Analysis and Differential Equations on Fractals

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Aslani, Shahriar | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51024 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar Motlagh, Alireza
Abstract:
In this thesis we consider dynamical aspects of fractals. More precisely, answering questions like how heat diffuses on fractals and how does a material with fractal structure vibrates? To give an answer to these questions we need a PDE theory on fractals. Since fractals do not have smooth structures, defining differential operators like Laplacian is not possible from a classical viewpoint of analysis, to overcome such a difficulty we also need a theory of analysis on fractals. So as a good instance of analysis on fractals we first define Laplacian on Sierpinsky gasket and we try to extend the concept on other finitely ramified self-similar fractals. We also construct Dirichlet forms, Green's functions and harmonic functions on these fractals. At last we are going to study the spectral analysis of self-similar structures
Keywords:
Fractals ; Partial Differential Equations ; Self-Similarity ; Analysis on Fractals ; Differential Equations Fractals