Related papers: A physically based connection between fractional c…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial…
A non--linear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additonal repulsive and attractive forces leading to a changed local mobiltiy. The…
Growth in crystals can be { usually } described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can be characterized by Euclidean geometry with its peculiar symmetries, the growth dynamics…
This study introduces a physics-based machine learning framework for modeling both brittle and ductile fractures. Unlike physics-informed neural networks, which solve partial differential equations by embedding physical laws as soft…
We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical…
This work presents a detailed analytical and geometrical investigation of the (2+1)-dimensional Boiti-Leon-Pempinelli system, a nonlinear dispersive model arising in the context of fluid and plasma dynamics. By employing a projective…
We combine classical continuum mechanics with the recently developed calculus for mixed-dimensional problems to obtain governing equations for flow in, and deformation of, fractured materials. We present models both in the context of finite…
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric…
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling…
This study investigates city dynamics employing a nonextensive diffusion equation suited for addressing diffusion within a fractal medium, where the nonadditive parameter, $q$, plays a relevant role. The findings demonstrate the efficacy of…
This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a…
An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted…
The curves of scaling behavior is a significant concept in fractal dimension analysis of complex systems. However, the underlying rationale of this kind of curves for fractal cities is not yet clear. The aim of this paper is at researching…
We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous model, where all characteristics and fields are defined everywhere in the volume but they follow some…
Based on the connection between Tsallis nonextensive statistics and fractional dimensional space, in this work we have introduced, with the aid of Verlinde's formalism, the Newton constant in a fractal space as a function of the…
An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo…
Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In…
The relationship between fracture aperture (maximum opening; dmax) and fracture width (w) has been the subject of debate over the past several decades. An empirical power law has been commonly applied to relate these two parameters. Its…
We present a number models describing the sequential deposition of a mixture of particles whose size distribution is determined by the power-law $p(x) \sim \alpha x^{\alpha-1}$, $x\leq l$ . We explicitly obtain the scaling function in the…
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of…