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

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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