Fractal Pennes and Cattaneo-Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

Foam drainage equation in fractal dimensions: breaking and instabilities

This paper is concerned with the construction of a phenomenological model for drainage of a liquid in foam in fractal dimensions. Our model is based on the concepts of "product-like fractal measure" introduced to model dynamics in porous media and "complex fractional transform" which converts a fractal space on a small scale to a smooth space with a large scale. The solution of the fractal foam drainage equation has been approximated using the He's homotopy perturbation method. Qualitative analysis shows that the behavior of the solitonic wave in fractal dimensions differ from the behavior in integer dimensions. This deformation generates instabilities in the foam dynamics, dispersion and spontaneous breaking of the solitonic wave.

Box dimension of the border of Kingdom of Saudi Arabia

Fractal dimension unlike topological dimension is (usually) a non-integer number which measures complexity, roughness, or irregularity of an object with respect to the space in which the set lies. It is used to characterize highly irregular objects in nature containing statistical self-similarity such as mountains, snowflakes, clouds, coastlines, borders etc. In this article, box dimension (a version of fractal dimension) of the border of Kingdom of Saudi Arabia (KSA) is computed using a multicore parallel processing algorithm based on the classical box-counting method. A power law relation is obtained from numerical simulations which relates the length of the border with the scale size and provides a very close estimate of the actual length of the KSA border within the scaling regions and scaling effects on the length of KSA border are considered. The algorithm presented in the article is shown to be highly scalable and efficient and the speedup of the algorithm is computed using Amdahl's and Gustafson's laws. For simulations, a high performance parallel computer is employed using Python codes and QGIS software.

Micropolar mechanics of product fractal media

Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.