  Anomalous diffusion has been investigated in many polymer and biological
systems. The analysis of PFG anomalous diffusion relies on the ability to
obtain the signal attenuation expression. However, the general analytical PFG
signal attenuation expression based on the fractional derivative has not been
previously reported. Additionally, the reported modified-Bloch equations for
PFG anomalous diffusion in the literature yielded different results due to
their different forms. Here, a new integral type modified-Bloch equation based
on the fractional derivative for PFG anomalous diffusion is proposed, which is
significantly different from the conventional differential type modified-Bloch
equation. 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. From the modified-Bloch
equation, the general solutions are derived, which includes the finite gradient
pulse width (FGPW) effect. The numerical evaluation of these PFG signal
attenuation expressions can be obtained either by the Adomian decomposition, or
a direct integration method that is fast and practicable. The theoretical
results agree with the continuous-time random walk (CTRW) simulations performed
in this paper. Additionally, the relaxation effect in PFG anomalous diffusion
is found to be different from that in PFG normal diffusion. The new
modified-Bloch equations and their solutions provide a fundamental tool to
analyze PFG anomalous diffusion in nuclear magnetic resonance (NMR) and
magnetic resonance imaging (MRI).
