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.