Abstract
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.
Highlights
IntroductionThe aim of this paper is to study the following Problem 1.1 on the non-classical heat equation, in the semi-n-dimensional space domain D with nonlocal sources, for which the internal energy supply depends on the average
Let us consider the domain D and its boundary S defined byD = Rn−1 × R+ = {(x, y) ∈ Rn : x = x1 > 0, y = (x2, . . . , xn) ∈ Rn−1}, S = ∂D = Rn−1 × {0} = {(x, y) ∈ Rn : x = 0, y ∈ Rn−1}.The aim of this paper is to study the following Problem 1.1 on the non-classical heat equation, in the semi-n-dimensional space domain D with nonlocal sources, for which the internal energy supply depends on the average1 t t 0 ux(0, y, s) ds of the heat flux on the boundary S.2010 Mathematics Subject Classification. 35C15, 35K05, 35K20, 35K60, 45D05, 45E10, 80A20.Key words and phrases
U(0, y, t) = 0, y ∈ Rn−1, t > 0, x > 0, y ∈ Rn−1, t > 0, u(x, y, 0) = h(x, y), x > 0, y ∈ Rn−1, where ∆ denotes the Laplacian in Rn. This problem is motivated by modeling the temperature in an isotropic medium with the average of non-uniform and nonlocal sources that provide a cooling or heating system, according to the properties of the function F with respect to the heat flow (y, s) → V (y, s) = ux(0, y, s) at the boundary S, see [11, 13]
Summary
The aim of this paper is to study the following Problem 1.1 on the non-classical heat equation, in the semi-n-dimensional space domain D with nonlocal sources, for which the internal energy supply depends on the average. U(0, y, t) = 0, y ∈ Rn−1, t > 0, x > 0, y ∈ Rn−1, t > 0, u(x, y, 0) = h(x, y), x > 0, y ∈ Rn−1, where ∆ denotes the Laplacian in Rn This problem is motivated by modeling the temperature in an isotropic medium with the average of non-uniform and nonlocal sources that provide a cooling or heating system, according to the properties of the function F with respect to the heat flow (y, s) → V (y, s) = ux(0, y, s) at the boundary S, see [11, 13].
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