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Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM)

Year 2022, Volume: 19 Issue: 2, 102 - 112, 01.11.2022

Abstract

This work focuses on presenting a reliable method, called fractional homotopy perturbation transform method (FHPTM) to solve nonlinear Naviers Stoks equations with the Caputo type fractional derivatives. The (FHPTM) is a combination of Laplace transform and homotopy perturbation method (He -Laplace method). He’s polynomial is used to simplify the nonlinearity which arise in our considered equation. Furthermore, three numerical examples are presented, it is supported by graphs and tables to compare solutions with little computational effort, which confirms the effectiveness and accuracy of the current method.

Thanks

The authors are, sincerely, very grateful to the editor and the entire staff of the journal, best wishes for your noble mission.

References

  • [1] K.B. Oldham and J. Spanier, ”The Fractional Calculus,” Academic Press, New York, 1974.
  • [2] M. Du, Z. Wang and H. Hu, ”Measuring memory with the order of fractional derivative,” Scientific Reports, 3, 3431; 2013.
  • [3] A.U. Rehman, M.B. Riaz, A. Akg¨ul, S.T. Saeed and D .Baleanu, ”Heat and mass transport impact on MHD second grade fluid: A comparative analysis of fractional operators,” Heat Transfer, vol. 50, no. 7, pp. 7042-7064, 2021.
  • [4] R.L. Bagley and P.J. Torvik, ”A theoretical basis for the application of fractional calculus to viscoelasticity,” J. Rheol., vol. 27, pp. 201–210, 1983.
  • [5] M. Aleem, M.I. Asjad and A. Akg¨ul, ”Heat transfer analysis of magnetohydrodynamic Casson fluid through a porous medium with constant proportional Caputo derivative,” Heat Transfer, vol. 50, no. 7, pp. 6444-6464, 2021.
  • [6] A. Carpinteri and F. Mainardi, ”Fractals and Fractional Calculus in Continuum Mechanics,” Springer Verlag,Wien, New York, 1997.
  • [7] K.S. Miller and B. Ross, ”An Introduction to the Fractional Calculus and Fractional Differential Equations,”Wiley, New York, 1993.
  • [8] I. Podlubny, ”Fractional Differential Equations, ” Academic Press, New York, 1999.
  • [9] A. Kilbas, H.M. Srivastava and J.J. Trujillo, ”Theory and Application of Fractional Differential Equations,” Amsterdam: Elsevier, 2006.
  • [10] A. Akgul, ”A novel method for a fractional derivative with non-local and non-singular kernel,” Chaos, Solitons &Fractals, vol. 114, pp. 478-482, 2018.
  • [11] M. Inc, E.G. Fan, ”Extended tanh-function method for finding travelling wave solutions of some nonlinear partial differential equations,” Z. Naturforsch, vol. A60, pp. 7-16, 2005.
  • [12] K. Pandey , L. Verma, Amit and K. Verma, ”On a finite difference scheme for Burgers’ equation,” Applied Mathematics and Computation, vol. 215, pp. 2206–2214, 2009.
  • [13] M. Zahid, M.I. Asjad, S. Hussain and A. Akg¨ul, ”Nonlinear magnetohydrodynamic flow of nanofluids across a porous matrix over an extending sheet with mass transpiration and bioconvection,” Heat Transfer, vol. 50, no. 8, pp. 7588-7603, 2021.
  • [14] K. S. Brajesh and K. Pramod, ”Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay,” Sociedad Espa˜nola de Matem´atica Aplicada, doi: 10.1007/s40324-017-0117-1, 2017.
  • [15] Y. Khan, N. Faraz, S. Kumar and A.Yildirim, ”A coupling method of homotopy perturbation and Laplace transformation for fractional models,” U. P. B. Science Bulletin, Series A, vol. 74, no. 1, pp. 57 68, 2012.
  • [16] S. Kumar, A. Yildirim, Y. Khan and L. Wei, ”A fractional model of the diffusion equation and its analytical solution using Laplace transform,” Sci. Iran. B, vol. 19, no. 4, pp. 1117–1123, 2012.
  • [17] L. Yanqin, ”Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method,” Abstract and Applied Analysis, vol. 2012, 2012.
  • [18] S. Dinkar, S. Prince, and C. Shubha, ”Homotopy Perturbation Transform Method with He’s Polynomial for Solution of Coupled Nonlinear Partial Differential Equations,” Nonlinear Engineering, vol. 5, no. 1, pp. 17–23, 2016.
  • [19] M. Zellal and K. Belghaba, ”Applications of homotopy perturbation transform method for solving Newell-Whitehead- Segel equation,” General Letters in Mathematics, vol. 3, no. 1, pp. 35-46, 2017.
  • [20] M. El-Shahed and A. Salem, ”On the generalized Navier–Stokes equations,” Appl Math Comput, vol. 156, no. 1, pp. 287–93, 2005.
  • [21] G.A. Birajdar, ”Numerical solution of time fractional Navier–Stokes equation by discrete Adomian decomposition method,” Nonlinear Eng, vol. 3, no. 1, pp. 21–26, 2014.
  • [22] K.S. Brajesh and K. Pramod, ”FRDTM for numerical simulation of multi-dimensional, time fractional model of Navier- Stokes equation,” Ain Shams Engineering Journal, vol. 9, no. 4, pp. 827-834, 2018.
  • [23] D. Kumar, J. Singh and S. Kumar, ”A Fractional Model of Navier-Stokes Equation Arising in Unsteady Flow of a Viscous Fluid,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 17, pp. 14-19, 2015.
  • [24] S. Kumar, D. Kumar, M. Abbasbandy and M. Rashidi, ”Analytical Solution of Fractional Navier-Stokes Equation by using Modified Laplace Decomposition Method, ” Ain Shams Engineering Journal, vol. 5, no. 2, pp. 569–574, 2014.
  • [25] S. Momani and Z. Odibat, ”Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006.
  • [26] A. A. Ragab, K. M. Hemida, M. S. Mohamed, and Abd El M. A.Salam, ”Solution of Time-Fractional Navier-Stokes Equation by Using Homotopy Analysis Method,” Gen. Math. Notes, vol. 13, no. 2, pp.13-21, 2012.
  • [27] M. Zellal and K. Belghaba, ”He’s variational iteration method for solving multi-dimensional of Navier Stokes equation,” International Journal of Analysis and Applications, vol. 18, no. 5, pp. 724-737, 2020.
  • [28] J. H. He, ”Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng, vol. 178, pp. 257–262, 1999.
  • [29] J. H. He, ”Homotopy perturbation method: a new nonlinear analytical technique,” App. Math. Comp, vol. 135, pp. 73–79, 2003.
  • [30] A. Ghorbani and J. S. Nadjafi, ” He’s homotopy Perturbation method for calculating Adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007.
  • [31] A. Ghorbani, ”Beyond Adomians polynomials: He polynomials,” Chaos Solitons Fractals, vol. 39, no. 3, pp. 1486– 1492, 2009.
Year 2022, Volume: 19 Issue: 2, 102 - 112, 01.11.2022

Abstract

References

  • [1] K.B. Oldham and J. Spanier, ”The Fractional Calculus,” Academic Press, New York, 1974.
  • [2] M. Du, Z. Wang and H. Hu, ”Measuring memory with the order of fractional derivative,” Scientific Reports, 3, 3431; 2013.
  • [3] A.U. Rehman, M.B. Riaz, A. Akg¨ul, S.T. Saeed and D .Baleanu, ”Heat and mass transport impact on MHD second grade fluid: A comparative analysis of fractional operators,” Heat Transfer, vol. 50, no. 7, pp. 7042-7064, 2021.
  • [4] R.L. Bagley and P.J. Torvik, ”A theoretical basis for the application of fractional calculus to viscoelasticity,” J. Rheol., vol. 27, pp. 201–210, 1983.
  • [5] M. Aleem, M.I. Asjad and A. Akg¨ul, ”Heat transfer analysis of magnetohydrodynamic Casson fluid through a porous medium with constant proportional Caputo derivative,” Heat Transfer, vol. 50, no. 7, pp. 6444-6464, 2021.
  • [6] A. Carpinteri and F. Mainardi, ”Fractals and Fractional Calculus in Continuum Mechanics,” Springer Verlag,Wien, New York, 1997.
  • [7] K.S. Miller and B. Ross, ”An Introduction to the Fractional Calculus and Fractional Differential Equations,”Wiley, New York, 1993.
  • [8] I. Podlubny, ”Fractional Differential Equations, ” Academic Press, New York, 1999.
  • [9] A. Kilbas, H.M. Srivastava and J.J. Trujillo, ”Theory and Application of Fractional Differential Equations,” Amsterdam: Elsevier, 2006.
  • [10] A. Akgul, ”A novel method for a fractional derivative with non-local and non-singular kernel,” Chaos, Solitons &Fractals, vol. 114, pp. 478-482, 2018.
  • [11] M. Inc, E.G. Fan, ”Extended tanh-function method for finding travelling wave solutions of some nonlinear partial differential equations,” Z. Naturforsch, vol. A60, pp. 7-16, 2005.
  • [12] K. Pandey , L. Verma, Amit and K. Verma, ”On a finite difference scheme for Burgers’ equation,” Applied Mathematics and Computation, vol. 215, pp. 2206–2214, 2009.
  • [13] M. Zahid, M.I. Asjad, S. Hussain and A. Akg¨ul, ”Nonlinear magnetohydrodynamic flow of nanofluids across a porous matrix over an extending sheet with mass transpiration and bioconvection,” Heat Transfer, vol. 50, no. 8, pp. 7588-7603, 2021.
  • [14] K. S. Brajesh and K. Pramod, ”Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay,” Sociedad Espa˜nola de Matem´atica Aplicada, doi: 10.1007/s40324-017-0117-1, 2017.
  • [15] Y. Khan, N. Faraz, S. Kumar and A.Yildirim, ”A coupling method of homotopy perturbation and Laplace transformation for fractional models,” U. P. B. Science Bulletin, Series A, vol. 74, no. 1, pp. 57 68, 2012.
  • [16] S. Kumar, A. Yildirim, Y. Khan and L. Wei, ”A fractional model of the diffusion equation and its analytical solution using Laplace transform,” Sci. Iran. B, vol. 19, no. 4, pp. 1117–1123, 2012.
  • [17] L. Yanqin, ”Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method,” Abstract and Applied Analysis, vol. 2012, 2012.
  • [18] S. Dinkar, S. Prince, and C. Shubha, ”Homotopy Perturbation Transform Method with He’s Polynomial for Solution of Coupled Nonlinear Partial Differential Equations,” Nonlinear Engineering, vol. 5, no. 1, pp. 17–23, 2016.
  • [19] M. Zellal and K. Belghaba, ”Applications of homotopy perturbation transform method for solving Newell-Whitehead- Segel equation,” General Letters in Mathematics, vol. 3, no. 1, pp. 35-46, 2017.
  • [20] M. El-Shahed and A. Salem, ”On the generalized Navier–Stokes equations,” Appl Math Comput, vol. 156, no. 1, pp. 287–93, 2005.
  • [21] G.A. Birajdar, ”Numerical solution of time fractional Navier–Stokes equation by discrete Adomian decomposition method,” Nonlinear Eng, vol. 3, no. 1, pp. 21–26, 2014.
  • [22] K.S. Brajesh and K. Pramod, ”FRDTM for numerical simulation of multi-dimensional, time fractional model of Navier- Stokes equation,” Ain Shams Engineering Journal, vol. 9, no. 4, pp. 827-834, 2018.
  • [23] D. Kumar, J. Singh and S. Kumar, ”A Fractional Model of Navier-Stokes Equation Arising in Unsteady Flow of a Viscous Fluid,” Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 17, pp. 14-19, 2015.
  • [24] S. Kumar, D. Kumar, M. Abbasbandy and M. Rashidi, ”Analytical Solution of Fractional Navier-Stokes Equation by using Modified Laplace Decomposition Method, ” Ain Shams Engineering Journal, vol. 5, no. 2, pp. 569–574, 2014.
  • [25] S. Momani and Z. Odibat, ”Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006.
  • [26] A. A. Ragab, K. M. Hemida, M. S. Mohamed, and Abd El M. A.Salam, ”Solution of Time-Fractional Navier-Stokes Equation by Using Homotopy Analysis Method,” Gen. Math. Notes, vol. 13, no. 2, pp.13-21, 2012.
  • [27] M. Zellal and K. Belghaba, ”He’s variational iteration method for solving multi-dimensional of Navier Stokes equation,” International Journal of Analysis and Applications, vol. 18, no. 5, pp. 724-737, 2020.
  • [28] J. H. He, ”Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng, vol. 178, pp. 257–262, 1999.
  • [29] J. H. He, ”Homotopy perturbation method: a new nonlinear analytical technique,” App. Math. Comp, vol. 135, pp. 73–79, 2003.
  • [30] A. Ghorbani and J. S. Nadjafi, ” He’s homotopy Perturbation method for calculating Adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007.
  • [31] A. Ghorbani, ”Beyond Adomians polynomials: He polynomials,” Chaos Solitons Fractals, vol. 39, no. 3, pp. 1486– 1492, 2009.
There are 31 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Mohamed Zellal 0000-0001-6903-6106

Kacem Belghaba 0000-0003-4372-6330

Publication Date November 1, 2022
Published in Issue Year 2022 Volume: 19 Issue: 2

Cite

APA Zellal, M., & Belghaba, K. (2022). Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM). Cankaya University Journal of Science and Engineering, 19(2), 102-112.
AMA Zellal M, Belghaba K. Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM). CUJSE. November 2022;19(2):102-112.
Chicago Zellal, Mohamed, and Kacem Belghaba. “Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM)”. Cankaya University Journal of Science and Engineering 19, no. 2 (November 2022): 102-12.
EndNote Zellal M, Belghaba K (November 1, 2022) Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM). Cankaya University Journal of Science and Engineering 19 2 102–112.
IEEE M. Zellal and K. Belghaba, “Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM)”, CUJSE, vol. 19, no. 2, pp. 102–112, 2022.
ISNAD Zellal, Mohamed - Belghaba, Kacem. “Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM)”. Cankaya University Journal of Science and Engineering 19/2 (November 2022), 102-112.
JAMA Zellal M, Belghaba K. Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM). CUJSE. 2022;19:102–112.
MLA Zellal, Mohamed and Kacem Belghaba. “Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM)”. Cankaya University Journal of Science and Engineering, vol. 19, no. 2, 2022, pp. 102-1.
Vancouver Zellal M, Belghaba K. Approximate Solutions to Fractional Multi-Dimensional Navier-Stokes Equation Using the (FHPTM). CUJSE. 2022;19(2):102-1.