Explicit solutions for a non-classical heat conduction problem for a   semi-infinite strip with a non-uniform heat source
  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 at the face $x=0$ is studied with the aim of finding explicit solutions. It is not a standard heat conduction problem because a heat source $-\Phi(x)F(V(t),t)$ is considered, where $V$ represents the heat flux at $x=0$. Explicit solutions independents of the space or temporal variables are given. Solutions with separated variables when the data functions are defined from the solution $X=X(x)$ of a linear initial value problem of second order and the solution $T=T(t)$ of a non-linear (in general) initial value problem of first order which involves the function $F$, are also given and explicit solutions corresponding to different definitions of $F$ are obtained. A solution by an integral representation depending on the heat flux at $x=0$ for the case in which $F=F(V,t)=\nu V$, $\nu>0$, is obtained and explicit expressions for the heat flux at $x=0$ and for its corresponding solution are calculated when $h=h(x)$ is a potential function and $\Phi=\Phi(x)$ is given by $\Phi(x)=\lambda x$, $\Phi(x)=-\mu\sinh{(\lambda x)}$ or $\Phi(x)=-\mu\sin{(\lambda x)}$, $\lambda>0$ and $\mu>0$. The limit when the temporal variable $t$ tends to $+\infty$ of each explicit solution obtained in this paper is studied and the "controlling" effects of the source term $-\Phi F$ are analysed by comparing the asymptotic behavior of each solution with the asymptotic behavior of the solution to the same problem but in absence of source term. Finally, a relationship between this problem with another non-classical initial and boundary value problem for the heat equation is established and explicit solutions for this second problem are also obtained. 
Andrea N. Ceretani, Domingo A. Tarzia and Luis T. Villa
math.AP