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).

已经在许多聚合物和生物系统中研究了异常扩散。PFG异常扩散的分析依赖于获得信号衰减表达式的能力。但是，以前尚未报道过基于分数导数的通用分析PFG信号衰减表达式。此外，由于其形式不同，文献中针对PFG异常扩散所报道的修改后的Bloch方程产生了不同的结果。在此，提出了一种基于分数阶导数的积分型改进的Bloch方程，用于PFG异常扩散，它与传统的微分型改进的Bloch方程明显不同。积分型修正的Bloch方程的优点在于，线性或非线性过程贡献的原始属性在组合的瞬间保持不变。从修正的布洛赫方程式中，可以得出一般解，其中包括有限梯度脉冲宽度（FGPW）效应。这些PFG信号衰减表达式的数值评估可以通过Adomian分解或快速实用的直接积分方法获得。理论结果与本文执行的连续时间随机游走（CTRW）模拟一致。此外，发现PFG异常扩散的弛豫效果与PFG正常扩散的弛豫效果不同。