Previous |  Up |  Next


magnetic resonance; spin exchange; delay differential equation; characteristic equation
Spin exchange with a time delay in NMR (nuclear magnetic resonance) was treated in a previous work. In the present work the idea is applied to a case where all magnetization components are relevant. The resulting DDE (delay differential equations) are formally solved by the Laplace transform. Then the stability of the system is studied using the real and imaginary parts of the determinant in the characteristic equation. Using typical parameter values for the DDE system, stability is shown for all relevant cases. Also non-oscillating terms in the solution were found by studying the same determinant using similar parameter values.
[1] Bellman, R., Cooke, K. L.: Differential-Difference Equations. Mathematics in Science and Engineering 6 Academic Press, New York (1963). MR 0147745 | Zbl 0105.06402
[2] Gamliel, D., Levanon, H.: Stochastic Processes in Magnetic Resonance. World Scientific, Singapore (1995).
[3] Gamliel, D.: Generalized exchange in magnetic resonance. Funct. Differ. Equ. 18 (2011), 201-227.
[4] Gamliel, D.: Using the Lambert function in an exchange process with a time delay. Electron. J. Qual. Theory Differ. Equ. Proc. 9th Coll. QTDE 7 (2012), 1-12.
[5] Gamliel, D., Domoshnitsky, A., Shklyar, R.: Time evolution of spin exchange with a time delay. Funct. Differ. Equ. 20 (2013), 81-113.
[6] Hadley, G.: Linear Algebra. Addison-Wesley Series in Industrial Management Addison Wesley Publishing Company, Reading (1961). MR 0121368 | Zbl 0108.01103
[7] Horn, R. A., Johnson, C. R.: Matrix Analysis. (2nd ed.), Cambridge University Press, Cambridge (2013). MR 2978290 | Zbl 1267.15001
[8] Kaplan, J. I., Fraenkel, G.: NMR of Chemically Exchanging Systems. Academic Press, New York (1980).
[9] Verheyden, K., Luzyanina, T., Roose, D.: Efficient computation of characteristic roots of delay differential equations using LMS methods. J. Comput. Appl. Math. 214 (2008), 209-226. DOI 10.1016/ | MR 2391684 | Zbl 1135.65349
[10] Vyhlídal, T., Zítek, P.: Mapping based algorithm for large scale computation of quasi-polynomial zeros. IEEE Trans. Autom. Control 54 (2009), 171-177. DOI 10.1109/TAC.2008.2008345 | MR 2478083
[11] Woessner, D. E., Zhang, S., Merritt, M. E., Sherry, A. D.: Numerical solutions of the Bloch equations provides insights into the optimum design of PARACEST agents for MRI. Mag. Reson. Med. 53 (2005), 790-799. DOI 10.1002/mrm.20408
[12] Wu, Z., Michiels, W.: Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method. J. Comput. Appl. Math. 236 (2012), 2499-2514. DOI 10.1016/ | MR 2879716 | Zbl 1237.65065
Partner of
EuDML logo