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Title: Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle

Abstract

The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods.

Authors:
;  [1]
  1. The University of Southern Mississippi, Department of Mathematics (United States)
Publication Date:
OSTI Identifier:
22769301
Resource Type:
Journal Article
Journal Name:
Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; BOUNDARY CONDITIONS; DIFFUSION; MATHEMATICAL MODELS; MATHEMATICAL SOLUTIONS; PARTIAL DIFFERENTIAL EQUATIONS; PERIODICITY; SIMULATION; UNCERTAINTY PRINCIPLE; WAVE FUNCTIONS

Citation Formats

Garon, Elyse M., E-mail: Elyse.Garon@usm.edu, and Lambers, James V., E-mail: James.Lambers@usm.edu. Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle. United States: N. p., 2018. Web. doi:10.1007/S40314-017-0465-6.
Garon, Elyse M., E-mail: Elyse.Garon@usm.edu, & Lambers, James V., E-mail: James.Lambers@usm.edu. Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle. United States. doi:10.1007/S40314-017-0465-6.
Garon, Elyse M., E-mail: Elyse.Garon@usm.edu, and Lambers, James V., E-mail: James.Lambers@usm.edu. Sun . "Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle". United States. doi:10.1007/S40314-017-0465-6.
@article{osti_22769301,
title = {Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle},
author = {Garon, Elyse M., E-mail: Elyse.Garon@usm.edu and Lambers, James V., E-mail: James.Lambers@usm.edu},
abstractNote = {The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods.},
doi = {10.1007/S40314-017-0465-6},
journal = {Computational and Applied Mathematics},
issn = {0101-8205},
number = 3,
volume = 37,
place = {United States},
year = {2018},
month = {7}
}