TY - JOUR
T1 - Towards a physics on fractals
T2 - Differential vector calculus in three-dimensional continuum with fractal metric
AU - Balankin, Alexander S.
AU - Bory-Reyes, Juan
AU - Shapiro, Michael
N1 - Publisher Copyright:
© 2015 Elsevier B.V.
PY - 2016/2/15
Y1 - 2016/2/15
N2 - One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.
AB - One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.
KW - Fractal
KW - Hausdorff derivative
KW - Local fractional calculus
KW - Metric derivative
KW - Quaternionic analysis
UR - http://www.scopus.com/inward/record.url?scp=84945922774&partnerID=8YFLogxK
U2 - 10.1016/j.physa.2015.10.035
DO - 10.1016/j.physa.2015.10.035
M3 - Artículo
SN - 0378-4371
VL - 444
SP - 345
EP - 359
JO - Physica A: Statistical Mechanics and its Applications
JF - Physica A: Statistical Mechanics and its Applications
ER -