Coarse-graining Disordered lattice Random media Wave equation Mathematics
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
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