Abstract
This article deals with the fractional Bloch equation by using Caputo-Fabrizio fractional derivative and Atangana-Baleanu fractional derivative with non-singular kernels. Bloch equation is extensively used in chemistry, physics, magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR). The nuclear magnetization M = (M x, M y, M z) is derived analytically, and its behaviour is discussed via plots for different fractional orders. A comparative study of the analytical solutions with Caputo-Fabrizio, Atangana-Baleanu and Caputo fractional derivatives is presented. Equilibrium stage is achieved faster via Atangana-Baleanu fractional derivative than other fractional derivatives.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aguilar, J.F.G., Martinez, H.Y. et al.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model. 40, 9079–9094 (2016)
Aguilar, J.F.G., Martinez, H.Y. et al.: Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy. 17, 6289–6303 (2015)
Almeida, R., Bastosa N.R.O., Monteiro M.T.T.: Modeling some real phenomena by fractional differential equations. Math. Meth. Appl. Sci. 39, 4846–4855 (2016)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal. Sci. 20, 763–769 (2016)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 2, 73–85 (2015)
Diethelm, K.: A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear. Dyn. 71, 613–619 (2013)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional Derivative. J. comput. Appl. Math. 264, 65–70 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier SanDiego (2006)
Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015)
Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Wiley. Inter. Sci. 34A(1), 16–23 (2009)
Petras, I.: Modelling and numerical analysis of fractional order Bloch equations. Comput. Math. Appl. 6, 341–356 (2011)
Podlubny, I.: Fractional Differential equations. Academic press San Diego (1999)
Singh, H.: Operational matrix approach for approximate solution of fractional model of Bloch equation. J. King. Saud. Univ-Sci. 29(2), 235–240 (2017)
Varalta, N., Gomes A.V., Camargo R.F.: A prelude to the fractional calculus applied to tumor dynamic. Tema. 15(2), 211–221 (2014)
Yu, Q., Liu, F., Turner, I., Burrage, K.: Numerical simulation of fractional Bloch Equations. J. Comput. Appl. Math. 255, 635–651 (2014)
Acknowledgements
The second author acknowledges the University Grants Commission of India for providing financial support for the above research (Sr.No. 2061440951, reference no.22/06/14(i)EU-V). The authors would like to thank the anonymous reviewers for their valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Ravi Kanth, A.S.V., Garg, N. (2019). Analytical Solutions of the Bloch Equation via Fractional Operators with Non-singular Kernels. In: Rushi Kumar, B., Sivaraj, R., Prasad, B., Nalliah, M., Reddy, A. (eds) Applied Mathematics and Scientific Computing. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01123-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-01123-9_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-01122-2
Online ISBN: 978-3-030-01123-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)