A heat conduction problem with sources depending on the average of the heat flux on the boundary

• Autores: Mahdi Boukrouche, Domingo Alberto Tarzia
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 61, Nº. 1, 2020, págs. 87-101
• Idioma: inglés
• DOI: 10.33044/revuma.v61n1a05
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• Resumen

  • Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain D=Rⁿˉ¹×R+ for which the internal energy supply depends on an average in the time variable of the heat flux (y,s)↦V(y,s)=ux(0,y,s)(y,s)↦V(y,s)=ux(0,y,s) on the boundary S=∂DS=∂D. The solution to the problem is found for an integral representation depending on the heat flux on SS which is an additional unknown of the considered problem. We obtain that the heat flux VV must satisfy a Volterra integral equation of the second kind in the time variable tt with a parameter in Rn−1Rn−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.

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