A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary
| Autor: | Boukrouche, Mahdi, Tarzia, Domingo A. |
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| Rok vydání: | 2019 |
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| Druh dokumentu: | Working Paper |
| Popis: | Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method. Comment: Accepted by Revista UMA, April 30 2019, in press. arXiv admin note: substantial text overlap with arXiv:1610.01680 |
| Databáze: | arXiv |
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