A self-similar field theory for 1D linear elastic continua and self-similar diffusion problem

TL;DR

This paper develops a self-similar field theory for 1D elastic continua and diffusion, using fractional integrals to derive explicit solutions and explore scale-invariant properties in elastic and diffusive systems.

Contribution

It introduces a novel self-similar continuous field approach with fractional integrals, providing explicit Green's functions and solutions for 1D elastic and diffusion problems with scale invariance.

Findings

Derived closed-form static Green's functions for 1D elasticity.

Solved the Cauchy problem and Green's functions for dynamic cases.

Identified self-similar solutions for diffusion with Lévi-stable distributions.

Abstract

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 $\delta$-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 L\'evi-stable distributions as solutions with infinite mean fluctuations describing the statistics L\'evi-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 $\delta$-function. The approach can be the starting point to tackle a variety of scale invariant interdisciplinary problems.