Some Properties on Koch Curve

Abstract

Many physical problems on fractal domains lead to nonlinear models involving reaction–diffusion equations, problems on elastic fractal media or fluid flow through fractal regions, etc. The prevalence of fractal-like objects in nature has led both mathematicians and physicists to study various processes on fractals. In recent years there has been an increasing interest in studying nonlinear partial differential equations on fractals, also motivated and stimulated by the considerable amount of literature devoted to the definition of a Laplace-type operator for functions on fractal domains. The energy of a function defined on a post critically finite (p.c.f) self-similar fractal can be written as a sum of directional energies. A general concept of graph energy defined on a finite connected graph is given. A work about the graph energy is mainly concerned on a Koch curve. First graphs on this Koch curve are built. These graphs produced from the initial graph by iteration repeatedly. Find the energy renormalization constant. Second we find the non-normalized and Normalized Laplacian of a Koch Curve. With the help of this we examine the Laplacian Renormalization constant and forbidden eigenvalues. Finally we develop the Spectral decimation function of Koch Curve.