Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source

Andrea N Ceretani,Domingo A Tarzia,Luis T Villa

doi:10.1186/s13661-015-0416-3

Abstract

A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition is studied. It is not a standard heat conduction problem because a non-uniform heat source dependent on the heat flux at the boundary is considered. The purpose of this article is to find explicit solutions and analyze how to control their asymptotic temporal behavior through the source term.

Highlights

We consider a one-dimensional isotropic and homogeneous medium with one inaccessible boundary under the effects of a temperature controller device which depends on the heat flux at the accessible boundary, when the initial distribution of temperature is known and the temperature at the accessible boundary is constant in time
We recall the relationship between Problems P and P given in [ ], and we find explicit solutions to Problem P through what we know about Problem P
5 Conclusions In this paper we consider a non-classical initial and boundary value problem for a nonhomogeneous one-dimensional heat equation which represents a temperature regulation problem for a semi-infinite homogeneous isotropic medium where the temperature controller device depends on the heat flux at the fixed boundary, an initial temperature distribution is known and the temperature at the fixed boundary is constant in time

Summary

Introduction

We consider a one-dimensional isotropic and homogeneous medium with one inaccessible boundary (semi-infinite material) under the effects of a temperature controller device which depends on the heat flux at the accessible boundary (fixed boundary), when the initial distribution of temperature is known and the temperature at the accessible boundary is constant in time. Proof An easy computation shows that the expressions given in ( ) and ( ) satisfy the integral equation ( ) They correspond to the heat flux ux( , t) at the boundary x = for the solution u of Problem P given in ( ). If σ ≤ , lim t→+∞ u (x, t) λμδn σ η –n if σ > , Proof By computing the integral in ( ) for the function h given in ( ), we obtain that the solution u to Problem P given in ( ) is defined by u (x, t) = h(x) exp |σ |t , ∀x > , t >.

Conclusions

Computation of