# Transient thermal boundary value problems in the half-space with mixed convective boundary conditions

## Abstract

Numerous problems in thermal engineering give rise to problems with abrupt changes in boundary conditions particularly as regards structural and construction applications. There is a continual need for improved analytical and numerical techniques for the study and analysis of such canonical problems. The present work considers transient heat propagation in a two-dimensional half-space with mixed boundary conditions of the Dirichlet and the Robin (convective) type. The exact temperature field is obtained in double integral form by applying the Wiener–Hopf technique, and this integral is then efficiently evaluated numerically with the help of numerical algorithms developed for the present problem and by exploiting the properties of the integrand. The results are then compared to the numerical solutions using finite element methods, and excellent agreement is shown. Some potential application areas of the theory are also discussed, specifically in the regime where edge effects can influence thermal convection processes.

## Keywords

Heat convection Mixed boundary conditions Transient heat problem Wiener–Hopf technique## Notes

### Acknowledgements

This work has benefited from a French Government Grant managed by ANR within the frame of the National Program of Investments for the Future ANR-11- LABX-022-01 (LabEx MMCD Project). W.J.P. is grateful to the EPSRC for his Fellowship Award EP/L018039/1.

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