Skip to main content
SpringerLink
Log in
Book cover

Conference on Non-integer Order Calculus and Its Applications

RRNR 2017: Non-Integer Order Calculus and its Applications pp 49–62Cite as

New Numerical Techniques for Solving Fractional Partial Differential Equations in Conformable Sense

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 496))

Abstract

This study adresses two new numerical techniques for solving some interesting one-dimensional time-fractional partial differential equations (PDEs). We have introduced modified homotopy perturbation method in conformable sense (MHPMC) and Adomian decomposition method in conformable sense (ADMC) which improve the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of three illustrative problems. Also, we have declared that the proposed models are very efficient and powerful techniques in finding approximate-analytical solutions for the PDEs of fractional order in conformable sense.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Zhang, Y.: A finite difference method for fractional partial differential equation. Appl. Math. Comput. 215, 524–529 (2009). https://doi.org/10.1016/j.amc.2009.05.018

  2. Özdemir, N., Yavuz, M.: Numerical solution of fractional Black-Scholes equation by using the multivariate Padé approximation. Acta Physica Polonica A 132, 1050–1053 (2017). https://doi.org/10.12693/APhysPolA.132.1050

  3. Ibrahim, R.W.: On holomorphic solutions for nonlinear singular fractional differential equations. Comput. Math. Appl. 62, 1084–1090 (2011). https://doi.org/10.1016/j.camwa.2011.04.037

  4. Odibat, Z., Momani, S.: A generalized differential transform method for linear partial differential equations of fractional order. Appl. Math. Lett. 21, 194–199 (2008). https://doi.org/10.1016/j.aml.2007.02.022

  5. Odibat, Z., Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 32, 28–39 (2008). https://doi.org/10.1016/j.apm.2006.10.025

  6. Bildik, N., Bayramoglu, H.: The solution of two dimensional nonlinear differential equation by the Adomian decomposition method. Appl. Math. Comput. 163, 519–524 (2005). https://doi.org/10.1016/j.amc.2004.03.029

  7. Bildik, N., Konuralp, A., Bek, F.O., Küçükarslan, S.: Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method. Appl. Math. Comput. 172, 551–567 (2006). https://doi.org/10.1016/j.amc.2005.02.037

  8. Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301, 508–518 (2005). https://doi.org/10.1016/j.jmaa.2004.07.039

  9. Elbeleze, A.A., Kılıçman, A., Taib, B.M.: Homotopy perturbation method for fractional Black-Scholes European option pricing equations using Sumudu transform. Math. Probl. Eng. Article ID 524852 (2013). https://doi.org/10.1155/2013/524852

  10. Turut, V., Güzel, N.: Comparing numerical methods for solving time–fractional reaction–diffusion equations. ISRN Math. Anal. Article ID 737206 (2012). https://doi.org/10.5402/2012/737206

  11. El-Wakil, S., Abdou, M., Elhanbaly, A.: Adomian decomposition method for solving the diffusion–convection–reaction equations. Appl. Math. Comput. 177, 729–736 (2006) https://doi.org/10.1016/j.amc.2005.09.105

  12. Gülkaç, V.: The homotopy perturbation method for the Black–Scholes equation. J. Statist. Comput. Simul. 80, 1349–1354 (2010). https://doi.org/10.1080/00949650903074603

  13. Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207, 96–110 (2007). https://doi.org/10.1016/j.cam.2006.07.015

  14. Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365, 345–350 (2007). https://doi.org/10.1016/j.physleta.2007.01.046

  15. Evirgen, F., Özdemir, N.: A fractional order dynamical trajectory approach for optimization problem with HPM. In: Baleanu D., Machado, J.A.T., Luo, A.C.J. (eds.) Fractional Dynamics and Control, pp. 145–155. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-0457-6-12

  16. Yavuz, M., Ozdemir, N., Okur, Y.Y.: Generalized differential transform method for fractional partial differential equation from finance. In: International Conference on Fractional Differentiation and Its Applications, pp. 778–785, Novi Sad, Serbia (2016)

    Google Scholar 

  17. Javidi, M., Ahmad, B.: Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. In: Advances in Difference Equations, vol. 2013, p. 375 (2013). https://doi.org/10.1186/1687-1847-2013-375

  18. Madani, M., Fathizadeh, M., Khan, Y., Yildirim, A.: On the coupling of the homotopy perturbation method and Laplace transformation. Math. Comput. Model. 53, 1937–1945 (2011) https://doi.org/10.1016/j.mcm.2011.01.023

  19. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014). https://doi.org/10.1016/j.cam.2014.01.002

  20. Anderson, D., Ulness, D.: Newly defined conformable derivatives. Adv. Dyn. Systems and Applications. 10, 109–137 (2015)

    MathSciNet  Google Scholar 

  21. Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Mathematics. 13, 889–898 (2015). https://doi.org/10.1515/math-2015-0081

  22. Avcı, D., Eroglu, B.I., Ozdemir, N.: Conformable heat problem in a cylinder. In: International Conference on Fractional Differentiation and Its Applications, pp. 572–558. Novi Sad, Serbia (2016)

    Google Scholar 

  23. Avcı, D., Eroğlu, B.B.İ., Özdemir, N.: Conformable fractional wave-like equation on a radial symmetric plate. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds.) Theory and Applications of Non-Integer Order Systems. LNEE, vol. 407, pp. 137–146. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-45474-0-13

  24. Avci, D., Iskender Eroglu, B.B., Ozdemir, N.: Conformable heat equation on a radial symmetric plate. Thermal Sci. 21, 819–826 (2017). https://doi.org/10.2298/TSCI160427302A

  25. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016

  26. Yavuz, M.: Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. Int. J. Optim. Control Theor. Appl. (IJOCTA) 8, 1–7 (2017). https://doi.org/10.11121/ijocta.01.2018.00540

  27. Acan, O., Baleanu, D.: A new numerical technique for solving fractional partial differential equations. arXiv preprint arXiv:170402575 (2017)

  28. Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135, 501–544 (1988). https://doi.org/10.1016/0022-247X(88)90170-9

  29. Xu, H., Liao, S.-J., You, X.-C.: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1152–1156 (2009). https://doi.org/10.1016/j.cnsns.2008.04.008

  30. Vanani, S.K., Aminataei, A.: Tau approximate solution of fractional partial differential equations. Comput. Math. Appl. 62, 1075–1083 (2011). https://doi.org/10.1016/j.camwa.2011.03.013

  31. Odibat, Z., Momani, S.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl. 58, 2199–2208 (2009). https://doi.org/10.1016/j.camwa.2009.03.009

Download references

Author information

Authors and Affiliations

  1. Necmettin Erbakan University, 42090, Konya, Turkey

    Mehmet Yavuz

  2. Balıkesir University, 10145, Balıkesir, Turkey

    Necati Özdemir

Authors
  1. Mehmet Yavuz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Necati Özdemir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehmet Yavuz .

Editor information

Editors and Affiliations

  1. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Piotr Ostalczyk

  2. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Dominik Sankowski

  3. Institute of Applied Computer Science, Lodz University of Technology, Lodz, Poland

    Jacek Nowakowski

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Yavuz, M., Özdemir, N. (2019). New Numerical Techniques for Solving Fractional Partial Differential Equations in Conformable Sense. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78458-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78457-1

  • Online ISBN: 978-3-319-78458-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation