Title: Fractional differential and fractional integral modified-Bloch equations for PFG anomalous diffusion and their general solutions,
Abstract: The studying of anomalous diffusion by pulsed field gradient (PFG) diffusion
technique still faces challenges. Two different research groups have proposed
modified Bloch equation for anomalous diffusion. However, these equations have
different forms and, therefore, yield inconsistent results. The discrepancy in
these reported modified Bloch equations may arise from different ways of
combining the fractional diffusion equation with the precession equation where
the time derivatives have different derivative orders and forms. Moreover, to
the best of my knowledge, the general PFG signal attenuation expression
including finite gradient pulse width (FGPW) effect for time-space fractional
diffusion based on the fractional derivative has yet to be reported by other
methods. Here, based on different combination strategy, two new modified Bloch
equations are proposed, which belong to two significantly different types: a
differential type based on the fractal derivative and an integral type based on
the fractional derivative. The merit of the integral type modified Bloch
equation is that the original properties of the contributions from linear or
nonlinear processes remain unchanged at the instant of the combination. The
general solutions including the FGPW effect were derived from these two
equations as well as from two other methods: a method observing the signal
intensity at the origin and the recently reported effective phase shift
diffusion equation method. The relaxation effect was also considered. It is
found that the relaxation behavior influenced by fractional diffusion based on
the fractional derivative deviates from that of normal diffusion. The general
solution agrees perfectly with continuous-time random walk (CTRW) simulations
as well as reported literature results. The new modified Bloch equations is a
valuable tool to describe PFG anomalous diffusion in NMR and MRI.