Skip to main content
SpringerLink
Log in

A reliable algorithm for fractional Bloch model arising in magnetic resonance imaging

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

Magnetic resonance imaging (MRI) is used in physics, chemistry, engineering and medicine to study complex materials. In this paper, numerical solution of fractional Bloch equations in MRI is obtained using fractional variation iteration method (FVIM) and fractional homotopy perturbation transform method (FHPTM). Sufficient conditions for the convergence of FVIM and its error estimate are established. The obtained results are compared with the existing as well as recently developed methods and with the exact solution. The obtained numerical results for different fractional values of time derivative are discussed with the help of figures and tables. Figures are drawn using the Maple package. Test examples are provided to illustrate the accuracy and competency of the proposed schemes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. J H He, Bull. Sci. Technol. Soc. 15(2), 86 (1999)

    Google Scholar 

  2. A D Robinson, Annu. Rev. Neurosci. 4, 463 (1981)

    Article  Google Scholar 

  3. M Mirzazadeh, Pramana – J. Phys. 86(5), 957 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  4. R L Bagley and P J Torvik, AIAA J. 23(6), 918 (1985)

    Article  ADS  Google Scholar 

  5. R L Magin, Crit. Rev. Biomed. Eng. 32(1), 1 (2004)

    Article  Google Scholar 

  6. G W Bohannan, J. Vib. Control 14(9–10), 1487 (2008)

    Article  MathSciNet  Google Scholar 

  7. N Engheia, IEEE Antennas Propag. Mag. 39(4), 35 (1997)

    Article  ADS  Google Scholar 

  8. G Cooper and D Cowan, Explor. Geophys. 34(1/2), 51 (2003)

    Article  Google Scholar 

  9. M F Ali, M Sharma and R Jain, Adv. Eng. Tech. Appl. 5(2), 41 (2016)

    Article  Google Scholar 

  10. R L Magin, Comput. Math. Appl. 59(5), 1586 (2010)

    Article  MathSciNet  Google Scholar 

  11. Q Xu, J Huang and L Zhou, Proceedings of the 34th Chinese Control Conference (2015) p. 4501

  12. H Fallahgoul, S Focardi and F Fabozzi, Fractional calculus and fractional processes with applications to financial economics (Academic Press, San Diego, 2016)

    MATH  Google Scholar 

  13. Y W Zhang, Pramana – J. Phys. 90, 34 (2018)

    ADS  Google Scholar 

  14. R Panda and M Dash, Signal Process. 86(9), 2340 (2006)

    Article  Google Scholar 

  15. O B Awojoyogber, Phys. A: Stat. Mech. Appl. 339(3–4), 437 (2004)

    Article  Google Scholar 

  16. K Murase and N Tanki, Magn. Reson. Imaging 29(1), 126 (2011)

    Article  ADS  Google Scholar 

  17. S Balac and L Chupin, Math. Comput. Model. 48(11–12), 1901 (2008)

  18. J C Leyte, Chem. Phys. Lett. 165(2–3), 213 (1990)

    Article  ADS  Google Scholar 

  19. D Matignon, IMACS – SMC Proceedings (Lille, France, 1996) p. 963

  20. R L Magin, O Abdullah, D Baleanu and X Zhou, J. Magn. Reson. 190(2), 255 (2008)

    Article  ADS  Google Scholar 

  21. I Petráš, Comput. Math. Appl. 61(2), 341 (2011)

    Article  MathSciNet  Google Scholar 

  22. S Kumar, N Faraz and K Sayevand, Walailak J. Sci. Tech. 11(4), 273 (2014)

    Google Scholar 

  23. Q Yu, F Liu, I Turner and K Burrage, J. Comput. Appl. Math. 255, 635 (2014)

    Article  MathSciNet  Google Scholar 

  24. S Qin, F Liu, I Turner, V Vegh and Q Yu, J. Comput. Appl. Math. 319, 308 (2017)

    Article  MathSciNet  Google Scholar 

  25. H Singh, Alexandria Eng. J. 55(3), 2863 (2016)

    Article  Google Scholar 

  26. H Singh and C S Singh, Alexandria Eng. J. (in press), https://doi.org/10.1016/j.aej.2017.07.002 (2017)

  27. H Singh, J. King Saud Univ. Eng. Sci. 29(2), 235 (2017)

    Article  MathSciNet  Google Scholar 

  28. S Qin, F Liu, I W Turner, Q Yang and Q Yu, Comput. Math. Appl. 75(1), 7 (2018)

    Article  MathSciNet  Google Scholar 

  29. Y Zhao, W Bu, X Zhao and Y Tang, J. Comput. Phys. 350, 117 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  30. J H He, Int. J. Nonlinear Mech. 34(4), 699 (1999)

    Article  ADS  Google Scholar 

  31. A Prakash and M Kumar, J. Appl. Anal. Comput. 6(3), 738 (2016)

    MathSciNet  Google Scholar 

  32. J H He and X H Wu, Comput. Math. Appl. 54(7–8), 881 (2007)

    Article  MathSciNet  Google Scholar 

  33. A Prakash, M Kumar and K K Sharma, Appl. Math. Comput. 260, 314 (2015)

    MathSciNet  Google Scholar 

  34. J H He, Comput. Method. Appl. M. 178(3–4), 257 (1999)

    Article  Google Scholar 

  35. J H He, Int. J. Nonlinear Mech. 35(1), 37 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  36. A Prakash, Nonlinear Eng. 5(2), 123 (2016)

    Article  ADS  Google Scholar 

  37. S Kumar, Appl. Math. Model. 38(13), 3154 (2014)

    Article  MathSciNet  Google Scholar 

  38. I Podlubny, Fractional differential equations (Academic Press, New York, 1999)

    MATH  Google Scholar 

  39. A A Elbeleze, A Kiliçman and B M Taib, Abstr. Appl. Anal. 2014, 518343 (2014)

    Google Scholar 

  40. M G Sakar and H Ergoren, Appl. Math. Model. 39(14), 3972 (2015)

    Article  MathSciNet  Google Scholar 

  41. Z M Odibat, Math. Comput. Model. 51(9–10), 1181 (2010)

    Article  Google Scholar 

  42. M Tatari and M Dehghan, J. Comput. Appl. Math. 207(1), 121 (2007)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the anonymous reviewers and editors for their valuable comments and suggestions to improve the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Kurukshetra,  136 119, India

    Amit Prakash

  2. Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University, Mathura,  281 406, India

    Manish Goyal & Shivangi Gupta

Authors
  1. Amit Prakash
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manish Goyal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shivangi Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amit Prakash.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prakash, A., Goyal, M. & Gupta, S. A reliable algorithm for fractional Bloch model arising in magnetic resonance imaging. Pramana - J Phys 92, 18 (2019). https://doi.org/10.1007/s12043-018-1683-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-018-1683-1

Keywords

PACS Nos

Search

Navigation