Abstract
We consider the non-classical heat conduction equation, in the domain D=mathbb{R}^{n-1}timesmathbb{R}^{+}, for which the internal energy supply depends on an integral function in the time variable of the heat flux on the boundary S=partial D, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable t with a parameter in mathbb{R}^{n-1}. The solution to this Volterra equation is the heat flux (y, s)mapsto V(y, t)= u_{x}(0, y, t) on S, which is an additional unknown of the considered problem. We show that a unique local solution, which can be extended globally in time, exists. Finally a one-dimensional case is studied with some simplifications. We obtain the solution explicitly by using the Adomian method, and we derive its properties.
Highlights
Let us consider the domain D and its boundary S defined byD = Rn– × R+ = (x, y) ∈ Rn : x = x >, y = (x, . . . , xn) ∈ Rn, ( . )S = ∂D = Rn– × { } = (x, y) ∈ Rn : x =, y ∈ Rn– .The aim of this paper is to study the following Problem . with a non-classical heat-flow feedback problem in the domain D with nonlocal source, for which the internal energy supply depends on the integral t ux (, y, s) ds on the boundary S.Problem
The aim of this paper is to study the following Problem . with a non-classical heat-flow feedback problem in the domain D with nonlocal source, for which the internal energy supply depends on the integral t ds on the boundary
2 Existence results we give first in Theorem . the integral representation ( . ) of the solution of considered Problem . , but it depends on the heat flow V on the boundary S, which satisfies Volterra integral equation ( . ) with initial condition ( . )
Summary
The goal of this paper is to obtain in Section the existence and uniqueness of the global solution of the non-classical heat conduction Problem . We recall here the Green’s function for the n-dimensional heat equation with homogeneous Dirichlet’s boundary conditions, given the following expression [ , ] But it depends on the heat flow V on the boundary S, which satisfies Volterra integral equation
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