Skip to main content
SpringerLink
Log in

On Asymptotic Stability Analysis and Solutions of Fractional-Order Bloch Equations

  • Conference paper
  • First Online:
Transactions on Engineering Technologies (IMECS 2018)

Included in the following conference series:

  • 225 Accesses

Abstract

The Bloch equations are a model for nuclear magnetic resonance (NMR), which is a physical phenomenon arising in engineering, medicine, and the physical sciences. The main contributions of this work are to obtain an asymptotic stability condition of Caputo fractional-order Bloch equations and to provide analytical solutions and numerical solutions of the mentioned fractional-order model. The asymptotic stability analysis is generalized for the incommensurate fractional-order Bloch equations. The standard two methods employed to solve the fractional-order Bloch equations are a revised variational iteration method and an Adams-Bashforth-Moulton type predictor-corrector scheme. The first method gives analytical solutions while the second scheme generates numerical solutions for the problem. Comparisons of the two types of obtained solutions are demonstrated via varying fractional orders of the model.

S. Sirisubtawee—A Researcher in the Centre of Excellence in Mathematics, Bangkok, 10400, Thailand.

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

Access this chapter

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

Institutional subscriptions

References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1989)

    MATH  Google Scholar 

  2. Obembe, A.D., Al-Yousef, H.Y., Hossain, M.E., Abu-Khamsin, S.A.: Fractional derivatives and their applications in reservoir engineering problems: a review. J. Petrol. Sci. Eng. 157, 312–327 (2017)

    Article  Google Scholar 

  3. Guner, O., Bekir, A.: The Exp-function method for solving nonlinear space-time fractional differential equations in mathematical physics. J. Assoc. Arab Universities Basic Appl. Sci. 24, 277–282 (2017)

    Google Scholar 

  4. Sirisubtawee, S., Koonprasert, S., Khaopant, C., Porka, W.: Two reliable methods for solving the (3 + 1)-dimensional space-time fractional Jimbo-Miwa equation. Math. Probl. Eng. (2017)

    Google Scholar 

  5. Carvalho, A., Pinto, C.M.: A delay fractional order model for the co-infection of malaria and HIV/AIDS. Int. J. Dyn. Control 5(1), 168–186 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dogan, N.: Numerical solution of chaotic Genesio system with multi-step Laplace Adomian decomposition method. Kuwait J. Sci. 40(1) (2013)

    Google Scholar 

  7. Sekson Sirisubtawee, S.K., Kaewta, S.: Duan-Rach modified decomposition method for solving some types of nonlinear fractional multi-point boundary value problems. J. Electr. Eng. 102(10), 2143–2176 (2017)

    MATH  Google Scholar 

  8. Ghazanfari, B., Veisi, F.: Homotopy analysis method for the fractional nonlinear equations. J. King Saud University Sci. 23(4), 389–393 (2011)

    Article  Google Scholar 

  9. Martin, O.: A modified variational iteration method for the analysis of viscoelastic beams. Appl. Math. Model. 40(17), 7988–7995 (2016)

    Article  MathSciNet  Google Scholar 

  10. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)

    Article  MathSciNet  Google Scholar 

  11. Petráš, I.: Fractional-order feedback control of a DC motor. J. Electr. Eng. 60(3), 117–128 (2009)

    Google Scholar 

  12. Zhao, E., Chao, T., Wang, S., Yang, M.: Finite-time formation control for multiple flight vehicles with accurate linearization model. Aerosp. Sci. Technol. 71, 90–98 (2017)

    Article  Google Scholar 

  13. Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Concepts Magn. Reson. Part A 34(1), 16–23 (2009)

    Article  Google Scholar 

  14. Qin, Y., Lu, C., Li, L.: Multi-scale cyclone activity in the Changjiang River-Huaihe River valleys during spring and its relationship with rainfall anomalie. Adv. Atmos. Sci. 34(2), 246–257 (2017)

    Article  Google Scholar 

  15. Baleanu, D., Magin, R.L., Bhalekar, S., Daftardar-Gejji, V.: Chaos in the fractional order nonlinear Bloch equation with delay. Commun. Nonlinear Sci. Numer. Simul. 25(1), 41–49 (2015)

    Article  MathSciNet  Google Scholar 

  16. Ünlü, C., Jafari, H., Baleanu, D.: Revised variational iteration method for solving systems of nonlinear fractional-order differential equations. Abstr. Appl. Anal. (2013)

    Google Scholar 

  17. Zayernouri, M., Matzavinos, A.: Fractional Adams-Bashforth/Moulton methods: an application to the fractional Keller-Segel chemotaxis system. J. Comput. Phys. 317, 1–14 (2016)

    Article  MathSciNet  Google Scholar 

  18. Matignon, D.: Stability properties for generalized fractional differential systems. In: Proceedings of Fractional Differential Systems: Models, Methods and Applications, vol. 5, pp. 145–158 (1998)

    Article  MathSciNet  Google Scholar 

  19. Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367, 102–113 (2007)

    Article  Google Scholar 

  20. Tavazoei, M.S., Haeri, M.: Limitations off requency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008)

    Article  MathSciNet  Google Scholar 

  21. Deng, W., Li, C., Lu, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)

    Article  MathSciNet  Google Scholar 

  22. He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167(1–2), 57–68 (1998)

    Article  MathSciNet  Google Scholar 

  23. Inokuti, M., Sekine, H., Mura, T.: General use of the Lagrange multiplier in nonlinear mathematical physics. In: Variational Method in the Mechanics of Solids, vol. 33, no. 5, pp. 156–162 (1978)

    Chapter  Google Scholar 

  24. Sirisubtawee, S.: Comparison of analytical and numerical solutions of fractional-order Bloch Equations using reliable methods. In: Lecture Notes in Engineering and Computer Science: Proceedings of the International Multi Conference of Engineers and Computer Scientists, Hong Kong, 14–16 March 2018, pp. 467–472 (2018)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editors and the anonymous referees for their valuable suggestions on the improvement of this work. The present work is extended and revised from the corresponding conference paper [24], which was financially supported by the Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (contract no. 5942106).

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

    Sekson Sirisubtawee

Authors
  1. Sekson Sirisubtawee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sekson Sirisubtawee .

Editor information

Editors and Affiliations

  1. International Association of Engineers, Hong Kong, Hong Kong

    Sio-Iong Ao

  2. Department of Computer and Communication, Dae Gu Catholic University, Daegu, Korea (Republic of)

    Haeng Kon Kim

  3. Graduate Division, Tijuana Institute of Technology, Tijuana, Mexico

    Oscar Castillo

  4. Department of SEEM, City University of Hong Kong, Kowloon, Hong Kong

    Alan Hoi-shou Chan

  5. Department of Industrial Engineering and Management, Kanagawa University, Yokohama, Japan

    Hideki Katagiri

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sirisubtawee, S. (2020). On Asymptotic Stability Analysis and Solutions of Fractional-Order Bloch Equations. In: Ao, SI., Kim, H., Castillo, O., Chan, As., Katagiri, H. (eds) Transactions on Engineering Technologies. IMECS 2018. Springer, Singapore. https://doi.org/10.1007/978-981-32-9808-8_21

Download citation

Publish with us

Policies and ethics

Search

Navigation