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One-dimensional wave equations defined by fractal Laplacians

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Abstract

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio and the three-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the post-critically finite condition or the open set condition. Using second-order self-similar identities introduced by Strichartz et al., we discretize the equations and use the finite element and central difference methods to obtain numerical approximations of the weak solutions. We prove that the numerical solutions converge to the weak solution and obtain estimates for the rate of convergence.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460-8093, USA

    John Fun-Choi Chan

  2. College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, China

    Sze-Man Ngai

  3. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Alexander Teplyaev

Authors
  1. John Fun-Choi Chan
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  2. Sze-Man Ngai
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  3. Alexander Teplyaev
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Corresponding author

Correspondence to John Fun-Choi Chan.

Additional information

The first two authors were supported in part by a Faculty Research Grant from Georgia Southern University.

The second author is also supported in part by an HKRGC grant and National Natural Science Foundation of China grant 11271122.

The third author is supported in part by NSF grant DMS-0505622.

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Chan, J.FC., Ngai, SM. & Teplyaev, A. One-dimensional wave equations defined by fractal Laplacians. JAMA 127, 219–246 (2015). https://doi.org/10.1007/s11854-015-0029-x

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  • DOI: https://doi.org/10.1007/s11854-015-0029-x

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