Numerical simulation of the fractional Bloch equations

In physics and chemistry, specifically in NMR (nuclear magnetic resonance) or MRI (magnetic resonance imaging), or ESR (electron spin resonance) the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx,My,Mz) as a function of time when relaxation times T1 and T2 are present. Recently, some fractional models have been proposed for the Bloch equations, however, effective numerical methods and supporting error analyses for the fractional Bloch equation (FBE) are still limited. In this paper, the time-fractional Bloch equations (TFBE) and the anomalous fractional Bloch equations (AFBE) are considered. Firstly, we derive an analytical solution for the TFBE with an initial condition. Secondly, we propose an effective predictor-corrector method (PCM) for the TFBE, and the error analysis for PCM is investigated. Furthermore, we derive an effective implicit numerical method (INM) for the anomalous fractional Bloch equations (AFBE), and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the AFBE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.