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On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure

1,*, 2, 3,4 and 5,6
1
Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania
2
Department of Mechanical Engineering, Transilvania University of Brasov, 500093 Brasov, Romania
3
Center for Modeling & Computer Simulation, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
4
Department of Mathematics & Statistics, Faculty of Basic and Applied Sciences (FBAS), International Islamic University (IIUI), Islamabad 44000, Pakistan
5
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
6
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 863; https://doi.org/10.3390/sym11070863
Received: 12 June 2019 / Revised: 22 June 2019 / Accepted: 26 June 2019 / Published: 2 July 2019
(This article belongs to the Special Issue Symmetry in Applied Continuous Mechanics)
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PDF [326 KB, uploaded 4 July 2019]

Abstract

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem. View Full-Text
Keywords: backward in time problem; dipolar thermoelastic body; uniqueness of solution; Cesaro means; partition of energies backward in time problem; dipolar thermoelastic body; uniqueness of solution; Cesaro means; partition of energies
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Marin, M.; Vlase, S.; Ellahi, R.; Bhatti, M. On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure. Symmetry 2019, 11, 863.

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