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Dissertation
Rigorous Approximation of Lattices with Random Materials by Wave Equations
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2023
DOI:
https://doi.org/10.17918/00001691
Abstract
Within the framework of "coarse-graining", we prove the macroscopic dynamics of a lattice composed of random materials (springs and masses) converges strongly with probability one (a.s.) to an effective wave equation as the initial conditions become long relative to the lattice spacing. More concretely, this means an effective wave equation approximates the dynamics of the lattice well, relative to the size of the initial conditions on time scales proportional to the length scaling of the initial conditions. We prove such a result for a one dimensional lattice with random masses and springs and a two dimensional square lattice with random masses. In addition, for the one dimensional lattice, we also prove a convergence in mean. For the two dimensional lattice, we discuss how the dimension of the lattice effects convergence rate in the case when the materials are random. We also give numerical results regarding a KdV approximation of an FPUT system with random masses where the masses are chosen with a technical condition we call transparency.
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Details
- Title
- Rigorous Approximation of Lattices with Random Materials by Wave Equations
- Creators
- Joshua Andrew McGinnis
- Contributors
- Jay Douglas Wright (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- 111 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991020879312304721