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Solution of Fractional Order Partial Differential Equations by Discrete Homotopy Perturbation Method

Year 2020, Volume: 20 Issue: 2, 213 - 221, 20.05.2020
https://doi.org/10.35414/akufemubid.685429

Abstract

This work is developed the discrete homotopy perturbation method with a space discrete version to solve the linear and nonlinear time derivative fractional partial differential equations. The fractional derivatives are considered in the sense of Caputo. The success and applicability of this method has been demonstrated by some sample problems. When fractional order is unit, obtained results are good agreement with the exact solutions. The method demonstrated in this study is expected to solve similar problems in fractional calculus.

References

  • Bratsos, A., Ehrhardt, M. and Famelis, I.T., 2008. A Discrete Adomian decomposition method for discrete nonlinear Schrödinger equations. Applied Mathematics and Computation, 197, 190—205.
  • Burgers, J.M., 1948. A Mathematical model illustration the theory of turbulence. Advances in Applied Mechanics, 1, 171—199.
  • Caputo, M., 1967. Linear models of dissipition whose Q is almost independent. II, Geophys. J. Roy. Astron., 13, 529—539.
  • Dhaigude, D.B. and Birajdar, G.A., 2014. Numerical solutions of fractional partial differential equations by discrete Adomian decomposition method. Advances in Applied Mathematics and Mechanics, 6, 107—119.
  • He, J.H., 1998. An approximate solution technique depending on an artifical parameter: a special example. Communications in Nonlinear Science and Numerical Simulation, 3, 92—97.
  • He, J.H., 2000. A coupling method of homotopy technique and perturbation technique for nonlinear problems. International Journal of Non-Linear Mechanic., 35, 37—43.
  • He, J.H., 2003. Homotopy perturbation method: A new nonlinear analytic technique. Applied Mathematics and Computation, 135, 73—79.
  • He, J.H., 2009. An elementary introduction to the homotopy perturbation method. Computers and Mathematics with Applications, 57, 410—412.
  • Hemeda, A.A., 2012. Homotopy perturbation method for solving partial differential equations of fractional order. International Journal of Mathematical Analysis, 6(49), 2431—2448.
  • Luchko, Y. and Gorenflo, R., 1999. An operational method for solving fractional differential equations with the Caputo derivative. Acta Mathmatica Vietnamica, 24, 207—233.
  • Özpınar F., 2018. Applying discrete homotopy analysis method for solving fractional partial differential equations. Entropy, 20(5), 332.
  • Özpınar F., 2018. Solving fractional difference equations by discrete Adomian decomposition method. Journal of Balıkesir University Institute of Science and Technology, 20(3), 15-22.
  • Özpınar F. and Belgacem F.B.M., 2019. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete Continuous Dynamical Systems - S, 12(3), 615-624.
  • Podlubny, I., , 1999. Fractional Differential Equations. Academic Press, San Diego.
  • Zhu, H., Shu, H. and Ding, M., 2010. Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method Computers and Mathematics with Applications, 60, 840—848.
  • Zhu, H., Shu, H. and Ding, M., 2010. Numerical solutions of partial differential equations by discrete homotopy analysis method. Applied Mathematics and Computation, 216, 3592—3605.
  • Zhu, H. and Ding, M., 2014. The discrete homotopy perturbation method for solving Burgers’ and heat equations. Journal of Information and Computing Science, 11(5), 1647—1657.

Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü

Year 2020, Volume: 20 Issue: 2, 213 - 221, 20.05.2020
https://doi.org/10.35414/akufemubid.685429

Abstract

Bu çalışma, lineer ve lineer olmayan zaman kesirli mertebeli kısmi diferensiyel denklemleri çözmek için ayrık uzak biçimli ayrık homotopi perturbasyon metodunu geliştirmiştir. Kesirli mertebe türevler Caputo anlamında göz önüne alınmıştır. Bu metodun başarısı ve uygulanabilirliği bazı örnek problemler ile gösterilmiştir. Elde edilen sonuçlar kesirli mertebe bir olduğunda, tam çözümler ile iyi bir uyumluluk göstermiştir. Bu çalışmada gösterilen metodun kesirli mertebe hesabındaki benzer problemleri çözmesi beklenmektedir.

References

  • Bratsos, A., Ehrhardt, M. and Famelis, I.T., 2008. A Discrete Adomian decomposition method for discrete nonlinear Schrödinger equations. Applied Mathematics and Computation, 197, 190—205.
  • Burgers, J.M., 1948. A Mathematical model illustration the theory of turbulence. Advances in Applied Mechanics, 1, 171—199.
  • Caputo, M., 1967. Linear models of dissipition whose Q is almost independent. II, Geophys. J. Roy. Astron., 13, 529—539.
  • Dhaigude, D.B. and Birajdar, G.A., 2014. Numerical solutions of fractional partial differential equations by discrete Adomian decomposition method. Advances in Applied Mathematics and Mechanics, 6, 107—119.
  • He, J.H., 1998. An approximate solution technique depending on an artifical parameter: a special example. Communications in Nonlinear Science and Numerical Simulation, 3, 92—97.
  • He, J.H., 2000. A coupling method of homotopy technique and perturbation technique for nonlinear problems. International Journal of Non-Linear Mechanic., 35, 37—43.
  • He, J.H., 2003. Homotopy perturbation method: A new nonlinear analytic technique. Applied Mathematics and Computation, 135, 73—79.
  • He, J.H., 2009. An elementary introduction to the homotopy perturbation method. Computers and Mathematics with Applications, 57, 410—412.
  • Hemeda, A.A., 2012. Homotopy perturbation method for solving partial differential equations of fractional order. International Journal of Mathematical Analysis, 6(49), 2431—2448.
  • Luchko, Y. and Gorenflo, R., 1999. An operational method for solving fractional differential equations with the Caputo derivative. Acta Mathmatica Vietnamica, 24, 207—233.
  • Özpınar F., 2018. Applying discrete homotopy analysis method for solving fractional partial differential equations. Entropy, 20(5), 332.
  • Özpınar F., 2018. Solving fractional difference equations by discrete Adomian decomposition method. Journal of Balıkesir University Institute of Science and Technology, 20(3), 15-22.
  • Özpınar F. and Belgacem F.B.M., 2019. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete Continuous Dynamical Systems - S, 12(3), 615-624.
  • Podlubny, I., , 1999. Fractional Differential Equations. Academic Press, San Diego.
  • Zhu, H., Shu, H. and Ding, M., 2010. Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method Computers and Mathematics with Applications, 60, 840—848.
  • Zhu, H., Shu, H. and Ding, M., 2010. Numerical solutions of partial differential equations by discrete homotopy analysis method. Applied Mathematics and Computation, 216, 3592—3605.
  • Zhu, H. and Ding, M., 2014. The discrete homotopy perturbation method for solving Burgers’ and heat equations. Journal of Information and Computing Science, 11(5), 1647—1657.
There are 17 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Articles
Authors

Figen Özpınar 0000-0002-7428-4988

Publication Date May 20, 2020
Submission Date February 7, 2020
Published in Issue Year 2020 Volume: 20 Issue: 2

Cite

APA Özpınar, F. (2020). Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 20(2), 213-221. https://doi.org/10.35414/akufemubid.685429
AMA Özpınar F. Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. May 2020;20(2):213-221. doi:10.35414/akufemubid.685429
Chicago Özpınar, Figen. “Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu Ile Çözümü”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20, no. 2 (May 2020): 213-21. https://doi.org/10.35414/akufemubid.685429.
EndNote Özpınar F (May 1, 2020) Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20 2 213–221.
IEEE F. Özpınar, “Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 20, no. 2, pp. 213–221, 2020, doi: 10.35414/akufemubid.685429.
ISNAD Özpınar, Figen. “Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu Ile Çözümü”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 20/2 (May 2020), 213-221. https://doi.org/10.35414/akufemubid.685429.
JAMA Özpınar F. Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2020;20:213–221.
MLA Özpınar, Figen. “Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu Ile Çözümü”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 20, no. 2, 2020, pp. 213-21, doi:10.35414/akufemubid.685429.
Vancouver Özpınar F. Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2020;20(2):213-21.