English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A non-classical initial and boundary value problem for a non-homogeneousone-dimensional heat equation for a semi-infinite material with a zerotemperature boundary condition at the face $x=0$ is studied with the aim offinding explicit solutions. It is not a standard heat conduction problembecause a heat source $-\Phi(x)F(V(t),t)$ is considered, where $V$ representsthe heat flux at $x=0$. Explicit solutions independents of the space ortemporal variables are given. Solutions with separated variables when the datafunctions are defined from the solution $X=X(x)$ of a linear initial valueproblem 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 alsogiven and explicit solutions corresponding to different definitions of $F$ areobtained. A solution by an integral representation depending on the heat fluxat $x=0$ for the case in which $F=F(V,t)=\nu V$, $\nu>0$, is obtained andexplicit expressions for the heat flux at $x=0$ and for its correspondingsolution are calculated when $h=h(x)$ is a potential function and$\Phi=\Phi(x)$ is given by $\Phi(x)=\lambda x$, $\Phi(x)=-\mu\sinh{(\lambdax)}$ or $\Phi(x)=-\mu\sin{(\lambda x)}$, $\lambda>0$ and $\mu>0$. The limitwhen the temporal variable $t$ tends to $+\infty$ of each explicit solutionobtained in this paper is studied and the "controlling" effects of the sourceterm $-\Phi F$ are analysed by comparing the asymptotic behavior of eachsolution with the asymptotic behavior of the solution to the same problem butin absence of source term. Finally, a relationship between this problem withanother non-classical initial and boundary value problem for the heat equationis established and explicit solutions for this second problem are alsoobtained.

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