一维线弹性连续体的自相似场论和自相似扩散问题
A self-similar field theory for 1D linear elastic continua and self-similar diffusion problem
摘要：
本文研究了具有自相似相互作用的一维连续体线弹性力学的一些基本问题。我们引入了一个自相似连续场的方法，其中的自相似性是由运动方程，这是空间非局部卷积与幂函数内核（分数积分）反映。我们得到了封闭形式的静态位移格林函数的表达式，由于一个单位$δ$力。在动力学框架下，我们导出了{ \it Cauchy问题}的解和滞后格林函数。我们推导出一个具有Lévi-stable分布的自相似扩散问题的解的分布，该解具有描述统计Lévi-flights的无穷均值波动。我们推导出一个层次的解决方案的自相似泊松方程，我们称之为“自相似的潜力”。这些非局部奇异势在某种意义上是一维狄拉克$δ$函数的自相似类似物。该方法可以作为解决各种尺度不变跨学科问题的起点。
英文摘要：
This paper is devoted to the analysis of some fundamental problems of linear elasticity in 1D continua with self-similar interparticle interactions. We introduce a self-similar continuous field approach where the self-similarity is reflected by equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit $δ$-force. In the dynamic framework we derive the solution of the {\it Cauchy problem} and the retarded Green's function. We deduce the distribution of a self-similar variant of diffusion problem with Lévi-stable distributions as solutions with infinite mean fluctuations describing the statistics Lévi-flights. We deduce a hierarchy of solutions for the self-similar Poisson's equation which we call "self-similar potentials". These non-local singular potentials are in a sense self-similar analogues to the 1D-Dirac's $δ$-function. The approach can be the starting point to tackle a variety of scale invariant interdisciplinary problems.
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