NUMERICAL ANALYSIS ON THE SIERPINSKI GASKET, WITH APPLICATIONS TO SCHRÖDINGER EQUATIONS, WAVE EQUATION, AND GIBBS' PHENOMENON

Fractals ◽  
2004 ◽  
Vol 12 (04) ◽  
pp. 413-449 ◽  
Author(s):  
KEVIN COLETTA ◽  
KEALEY DIAS ◽  
ROBERT S. STRICHARTZ
Keyword(s):  
Finite Element Method ◽  
Wave Equation ◽  
Fourier Series ◽  
Sierpinski Gasket ◽  
Gibbs Phenomenon ◽  
The Finite Element Method ◽  
Sierpiński Gasket ◽  
Jump Discontinuities ◽  
Self Similar ◽  
Series Type

We show how to improve the finite element method on the Sierpinski gasket (SG) to allow arbitrary partitions of the space. We use this method to study numerically solutions of the Schrödinger equation with well-type potentials, and the wave equation. We also show that Fourier series-type expansions on SG of functions with jump discontinuities appear to exhibit a self-similar Gibbs' phenomenon.

The Finite Element Method on the Sierpinski Gasket

2001 ◽  
Vol 17 (4) ◽  
pp. 561-588 ◽  
Author(s):  
Michael Gibbons ◽  
Arjun Raj ◽  
Robert S. Strichartz
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Sierpinski Gasket ◽  
The Finite Element Method ◽  
Sierpiński Gasket ◽  
Element Method

GAPS IN THE RATIOS OF THE SPECTRA OF LAPLACIANS ON FRACTALS

Fractals ◽  
2009 ◽  
Vol 17 (04) ◽  
pp. 523-535 ◽  
Author(s):  
KATHRYN E. HARE ◽  
DENGLIN ZHOU
Keyword(s):  
Fourier Series ◽  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Classical Setting ◽  
Counting Formula ◽  
Laplacian Operators ◽  
Surprising Fact ◽  
Better Than

In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectrum. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there are intervals which contain no ratios of eigenvalues. In this paper we give general criteria for this phenomena and show that Laplacians on many interesting classes of fractals satisfy our criteria.

AVERAGE GEODESIC DISTANCE OF NODE-WEIGHTED SIERPINSKI NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950110
Author(s):  
LIFENG XI ◽  
QIANQIAN YE ◽  
JIANGWEN GU
Keyword(s):  
Asymptotic Formula ◽  
Geodesic Distance ◽  
Sierpinski Gasket ◽  
Weight Vector ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar

This paper discusses the asymptotic formula of average distances on node-weighted Sierpinski skeleton networks by using the integral of geodesic distance in terms of self-similar measure on the Sierpinski gasket with respect to the weight vector.

scholarly journals Spectral decimation for families of self-similar symmetric Laplacians on the Sierpiński gasket

2019 ◽  
Vol 7 (1) ◽  
pp. 1-62
Author(s):  
Sizhen Fang ◽  
Dylan King ◽  
Eun Bi Lee ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

scholarly journals A CONVEX SURFACE WITH FRACTAL CURVATURE

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050059
Author(s):  
IANCU DIMA ◽  
RACHEL POPP ◽  
ROBERT S. STRICHARTZ ◽  
SAMUEL C. WIESE
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convex Surface ◽  
The Self ◽  
The Finite Element Method ◽  
Similar Measure ◽  
Entire Spectrum ◽  
Self Similar ◽  
Polyhedral Approximations ◽  
Element Method

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

Unified Solution on Skew Plate Bending with Four Edges Supported under Arbitrary Load

2011 ◽  
Vol 243-249 ◽  
pp. 5994-5998
Author(s):  
Lang Cao ◽  
Xing Jie Xing ◽  
Feng Guang Ge
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Engineering Design ◽  
Plate Bending ◽  
The Finite Element Method ◽  
Arbitrary Load ◽  
Skew Plate ◽  
Good Agreement

According to the bending equation and boundary conditions of skew plate in the oblique coordinates system parallel to the edge of the plate, expanding deflection and load into form of Fourier series, the paper derives and obtains unified solution of bending problem for the four-edge-supported skew plate under arbitrary load. Programmed and calculated by mathematica language, the paper also comes with deflections and moments under the condition of any oblique angles, ratios of side length and Poisson ratios. The results of the paper is compared with those by the finite element method in the example, and they’re in good agreement with each other. The paper extends the bending theory of rectangular plate to the skew plate of any angle. The theory being reliable and the result being accurate, the research of the paper can provide reference for engineering design.

Self–similar Energy Forms on the Sierpinski Gasket with Twists

2007 ◽  
Vol 27 (1) ◽  
pp. 45-60 ◽  
Author(s):  
Mihai Cucuringu ◽  
Robert S. Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar

Effective resistances for harmonic structures on p.c.f. self-similar sets

1994 ◽  
Vol 115 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Jun Kigami
Keyword(s):  
Diffusion Processes ◽  
Difference Operator ◽  
Scaling Limit ◽  
Sierpinski Gasket ◽  
Dirichlet Forms ◽  
Sierpiński Gasket ◽  
Know How ◽  
Analysis On Fractals ◽  
Topological Description ◽  
Self Similar

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

scholarly journals Polynomials on the Sierpiński gasket with respect to different Laplacians which are symmetric and self-similar

2020 ◽  
Vol 7 (4) ◽  
pp. 387-444
Author(s):  
Christian Loring ◽  
W. Jacob Ogden ◽  
Ely Sandine ◽  
Robert Strichartz
Keyword(s):  
Sierpinski Gasket ◽  
Sierpiński Gasket ◽  
Self Similar
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