Abstract
Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian. For harmonic functions, biharmonic functions, and Dirichlet eigenfunctions of Δ, we give efficient algorithms to compute the solutions exactly, we display the results of implementing these algorithms, and we prove various properties of the solutions that are suggested by the data. Completing the work of Fukushima and Shima [8] who computed the Dirichlet eigenvalues and their multiplicities, we show how to construct a basis (but not orthonormal) for the eigenspaces, so that we have the analog of Fourier sine series on the unit interval. We also show that certain eigenfunctions have the property that they are a nonzero constant along certain lines contained in SG. For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms. We give strong evidence that the analog of finite propagation for the wave equation does not hold because of inconsistent scaling behavior in space and time.
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Communicated by Stephane Jaffard
Research supported in part by the National Science Foundation, Grant DMS-9623250.
Research supported by the National Science Foundation through the Research Experiences for Undergraduates (REU) Program.
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Dalrymple, K., Strichartz, R.S. & Vinson, J.P. Fractal differential equations on the Sierpinski gasket. The Journal of Fourier Analysis and Applications 5, 203–284 (1999). https://doi.org/10.1007/BF01261610
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DOI: https://doi.org/10.1007/BF01261610