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The time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases a = 1 with arbitrary n, or n = 1 with arbitrary a. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

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We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variabletandx,respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.

Puerarin, an isoflavone component extracted from Kudzu (Pueraria lobata), has been demonstrated to alleviate alcoholrelated disorders. Our study examined whether puerarin ameliorates chronic alcoholic liver injury through inhibition of endotoxin gut leakage, the subsequent Kupffer cell activation, and endotoxin receptors expression. Rats were provided with the Liber-DeCarli liquid diet for 8 weeks. Puerarin (90 mg/kg or 180 mg/kg daily) was orally administered from the beginning of the third week until the end of the experiment. Chronic alcohol intake caused increased serum alanine aminotransferase, aspartate aminotransferase, hepatic gamma-glutamyl transpeptidase, and triglyceride levels as well as fatty liver and neutrophil infiltration in hepatic lobules as determined by biochemical and histologic assays. A significant increase of liver tumor necrosis factor a was detected by enzyme-linked immunosorbent assay. These pathologic effects correlated with increased endotoxin level in portal vein and upregulated protein expression of hepatic CD68, lipopolysaccharide-binding protein, CD14, Toll-like receptor 2, and Toll-like receptor 4. Meanwhile, the intestinal microvilli were observed to be sparse, shortened, and irregularity in distribution under the transmission electron microscope in conjunction with the downregulated intestinal zonula occludens-1 protein expression. These hepatic pathologic changes were significantly inhibited in puerarintreated animals as were the endotoxin levels and hepatic CD68 and endotoxin receptors. Moreover, the pathologic changes in intestinal microvillus and the decreased intestinal zonula occludens-1 were also ameliorated with puerarin treatment. These results thus demonstrate that puerarin inhibition of endotoxin gut leakage, Kupffer cell activation, and endotoxin receptors expression is involved in the alleviation of chronic alcoholic liver injury in rats.

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Abstract. We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or RiemannLiouville) derivative of order β ∈ (0, 2] and the first-order time derivative with Caputo derivative of order α ∈ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.