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 $\delta$-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. 
Thomas Michelitsch (IJLRA), G\'erard A. Maugin (IJLRA), Rahman   Mujibur, Shahram Derogar, Andrzej F. Nowakowski, Franck C.G...
math-ph math.MP math.PR physics.class-ph