Journal
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
Volume 30, Issue 3, Pages -Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218348X22500736
Keywords
Fractal; Laplacian; Heat Equation; Self-Similar Measure with Overlaps
Funding
- National Natural Science Foundation of China [11901187, 11771136]
- Construct Program of the Key Discipline in Hunan Province
- Hunan Province Hundred Talents Program
- Faculty Research Scholarly Pursuit Award from Georgia Southern University
Ask authors/readers for more resources
We study the heat equation on a bounded open set U subset of R-d supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. Important information about the structure of these measures can be obtained for a class of measures that we call essentially of finite type. We make use of this information to set up a framework to study the associated heat equations in one dimension. We discretize the heat equation and apply the finite element method to yield a system of linear differential equations. We show the convergence of numerical solutions to the actual solution and provide the rate of convergence. We also investigate the propagation speed problem.
We study the heat equation on a bounded open set U subset of R-d supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available