method for the N-soliton solutions for a generalized FmKdV equation with time-dependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16(6), 259–269 (2015)MathSciNet Alquran, M., Jaradat, H.M., Al-Shara, S., Awawdeh, F.: A new simplified bilinear method for the N-soliton solutions for a generalized FmKdV equation with time-dependent variable coefficients. Int. J. Nonlinear Sci. Numer. Simul. 16(6), 259–269 (2015)MathSciNet
10.
Zurück zum Zitat Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)MathSciNetCrossRef Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)MathSciNetCrossRef
11.
Zurück zum Zitat Abu Arqub, O., El-Ajou, A., Bataineh, A., Hashim, I.: A representation of the exact solution of generalized LaneEmden equations using a new analytical method. Abstr. Appl. Anal. 2013, 10 (2013). doi:10.1155/2013/378593 CrossRefMATH Abu Arqub, O., El-Ajou, A., Bataineh, A., Hashim, I.: A representation of the exact solution of generalized LaneEmden equations using a new analytical method. Abstr. Appl. Anal. 2013, 10 (2013). doi:10.​1155/​2013/​378593 CrossRefMATH
12.
Zurück zum Zitat Abu Arqub, A., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)MathSciNetCrossRefMATH Abu Arqub, A., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)MathSciNetCrossRefMATH
13.
Zurück zum Zitat El-Ajou, A., Abu Arqub, A., Momani, S.: Approximate analytical solution of the nonlinear fractional KdVBurgers equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)MathSciNetCrossRefMATH El-Ajou, A., Abu Arqub, A., Momani, S.: Approximate analytical solution of the nonlinear fractional KdVBurgers equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)MathSciNetCrossRefMATH
14.
Zurück zum Zitat Alquran, M.: Analytical solutions of fractional foam drainage equation by residual power series method. Math. Sci. 8(4), 153–160 (2014)MathSciNetCrossRef Alquran, M.: Analytical solutions of fractional foam drainage equation by residual power series method. Math. Sci. 8(4), 153–160 (2014)MathSciNetCrossRef
15.
Zurück zum Zitat El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A.: A novel expansion iterative method for solving linear partial differential equations of fractional order. Appl. Math. Comput. 257, 119–133 (2015)MathSciNetMATH El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A.: A novel expansion iterative method for solving linear partial differential equations of fractional order. Appl. Math. Comput. 257, 119–133 (2015)MathSciNetMATH
17.
Zurück zum Zitat Jaradat, H.M., Al-Shara, S., Khan, Q.J.A., Alquran, M., Al-Khaled, K.: Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method. IAENG Int. J. Appl. Math. 46(1), 64–70 (2016)MathSciNet Jaradat, H.M., Al-Shara, S., Khan, Q.J.A., Alquran, M., Al-Khaled, K.: Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method. IAENG Int. J. Appl. Math. 46(1), 64–70 (2016)MathSciNet
18.
Zurück zum Zitat Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled BoussinesqBurger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. doi:10.1007/s11071-016-2716-2 Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled BoussinesqBurger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. doi:10.​1007/​s11071-016-2716-2
19.
Zurück zum Zitat El-Ajou, A., Abu arqub, O., Al Zhour, Z., Momani, S.: New results on fractional power series: theories and applications. Entropy 15, 5305–5323 (2013)MathSciNetCrossRefMATH El-Ajou, A., Abu arqub, O., Al Zhour, Z., Momani, S.: New results on fractional power series: theories and applications. Entropy 15, 5305–5323 (2013)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Lin, S.P.: Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429 (1974)CrossRefMATH Lin, S.P.: Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429 (1974)CrossRefMATH
22.
Zurück zum Zitat Seplveda, M., Vera, O.: Numerical method for the KdVKawahara and BenneyLin equations. PAMM 7(1), 2020033–2020034 (2007)CrossRef Seplveda, M., Vera, O.: Numerical method for the KdVKawahara and BenneyLin equations. PAMM 7(1), 2020033–2020034 (2007)CrossRef
23.
Zurück zum Zitat Safari, M., Ganji, D.D., Sadeghi, E.M.M.: Application of He’s homotopy perturbation method and He’s variational iteration methods for solution of Benney-Lin equation. Int. J. Comput. Math. 87, 1872–1884 (2009)MathSciNetCrossRefMATH Safari, M., Ganji, D.D., Sadeghi, E.M.M.: Application of He’s homotopy perturbation method and He’s variational iteration methods for solution of Benney-Lin equation. Int. J. Comput. Math. 87, 1872–1884 (2009)MathSciNetCrossRefMATH
24.
Zurück zum Zitat Song, L.: Analytical approximate solutions of the Benney-Lin equation with symbolic computation. Int. J. Adv. Comput. Technol. 5, 5 (2013) Song, L.: Analytical approximate solutions of the Benney-Lin equation with symbolic computation. Int. J. Adv. Comput. Technol. 5, 5 (2013)
25.
Zurück zum Zitat Gupta, P.K.: Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method. Comput. Math. Appl. 61, 2829–2842 (2011)MathSciNetCrossRefMATH Gupta, P.K.: Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method. Comput. Math. Appl. 61, 2829–2842 (2011)MathSciNetCrossRefMATH
26.
Zurück zum Zitat Berloff, N.G., Howard, L.N.: Solitary and periodic solutions of nonlinear non integrable equations. Stud. Appl. Math. 99, 1–24 (1997)MathSciNetCrossRefMATH Berloff, N.G., Howard, L.N.: Solitary and periodic solutions of nonlinear non integrable equations. Stud. Appl. Math. 99, 1–24 (1997)MathSciNetCrossRefMATH
27.
Zurück zum Zitat Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)MathSciNetMATH Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)MathSciNetMATH
Metadaten
Titel
Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems
verfasst von
Hira Tariq
Ghazala Akram
Publikationsdatum
06.10.2016
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2017
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-016-1056-1

Weitere Artikel der Ausgabe 1-2/2017

Journal of Applied Mathematics and Computing 1-2/2017 Zur Ausgabe