On the Riemann-Liouville fractional calculus and some recent applications
On the Riemann-Liouville fractional calculus and some recent applications
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DOI：
10.1142/s0218348x95000497
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发表时间：
1995-09-01
期刊：
影响因子：
--
通讯作者：
Metzler, R
中科院分区：
文献类型：
--
作者：
Nonnenmacher, TF;Metzler, R
当Benoit Mandelbrot在他的经典著作《自然的分形几何》中讨论分数布朗运动问题时，他已经指出了与黎曼-刘维尔分数积分和微分学的一些密切关系。在过去的十年中，出现了几篇论文，其中将松弛、振荡、扩散和波传播的整数阶标准微分方程建模过程推广到分数阶微分方程。所有这一切背后的基本思想是微分阶数不必是整数而是小数（即 d(q)/dt(q) 且 0 < q < 1）。将讨论在聚合物甚至生物组织等复杂系统中减缓松弛过程以及自相似蛋白质动力学的应用。此外，我们研究了分数扩散方程，并提出了分形物体上随机游走者位置的相应概率密度函数。狐狸功能发挥着主导作用。
When Benoit Mandelbrot discussed the problem of fractional Brownian motion in his classic book The Fractal Geometry of Nature, he already pointed out some strong relations to the Riemann-Liouville fractional integral and differential calculus. Over the last decade several papers have appeared in which integer-order, standard differential equations modeling processes of relaxation, oscillation, diffusion and wave propagation are generalized to fractional order differential equations. The basic idea behind all that is that the order of differentiation need not be an integer but a fractional number (i.e. d(q)/dt(q) with 0 < q < 1). Applications to slow relaxation processes in complex systems like polymers or even biological tissue and to selfsimilar protein dynamics will be discussed. In addition, we investigate a fractional diffusion equation and we present the corresponding probability density function for the location of a random walker on a fractal object. Fox-functions play a dominant part.