Modeling and numerical analysis of fractional-order Bloch equations

This paper deals with the Bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time. These equations were introduced by Felix Bloch in 1946 and they are used for a description of the Nuclear Magnetic Resonance (NMR). This physical phenomenon is used in medicine, chemistry, physics, and engineering to study complex material. Fractional-order generalization of the Bloch equations was presented by Richard Magin et al. in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous, porous, or composite materials. In this paper we describe numerical and simulation models (created for Matlab/Simulink) of the classical and the fractional-order Bloch equations. The behaviour and stability analysis of the Bloch equations are presented as well.

[1]  Yangquan Chen,et al.  Two direct Tustin discretization methods for fractional-order differentiator/integrator , 2003, J. Frankl. Inst..

[2]  Maria da Graça Marcos,et al.  Fractional dynamics in the trajectory control of redundant manipulators , 2008 .

[3]  Le Page,et al.  Complex Variables and the Laplace Transform for Engineers , 2010 .

[4]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[5]  I. Podlubny.  Fractional differential equations , 1998 .

[6]  W. W. Hansen,et al.  Nuclear Induction , 2011 .

[7]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[8]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[9]  Richard L. Magin,et al.  Solving the fractional order Bloch equation , 2009 .

[10]  D. Matignon.  Stability properties for generalized fractional differential systems , 1998 .

[11]  Chyi Hwang,et al.  A note on time-domain simulation of feedback fractional-order systems , 2002, IEEE Trans. Autom. Control..

[12]  R. Bagley,et al.  On the Appearance of the Fractional Derivative in the Behavior of Real Materials , 1984 .

[13]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[14]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[15]  W. Deng,et al.  Chaos synchronization of the fractional Lü system , 2005 .

[16]  I. Podlubny.  Fractional-Order Systems and -Controllers , 1999 .

[17]  S. Westerlund.  Dead matter has memory , 1991 .

[18]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[19]  B. Onaral,et al.  Fractal system as represented by singularity function , 1992 .

[20]  J. C. Wang.  Realizations of Generalized Warburg Impedance with RC Ladder Networks and Transmission Lines , 1987 .

[21]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[22]  Mohammad Saleh Tavazoei,et al.  Limitations of frequency domain approximation for detecting chaos in fractional order systems , 2008 .

[23]  I. Podlubny.  Matrix Approach to Discrete Fractional Calculus , 2000 .

[24]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[25]  J. Álvarez-Ramírez,et al.  Effective medium equations for fractional Fick's law in porous media , 2007 .

[26]  L. Dorcak.  Numerical Models for the Simulation of the Fractional-Order Control Systems , 2002 .

[27]  I. Petráš.  Stability of Fractional-Order Systems with Rational Orders , 2008, 0811.4102.

[28]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[29]  A. Oustaloup.  La dérivation non entière , 1995 .

[30]  Weihua Deng,et al.  Numerical algorithm for the time fractional Fokker-Planck equation , 2007, J. Comput. Phys..

[31]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[32]  Yu-Chung N. Cheng,et al.  Magnetic Resonance Imaging: Physical Principles and Sequence Design , 1999 .

[33]  C. Halijak,et al.  Approximation of Fractional Capacitors (1/s)^(1/n) by a Regular Newton Process , 1964 .

[34]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[35]  Mohammad Saleh Tavazoei,et al.  Chaotic attractors in incommensurate fractional order systems , 2008 .

[36]  K. Diethelm.  Mittag-Leffler Functions , 2010 .

[37]  M. Haeri,et al.  Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems , 2007 .

[38]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[39]  M. Nakagawa,et al.  Basic Characteristics of a Fractance Device , 1992 .

[40]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[41]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[42]  Chien-Cheng Tseng,et al.  Design of FIR and IIR fractional order Simpson digital integrators , 2007, Signal Process..

[43]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[44]  Ivo Petráš,et al.  Chaos in the fractional-order Volta’s system: modeling and simulation , 2009 .

[45]  Ivo Petráš,et al.  Method for simulation of the fractional order chaotic systems , 2006 .