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convective heat-transport; two-point convection-diffusion boundary-value problem; optimization of the amount of heat
The one-dimensional steady-state convection-diffusion problem for the unknown temperature $y(x)$ of a medium entering the interval $(a,b)$ with the temperature $y_{\min }$ and flowing with a positive velocity $v(x)$ is studied. The medium is being heated with an intensity corresponding to $y_{\max }-y(x)$ for a constant $y_{\max }>y_{\min }$. We are looking for a velocity $v(x)$ with a given average such that the outflow temperature $y(b)$ is maximal and discuss the influence of the boundary condition at the point $b$ on the “maximizing” function $v(x)$.
[1] Deuflhard, P., Weiser, M.: Numerische Matematik 3, Adaptive Lösung partieller Differentialgleichungen. De Gruyter, Berlin, 2011. MR 2779847
[2] Ferziger, J. H., Perić, M.: Computational Methods for Fluid Dynamics. Springer, Berlin, 2002, 3rd Edition. MR 1384758 | Zbl 0998.76001
[3] Kamke, E.: Handbook on Ordinary Differential Equations. Nauka, Moscow, 1971, (in Russian).
[4] Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin, 1996. MR 1477665 | Zbl 0844.65075
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